diff --git a/.gitattributes b/.gitattributes index 1ef325f1b111266a6b26e0196871bd78baa8c2f3..154244c7b198ec3c4b150e11f5cf684f616b5f7b 100644 --- a/.gitattributes +++ b/.gitattributes @@ -57,3 +57,21 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text # Video files - compressed *.mp4 filter=lfs diff=lfs merge=lfs -text *.webm filter=lfs diff=lfs merge=lfs -text +GeoCoq/eval/*.json filter=lfs diff=lfs merge=lfs -text +GeoCoq/test/*.json filter=lfs diff=lfs merge=lfs -text +multilingual/eval/*.json filter=lfs diff=lfs merge=lfs -text +multilingual/test/*.json filter=lfs diff=lfs merge=lfs -text +multilingual/train/*.json filter=lfs diff=lfs merge=lfs -text +lean/eval/*.json filter=lfs diff=lfs merge=lfs -text +lean/test/*.json filter=lfs diff=lfs merge=lfs -text +lean/train/*.json filter=lfs diff=lfs merge=lfs -text +coq/eval/*.json filter=lfs diff=lfs merge=lfs -text +coq/test/*.json filter=lfs diff=lfs merge=lfs -text +coq/train/*.json filter=lfs diff=lfs merge=lfs -text +GeoCoq/train/*.json filter=lfs diff=lfs merge=lfs -text +math-comp/eval/*.json filter=lfs diff=lfs merge=lfs -text +math-comp/test/*.json filter=lfs diff=lfs merge=lfs -text +math-comp/train/*.json filter=lfs diff=lfs merge=lfs -text +category-theory/eval/*.json filter=lfs diff=lfs merge=lfs -text +category-theory/test/*.json filter=lfs diff=lfs merge=lfs -text +category-theory/train/*.json filter=lfs diff=lfs merge=lfs -text diff --git a/GeoCoq/eval/GeoCoq_eval.yaml b/GeoCoq/eval/GeoCoq_eval.yaml new file mode 100644 index 0000000000000000000000000000000000000000..0216def0700a4a418f5059b301819322f62c1224 --- /dev/null +++ b/GeoCoq/eval/GeoCoq_eval.yaml @@ -0,0 +1,403 @@ +name: GeoCoq_eval +num_files: 92 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 208 +datasets: +- project: /GeoCoq + files: + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_pseudo_reflexivity.v + theorems: + - cong_inner_transitivity + - segment_construction + - path: theories/Main/Annexes/quadrilaterals_inter_dec.v + theorems: + - rmb_perp + - half_plgs + - par_strict_cong_mid + - Plg_perm + - path: theories/Main/Tarski_dev/Ch05_bet_le.v + theorems: + - lt_comm + - le_reflexivity + - path: theories/Coinc/Utils/sets.v + theorems: + - leb_dec + - lt + - sortOK + - path: theories/Main/Annexes/coplanar.v + theorems: + - reflectl__coplanar + - coplanar_perm_3 + - path: theories/Main/Meta_theory/Models/beeson_to_tarski.v + theorems: + - IT_dec + - BetT_dec + - path: theories/Main/Tarski_dev/Ch11_angles.v + theorems: + - in_angle_out + - cong_lt_per2__lt_1 + - cong3_diff2 + - cong__acute + - t18_18_aux + - suppa_sym + - suppa__per + - angle_construction_3 + - col_orth__orth + - orth_at_dec + - out_suppa__bet + - cop3_orth__orth + - acute_sym + - orth_at__ncop2 + - bet__obtuse + - lta_suppa2__lta + - l11_62_existence_bis + - l11_18_1 + - path: theories/Main/Tarski_dev/Ch12_parallel.v + theorems: + - par_col4__par + - par_not_col + - par_strict_all_one_side + - l12_17 + - line_dec + - par_col2_par + - path: theories/Main/Tarski_dev/Ch10_line_reflexivity.v + theorems: + - osym_not_col + - midpoint_preserves_image + - l10_4 + - path: theories/Main/Meta_theory/Models/hilbert_to_tarski.v + theorems: + - betH_trans + - congH_perms + - betH_colH + - addition_betH + - same_side_prime__plane + - playfair_s_postulateH + - betH_to_bet + - path: theories/Main/Tarski_dev/Ch16_coordinates.v + theorems: + - l16_9_1 + - path: theories/Main/Tarski_dev/Ch13_4_cos.v + theorems: + - lcos3_permut1 + - lcos_not_lg_null + - lcos2_uniqueness + - lcos3_eq_trans + - path: theories/Coinc/ColR.v + theorems: + - positive_dec + - proper_7 + - path: theories/Main/Annexes/sums.v + theorems: + - sums__eq34 + - sums_sym + - path: theories/Main/Tarski_dev/Ch03_bet.v + theorems: + - lower_dim_ex + - path: theories/Main/Annexes/quadrilaterals.v + theorems: + - Rectangle_Parallelogram + - col_cong_mid2 + - col_cong_mid1 + - mid_plg_2 + - parallelogram_strict_not_col_2 + - plg_to_parallelogram + - path: theories/Algebraic/Counter_models/nD/counter_model_upper_dim.v + theorems: + - bet_nD'_2D + - path: theories/Elements/OriginalProofs/euclidean_tactics.v + theorems: + - neq_or_eq + - path: theories/Algebraic/Counter_models/nD/counter_model_euclid.v + theorems: + - cong_abab + - i2'_in_unit_disk + - one_side_cong'_eq + - p_scalar_point_lt1 + - d'_in_unit_disk + - lower_dim + - nbet_bca + - path: theories/Main/Annexes/suma.v + theorems: + - bet2_suma__out + - sams_nos__nts + - inangle__sams + - conga3_suma__suma + - suma_assoc_2 + - out213__suma + - sams_suma__out546 + - trisuma_perm_231 + - path: theories/Algebraic/Counter_models/nD/counter_model_five_segment.v + theorems: + - bet_symmetry + - path: theories/Main/Tarski_dev/Ch08_orthogonality.v + theorems: + - per_distinct_1 + - l8_16_1 + - l8_16_2 + - perp_exists + - col_per2_cases + - perp_proj + - l8_8 + - path: theories/Main/Tarski_dev/Ch02_cong.v + theorems: + - cong_symmetry + - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_inner_transitivity.v + theorems: + - segment_construction + - path: theories/Main/Tarski_dev/Ch14_sum.v + theorems: + - sum_exists + - path: theories/Algebraic/Counter_models/nD/gupta_inspired_to_independent_version.v + theorems: + - segment_constructionG_f + - path: theories/Main/Meta_theory/Continuity/completeness.v + theorems: + - extension_to_plane__plane + - path: theories/Main/Meta_theory/Models/tarski_to_euclid.v + theorems: + - Euclid_neutral_ruler_compass_follows_from_Tarski_ruler_and_compass + - inside + - path: theories/Main/Tarski_dev/Ch16_coordinates_with_functions.v + theorems: + - signed_area_perm + - twice_signed_area_perm + - Col_AM_Col + - path: theories/Main/Meta_theory/Continuity/first_order.v + theorems: + - fof__fof1 + - fof1__fof + - path: theories/Main/Tarski_dev/Ch13_1.v + theorems: + - symmetry_preserves_per + - path: theories/Main/Annexes/Tagged_predicates.v + theorems: + - NCol_tagged_NCol + - Diff_tagged_Diff + - Par_Par_tagged + - path: theories/Main/Elements_statements/Book_1.v + theorems: + - prop_27 + - path: theories/Main/Tarski_dev/Ch13_2_length.v + theorems: + - lg_null_instance + - path: theories/Main/Tarski_dev/Ch14_order.v + theorems: + - neg_not_pos + - col_2_le_or_ge + - path: theories/Main/Highschool/concyclic.v + theorems: + - concyclic_perm_17 + - path: theories/Algebraic/POF_to_Tarski.v + theorems: + - upper_dim_aux + - extension_contraction + - c2D_neq0 + - betS_inner_transitivity + - bet_colF + - mulmx_tr + - cong_perp_aux1 + - euclid'' + - contraction_betS + - path: theories/Main/Tarski_dev/Ch15_lengths.v + theorems: + - signEq_refl + - pyth_context + - bet_leP + - ltP_pos + - path: theories/Algebraic/coplanarity.v + theorems: + - l9_33 + - path: theories/Main/Tarski_dev/Ch13_5_Pappus_Pascal.v + theorems: + - lcos_per_anga + - l13_10_aux4 + - path: theories/Main/Annexes/vectors.v + theorems: + - eqv_refl + - path: theories/Elements/OriginalProofs/proposition_47A.v + theorems: + - proposition_47A + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_inner_transitivity.v + theorems: + - euclid + - path: theories/Elements/OriginalProofs/proposition_30B.v + theorems: + - proposition_30B + - path: theories/Main/Tarski_dev/Ch13_3_angles.v + theorems: + - is_anga_distinct + - anga_not_null_lg + - eqA_preserves_ang + - anga_conga_anga + - path: theories/Main/Annexes/half_angles.v + theorems: + - col_halfa__out + - ghalfa_preserves_conga_1 + - ghalfa__acute + - path: theories/Main/Annexes/project.v + theorems: + - project_preserves_eqv + - projp_col + - projp2_col + - path: theories/Main/Meta_theory/Continuity/first_order_dedekind_circle_circle.v + theorems: + - fod__circle_circle + - path: theories/Algebraic/Counter_models/Planar/counter_model_lower_dim.v + theorems: + - euclid + - bet_inner_transitivity + - path: theories/Algebraic/Counter_models/nD/counter_model_bet_inner_transitivity.v + theorems: + - col__Col_13 + - cong_inner_transitivity + - path: theories/Algebraic/Counter_models/Planar/counter_model_euclid.v + theorems: + - betR_abc + - c'_norm + - path: theories/Main/Meta_theory/Continuity/dedekind_cantor.v + theorems: + - nested__diff + - path: theories/Main/Tarski_dev/Ch09_plane.v + theorems: + - col_cop2__cop + - l9_38 + - coplanar_pseudo_trans + - one_side_symmetry + - per_col_eq + - l9_19 + - path: theories/Main/Annexes/circles.v + theorems: + - circle_circle_os + - outc_eq + - ray_cut_chord + - bet_inc2__inc + - onc2_per__mid + - Prop_7_8_uniqueness + - path: theories/Main/Meta_theory/Decidability/equivalence_between_decidability_properties_of_basic_relations.v + theorems: + - bet_dec_eq_dec + - path: theories/Main/Meta_theory/Models/tarski_to_hilbert.v + theorems: + - between_one + - OS_same_side' + - path: theories/Main/Tarski_dev/Ch04_col.v + theorems: + - col_permutation_5 + - path: theories/Algebraic/Counter_models/nD/counter_model_cong_pseudo_reflexivity.v + theorems: + - segment_construction + - bet_inner_transitivity + - path: theories/Main/Meta_theory/Parallel_postulates/parallel_postulates.v + theorems: + - equivalent_postulates_without_decidability_of_intersection_of_lines_bis + - path: theories/Main/Tarski_dev/Ch14_prod.v + theorems: + - prod_x_axis_unit_change + - path: theories/Elements/OriginalProofs/lemma_ray5.v + theorems: + - lemma_ray5 + - path: theories/Main/Meta_theory/Parallel_postulates/par_trans_NID.v + theorems: + - playfair__playfair_quater + - path: theories/Main/Meta_theory/Parallel_postulates/rah_existential_saccheri.v + theorems: + - rah__existential_saccheri + - path: theories/Main/Tarski_dev/Ch07_midpoint.v + theorems: + - Mid_cases + - col_bet2_cong2 + - bet2_lt2__lt + - midpoint_distinct_1 + - midpoint_out_1 + - path: theories/Algebraic/Counter_models/counter_model_bet_identity.v + theorems: + - cong_aux_2 + - path: theories/Algebraic/Counter_models/nD/counter_model_bet_symmetry.v + theorems: + - onemx_neq0 + - path: theories/Elements/OriginalProofs/proposition_05b.v + theorems: + - proposition_05b + - path: theories/Main/Meta_theory/Models/tarski_to_gupta_inspired.v + theorems: + - T_euclidean_to_TG_euclidean + - path: theories/Main/Tactics/CoincR_for_concy.v + theorems: + - st_ok_empty_for_concy + - path: theories/Main/Meta_theory/Continuity/dedekind_completeness.v + theorems: + - extension_image_density + - path: theories/Algebraic/Counter_models/nD/independent_version_to_beeson.v + theorems: + - ITbetween_inner_transitivity + - path: theories/Coinc/Utils/arity.v + theorems: + - nthCircPerm1Eq + - pred_conj_aux + - path: theories/Algebraic/Counter_models/nD/bet_identity.v + theorems: + - g2_2 + - g2_6_2 + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_cop.v + theorems: + - not_col_perm_1 + - path: theories/Coinc/CoincR.v + theorems: + - interp_CPHdOK + - path: theories/Main/Highschool/incenter.v + theorems: + - incenter_permut213 + - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_symmetry.v + theorems: + - parP + - path: theories/Elements/OriginalProofs/lemma_samesidereflexive.v + theorems: + - lemma_samesidereflexive + - path: theories/Main/Meta_theory/Continuity/elementary_continuity_props.v + theorems: + - segment_circle__one_point_line_circle + - path: theories/Elements/OriginalProofs/proposition_31short.v + theorems: + - proposition_31short + - path: theories/Main/Tarski_dev/Ch13_6_Desargues_Hessenberg.v + theorems: + - l13_19 + - path: theories/Main/Meta_theory/Continuity/angle_archimedes.v + theorems: + - archi_in_acute_angles + - angles_archi_aux + - path: theories/Main/Highschool/orientation.v + theorems: + - symmetry_preserves_bet + - path: theories/Main/Annexes/inscribed_angle.v + theorems: + - suma__suppa + - path: theories/Elements/OriginalProofs/proposition_23C.v + theorems: + - proposition_23C + - path: theories/Elements/OriginalProofs/lemma_collinearbetween.v + theorems: + - lemma_collinearbetween + - path: theories/Elements/OriginalProofs/lemma_RTsymmetric.v + theorems: + - lemma_RTsymmetric + - path: theories/Main/Highschool/SegmTrisect.v + theorems: + - SegmTrisectSecondThirdExistence + - path: theories/Main/Annexes/defect.v + theorems: + - defect_perm_312 + - path: theories/Elements/OriginalProofs/lemma_3_6b.v + theorems: + - lemma_3_6b + - path: theories/Algebraic/Counter_models/Planar/counter_model_upper_dim.v + theorems: + - bc_neq + - path: theories/Elements/OriginalProofs/lemma_TCreflexive.v + theorems: + - lemma_TCreflexive diff --git a/GeoCoq/eval/local.meta.json b/GeoCoq/eval/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..aba3dd3bf6116d893521c36a772748895548afa4 --- /dev/null +++ b/GeoCoq/eval/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ff7b268c4284b95fd0df83e54c6d2c4502727a0d23f513cd555096497a94e646 +size 388 diff --git a/GeoCoq/eval/local_data_0000004928.json b/GeoCoq/eval/local_data_0000004928.json new file mode 100644 index 0000000000000000000000000000000000000000..752cc174ad96c72b85a69b92e8e25f38b4d93018 --- /dev/null +++ b/GeoCoq/eval/local_data_0000004928.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:172135036cf1e0a9657bb6088150e25ebf1ab8e6e68d417017bd4d5d01ae257b +size 18886874 diff --git a/GeoCoq/eval/local_lemma_0000000000.json b/GeoCoq/eval/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/GeoCoq/eval/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/GeoCoq/test/GeoCoq_test.yaml b/GeoCoq/test/GeoCoq_test.yaml new file mode 100644 index 0000000000000000000000000000000000000000..88d2edb0cd9092ba413aafe497a0f51e8277409a --- /dev/null +++ b/GeoCoq/test/GeoCoq_test.yaml @@ -0,0 +1,788 @@ +name: GeoCoq_test +num_files: 136 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 505 +datasets: +- project: /GeoCoq + files: + - path: theories/Main/Meta_theory/Models/hilbert_to_tarski.v + theorems: + - OS2__TS + - betH_expand + - strong_pasch + - bet_identity + - isosceles_congaH + - out2_out1 + - betH_outH2__betH + - pars__para + - same_side_not_incid + - bet_colH + - same_side_comm + - line_col + - congaH_commr + - congH_refl + - cong_trans + - outH_trivial + - colH_trans + - outH_dec + - H_euclidean_to_T_euclidean + - other_point_exists + - point3_online_exists + - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_inner_transitivity.v + theorems: + - continuity + - point_equality_decidability + - proclus + - path: theories/Main/Meta_theory/Continuity/elementary_continuity_props.v + theorems: + - triangle_inequality1 + - euclid_22_aux + - one_point_line_circle__two_points_line_circle + - path: theories/Main/Annexes/suma.v + theorems: + - sams_lea2_suma2__lea456 + - sams_lea_lta789_suma2__lta456 + - sams_lea2_suma2__lea + - bet_suppa__suma + - trisuma_distincts + - col2_suma__col + - suma_suppa2__suma + - suma_assoc + - col_trisuma__bet + - acute_per_suma__nbet + - os2__sams + - bet__sams + - conga2_sams__sams + - sams_chara + - ts__suma_1 + - path: theories/Coinc/CongR.v + theorems: + - CTcong_comm + - CTcong_pseudo_refl + - eq_le2__eq2 + - path: theories/Main/Elements_statements/Book_1.v + theorems: + - prop_46 + - prop_2 + - prop_30 + - prop_28_2 + - prop_8 + - prop_3 + - prop_21_2 + - prop_29_1 + - prop_32_2 + - prop_20 + - prop_31 + - prop_5_2 + - path: theories/Main/Annexes/project.v + theorems: + - cong_conga3_cong3 + - project_to_projp + - projp_project_to_perp + - projp_preserves_bet + - projp_is_project + - path: theories/Algebraic/Counter_models/nD/counter_model_bet_symmetry.v + theorems: + - segment_construction + - path: theories/Main/Annexes/coplanar.v + theorems: + - ncoplanar_perm_11 + - reflect__coplanar + - coplanar_perm_11 + - ncoplanar_perm_10 + - coplanar_perm_9 + - ncop__ncols + - ncoplanar_perm_13 + - ncoplanar_perm_17 + - ncoplanar_perm_4 + - midpoint__coplanar + - ts__coplanar + - path: theories/Coinc/CoincR.v + theorems: + - CoappDup + - memCPAuxOK + - interp_CP + - path: theories/Main/Annexes/saccheri.v + theorems: + - t22_14__oah + - sac__pars1234 + - t22_14__bet_aux + - mid2_sac__lam6521 + - t22_7__cong + - sac__par1234 + - path: theories/Algebraic/Counter_models/nD/counter_model_cong_identity.v + theorems: + - continuity + - five_segment + - euclid + - cong_pseudo_reflexivity + - path: theories/Main/Highschool/orientation.v + theorems: + - per_preserves_bet + - proj3_col + - cong_le_le + - proj_par + - midpoint_col + - path: theories/Main/Meta_theory/Models/tarski_to_euclid.v + theorems: + - Euclid_follows_from_Tarski_euclidean + - InCirc_not_OnCirc + - nCol_not_Col + - path: theories/Main/Meta_theory/Continuity/grad.v + theorems: + - grad_neq__neq12 + - gradexp_le__reach + - grad_sum + - path: theories/Algebraic/POF_to_Tarski.v + theorems: + - cong_eq + - bet_betS + - ratioP_aux + - ratio_cp'_aux_2 + - dot_ge0 + - betS_ratio + - ratio_bet' + - addrBBB + - bet_axx + - betEP' + - ratiop0 + - bet_symmetry + - extension_eq + - quad_eq0 + - row2_eq0 + - eq_div_scale + - betS_sym + - ratioNN + - betE_xax + - path: theories/Algebraic/Counter_models/Planar/counter_model_lower_dim.v + theorems: + - continuity + - bet_symmetry + - path: theories/Main/Meta_theory/Models/tarski_to_hilbert.v + theorems: + - incident_col + - incidentp_eqp + - cop_plane_aux + - col_incident + - eqp_incidentp + - incidentp_Proper + - Hilbert_2D_follows_from_Tarski_2D + - path: theories/Main/Annexes/circles.v + theorems: + - bet_chord__diam_or_ncol + - bet_incs2__incs + - onc__inc + - outcs__outc + - diam_uniqueness + - path: theories/Elements/OriginalProofs/lemma_congruencetransitive.v + theorems: + - lemma_congruencetransitive + - path: theories/Main/Annexes/quadrilaterals_inter_dec.v + theorems: + - perp_rmb + - plg_par + - plg_cong_2 + - plgs_mid + - parallelogram_to_plg + - plg_not_comm_1 + - perp3__perp + - degenerated_rect_eq + - plg_mid + - plg_per_rect4 + - cop_perp3__rect + - path: theories/Main/Highschool/SegmTrisect.v + theorems: + - SegmTrisectSecondThirdUniqueness + - path: theories/Algebraic/Counter_models/nD/counter_model_euclid.v + theorems: + - cauchy_schwartz' + - cong_abdc + - E_in_disk_aux + - dist_lt1 + - cauchy_schwartz_eq' + - a'nD_eq0 + - bc_neq_nD + - cong'_cdab + - five_segment_holds_aux2 + - norm_point_subr_gt0 + - segment_addition + - triangle_ineq + - seg_aux7 + - bet_symmetry_vector + - c'nD_in_unit_disk + - i1'_in_unit_disk + - dist_eq1 + - cong_identity_holds + - segment_construction_holds_aux + - i_neq_i1 + - Rcf_to_GI_PED + - omd_i1i1 + - path: theories/Main/Tarski_dev/Ch14_sum.v + theorems: + - sum_cong + - opp0 + - cong_sum + - sum4_equiv + - proj_pars + - grid_not_par + - diff_uniquenessA + - sum_stable + - change_grid_sum + - double_not_null_not_nul + - pj_trivial + - sum_diff + - path: theories/Elements/OriginalProofs/lemma_ray.v + theorems: + - lemma_ray + - path: theories/Main/Tarski_dev/Ch11_angles.v + theorems: + - l11_19 + - l11_61 + - triangle_strict_reverse_inequality + - not_and_lta + - conga_dec + - l11_61_bis + - triangle_strict_inequality_2 + - l11_62_unicity + - gta_distincts + - col_orth_at__orth_at + - conga_obtuse__obtuse + - l11_21_a + - suppa2__conga123 + - inangle_one_side2 + - conga_preserves_in_angle + - l11_46 + - l11_60_bis + - angle_partition + - cong2_conga_obtuse__cong_conga2 + - lea_line + - conga_sym + - obtuse_chara + - conga_preserves_lta + - symmetry_preserves_conga + - in_angle_reverse + - out2__conga + - cong2_per2__cong + - l11_3 + - l11_31_2 + - cong_lt_per2__lt + - path: theories/Main/Meta_theory/Models/tarski_to_beeson.v + theorems: + - inner_pasch_B + - ColB_Col + - five_segment_B + - path: theories/Main/Tarski_dev/Ch12_parallel.v + theorems: + - par_strict_par + - col_perp2_ncol__col + - not_par_strict_id + - Par_strict_cases + - Par_perm + - par_strict_not_col_2 + - par_strict_right_comm + - path: theories/Main/Tarski_dev/Ch13_5_Pappus_Pascal.v + theorems: + - l13_10 + - l13_10_aux2 + - lcos2_lcos + - path: theories/Main/Annexes/half_angles.v + theorems: + - halfa_chara2 + - out_preserves_halfa + - ghalfa_right_comm + - ghalfa_exists + - cop_halfa_perp__os + - acute_ghalfa2_sams_suma2__ghalfa456 + - path: theories/Algebraic/Counter_models/nD/bet_identity.v + theorems: + - g2_5 + - g2_11 + - g2_1 + - path: theories/Main/Tarski_dev/Ch02_cong.v + theorems: + - not_cong_3421 + - cong_4312 + - cong_reflexivity + - distinct + - path: theories/Main/Annexes/quadrilaterals.v + theorems: + - Square_not_triv + - Rhombus_Rectangle_Square + - Square_Rectangle + - Square_not_triv_3 + - Square_not_triv_2 + - bet_cong_bet + - sym_par + - bet_half_bet + - mid_plg_1 + - path: theories/Main/Tarski_dev/Ch10_line_reflexivity.v + theorems: + - midpoint_preserves_per + - l10_2_existence_spec + - is_image_is_image_spec + - ex_sym + - l10_4_spec + - image_in_gen_is_image + - is_image_spec_col_cong + - path: theories/Main/Highschool/incenter.v + theorems: + - incenter_permut312 + - path: theories/Main/Tactics/CoincR_for_cop.v + theorems: + - ss_ok_empty_for_cop + - path: theories/Main/Highschool/triangles.v + theorems: + - equilateral_strict_swap_3 + - equilateral_strict_swap_2 + - equilateral_isosceles_1 + - isosceles_sym + - equilateral_swap_2 + - equilateral_strict_swap_5 + - equilateral_strict__not_col + - path: theories/Main/Annexes/sums.v + theorems: + - sums_assoc_2 + - sums_diff_1 + - le2_sums2__le34 + - path: theories/Main/Annexes/tangency.v + theorems: + - TangentCC_Col + - interccat__neq + - tangent_construction + - path: theories/Main/Tarski_dev/Ch16_coordinates_with_functions.v + theorems: + - signed_area_AAB + - characterization_of_collinearity_F + - put_neg_in_goal + - twice_signed_area_AAB + - SubF + - twice_signed_area_ABA + - characterization_of_perpendicularity_F + - AM_lower_dim + - signed_area_sum + - signed_area_ABA + - characterization_of_equality_F + - OF + - path: theories/Main/Tarski_dev/Ch06_out_lines.v + theorems: + - bet2_le2__le1346 + - l6_11_existence + - bet_out + - out_bet__out + - bet2_le2__le2356 + - l6_4_2 + - path: theories/Main/Tarski_dev/Ch12_parallel_inter_dec.v + theorems: + - parallel_uniqueness + - l12_23 + - not_par_one_not_par + - par_dec + - l12_20_bis + - path: theories/Algebraic/coplanarity.v + theorems: + - Cop_3412 + - Cop_3214 + - Cop_4213 + - Copl_4312 + - Copl_3142 + - Cop_2413 + - Copl_1324 + - Cop_1243 + - Cop_4123 + - Copl_1432 + - Copl_2134 + - Copl_1342 + - path: theories/Main/Tarski_dev/Ch10_line_reflexivity_2.v + theorems: + - cong_cop_mid_perp__col + - ex_perp_cop + - image_gen_preserves_inter + - image_preserves_midpoint + - path: theories/Elements/OriginalProofs/proposition_37.v + theorems: + - proposition_37 + - path: theories/Main/Tarski_dev/Ch15_lengths.v + theorems: + - thales + - l15_7 + - sign_dec + - length_id_2 + - cong2_lea__le + - signeq__prod_pos + - image_preserves_out + - pythagoras_obtuse_or_per + - path: theories/Main/Tarski_dev/Ch13_3_angles.v + theorems: + - anga_col_out + - ang_distincts + - not_conga_is_anga + - anga_acute + - is_null_ang_out + - is_ang_distinct + - is_anga_conga_is_anga + - null_ang + - anga_distinct + - not_null_not_col + - all_eqaa + - path: theories/Main/Meta_theory/Continuity/archimedes.v + theorems: + - t22_23_aux + - t22_18_aux2 + - t22_18_aux1 + - path: theories/Main/Tarski_dev/Ch03_bet.v + theorems: + - bet_neq23__neq + - path: theories/Main/Tarski_dev/Ch13_4_cos.v + theorems: + - lcos_eq_refl + - lcos2_comm + - lcos3_lg_anga + - lcos_eq_sym + - is_null_anga_dec + - lcos2_exists + - null_lcos + - l13_6 + - obtuse_not_acute + - path: theories/Elements/OriginalProofs/proposition_44.v + theorems: + - proposition_44 + - path: theories/Main/Tarski_dev/Ch16_coordinates.v + theorems: + - bet_betCood_aux + - exists_grid + - cong_3_2_cong_4 + - square_distance_formula + - characterization_of_congruence + - coordinates_of_point + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_pseudo_reflexivity.v + theorems: + - point_equality_decidability + - path: theories/Elements/OriginalProofs/proposition_07.v + theorems: + - proposition_07 + - path: theories/Elements/OriginalProofs/lemma_21helper.v + theorems: + - lemma_21helper + - path: theories/Algebraic/tcp_ndc.v + theorems: + - tcp_os_mab_mmac_b_ssa + - tcp_os_b_ssa_mab_mac + - tcp_os_b_sa_mmab_mac + - tcp_ncol_inplane_2_4 + - path: theories/Main/Highschool/bisector.v + theorems: + - bisector_foot_out_out + - path: theories/Main/Meta_theory/Continuity/completeness.v + theorems: + - extension_reverse_bet + - col2_diff_inj_line__diff + - path: theories/Algebraic/Counter_models/counter_model_bet_identity.v + theorems: + - segment_construction + - not_between_identity + - cong_aux_4 + - euclid + - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_symmetry.v + theorems: + - cong_inner_transitivity + - path: theories/Main/Tarski_dev/Ch09_plane.v + theorems: + - os_out_os + - osp_transitivity + - cop_out__osp + - l9_17 + - two_sides_not_col + - ex_ncol_cop2 + - mid_preserves_col + - cop3_osp__osp + - ts2__ex_bet2 + - invert_two_sides + - cop__one_or_two_sides + - one_side_reflexivity + - bet_cop2__cop + - cop_osp__ex_cop2 + - path: theories/Algebraic/Counter_models/nD/counter_model_cong_inner_transitivity.v + theorems: + - euclid + - cong_identity + - path: theories/Algebraic/Counter_models/Planar/counter_model_upper_dim.v + theorems: + - lower_dim + - path: theories/Main/Highschool/gravityCenter.v + theorems: + - intersection_two_medians_exist_unique + - is_gravity_center_third + - path: theories/Main/Highschool/sesamath_exercises.v + theorems: + - sesamath_4ieme_G2_ex42 + - sesamath_4ieme_G2_ex45 + - sesamath_4ieme_G2_ex44_1 + - sesamath_4ieme_G2_ex37 + - sesamath_4ieme_G2_ex40 + - path: theories/Main/Tarski_dev/Ch13_1.v + theorems: + - perp2_preserves_bet23 + - triangle_mid_par + - path: theories/Coinc/Utils/arity.v + theorems: + - InNth + - InCPOK + - listInd + - consTailCPOK + - circPermNIdAux + - CPLRec + - lastCPIn + - consHeadCP + - path: theories/Main/Meta_theory/Parallel_postulates/thales_converse_postulate_thales_existence.v + theorems: + - thales_converse_postulate__thales_existence + - path: theories/Main/Annexes/vectors.v + theorems: + - one_side_col_out + - same_dir_id + - sum_sym + - plg_plg_bet + - is_to_ise + - path: theories/Algebraic/Counter_models/nD/counter_model_upper_dim.v + theorems: + - oP + - path: theories/Elements/OriginalProofs/lemma_linereflectionisometry.v + theorems: + - lemma_linereflectionisometry + - path: theories/Main/Tarski_dev/Ch14_order.v + theorems: + - pos_inv_pos + - path: theories/Main/Meta_theory/Models/gupta_inspired_to_tarski.v + theorems: + - g2_5 + - path: theories/Main/Meta_theory/Models/tarski_to_gupta_inspired.v + theorems: + - inner_pasch' + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_concyclic.v + theorems: + - not_col_perm_1 + - concy_3 + - path: theories/Main/Annexes/Tagged_predicates.v + theorems: + - Mid_tagged_Mid + - path: theories/Main/Meta_theory/Parallel_postulates/TCP_tarski.v + theorems: + - impossible_case_4_2 + - impossible_two_sides_not_col + - path: theories/Algebraic/Counter_models/nD/counter_model_lower_dim.v + theorems: + - lower_dim + - path: theories/Main/Tarski_dev/Ch08_orthogonality.v + theorems: + - perp_cong + - perp_in_perp + - l8_14_2_1b_bis + - per_dec + - perp_col2 + - perp_in_col_perp_in + - l8_3 + - per_perp_in + - path: theories/Elements/OriginalProofs/proposition_30.v + theorems: + - proposition_30 + - parnotmeet + - path: theories/Main/Meta_theory/Parallel_postulates/rah_rectangle_principle.v + theorems: + - rah__rectangle_principle + - path: theories/Elements/OriginalProofs/lemma_collinearitypreserved.v + theorems: + - lemma_collinearitypreserved + - path: theories/Elements/OriginalProofs/proposition_43.v + theorems: + - proposition_43 + - path: theories/Main/Meta_theory/Models/beeson_to_tarski.v + theorems: + - ICol_chara + - lower_dim + - ICol_dec + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_cop.v + theorems: + - Tarski_is_a_Coinc_predicates_for_cop + - path: theories/Algebraic/Counter_models/Planar/counter_model_five_segment.v + theorems: + - bet_inner_transitivity + - cong_identity + - nbetS_cab + - bet_1Dd + - path: theories/Main/Annexes/midpoint_theorems.v + theorems: + - col_permut132 + - col123_124__col134 + - path: theories/Main/Tactics/CoincR_for_col.v + theorems: + - ss_ok_empty_for_col + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_inner_transitivity.v + theorems: + - cong_pseudo_reflexivity + - path: theories/Main/Tarski_dev/Ch14_prod.v + theorems: + - prod_0_r + - prod_assoc1 + - prod_y_axis_change + - path: theories/Elements/OriginalProofs/lemma_erectedperpendicularunique.v + theorems: + - lemma_erectedperpendicularunique + - path: theories/Main/Meta_theory/Parallel_postulates/legendre.v + theorems: + - legendre_aux2 + - legendre_s_third_theorem + - path: theories/Algebraic/Counter_models/Planar/counter_model_euclid.v + theorems: + - lower_dim + - nbet_bca + - cong_identity + - path: theories/Main/Meta_theory/Parallel_postulates/thales_existence_rah.v + theorems: + - thales_existence__rah + - path: theories/Algebraic/Counter_models/nD/independent_version_to_tarski.v + theorems: + - altIT2D_to_T2D + - Estability + - path: theories/Main/Tarski_dev/Ch13_2_length.v + theorems: + - ex_points_lg_not_col + - is_len_cong + - lg_eql_lg + - path: theories/Main/Highschool/exercises.v + theorems: + - quadrileral_midpoints + - path: theories/Elements/OriginalProofs/proposition_24.v + theorems: + - proposition_24 + - path: theories/Main/Tarski_dev/Ch05_bet_le.v + theorems: + - cong__le3412 + - between_cong_2 + - nlt + - cong__nlt + - le__nlt + - path: theories/Elements/OriginalProofs/proposition_29B.v + theorems: + - proposition_29B + - path: theories/Elements/OriginalProofs/lemma_TGflip.v + theorems: + - lemma_TGflip + - path: theories/Coinc/Utils/sets.v + theorems: + - eq_equiv + - lt + - lt_trans + - STexists_mem_4 + - InCPOCP + - path: theories/Main/Meta_theory/Parallel_postulates/tarski_euclid.v + theorems: + - tarski_s_euclid_implies_euclid_5 + - path: theories/Main/Annexes/inscribed_angle.v + theorems: + - cop_obtuse_onc3__ts + - bet_cong__ghalfa + - path: theories/Elements/OriginalProofs/lemma_legsmallerhypotenuse.v + theorems: + - lemma_legsmallerhypotenuse + - path: theories/Main/Highschool/varignon.v + theorems: + - varignon_aux_aux + - path: theories/Elements/OriginalProofs/proposition_28C.v + theorems: + - proposition_28C + - path: theories/Algebraic/Counter_models/nD/independent_version_to_beeson.v + theorems: + - ITcong_inner_transitivity + - path: theories/Coinc/ColR.v + theorems: + - CTcol_permutation_5 + - proper_3 + - path: theories/Main/Meta_theory/Parallel_postulates/rah_posidonius_postulate.v + theorems: + - rah__posidonius_aux + - path: theories/Main/Meta_theory/Parallel_postulates/parallel_postulates.v + theorems: + - equivalence_of_aristotle_greenberg_and_decidability_of_intersection + - equivalent_postulates_with_decidability_of_intersection_of_lines + - path: theories/Main/Annexes/perp_bisect.v + theorems: + - perp_bisect_is_on_perp_bisect + - perp_bisect_perp + - perp_bisect_existence + - path: theories/Elements/OriginalProofs/lemma_twolines.v + theorems: + - lemma_twolines + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_identity.v + theorems: + - five_segment + - path: theories/Coinc/Utils/general_tactics.v + theorems: + - ltac_something_show + - path: theories/Main/Tarski_dev/Ch07_midpoint.v + theorems: + - midpoint_def + - path: theories/Main/Meta_theory/Parallel_postulates/weak_tarski_s_parallel_postulate_weak_inverse_projection_postulate.v + theorems: + - weak_tarski_s_parallel_postulate__weak_inverse_projection_postulate + - path: theories/Elements/OriginalProofs/proposition_08.v + theorems: + - proposition_08 + - path: theories/Elements/OriginalProofs/proposition_02.v + theorems: + - proposition_02 + - path: theories/Elements/OriginalProofs/lemma_pointreflectionisometry.v + theorems: + - lemma_pointreflectionisometry + - path: theories/Elements/OriginalProofs/lemma_planeseparation.v + theorems: + - lemma_planeseparation + - path: theories/Algebraic/Counter_models/nD/counter_model_five_segment.v + theorems: + - row_mx_head + - path: theories/Main/Meta_theory/Parallel_postulates/SPP_tarski.v + theorems: + - impossible_case_7 + - strong_parallel_postulate_implies_tarski_s_euclid + - path: theories/Algebraic/Counter_models/nD/independence.v + theorems: + - big_pairs_andb_and + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_col.v + theorems: + - col_perm_1 + - path: theories/Main/Tactics/CoincR_for_concy.v + theorems: + - ss_ok_empty_for_concy + - path: theories/Main/Meta_theory/Models/tarski_to_makarios.v + theorems: + - five_segment' + - path: theories/Elements/OriginalProofs/proposition_12.v + theorems: + - proposition_12 + - path: theories/Elements/OriginalProofs/lemma_lessthancongruence2.v + theorems: + - lemma_lessthancongruence2 + - path: theories/Main/Tarski_dev/Ch15_pyth_rel.v + theorems: + - Ps_Col + - path: theories/Algebraic/Counter_models/nD/dimensional_axioms.v + theorems: + - nth_basis + - path: theories/Elements/OriginalProofs/lemma_collinear5.v + theorems: + - lemma_collinear5 + - path: theories/Elements/OriginalProofs/lemma_samenotopposite.v + theorems: + - lemma_samenotopposite + - path: theories/Elements/OriginalProofs/euclidean_tactics.v + theorems: + - nCol_notCol + - path: theories/Algebraic/Counter_models/nD/counter_model_segment_construction.v + theorems: + - ncol__nColT + - path: theories/Elements/OriginalProofs/lemma_samesidecollinear.v + theorems: + - lemma_samesidecollinear diff --git a/GeoCoq/test/local.meta.json b/GeoCoq/test/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..d6cb790d995c5bf29005779bccd986688dbcdd11 --- /dev/null +++ b/GeoCoq/test/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c37442c88d461507f1fe63e4907b411c37e02cc90110acbacbc0675f2e4dbb59 +size 390 diff --git a/GeoCoq/test/local_data_0000010000.json b/GeoCoq/test/local_data_0000010000.json new file mode 100644 index 0000000000000000000000000000000000000000..c5303e4f6dda5235b457331139a214b4538bd20b --- /dev/null +++ b/GeoCoq/test/local_data_0000010000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bb1436aa8520714cb2acc07b83291066eb880936bd823d5a6520cbbf0affd522 +size 38486203 diff --git a/GeoCoq/test/local_data_0000012495.json b/GeoCoq/test/local_data_0000012495.json new file mode 100644 index 0000000000000000000000000000000000000000..af1b49d593a166f055921b211c3be77764ce3ffe --- /dev/null +++ b/GeoCoq/test/local_data_0000012495.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:02631242127130a33ad1f2094dfb2d0d2c6bb1ed420948f1e5dae35c08951c4c +size 17108362 diff --git a/GeoCoq/test/local_lemma_0000000000.json b/GeoCoq/test/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/GeoCoq/test/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/GeoCoq/train/GeoCoq_train.yaml b/GeoCoq/train/GeoCoq_train.yaml new file mode 100644 index 0000000000000000000000000000000000000000..d85c36fcc71a3b8031df7298860b9431bd2c587a --- /dev/null +++ b/GeoCoq/train/GeoCoq_train.yaml @@ -0,0 +1,4835 @@ +name: GeoCoq_train +num_files: 394 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 4036 +datasets: +- project: /GeoCoq + files: + - path: theories/Coinc/Utils/arity.v + theorems: + - minus_n_0 + - app_2_n + - nthCircPerm2 + - app_n_1_app_eq + - app_app_2_n + - circPermNIdFirst + - app_2_n_app + - app_1_n + - PermOK + - circPermNConsTlOK + - fixLastCPOK + - pred_conj + - app_n_1 + - lastTailOK + - app + - allButLastCPTl + - CPToListTl1 + - NoDupOK + - tailCP + - headCPbis + - consHeadCPTl + - minus_n1_n2_0 + - pcaHdTl + - circPermNCPOK + - app_2_n_app_eq + - nthCP01 + - consHeadCPHd + - lastCP + - CPPair + - CPLOK + - nthCircPermNAny + - lengthNilOK + - circPermNId + - nthFirst + - plus_0_n + - tailDefaultCP + - ListToCP + - plus_n_0 + - CPLHdTlOK + - fixLastCP + - CP_ind + - CPLHd + - NoDup_dec + - lengthOfCPToList + - consTailOK + - circPermNCP0 + - circPermNConsOK + - CPLCP + - allButLastCP + - CPToListTl2 + - consHeadCPOK + - consTailCPLast + - PermOKAux + - lastCPTl + - plus_n_1 + - headCP + - consTlHdHdTl + - ListToCPTl + - nthCircPerm2Eq + - app_app_n_1 + - app_n_1_app + - app_1_n_app_eq + - app_1_n_app + - CPCPL + - app_app_1_n + - nthLast + - nthCPTlOK + - NotNoDupDup + - circPermPerm + - nthCircPerm1 + - tailCPbis + - consTailCPTlD + - app_app_2_n_default + - app_2_n_app_default + - consTailCP + - consTailCPTl + - consTailCPAbl + - nthEqOK + - CPToList + - CPToListOK + - consHdTlTlHd + - consTailPerm + - path: theories/Main/Tarski_dev/Ch11_angles.v + theorems: + - l11_50_1 + - lea_distincts + - lea_out4__lea + - lta_not_conga + - lta__lea + - acute_not_per + - orth_symmetry + - bet_conga__bet + - l11_49 + - ex_suppa + - segment_construction_0 + - l11_25 + - cop3_orth_at__orth_at + - conga_inangle2_per__acute + - lta_dec + - col_conga__conga + - orth_at_symmetry + - conga__lea + - l11_17 + - conga_line + - l11_4_2 + - conga_ex_cong3 + - l11_31_1 + - lta_trans + - col2_orth_at__orth_at + - t18_19 + - l11_22 + - in_angle_line + - lea_asym + - l11_22a + - col_conga_col + - conga_pseudo_refl + - orth_at2__eq + - lta__nlea + - le2_per2__le + - acute_per__lta + - l11_4_1 + - t18_18 + - l11_18_2 + - conga_right_comm + - l11_44_2bis + - lta_right_comm + - reflectl__conga + - l11_47 + - inangle_dec + - suppa2__conga456 + - l11_24 + - l11_29_b + - col_conga_cop_reflectl__col + - angle_bisector + - triangle_inequality + - inangle__lta + - conga_distinct + - col_in_angle_out + - acute_conga__acute + - cong3_diff + - acute_cop_perp__one_side + - l11_30 + - cong2_per2__cong_3 + - conga__acute + - in_angle_one_side + - eq_conga_out + - bet_in_angle_bet + - l11_41 + - l11_10 + - inangle_distincts + - acute_one_side_aux + - acute_distincts + - conga_diff45 + - acute_one_side_aux0 + - lt2_per2__lt + - obtuse_distincts + - inangle3123 + - conga2_suppa__suppa + - out341__inangle + - per2__suppa + - acute_obtuse__lta + - lea_total + - acute_trivial + - conga_os__out + - l11_15 + - conga_comm + - ncol_conga_ncol + - lta_os__ts + - conga_inangle_per__acute + - l11_57 + - bet_le__lt + - bet_out__suppa + - ex_conga_ts + - acute_suppa__obtuse + - inangle2__lea + - l11_50_2 + - bet_suppa__out + - conga_refl + - l11_13 + - l11_14 + - lea_in_angle + - cong2_per2__cong_conga2 + - in_angle_asym + - triangle_inequality_2 + - l11_29_a + - nlta__lea + - acute_chara + - conga__lea456123 + - suppa_distincts + - cong4_cop2__eq + - angle_construction_1 + - nlta + - ts2__inangle + - l11_36 + - orth_dec + - tsp_dec + - l11_3_bis + - l11_25_aux + - out_lea__out + - lea_suppa2__lea + - conga_trivial_1 + - inangle1123 + - lea_refl + - l11_43_aux + - angle_construction_2 + - inangle_one_side + - l11_28 + - in_angle_trans + - orth_at__ncop + - suppa_comm + - orth_at__ncop1 + - conga2_cop2__col_1 + - obtuse_out2__obtuse + - col2_orth__orth + - orth_at_distincts + - l11_44_2_b + - l11_44_2 + - conga_sym_equiv + - col_inangle2__out + - mid2_orth_at2__cong + - lea_obtuse_obtuse + - conga_diff2 + - bet_obtuse__acute + - lea121345 + - lea__nlta + - obtuse_suppa__acute + - per_suppa__per + - lta_comm + - l11_44_2_a + - col_in_angle + - conga_diff56 + - lea_dec + - osp_dec + - lea_left_comm + - col_lta__out + - inangle__lea + - two_sides_in_angle + - conga_diff1 + - cong_le_per2__le + - bet2_lta__lta + - conga_cop__or_out_ts + - conga2_cop2__col + - bet__suppa + - col_lta__bet + - lta_distincts + - cong2_conga_cong + - l11_52 + - out__acute + - l11_63_existence + - inangle__lea_1 + - conga_left_comm + - not_conga + - os_ts__inangle + - orth_at_chara + - obtuse_per__lta + - acute_lea_acute + - cop_inangle__ex_col_inangle + - in_angle_two_sides + - cong2__ncol + - lea_right_comm + - triangle_strict_inequality + - col_cop_orth__orth_at + - l11_41_aux + - cong3_preserves_out + - lea_trans + - cong3_conga + - l11_44_1 + - col_cop_orth_at__eq + - in_angle_trans2 + - orth_distincts + - conga_trans + - l11_44_1_b + - l11_51 + - l11_43 + - l11_22b + - l8_21_3 + - lea123456_lta__lta + - out321__inangle + - suppa_left_comm + - lea_comm + - acute_bet__obtuse + - acute_col__out + - lta_left_comm + - triangle_reverse_inequality + - obtuse_sym + - l11_60_aux + - l11_53 + - col_obtuse__bet + - l11_21_b + - orth_at2_tsp__ts + - os2__inangle + - not_conga_sym + - suppa_right_comm + - bet_lea__bet + - l11_62_existence + - l11_22_bet + - l11_60 + - l11_16 + - suppa_dec + - l11_63_aux + - l11_44_1_a + - l11_aux + - acute__not_obtuse + - lea456789_lta__lta + - nlea__lta + - not_lta_and_conga + - os3__lta + - angle_construction_4 + - l11_62_unicity_bis + - or_lta2_conga + - conga_cop_out_reflectl__out + - cong3_conga2 + - acute_out2__acute + - le_lt_per2__lt + - path: theories/Main/Tarski_dev/Ch13_4_cos.v + theorems: + - lcos_morphism + - perp_acute_out + - lcos_per + - lcos_const0 + - lcos_eqa_lcos + - lcos2_eq_lcos3_eq + - lcos_lg_distincts + - acute_not_obtuse + - lcos3_eq_sym + - lcos3_permut2 + - lcos_lg + - lcos2_eq_trans + - out_acute + - lcos_const_o + - lcos_lg_anga + - lcos3_eql_lcos3 + - lcos_eql_lcos + - perp_out__acute + - acute_comp_not_acute + - perp_acute + - eql_lcos_null + - lcos2_lg_not_null + - lcos2_eq_sym + - lcos_eq_lcos3_eq + - lcos_const_cb + - lcos_eq_lcos2_eq + - lcos_const + - lcos3_lcos_1_2 + - lcos2_exists' + - null_lcos_eql + - lcos_lg_not_null + - lcos_const1 + - l13_7 + - lcos_const_ab + - lcos3_lg_not_null + - lcos3_permut3 + - lcos_exists + - lcos2_eq_refl + - flat_not_acute + - perp_obtuse_bet + - lcos2_lg_anga + - lcos3_exists + - lcos_uniqueness + - lcos_const_a + - lcos2_eql_lcos2 + - lcos3_uniqueness + - lcos_eq_trans + - lcos3_eq_refl + - lcos3_lcos_2_1 + - path: theories/Main/Tarski_dev/Ch06_out_lines.v + theorems: + - l6_2 + - l6_7 + - diff_col_ex3 + - colx + - l6_3_2 + - out2_bet_out + - out_diff2 + - not_bet_and_out + - out_cong_cong + - l6_21 + - out2__bet + - l6_13_1 + - segment_reverse + - cong_preserves_bet + - col_transitivity_1 + - l6_16_1 + - Out_cases + - bet2__out + - out_diff1 + - segment_construction_3 + - out_distinct + - bet2_out_out + - out_bet_out_1 + - out_col + - diff_col_ex + - out_bet_out_2 + - l6_3_1 + - bet_out_1 + - out_dec + - out_to_bet + - out_trivial + - not_bet_out + - not_out_bet + - l6_13_2 + - or_bet_out + - l6_11_uniqueness + - bet2_le2__le1245 + - bet_out__bet + - l6_6 + - bet_out_out_bet + - col_transitivity_2 + - col3 + - not_col_exists + - col2__eq + - diff_bet_ex3 + - col_out2_col + - l6_4_1 + - path: theories/Algebraic/POF_to_Tarski.v + theorems: + - dotC' + - markov_betE + - dot2B + - col_2D_aux + - betS_gt0 + - b2D_neq0 + - col_2D_aux_1 + - contraction_betR + - betE_axx + - Rcf_to_GI_PED + - ratio_cp'_aux_1 + - Rcf_to_GI_euclidean + - betSP' + - cong_identity + - ratio_inv + - addrDBB + - inner_pasch + - congP_aux + - upper_dim_dgc1 + - quad_ge0 + - point_equality_decidability + - ratio_eq + - ratiovv + - b_neq0 + - upper_dim_dgc_aux0 + - ratio_cp + - add2r_eq + - bet_betE + - ab_neq + - bet_inner_transitivity + - betS_neq12 + - Rcf_to_T_PED + - betSP + - cong_perp + - bet_neq0 + - extension_col + - add_ratio_1 + - lower_dim + - bet_cong_ratio_eq + - euclid' + - dotE + - ratio_e0_n1 + - ratio_bet'' + - cong_perp_aux2 + - inner_pasch' + - vector2_eq0 + - bet_gt0 + - betEP + - congP + - col_2D + - contraction_bet + - inner_pasch_gen + - tcp + - betR_abc + - thing_gt0 + - upper_dim_dgc1_aux + - contraction_eq + - bet_lt + - segment_construction + - ratioNr + - bet_sym + - row2_eq_nD + - cong_eq' + - add11_neq0 + - bet_opp + - ratio_betS + - all_v_neq0 + - ratio_bet + - a2D_eq0 + - ratioP + - upper_dim_dgc_aux1 + - stuff_gt0 + - betE_sym + - dotvN + - dotNv + - cosine_rule + - congP_aux' + - Rcf_to_T2D + - addrBDD + - add_ratio + - five_segment + - stuff_neq0 + - ca_neq + - addf_divrr + - ratiorN + - bet_lt1 + - dot2N + - betS_neq23 + - betS_neq13 + - thing_neq0 + - ratio0p + - bet_col + - bet_neq0' + - cong_pseudo_reflexivity + - bet_gt0' + - addrBDB + - betS_extension + - upper_dim + - quad_eq0' + - quad_neq0 + - proclus + - euclid + - ratio_lt0_v2_neq0 + - extension_bet + - cosine_rule' + - dot2_eq0 + - upper_dim_dgc2_aux + - betS_id + - dotC + - ratio_lt0_v1_neq0 + - Rcf_to_T_euclidean + - row2_eq0_nD + - col_2D' + - Rcf_to_GI2D + - betSP'_aux + - extensionP + - bet_xxa + - funmxN + - betE_xxa + - bet_xax + - Rcf_to_T + - sub_1_ratio + - eq_pick_neq0 + - eq_inv_scale + - ratio_e1_n0 + - cong_inner_transitivity + - ab_neq_2D + - c_neq0 + - ca_neq_2D + - bc_neq_2D + - vector2_eq + - congC + - addrDBD + - col_2D_aux_2 + - betR_bca + - upper_dim_dgc2 + - col_2D_aux_3 + - bet_ratio + - bc_neq + - inner_pasch'' + - ratio_cp' + - dot2D + - a_eq0 + - ratio_neq0 + - row2_eq + - vector2_neq0 + - bet_trans + - betR_cab + - path: theories/Main/Tarski_dev/Ch14_sum.v + theorems: + - diff_stable + - pj_left_comm + - sum3_col + - sum_uniquenessB + - sum_to_sum3 + - opp_uniqueness + - proj_preserves_sum + - sum_x_axis_unit_change + - proj_col + - sum_cong2 + - project_trivial + - sum_ar2 + - midpoint_opp + - sum_to_sump + - project_col_project + - sum_assoc + - sum22_comm + - change_grid_sum_0 + - sum_A_B_A + - sum_opp + - sum_diff2_diff_sum2_b + - sum_diff_diff_a + - opp_comm + - sum_A_B_B + - sum_O_B_eq + - diff_opp + - sum_plg + - sum_par_strict + - pj_comm + - Pj_exists + - opp_exists + - sum_A_exists + - sum_B_exists + - opp0_uniqueness + - sum_A_null + - diff_uniqueness + - diff_to_plg + - sum_diff_diff + - sum_y_axis_change + - pj_right_comm + - sum_A_O + - sum3_to_sum4 + - sum_uniquenessA + - double_null_null + - sum22_col + - sum3_comm_1_2 + - diff_A_O + - sum22_permut + - sum_O_B + - sum_abcd + - diff_null_eq + - sum_comm + - pj_uniqueness + - pj_col_project + - sum_iff_cong + - sum_A_O_eq + - sum_assoc_2 + - sum_O_O + - sum4_comm + - opp_midpoint + - plg_to_sum + - diff_exists + - diff_null + - not_null_double_not_null + - diff_sum + - sum3_comm_2_3 + - sum4_col + - sum3_exists + - sum_B_null + - sum3_permut + - diff_ar2 + - sum_diff_diff_b + - sum_assoc_1 + - sump_to_sum + - proj_id + - diff_O_A_opp + - sum3_uniqueness + - sum_uniqueness + - diff_uniquenessB + - diff_O_A + - sum4_permut + - sum_diff2_diff_sum2_a + - path: theories/Main/Meta_theory/Continuity/angle_archimedes.v + theorems: + - grada_distincts + - archi__gradaexp_destruction + - grada_out__out + - grada__lea + - gradaexp_destruction_aux + - grada2_sams_suma__grada + - archi__grada_destruction + - acute_archi_aux1 + - acute_archi_aux + - conga2_grada__grada + - acute_archi_aux2 + - angles_archi_aux1 + - archi_in_angles + - gradaexp__grada + - path: theories/Main/Annexes/saccheri.v + theorems: + - per2_os__pars + - acute_sac__aah + - sac__perp3414 + - t22_17__oah + - t22_7__per + - lam6534_mid2__sac + - lam__pars1234 + - sac__perp1214 + - t22_12__rah + - lt_os_per2__lta + - conga_per2_os__cong + - t22_14__rah + - t22_8__lt5612 + - t22_7__lt5612 + - t22_12__aah + - acute_lam__lt + - t22_9_aux + - mid2_sac__perp_lower + - absolute_exterior_angle_theorem + - t22_7__obtuse + - lam_per__rah + - lam_obtuse__lt + - cong_mid__suma + - t22_20 + - cong2_sac2__cong + - t22_8__acute + - t22_14__aah + - t22_14__sams_nbet_aux + - ex_lambert + - cong_sac__per + - lta123234_os_per2__lt + - not_rah + - obtuse_sac__oah + - mid2_sac__lam6534 + - lam_obtuse__oah + - lt4321_os_per2__lta + - t22_14__nsams + - t22_9__acute + - mid2_sac__perp_upper + - t22_11__obtuse + - per2_os__ncol234 + - lt_sac__obtuse + - lam__par1423 + - lam_perm + - per__ex_saccheri + - sac_perm + - t22_14_aux + - t22_17__rah + - t22_12__oah + - t22_8__obtuse + - t22_14__bet + - lta_os_per2__lt + - t22_14__sams_nbet + - lam_acute__aah + - t22_14__nsams_aux + - t22_7__lt1256 + - t22_9__per + - not_aah + - t22_8_aux + - sac__cong + - lam__pars1423 + - sac__par1423 + - not_oah + - t22_11__per + - cong_lam__per + - sac__conga + - t22_11__acute + - per_sac__rah + - lam__os + - t22_8__per + - t22_9__obtuse + - lt_sac__acute + - cong2_lam2__cong_conga + - lam6521_mid2__sac + - sac__pars1423 + - saccheri_s_three_hypotheses + - three_hypotheses_aux + - sac_distincts + - lam_lt__obtuse + - lam_per__cong + - t22_8__lt1256 + - lam__par1234 + - ex_saccheri + - t22_17__aah + - t22_8__cong + - t22_7__acute + - per2_os__ncol123 + - lam_lt__acute + - cop_sac2__sac + - path: theories/Main/Tarski_dev/Ch12_parallel_inter_dec.v + theorems: + - par_trans + - not_par_inter + - cop_npar__inter + - l12_19 + - col_par_par_col + - l12_21_a + - not_par_inter_exists + - par_not_par + - cop_npars__inter_exists + - not_par_strict_inter_exists + - l12_20 + - l12_22_a + - l12_16_2D + - cop2_npar__inter + - cop_par__inter + - cop_npar__inter_exists + - cop_par_perp__perp + - l12_21 + - cop4_par_perp2__par + - par_perp__perp + - par_perp2__par + - par_inter + - l12_16 + - l12_22 + - inter__npar + - path: theories/Main/Tarski_dev/Ch16_coordinates_with_functions.v + theorems: + - centroid_theorem + - twice_signed_area_ABB + - sum_f + - characterization_of_perpendicularity_F_bis + - field_prop + - characterization_of_neq_F + - characterization_of_neq_F_bis + - MulF + - characterization_of_right_triangle_F + - characterization_of_midpoint_F + - triangle_area + - leF_Transitive + - coordinates_of_point_F + - sqrt3_square + - characterization_of_equality_F_bis + - axiom_A2b + - DivF + - ringF + - diff_col + - characterization_of_parallelism_F_bis + - eqF_Equivalence + - field_prop_1 + - neq20 + - oppF_neq0 + - characterization_of_parallelism_F_aux + - signed_area_ABB + - triangles_same_base + - Cd_Cd_EqF + - eqF_Transitive + - prod_col + - twice_signed_area_cyclic + - characterization_of_equality_F_aux + - pythrel_null + - sum_col + - inv_f + - Perp_AM_Perp + - div_uniqueness + - neqO_mul_neqO + - InvF + - diff_f + - AM_Perp_AM_Perp_AM_Par + - neg_and_eqF + - fieldF + - co_side + - FRing + - div_f + - chasles_ratios + - Par_AM_Par + - inv_uniqueness + - Fcri + - AddF + - inv_col + - ratio_zero + - prod_f + - pythrel_not_null + - characterization_of_congruence_F + - field_prop_zero + - eqF_Reflexive + - Per_AM_Per + - eq_dec_F + - Py_triv_ABB + - Cong_AM_Cong + - characterization_of_betweenness_F + - AM_Perp_triv1 + - PythF + - signed_area_cyclic + - Pyth_f + - mulF_morphism + - exists_equilateral_triangle + - ncolOEE' + - axiom_A2a + - AM_Perp_triv2 + - OneF + - divF_morphism + - addF_morphism + - invF_morphism + - Fintegral + - AM_Par_ratio_AM_Par + - opp_f + - div_exists + - mulF__eq0 + - div_col + - leF_Antisymmetric + - subF__eq0 + - AM_perp_AM_Par_AM_perp + - subF_morphism + - coordinates_of_point_f + - opp_col + - Ps_One + - eqF_Symmetric + - PythFOk + - pythrelOO + - Fmult_integral + - characterization_of_parallelism_F + - OppF + - perp_triangle_area + - inv_exists_with_notation + - opp_pythrel + - oppF_morphism + - supplement_AM_CongA + - path: theories/Main/Highschool/circumcenter.v + theorems: + - midpoint_thales_reci_circum + - is_circumcenter_uniqueness + - is_circumcenter_coplanar + - is_circumcenter_perm_4 + - is_circumcenter_perm_3 + - circumcenter_per + - is_circumcenter_perm_5 + - is_circumcenter_perm + - circumcenter_cong + - circumcenter_perp_all + - circumcenter_perp + - is_circumcenter_cases + - is_circumcenter_perm_1 + - exists_circumcenter + - circumcenter_intersect + - is_circumcenter_perm_2 + - path: theories/Algebraic/Counter_models/Planar/counter_model_five_segment.v + theorems: + - euclid + - bet_1D + - five_segment + - all_lines_meet + - col_xxa + - bet'P + - bet'_sym + - segment_construction + - bet_1Di + - nbet_cab + - nbet_bca + - bet_symmetry + - nbetS_bca + - cong_inner_transitivity + - col_213 + - neq_mx10 + - col_axx + - inner_pasch + - line_intersection_1 + - nbetS_abc + - permP + - col_xax + - cong_pseudo_reflexivity + - neq_mx + - point_equality_decidability + - dup_meet + - upper_dim + - nbet_abc + - lower_dim + - line_intersection_2 + - line_intersection_3 + - path: theories/Main/Elements_statements/Book_3.v + theorems: + - prop_11_12 + - prop_3_1 + - prop_6 + - prop_18 + - prop_5 + - prop_4 + - prop_2 + - prop_9 + - prop_3_2 + - path: theories/Elements/OriginalProofs/lemma_8_3.v + theorems: + - lemma_8_3 + - path: theories/Algebraic/Counter_models/nD/bet_identity.v + theorems: + - g2_16 + - g2_3 + - bet_identity_aux + - g2_14 + - g2_13 + - g2_7_2 + - g2_4 + - bet_outer_trans + - g2_6_1 + - g2_10 + - g2_9 + - g2_7_1 + - g2_9_aux + - g2_8 + - g2_12 + - g2_15 + - another_point + - path: theories/Main/Tarski_dev/Ch08_orthogonality.v + theorems: + - diff_per_diff + - perp_in_left_comm + - l8_22 + - l8_14_2_1a + - perp_in_comm + - perp_col4 + - Perp_in_cases + - perp_right_comm + - Per_perm + - perp_in_per_1 + - per_cong + - perp_perp_in + - perp_in_right_comm + - perp_not_eq_2 + - perp_per_1 + - l8_10 + - perp_in_per_4 + - per_perp + - l8_24 + - l8_13_2 + - perp_in_perp_bis + - l8_5 + - l8_9 + - l8_18_uniqueness + - perp_in_id + - perp_in_per + - l8_2 + - l8_14_3 + - l8_4 + - perp_not_eq_1 + - l8_20_1 + - l8_21_aux + - perp_col2_bis + - midpoint_distinct + - l8_18_existence + - l8_15_2 + - perp_per_2 + - perp_col1 + - col_col_per_per + - perp_in_sym + - l8_15_1 + - perp_distinct + - midpoint_existence_aux + - perp_in_per_2 + - Perp_in_perm + - perp_not_col2 + - Perp_cases + - per_not_colp + - perp_col0 + - l8_12 + - perp_vector + - perp_left_comm + - col_per2__per + - perp_not_col + - Per_cases + - Perp_perm + - per_distinct + - perp_in_distinct + - per_cong_mid + - l8_14_2_2 + - l8_14_2_1b + - cong_perp_or_mid + - l8_6 + - perp_in_col + - per_col + - l8_7 + - perp_col + - perp_in_dec + - col_per_perp + - l8_22_bis + - per_double_cong + - perp_sym + - perp_in_per_3 + - midpoint_existence + - l8_20_2 + - l8_21 + - l8_14_1 + - perp_comm + - per_not_col + - path: theories/Main/Tarski_dev/Ch13_3_angles.v + theorems: + - anga_out_anga + - acute_not_bet + - eqA_equivalence + - ang_not_null_lg + - ang_const_o + - not_conga_is_ang + - ang_cong_ang + - ang_sym + - ex_eqaa + - ang_const + - is_ang_conga + - is_null_all + - null_anga_null_anga' + - null_anga + - ex_eqa + - anga_col_null + - not_flat_ang_def_equiv + - not_null_ang_ang + - flat_ang + - not_null_ang_def_equiv + - is_anga_to_is_ang + - ex_ang + - anga_distincts + - anga_sym + - ang_not_lg_null + - is_anga_conga + - out_out_anga + - out_is_len_eq + - not_cong_is_ang1 + - is_null_anga_out + - anga_exists + - out_null_anga + - ang_exists + - anga_not_flat + - not_conga_not_ang + - bet_flat_ang + - is_ang_conga_is_ang + - out_null_ang + - ex_points_ang + - all_eqa + - anga_const + - eqA_preserves_anga + - not_cong_is_anga1 + - out_len_eq + - anga_conga + - ang_conga + - ex_anga + - anga_const_o + - ang_distinct + - ex_points_anga + - anga_is_ang + - anga_not_lg_null + - path: theories/Algebraic/Counter_models/nD/counter_model_euclid.v + theorems: + - t'_in_unit_disk + - segment_addition' + - omd_ge0 + - point_equality_decidability + - omd_i1i2 + - cong_inner_transitivity + - euclid + - dist_neq1 + - point_vector_neq + - p_scalar_lt1 + - omd_i2i2 + - cauchy_schwartz + - E_in_disk + - cong_v_bcaa + - five_segment_holds + - segment_construction_holds_aux4 + - cong_aaxx + - omd_ii + - bet_symmetry + - cong_pseudo_reflexivity_vector + - bet_outer_transitivity' + - bet_intersect + - inner_pasch_holds + - segment_construction_holds_aux3 + - five_segment_holds_aux + - dist_eq1' + - addcb_eqt + - segment_construction_holds_aux6 + - b'_in_unit_disk + - nbet_cab + - i'_in_unit_disk + - bet_b1d1c + - b'c'_neq + - cong_id + - bet_in_disk + - b'nD_in_unit_disk + - cong_cdab + - divf_le1 + - omd_oi + - pasch + - five_segment_holds_aux3 + - roots_poly2 + - cong_v_aabc + - betR_abk + - bet_adt + - bet_abx_x01_lt1 + - omd_cb + - one_half_gt0 + - cong_identity + - c'_in_unit_disk + - three_non_collinear_points + - cong_pseudo_reflexivity + - sqrt_invf + - b'nD_neq0 + - a'_in_unit_disk + - inner_pasch_vector + - scaler_eq + - bet_abx_x01_gt + - inner_pasch + - five_segment_holds_aux4 + - omd_eq0 + - one_half_neq0 + - segment_construction + - quad_kNN + - divf_eq1 + - dist_reflexivity + - cong_inner_transitivity' + - one_sub_half + - omd_oi2 + - betR_abc + - one_half_ge0 + - dist_cong' + - euclid_fails + - norm_sqr + - betR_cab + - omd_reflexivity + - Rcf_to_GI2D + - o'_in_unit_disk + - quad_NN + - cong_inner_transitivity_vector + - invf_eq + - bet_dist_bet' + - betR_akb + - invf_lt_invf + - bet_inner_transitivity + - cong_bacd + - Rcf_to_GI_PED' + - one_quarter_lt_one_half + - lower_dim_holds + - omd_ca + - addr_ge_gt0 + - norm_point_lt1 + - c'_norm + - five_segment + - dist'_ge0 + - segment_construction_holds_aux2 + - omd_oo + - one_eighth_lt_one_quarter + - xt_neq + - a'_eq0 + - bet_2D_extension_2D + - omd_cc + - b'_neq0 + - omd_v_reflexivity + - norm_point_le1 + - bet_inner_transitivity_vector + - a'nD_in_unit_disk + - five_segment_holds_aux5 + - omd_oi1 + - upper_dim_aux + - betR_bca + - add_halfhalf + - nbet_abc + - three_non_collinear_points_vector + - upper_dim_aux_2 + - one_half_lt1 + - unit_disk_nD_2D + - dist_gt0 + - euclid_aux + - point_vector_eq + - congA_axt_ab1d1 + - omd_gt0 + - euclid_fails_2D + - dist_v_reflexivity + - betR_neq0 + - cong'_abab + - one_side_cong_eq + - Col_Col + - bet_outer_transitivity + - quarter + - segment_construction_holds_aux5 + - norm_ge0 + - omd_ii2 + - cong_identity_holds_aux + - dist_le1 + - euclid_fails_aux + - segment_construction_holds_aux1 + - divf_lt1 + - dist_aa_eq1 + - x_neq0 + - poly_cong + - bet_bdc + - cauchy_schwartz_eq + - segment_construction_holds + - seg_aux8 + - X_in_disk + - bet_abx_x00_eq0 + - path: theories/Elements/OriginalProofs/lemma_lessthantransitive.v + theorems: + - lemma_lessthantransitive + - path: theories/Main/Tarski_dev/Ch04_cong_bet.v + theorems: + - l4_3 + - cong3_bet_eq + - l4_5 + - l4_2 + - l4_6 + - l4_3_1 + - path: theories/Main/Tarski_dev/Ch15_lengths.v + theorems: + - pythagoras_obtuse + - not_triangular_equality1 + - square_increase_rev + - ltp_to_lep + - inter_circle_per + - pos_neg__prod_neg + - length_uniqueness + - length_sym + - length_id_1 + - ltp__ltps + - length_cong + - inter_circle + - square_increase_strict + - prod_pos__signeq + - l15_7_1 + - lt_to_ltp + - length_existence + - image_preserves_col + - not_signEq_prod_neg + - l15_3 + - length_out + - conga_bet_conga + - ltp__lep_neq + - length_leP_le_2 + - triangular_equality_equiv + - sum_preserves_lep_rev + - inter_circle_obtuse + - length_not_neg + - square_increase + - lta_out_lta + - lep_neq__ltp + - leP_bet + - circle_projp_between + - pythagoras + - triangular_equality_bis + - ltp__diff_pos + - length_id + - sum_pos_null + - prod_col + - length_pos_or_null + - sum_preserves_ltp_rev + - length_eq_cong_1 + - prod_ng___not_signeq + - lea_out_lea + - ltps__ltp + - inter_tangent_circle + - length_Ps + - root_uniqueness + - not_neg_pos + - length_eq_cong_2 + - length_O + - ltp_to_lt + - length_Ar2 + - is_length_exists + - length_pos + - pythagoras_acute + - diff_pos__ltp + - project_preserves_out + - triangular_equality + - length_not_col_null + - image_preserves_bet1 + - sum_preserves_lep + - l15_7_2 + - projp_lt + - sum_preserves_ltp + - length_leP_le_1 + - square_not_neg + - path: theories/Main/Highschool/concyclic.v + theorems: + - concyclic_perm_2 + - concyclic_perm_9 + - concyclic_perm_5 + - concyclic_perm_7 + - concyclic_perm_3 + - concyclic_perm_19 + - concyclic_perm_10 + - concyclic_trans + - concyclic_perm_8 + - concyclic_perm_18 + - concyclic_perm_11 + - concyclic_perm_4 + - concyclic_perm_21 + - concyclic_perm_16 + - concyclic_aux + - concyclic_perm_15 + - concyclic_perm_23 + - concyclic_1123 + - concyclic_perm_20 + - concyclic_perm_6 + - concyclic_perm_1 + - concyclic_perm_13 + - concyclic_perm_14 + - concyclic_perm_12 + - concyclic_perm_22 + - path: theories/Elements/OriginalProofs/lemma_collinear1.v + theorems: + - lemma_collinear1 + - path: theories/Main/Annexes/circles.v + theorems: + - bet_cop_onc2__ex_onc_os_out + - inc112 + - mid_chord__diam_or_ncol + - outcs_exists1 + - circle_cases + - line_circle_two_points + - eqc_chara + - tree_points_onc_cop + - cong2_onc3__eq + - onc2__cong + - concyclic_pseudo_trans + - col_onc2__mid + - onc_not_center + - mid2_onc4__eq + - outcs_exists + - incs_exists + - inc_onc2_out__eq + - tree_points_onc_cop2 + - diam_points + - symmetric_oncircle + - onc212 + - chord_completion + - chord_lt_diam + - bet_onc_le_a + - onc_exists + - bet_onc_lt_b + - col_onc2_perp__mid + - ninc__outcs + - prop_7_8 + - bet_onc_lt_a + - inc__radius + - onc2_mid__incs + - center_onc2_mid__ncol + - mid_onc2__per + - chords_midpoints_col_par + - inc__noutcs + - incs__inc + - inc__outc + - inc_eq + - bet_inc2__incs + - inc2_le_diam + - bet_cong_onc3_cases + - mid_onc2_perp__col + - onc3_mid2__ncol + - eqc_trans + - incs2_lt_diam + - inc_outc__onc + - chord_le_diam + - onc_sym + - diam_cong_incs__outcs + - onc__outc + - bet_onc_le_b + - outc__nincs + - onc2_out__outcs + - chord_intersection + - concyclic_gen_pseudo_trans + - mid_onc__diam + - diam__midpoint + - concyclic_gen_perm_2 + - concyclic_gen_trans_1 + - onc2_mid_cong_col + - concyclic_perm_2 + - cop2_onc6__eqc + - outcs__ninc + - onc4_cong2__eq + - neqc_chara + - concyclic_trans_1 + - cop_mid_onc2_perp__col + - col_inc2_outcs__out + - circle_circle_cop + - eqc_refl + - incs_exists1 + - diam_end_uniqueness + - cong2_cop2_onc3__eq + - diam_sym + - cong_chord_cong_center1 + - incs__outcs + - concyclic_aux + - cong_chord_cong_center + - col_inc_onc2__bet + - concyclic_perm_1 + - diam_exists + - cong_onc3_cases + - onc_col_diam__eq + - mid_onc2__perp + - onc3__ncol + - onc_sym__onc + - incs__noutc + - eqc_sym + - incs_onc_diam__lt + - tree_points_onc + - concyclic_gen_perm_1 + - center_col__diam + - path: theories/Main/Tarski_dev/Ch09_plane.v + theorems: + - sym_preserve_diff + - col2_cop__cop + - bet_cop__tsp + - tsp__nosp + - two_sides_dec + - l9_41_1 + - out_one_side + - tsp_exists + - l9_9 + - l9_10 + - col_one_side_out + - l9_4_2 + - col_preserves_two_sides + - l9_3 + - out_two_sides_two_sides + - tsp__ncop2 + - cop2_os__osp + - l8_21_bis + - l9_19_3 + - cop3_tsp__tsp + - osp__ncop2 + - l9_4_2_aux + - l9_18 + - l9_9_bis + - tsp_distincts + - sac__coplanar + - one_side_transitivity + - os_ts1324__os + - os__coplanar + - col2_os__os + - cop3_tsp__ts + - l9_30 + - bet_ts__os + - one_or_two_sides_aux + - col_one_side + - col123__nos + - ts__ncol + - l9_18_3 + - bet__ts + - one_side_dec + - bet_ts__ts + - cop_tsp__ex_cop2 + - ex_ncol_cop + - not_two_sides_id + - os_distincts + - cong3_cop2__col + - l9_8_2 + - invert_one_side + - cop_nts__os + - l9_31 + - per_mid_per + - out_out_one_side + - osp__ncop1 + - ts_ts_os + - cop_perp2__col + - osp_bet__osp + - cop3_osp__os + - tsp__ncop1 + - one_side_chara + - col_cop__cop + - coplanar_trans_1 + - l9_2 + - ex_diff_cop + - one_side_not_col124 + - col2_cop2__eq + - mid_two_sides + - col_two_sides + - l9_5 + - ncop_distincts + - out_out_two_sides + - osp_symmetry + - osp_reflexivity + - out_one_side_1 + - ts_distincts + - outer_pasch + - cop_nos__ts + - cop_per2__col + - l9_39 + - cop_dec + - l9_41_2 + - l9_4_1_aux + - l9_8_1 + - one_side_not_col123 + - osp__ntsp + - bet_cop__cop + - col124__nos + - cop2_ts__tsp + - col_two_sides_bet + - l9_4_1 + - osp_distincts + - path: theories/Main/Annexes/inscribed_angle.v + theorems: + - conga_cop2_onc4__os + - cop_onc4_ts__suppa + - high_school_exterior_angle_theorem + - onc3_ts__obtuse + - inscribed_angle_aux1 + - not_obtuse_saccheris + - suma123231__sams + - inscribed_angle_aux + - trisuma__bet + - cop2_onc4__or_conga_suppa + - thales_theorem + - right_saccheris + - onc3_os__acute + - cop2_onc4_suppa__ts + - conga_cop_onc6_os__eqc + - chord_par_diam + - thales_converse_theorem_1 + - bet__trisuma + - diam_onc2_ts__suppa + - cop_onc4_os__conga + - conga_cop_onc3_os__onc + - bet_suma__suma + - acute_cop_onc3__os + - suppa_ts2__suppa + - thales_converse_theorem + - inscribed_angle + - triangle_circumscription + - inscribed_angle_1 + - conga_os__concyclic + - suppa_ts__concyclic + - path: theories/Elements/OriginalProofs/lemma_parallelcollinear1.v + theorems: + - lemma_parallelcollinear1 + - path: theories/Algebraic/Counter_models/nD/dimensional_axioms.v + theorems: + - iP + - nth_new_basis + - eq_nD_2D + - betR_ud_2D + - lower_dim_all1 + - lower_dim + - to_nD0 + - betR_o_i_basis_nth0 + - dist_10_01 + - upper_dimS + - i_neq_basis_nth0 + - betS_nD_2D + - oP + - to_nD_scale + - lower_dim_all2 + - upper_dim + - to_nDB + - pick_to_nD + - bet_nD_2D + - bet_o_i_basis_nth0 + - betR_nD_2D + - basis_nth1 + - basis_nth0 + - path: theories/Main/Annexes/quadrilaterals.v + theorems: + - plg_trivial + - Rectangle_not_triv + - not_col_sym_not_col + - plgf3_mid + - col_not_plgs + - plgs_irreflexive + - plg_existence + - parallelogram_strict_not_col + - plgf_permut + - plgf_trivial1 + - Rectangle_triv + - plg_trivial1 + - plgf_not_comm + - rhombus_cong_square + - midpoint_par_strict + - cong3_id + - plgs_one_side + - plgf_cong + - plg_col_plgf + - plgf_trivial + - bet_double_bet + - plgs_diff + - cong_identity_inv + - midpoint_preserves_bet + - mid_plgf_aux + - plgf_trivial_trans + - plg_bet1 + - bet3_cong3_bet + - symmetry_preserves_two_sides + - symmetry_preserves_one_side + - plg_sym + - symmetry_preseves_bet2 + - plgs__pars + - Square_Parallelogram + - plgf_mid + - Parallelogram_strict_Parallelogram + - mid_par_cong2 + - Rhombus_Plg + - Rectangle_not_triv_2 + - Kite_comm + - Rectangle_Plg + - plgf_irreflexive + - col_cong_mid + - symmetry_preserves_bet + - plgf_not_point + - parallelogram_strict_not_col_3 + - plgs_not_col + - parallelogram_strict_not_col_4 + - plg_cong_rectangle + - symmetry_preseves_bet1 + - plgf_bet + - plgs_sym + - plgf_sym + - mid_par_cong + - plgf_trivial2 + - mid_par_cong1 + - mid_plgf + - plgf_trivial_neq + - midpoint_midpoint_col + - mid_plgs + - plg_irreflexive + - midpoint_cong_uniqueness + - mid_plg + - plgs_existence + - path: theories/Main/Meta_theory/Models/hilbert_to_tarski.v + theorems: + - outH_sym + - Gupta_inspired_variant_neutral_dimensionless_follows_from_Hilbert + - Hilbert_is_a_Col_theory + - between_one + - not_cut3 + - plane_separation + - lower_dim_l + - ncolH_distincts + - ncolH_expand + - cut_morphism + - plane_separation_2D + - cong_distincts + - soustraction_betH + - H_to_T_PED + - col_colh + - betH_congH3_outH_betH + - cut_distinct + - colH_trivial111 + - cut2_not_cut + - TS_upper_dim + - cong_transitivity + - tarski_s_euclid + - morph + - inter_uniquenessH + - cong_preserves_bet + - H_to_T + - coplanar_plane1 + - betH_distincts + - cut_comm + - outH_expand + - ColH_bets + - IncidL_morphism' + - axiom_five_segmentsH + - congH_sym + - congaH_existence_congH + - IncidL_not_IncidL__not_colH + - five_segment + - cong_identity + - line_on_plane' + - inner_pasch_aux + - TS_upper_dim_bis + - betH_trans1 + - segment_construction + - bet_trans + - colH_trivial112 + - IncidP_morphism' + - congH_perm + - same_side_refl + - colH_dec + - colH_permut_312 + - cong_preserves_col_stronger + - ncolH_exists + - decidability_of_intersectionH + - same_side_trans + - out_same_side + - segment_constructionH + - same_side_morphism + - tarski_upper_dim + - betH_congH2__False + - betH_outH__outH + - th18_aux + - th12 + - ColH__Col + - outH_col + - colH_permut_213 + - same_side_not_cut + - cut'_comm + - cong_inner_transitivity + - cut_exists + - not_betH121 + - out_construction + - th15 + - colH_permut_231 + - Col__ColH + - other_point_on_line + - out_distinct + - cong_permT + - inter_incid_uniquenessH + - betH_trans0 + - col_upper_dim + - betH2_out + - lower_dim_e + - H2D_to_T2D + - upper_dim + - colH_trivial121 + - colH_trivial122 + - th17 + - betH_line + - th19 + - IncidLP_morphismr + - coplanar_plane + - par__or_eq_para + - congH_colH_betH + - colH_permut_321 + - out_same_side' + - out2_out + - bet__beth + - Tarski_3D_follows_from_Hilbert_3D + - th14 + - same_side_prime_refl + - betH_dec + - between_only_one' + - congH_perml + - construction_uniqueness + - coplanar_plane0 + - pasch_general_case + - colH_IncidL__IncidL + - EqL_dec + - betH_trans2 + - betH_not_congH + - IncidLP_morphisml + - plane_coplanar + - colH_permut_132 + - congH_permlr + - cong_sym + - same_side_prime_not_colH + - congaH_sym + - bet_cong3_bet + - cut_same_side_cut + - cong_existence' + - IncidLP_morphism + - same_side__plane + - bet_disjoint + - bet_comm + - cong_preserves_col + - path: theories/Main/Tarski_dev/Ch05_bet_le.v + theorems: + - le_comm + - bet3__bet + - nlt__le + - lt_diff + - le_transitivity + - lt1123 + - lt_right_comm + - l5_2 + - bet2_le2__le + - Lt_cases + - l5_6 + - le1221 + - col_dec + - le_trivial + - bet__lt1213 + - l5_3 + - l5_12_b + - cong3_symmetry + - bet_le_eq + - nle__lt + - le_zero + - bet_cong_eq + - lt_transitivity + - segment_construction_2 + - l5_12_a + - le3456_lt__lt + - le_anti_symmetry + - or_lt_cong_gt + - cong2_lt__lt + - fourth_point + - le_bet + - between_cong + - bet__le2313 + - third_point + - le_right_comm + - gt_comm + - between_cong_3 + - lt_left_comm + - le_diff + - ge_left_comm + - l5_5_1 + - le1234_lt__lt + - lt__nle + - bet_dec + - lt__le + - not_and_lt + - cong_dec + - ge_comm + - le_cases + - Le_cases + - l5_5_2 + - bet__le1213 + - cong__le + - l5_1 + - ge_right_comm + - bet__lt2313 + - gt_right_comm + - le_left_comm + - gt_left_comm + - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_symmetry.v + theorems: + - col_xxa + - bet_inner_transitivity + - point_equality_decidability + - bet_col' + - coplanarP + - inner_pasch + - cong_pseudo_reflexivity + - euclid + - five_segment + - col_perm_2 + - A10_segment_construction + - lower_dim + - col_2D_bet' + - bet_on_abscissa + - col_trans + - colP + - upper_dim + - ncolP + - par_strictP + - cong_identity + - col_perm_1 + - bet'_col_2D + - bet_symmetry + - path: theories/Main/Annexes/sums.v + theorems: + - le_lt12_sums2__lt34 + - le_lt56_sums2__lt12 + - sums__eq12 + - le2_sums2__le + - sums112323 + - sums_comm + - sums_assoc + - sums2__cong34 + - sums_right_comm + - eq_sums__eq + - lt2_sums2__lt12 + - le2_sums2__le12 + - sums__cong1245 + - sums__le3456 + - le2_sums2__cong34 + - sums2__cong56 + - le_lt56_sums2__lt34 + - sums123312 + - lt2_sums2__lt34 + - sums__cong2345 + - lt2_sums2__lt + - sums_left_comm + - le_lt34_sums2__lt12 + - le2_sums2__cong12 + - cong3_sums__sums + - sums_middle_comm + - sums2__cong12 + - bet__sums + - sums__le1256 + - ex_sums + - le_lt12_sums2__lt + - le_lt34_sums2__lt + - sums_diff_2 + - sums_assoc_1 + - path: theories/Algebraic/Counter_models/Planar/counter_model_segment_construction.v + theorems: + - point_equality_decidability + - upper_dim + - bet_symmetry + - inner_pasch + - segment_construction + - cong_inner_transitivity + - cong_pseudo_reflexivity + - cong_identity + - lower_dim + - five_segment + - continuity + - euclid + - bet_inner_transitivity + - path: theories/Main/Highschool/gravityCenter.v + theorems: + - intersection_two_medians_exist + - is_gravity_center_perm_4 + - is_gravity_center_perm_1 + - is_gravity_center_exist_unique + - is_gravity_center_coplanar + - is_gravity_center_diff_2 + - three_medians_intersect + - is_gravity_center_third_reci + - is_gravity_center_diff_3 + - is_gravity_center_perm_3 + - is_gravity_center_col + - is_gravity_center_perm + - is_gravity_center_perm_2 + - is_gravity_center_cases + - is_gravity_center_diff_1 + - is_gravity_center_perm_5 + - path: theories/Main/Annexes/coplanar.v + theorems: + - col__coplanar + - coplanar_perm_15 + - coplanar_perm_18 + - coplanar_perm_2 + - ncoplanar_perm_5 + - ncoplanar_perm_7 + - coplanar_perm_4 + - coplanar_trivial + - inangle__coplanar + - ncoplanar_perm_3 + - ncoplanar_perm_19 + - ncop__ncol + - pars__coplanar + - plgf__coplanar + - out__coplanar + - square__coplanar + - coplanar_perm_20 + - coplanar_perm_1 + - ncoplanar_perm_14 + - ncoplanar_perm_1 + - bet__coplanar + - coplanar_perm_7 + - rectangle__coplanar + - ncoplanar_perm_2 + - coplanar_perm_23 + - ncoplanar_perm_23 + - lambert__coplanar + - coplanar_perm_10 + - ncoplanar_perm_9 + - coplanar_perm_14 + - ncoplanar_perm_22 + - ncoplanar_perm_18 + - ncoplanar_perm_21 + - coplanar_perm_5 + - coplanar_perm_16 + - coplanar_perm_12 + - coplanar_perm_13 + - coplanar_perm_6 + - par__coplanar + - ncoplanar_perm_16 + - coplanar_perm_19 + - parallelogram__coplanar + - coplanar_perm_8 + - ncoplanar_perm_15 + - perp__coplanar + - coplanar_perm_21 + - rhombus__coplanar + - ncoplanar_perm_6 + - ncoplanar_perm_20 + - coplanar_perm_22 + - coplanar_perm_17 + - ncoplanar_perm_8 + - plg__coplanar + - plgs__coplanar + - ncoplanar_perm_12 + - path: theories/Main/Tarski_dev/Ch10_line_reflexivity.v + theorems: + - l10_6_uniqueness_spec + - l10_15 + - is_image_rev + - l10_6_uniqueness + - l10_14 + - col_image_spec__eq + - is_image_spec_dec + - image_spec_triv + - image_id + - is_image_col_cong + - l10_2_uniqueness + - l10_6_existence + - col__image_spec + - l10_2_existence + - image_in_col + - col__refl + - image_in_is_image_spec + - l10_2_uniqueness_spec + - l10_7 + - l10_8 + - image__midpoint + - image_image_in + - ex_sym1 + - l10_6_existence_spec + - ex_per_cong + - image_triv + - l10_5 + - exists_cong_per + - image_spec__eq + - cong_midpoint__image + - is_image_spec_rev + - path: theories/Main/Annexes/project.v + theorems: + - project_par_project + - projp_idem + - project_not_id + - project_par + - project_col_eq + - project_par_dir + - project_not_col + - project_preserves_bet + - eqv_eq_project + - projp_id + - triangle_par + - project_id + - proj_distinct + - project_idem + - par_col_project + - ker_col + - perp_projp2_eq + - project_par_eqv + - projp_preserves_eqv + - projp_is_project_perp + - project_project_par + - col_2_par_projp2_cong + - col_par_projp2_eq + - eqv_project_eq_eq + - project_uniqueness + - eqv_cong + - perp_projp + - par3_conga3 + - projp_projp_perp + - ker_par + - col_projp_eq + - project_col + - project_existence + - projp_to_project + - path: theories/Elements/OriginalProofs/lemma_TTflip.v + theorems: + - lemma_TTflip + - path: theories/Main/Meta_theory/Dimension_axioms/upper_dim_2.v + theorems: + - upper_dim_implies_not_one_side_two_sides + - upper_dim_implies_all_coplanar + - upper_dim_stab + - all_coplanar_implies_upper_dim + - all_coplanar_upper_dim + - upper_dim_implies_one_or_two_sides + - upper_dim_implies_perp2__col + - upper_dim_implies_not_two_sides_one_side + - upper_dim_implies_per2__col + - upper_dim_implies_col_perp2__col + - upper_dim_implies_not_two_sides_one_side_aux + - path: theories/Coinc/CoincR.v + theorems: + - CoappDupPerm + - CPToSSHdTl + - pick_varieties_ok_2 + - memCPProper + - memCPAuxProperOK + - proper_0 + - consHdInterpOK + - ss_ok_inter_ok1 + - memMemCPAuxOK + - pick_variety_aux_memCPAux2 + - identify_varieties + - memCPAux + - proper_2 + - CPToSS + - CPToSSOK + - interp_CPTlOK + - collect_wds + - pick_variety_aux_memCPAux1 + - memCPToSSOK + - pick_variety_auxCP + - memCPAuxTlOK + - pick_variety_auxCP_existsF + - CoappDupAux + - ss_ok_empty + - proper_00 + - positive_dec + - pick_varieties_ok_1 + - interp_CPOK + - pick_variety_auxCP_forallT + - proper_3 + - test_coinc_ok + - PropToTagged + - identify_varieties_ok + - st_ok_empty + - mca_pick_variety_aux_pca + - ss_ok_inter_ok2 + - CPToSSOKAux + - exists_witness_ok + - memMemCPOK + - proper_1 + - memCPAuxHdTl + - collect_coincs + - memCPConsHd + - path: theories/Main/Utils/all_equiv.v + theorems: + - stronger2__stronger_right + - stronger2__stronger_left + - stronger_transitivity + - all_equiv'_aux + - all_equiv2_impl2__all_equiv + - incl_preserves_stronger + - all_equiv2_impl__stronger + - all_equiv2_stronger2__all_equiv + - all_equiv_under_chara + - all_equiv' + - all_equiv3_stronger3__all_equiv + - all_equiv_chara + - all_equiv'_auxP + - all_equiv_trivial + - all_equiv__equiv + - incl_preserves_all_equiv + - all_equiv3_impl3__all_equiv + - path: theories/Main/Meta_theory/Models/tarski_to_beeson.v + theorems: + - between_identity_B + - T_dec + - NColB_NDiffCol + - cong_stability + - BetH_Bet + - segment_construction_B + - NT_NBet + - T_Bet + - Bet_T + - between_symmetry_B + - Diff_Col_ColB + - NColB_NColOrEq + - bet_stability + - between_inner_transitivity_B + - lower_dim_B + - Beeson_follows_from_Tarski + - path: theories/Main/Highschool/orientation.v + theorems: + - out_preserves_eq_o + - eqo_refl + - proj_comm + - proj_not_col + - per_preserves_bet_aux1 + - proj_preserves_bet1 + - proj_diff_not_col_inv + - proj_diff_not_col + - proj_eq_col + - proj_preserves_bet + - bet_le_le + - proj_per + - proj_inv_exists + - per_id + - col_proj_proj + - proj_perp_id + - eq_o_one_side + - proj_id + - symmetry_preseves_bet2 + - proj_exists + - bet_half_bet + - bet_double_bet + - one_side_eqo + - perp_not_eq_3 + - eq_o_eqo + - proj3_id + - proj_col_proj + - eq_o_refl + - per_preserves_bet_aux2 + - one_side_eq_o + - eqo_one_side + - midpoint_preserves_bet + - out_preserves_eqo + - le_left_comm + - cong_identity_inv + - proj_uniqueness + - midpoint_par + - proj_col + - le_right_comm + - symmetry_preseves_bet1 + - out_preserves_eqo1 + - col_proj_col + - eqo_eq_o + - midpoint_par_strict + - per_diff + - proj_one_side + - proj_par_strict + - proj_diff + - proj_not_eq_not_col + - per_one_side + - par_cong_mid + - le_cong_le + - per_proj + - le_comm + - proj_not_eq + - path: theories/Main/Annexes/quadrilaterals_inter_dec.v + theorems: + - par_preserves_conga_os + - plgs_permut + - par_cong_plg + - plg_not_comm + - per_rmb + - plg_per_rect2 + - plgf_rect_id + - plgf_plgs_trans + - cop_perp3__perp + - ncol123_plg__pars1234 + - pars_par_plg + - ncol124_plg__pars1234 + - ts_cong_par_cong_par + - par_cong_mid + - rect_per + - plg_not_comm_2 + - plg_conga1 + - par_cong3_rect + - col_cong_cong + - plg_uniqueness + - rect_comm2 + - Square_Rhombus + - rmb_per + - par_2_plg + - conga_to_par_os + - pars_par_pars + - plgs_two_sides + - ncol234_plg__pars1423 + - par_cong_mid_ts + - plgs_comm2 + - rect_permut + - plg_mid_2 + - os_cong_par_cong_par + - not_par_pars_not_cong + - ncol123_plg__plgs + - plgs_pseudo_trans + - rect_per4 + - plgs_cong_1 + - plg_per_rect3 + - plgs_pars_2 + - plgs_trans_trivial + - parallelogram_equiv_plg + - plg_cong + - parallel_2_plg + - square_perp_rectangle + - ncol124_plg__pars1423 + - rect_per2 + - par_cong_cop + - par_cong_mid_os + - plg_par_2 + - sac_plg + - plgs_pars_1 + - parallelogram_strict_midpoint + - plg_permut + - plg_per_rect + - rect_2_rect + - perp3__rect + - plg_cong_1 + - plgf_plgf_plgf + - rmb_cong + - plg_conga + - plg_comm2 + - cong3_par2_par + - ncol134_plg__pars1423 + - par_par_cong_cong_parallelogram + - ncol134_plg__pars1234 + - plgs_par_strict + - exists_square + - sac_rectangle + - plgs_half_plgs + - plgs_in_angle + - rect_per3 + - rect_per1 + - plg_par_1 + - plg_pseudo_trans + - plg_per_rect1 + - par_cong_plg_2 + - plgs_not_comm + - plgs_half_plgs_aux + - plgs_cong + - plgs_cong_2 + - ncol234_plg__pars1234 + - ncol134_plg__plgs + - ncol123_plg__pars1423 + - ncol234_plg__plgs + - plgf_comm2 + - ncol124_plg__plgs + - par_strict_cong_mid1 + - par_cong_cong + - par_strict_trans + - path: theories/Main/Tarski_dev/Ch14_prod.v + theorems: + - prod_to_prodp + - distr_r + - prodp_to_prod + - eq_squares_eq_or_opp + - prod_sym + - change_grid_prod1 + - proj_preserves_prod + - prod_O_l_eq + - prod_O_r_eq + - prod_comm + - change_grid_prod + - prod_0_l + - l14_31_2 + - diff_of_squares + - prod_1_r + - prod_1_l + - distr_l_diff + - project_pj + - prod_exists + - prod_uniqueness + - inv_exists + - change_grid_prod_l_O + - distr_l + - opp_prod + - prod_1_l_eq + - prod_assoc + - prod_assoc2 + - prod_null + - prod_uniquenessB + - diff_2_prod + - prod_uniquenessA + - l14_31_1 + - prod_1_r_eq + - path: theories/Elements/OriginalProofs/proposition_10.v + theorems: + - proposition_10 + - path: theories/Main/Elements_statements/Book_1.v + theorems: + - prop_29_2 + - prop_19 + - prop_9 + - prop_33 + - prop_16 + - prop_12 + - prop_28_1 + - prop_21_1 + - prop_17 + - prop_22 + - prop_25 + - prop_7 + - prop_14 + - prop_26_2 + - prop_1_circle_circle + - prop_5_1 + - prop_24 + - prop_34_1 + - prop_47 + - prop_32_1 + - prop_13 + - prop_1_euclidean + - prop_4 + - prop_6 + - prop_15 + - prop_18 + - prop_23 + - prop_29_3 + - prop_11 + - prop_10 + - prop_26_1 + - path: theories/Main/Meta_theory/Models/tarski_to_hilbert.v + theorems: + - axiom_cong_5' + - Hilbert_3D_follows_from_Tarski_3D + - EqP_Equiv + - cop_plane + - axiom_hcong_1_existence + - axiom_two_points_on_line + - lower_dim_3' + - axiom_plane_uniqueness + - out_outH + - axiom_between_one + - lower_dim' + - axiom_hcong_scott + - axiom_between_only_one + - axiom_between_out + - same_side_sym + - axiom_Incidp_morphism + - Bet_Between_H + - same_side_OS + - axiom_conga_comm + - same_side_one_side + - EqL_Equiv + - plane_cop + - axiom_line_existence + - eqp_symmetry + - axiom_line_on_plane + - cut_two_sides + - incident_Proper + - axiom_conga_permlr + - Para_Par + - outH_out + - eq_reflexivity + - ncols_coincide + - eqp_transitivity + - eqp_reflexivity + - axiom_hcong_4_uniqueness + - incident_eq + - axiom_hcong_1_uniqueness + - axiom_hcong_3 + - cols_coincide_1 + - axiom_line_uniqueness + - axiom_Incid_morphism + - col_disjoint_bet + - axiom_between_diff + - axiom_one_point_on_plane + - exists_not_incident + - axiom_euclid_uniqueness + - axiom_plane_existence + - OS_distinct + - Hilbert_neutral_follows_from_Tarski_neutral + - axiom_Incid_dec + - same_side_trans + - axiom_Incidp_dec + - axiom_conga_refl + - Hilbert_euclidean_follows_from_Tarski_euclidean + - eq_incident + - axiom_congaH_outH_congaH + - Hilbert_euclidean_ID_follows_from_Tarski_euclidean + - eq_symmetry + - axiom_hcong_4_existence + - axiom_between_comm + - cols_coincide_2 + - axiom_pasch + - axiom_between_col + - eq_transitivity + - one_side_same_side + - Incid_line + - cols_coincide + - path: theories/Main/Annexes/vectors.v + theorems: + - vector_construction_uniqueness + - null_sum + - plg_out_plg + - same_dir_to_null + - eqv_permut + - same_dir_null + - bet_same_dir1 + - eqv_trans + - opp_dir_id + - eqv_sym + - same_dir_refl + - same_dir_dec + - opp_dir_to_null + - mid_eqv + - bet_same_dir2 + - par_ts_same_dir + - ise_to_is + - opp_not_same_dir + - eqv_comm + - plgf_out_plgf + - same_dir_ts + - same_or_opp_dir + - same_not_opp_dir + - null_sum_eq + - sum_exists + - sum_uniqueness + - eqv_mid + - same_dir_out + - same_dir_trans + - vector_construction + - same_dir_comm + - plg_opp_dir + - vector_uniqueness + - vector_same_dir_cong + - chasles + - eqv_opp_null + - plgf_plgf_bet + - plgs_out_plgs + - eqv_par + - eqv_trivial + - same_dir_out1 + - null_vector + - eqv_sum + - plgs_plgs_bet + - same_dir_sym + - opposite_sum + - path: theories/Main/Tarski_dev/Ch14_order.v + theorems: + - diff_2_le_le + - prod_pos_pos + - pos_null_neg + - ltP_sum_pos + - bet_lt21_le32 + - ltP_neg + - l14_36_b + - pos_not_neg + - leP_trans + - not_pos_and_neg + - l14_36_a + - square_pos + - bet_lt12_le13 + - bet_lt21_le31 + - leP_asym + - compatibility_of_sum_with_order + - opp_pos_neg + - leP_refl + - ltP_ar2 + - le_pos_prod_le + - pos_opp_neg + - diff_pos_diff_neg + - ltP_neq + - bet_lt12_le23 + - ps_le + - opp_neg_pos + - lt_diff_ps + - opp_2_le_le + - O_not_positive + - col_pos_or_neg + - sum_pos_pos + - compatibility_of_prod_with_order + - leP_sum_leP + - path: theories/Algebraic/Counter_models/Planar/counter_model_euclid.v + theorems: + - five_segment + - cong_inner_transitivity + - euclid + - nbet_abc + - c'_in_unit_disk + - upper_dim_aux + - inner_paschP + - upper_dim + - b'_in_unit_disk + - a'_eq0 + - segment_construction + - cong_pseudo_reflexivity + - point_equality_decidability + - nbet_cab + - bet_symmetry + - bet_inner_transitivity + - betR_bca + - upper_dim_aux_2 + - a'_in_unit_disk + - Rcf_to_GI_PED' + - betR_cab + - path: theories/Main/Meta_theory/Continuity/dedekind_cantor.v + theorems: + - nested__ex_right + - nested__ex_left + - nested_aux1 + - nested_sym + - dedekind__cantor + - nested__diff0 + - nested__bet + - nested_aux2 + - path: theories/Main/Meta_theory/Models/gupta_inspired_to_tarski.v + theorems: + - g2_15 + - inner_paschT + - g2_9 + - between_identityT + - col_decG + - g2_7 + - g2_4 + - g2_2 + - g2_10 + - bet_decG + - g2_1 + - g2_16 + - between_trivialT + - g2_3 + - GI_to_T_PED + - l2_11T + - GI2D_to_T2D + - construction_uniquenessT + - upper_dimT + - cong_trivial_identityT + - g2_8 + - g2_6 + - cong_inner_transitivityT + - g2_14 + - g2_13 + - g2_11 + - euclidT + - g2_12 + - GI_to_T + - GI_euclidean_to_T_euclidean + - path: theories/Main/Annexes/half_angles.v + theorems: + - conga2_ghalfa__ghalfa + - halfa_not_null + - halfa__ts + - halfa_chara1 + - halfa_uniqueness + - ghalfa_out4__ghalfa + - halfa_sym + - halfa1123__out + - halfa__nbet2 + - halfa__suma + - ghalfa__suma + - ghalfa2__conga_2 + - ghalfa_left_comm + - inangle_halfa2__inangle + - halfa_exists + - halfa2_lta__lta2 + - ghalfa_preserves_conga_2 + - suma_preserves_ghalfa + - halfa_distincts + - halfa__lea + - halfa3123__out + - halfa__nbet + - halfa2_lea__lea1 + - ghalfa2__conga_1 + - halfa__coplanar + - acute_ghalfa2_sams_suma2__ghalfa123 + - ghalfa_distincts + - ghalfa_comm + - null_halfa__null + - ghalfa__out + - ghalfa_chara + - conga_halfa__conga2 + - halfa__ghalfa + - conga_halfa__conga1 + - halfa2_lta__lta1 + - halfa__acute + - cop_halfa_perp__os2 + - ghalfa_preserves_lta + - halfa__sams + - halfa2_lea__lea2 + - path: theories/Main/Meta_theory/Parallel_postulates/parallel_postulates.v + theorems: + - aristotle + - equivalent_postulates_assuming_greenberg_s_axiom + - stronger_postulates_ter + - greenberg + - stronger_postulates_bis + - equivalent_postulates_without_any_continuity_bis + - equivalent_postulates_without_decidability_of_intersection_of_lines + - equivalent_postulates_assuming_archimedes_axiom + - postulates_in_euclidean_context + - stronger_postulates + - equivalent_postulates_without_any_continuity + - inter_dec + - path: theories/Elements/OriginalProofs/lemma_rectangleparallelogram.v + theorems: + - lemma_rectangleparallelogram + - path: theories/Main/Highschool/orthocenter.v + theorems: + - is_orthocenter_perm_3 + - altitude_intersect + - is_orthocenter_perm_4 + - construct_intersection + - is_orthocenter_perm + - orthocenter_per + - diff_not_col_col_par4_mid + - is_orthocenter_cases + - not_col_par_col_diff + - orthocenter_col + - is_orthocenter_perm_5 + - is_orthocenter_perm_2 + - is_orthocenter_coplanar + - is_orthocenter_perm_1 + - altitude_is_perp_bisect + - construct_triangle + - path: theories/Main/Meta_theory/Models/tarski_to_euclid.v + theorems: + - InCircCenter + - InCirc_InCirc + - BetS_BetS + - outside + - bet_cases + - Euclid_neutral_follows_from_Tarski_neutral + - circle_circle' + - 'on' + - Euclid5 + - circle_line + - OnCirc_OnCirc + - Col_Col + - eOutCirc_OutCirc + - eOnCirc_OnCirc + - path: theories/Algebraic/Counter_models/counter_model_bet_identity.v + theorems: + - upper_dim + - point_eq_dec + - inner_pasch + - cong_identity + - not_bet_diff + - cong_aux_3 + - lower_dim + - five_segment + - cong_pseudo_reflexivity + - cong_aux + - cong_inner_transitivity + - cong_aux_6 + - cong_aux_5 + - cong_sym + - path: theories/Main/Tarski_dev/Ch13_1.v + theorems: + - is_image_perp_in + - per13_preserves_bet + - perp2_par + - per3_preserves_bet2_aux + - l13_2_1 + - per3_preserves_bet2 + - cop4_perp_in2__col + - per2_col_eq + - perp2_preserves_bet13 + - cong_perp_conga + - perp_in2__col + - perp2_pseudo_trans + - perp2_trans + - perp2_comm + - perp_in_rewrite + - perp2_perp_in + - l13_1_aux + - l13_8 + - col_cop_perp2__pars_bis + - perp_out_acute + - perp2_sym + - perp2_left_comm + - perp2_refl + - per_lt + - l13_2 + - per23_preserves_bet_inv + - perp_per_bet + - per3_preserves_bet1 + - perp_inter_perp_in_n + - perp_bet_obtuse + - per13_preserves_bet_inv + - l13_1 + - perp2_right_comm + - ts_per_per_ts + - per2_preserves_diff + - per23_preserves_bet + - path: theories/Main/Tarski_dev/Ch12_parallel.v + theorems: + - par_not_col_strict + - par_col_par + - par_distincts + - l12_22_b + - conga_cop_inangle_per2__inangle + - par_strict_distinct + - par_strict_irreflexivity + - par_col2_par_bis + - l12_9 + - par_strict_col2_par_strict + - par_comm + - col_cop2_perp2__col + - par_distinct + - inter_distincts + - l12_18_b + - perp_inter_exists + - l12_6 + - par_neq1 + - col_par + - par_one_or_two_sides + - l12_9_2D + - not_par_not_col + - inter_uniqueness_not_par + - par_strict_not_col_1 + - par_strict_col_par_strict + - perp_inter_perp_in + - par_id + - col_cop_perp2__pars + - par_reflexivity + - par_left_comm + - perp_not_par + - inter_right_comm + - l12_18_d + - parallel_existence1 + - Par_strict_perm + - l12_18_a + - cong_conga_perp + - par_col_par_2 + - par_strict_comm + - par_right_comm + - par_strict_symmetry + - not_par_inter_uniqueness + - par_strict_not_col_3 + - inter_left_comm + - pars__os3412 + - par_strict_not_cols + - not_strict_par2 + - col2_par__col4 + - par_strict_left_comm + - parallel_existence + - par_neq2 + - perp_dec + - not_strict_par + - inter_comm + - col_not_col_not_par + - Par_cases + - l12_21_b + - par_strict_one_side + - l12_18_c + - all_one_side_par_strict + - acute_col_perp__out + - par_two_sides_two_sides + - par_strict_not_col_4 + - inter_trivial + - par_symmetry + - l12_22_aux + - inter_sym + - l12_18 + - not_strict_par1 + - acute_col_perp__out_1 + - par_strict_col4__par_strict + - path: theories/Algebraic/Counter_models/Planar/counter_model_upper_dim.v + theorems: + - ab_neq + - congP_aux' + - oner_eqm1 + - neq + - congP_aux + - betR_bca + - eq3 + - de_neq + - upper_dim + - betR_cab + - a_eq0 + - row3_eq + - oner_neqm1 + - ca_neq + - ac_neq + - cong_bdbe + - cong_adae + - betR_abc + - cong_cdce + - negb_and3 + - path: theories/Coinc/ColR.v + theorems: + - CTcol_permutation_4 + - collect_diffs + - proper_1 + - proper_9 + - test_col_ok + - pick_lines_ok_2 + - CTcol_trivial_1 + - CTcol_permutation_3 + - subst_sp_ok + - strong_induction + - proper_8 + - exists_witness_ok + - collect_cols + - ss_ok_empty + - proper_4 + - identify_lines_ok + - CTcol_permutation_2 + - proper_2 + - proper_00 + - pick_lines_ok_1 + - proper_0 + - proper_5 + - sp_ok_empty + - proper_6 + - subst_ss_ok + - eq_eq_tagged + - CTcol_trivial_2 + - path: theories/Algebraic/Counter_models/nD/counter_model_upper_dim.v + theorems: + - my_enum_ordSr + - upper_dim + - nth_basis + - betS_nD'_2D + - b2DE + - betR_nD'_2D + - to_nD'B + - lower_dim + - last_new_basis + - to_nD'_scale + - pick_to_nD' + - upper_dim_all2 + - lower_dim_all2 + - lower_dim_all1 + - upper_dim_all1 + - eq_nD'_2D + - a2DE + - to_nD'E + - c2DE + - iP + - nth_new_basis + - path: theories/Algebraic/coplanarity.v + theorems: + - InPlane'__Coplanar + - Cop_4321 + - Cop_2341 + - Cop_2134 + - Cop__Coplanar + - Copl_2431 + - Cop_3421 + - Copl_3412 + - Copl_2341 + - Copl_4123 + - Copl_3421 + - Cop_1324 + - Copl_1423 + - Cop_4132 + - Copl_1243 + - Copl_3214 + - Cop_1342 + - Cop_aabc + - Copl_4231 + - Cop_2143 + - Cop_3241 + - Cop_2431 + - Copl_2413 + - Cop_4231 + - InPlane__Coplanar + - Cop_2314 + - Copl_2314 + - Copl_4321 + - Cop_3142 + - Copl_3241 + - Copl_2143 + - Cop_4312 + - Cop_1423 + - Copl_4213 + - Cop_1432 + - Copl_4132 + - Copl_3124 + - Cop_3124 + - path: theories/Main/Tarski_dev/Ch07_midpoint.v + theorems: + - col_cong2_bet2 + - symmetry_preserves_midpoint + - l7_20_bis + - swap_diff + - is_midpoint_id + - midpoint_not_midpoint + - l7_3_2 + - l7_20 + - midpoint_distinct_2 + - le_mid2__le12 + - l7_15 + - lt_mid2__lt12 + - l7_9_bis + - midpoint_distinct_3 + - l7_9 + - l7_17_bis + - bet_col1 + - col_cong2_bet3 + - cong_mid2__cong + - bet2_lt_le__lt + - l7_22 + - midpoint_out + - is_midpoint_id_2 + - symmetric_point_construction + - l7_2 + - midpoint_bet + - midpoint_col + - l7_17 + - mid__lt + - col_cong_bet + - col_cong2_bet1 + - cong_cong_half_1 + - symmetric_point_uniqueness + - Mid_perm + - midpoint_cong + - cong_cong_half_2 + - lt_mid2__lt13 + - l7_25 + - le_mid2__le13 + - l7_21 + - cong_col_mid + - l7_3 + - l7_22_aux + - col_cong2_bet4 + - l7_16 + - col_bet2_cong1 + - midpoint_preserves_out + - midpoint_dec + - l7_13 + - path: theories/Elements/OriginalProofs/lemma_9_5.v + theorems: + - lemma_9_5 + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_identity.v + theorems: + - cong_identity + - cong_pseudo_reflexivity + - bet_symmetry + - col_alt_def + - segment_construction + - euclid + - continuity + - bet_inner_transitivity + - point_equality_decidability + - inner_pasch + - cong_inner_transitivity + - lower_dim + - upper_dim + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_inner_transitivity.v + theorems: + - col_alt_def + - bet_inner_transitivity + - point_equality_decidability + - upper_dim + - continuity + - segment_construction + - lower_dim + - cong_inner_transitivity + - cong_identity + - five_segment + - bet_symmetry + - inner_pasch + - path: theories/Algebraic/tcp_ndc.v + theorems: + - tcp_bet_sa_b_inter_mab_mac_b_sa + - tcp_aligned_inplane + - tcp_ts_mab_mac_a_ssa + - tcp_ts_mmab_mac_a_sa + - tcp_npars_mab_mmac_b_sa + - tcp_os_b_sa_mab_mac + - tcp_pars + - tcp_ncol_inplane_3_3 + - tcp_ts_a_ssa_mab_mac + - tcp_ncol_inplane_1_4 + - tcp_npars_mab_mac_b_ssa + - tcp_ncol_inplane_3_1 + - tcp_ts_mmab_mac_a_mbc + - tcp_ts_a_sa_mab_mac + - tcp_ncol_inplane_3_6 + - tcp_os_mmab_mac_b_sa + - tcp_npars_mab_mmac_b_ssa + - tcp_os_mab_mac_b_sa + - tcp_ts_mab_mac_a_sa + - tcp_ts_a_mbc_mab_mac + - tcp_ncol_inplane_3_5 + - tcp_ncol_inplane_2_5 + - tcp_ncol_midpoints + - tcp_ncol_inplane_1_1 + - tcp_os_mmab_mac_b_ssa + - tcp_os_b_ssa_mmab_mac + - tcp_os_b_ssa_mab_mmac + - tcp_npars_mab_mac_b_sa + - tcp_npars_mmab_mac_b_sa + - tcp_ncol_inplane_1_3 + - tcp_ncol_inplane + - tcp_ncol_inplane_3_4 + - tcp_ncol_inplane_2_6 + - tcp_ncol_inplane_1_6 + - tcp_ncol_inplane_2_1 + - tcp_ncol_inplane_1_2 + - tcp_ncol_ndc_choice_col + - tcp_os_mab_mmac_b_sa + - tcp_aligned_plane_exists + - tcp_ncol_ndc_choice + - tcp_os_mab_mac_b_ssa + - tcp_ts_mab_mac_a_mbc + - tcp_ncol_inplane_2_2 + - tcp_ncol_inplane_aux + - tcp_ncol_inplane_1_5 + - tcp_ncol_inplane_2_3 + - tcp_ncols_ndc + - tcp_ts_mab_mmac_a_sa + - tcp_ncol_inplane_4 + - tcp_ncol_inplane_3_2 + - tcp_os_b_sa_mab_mmac + - tcp_ncol_ndc_ncol + - path: theories/Main/Highschool/triangles.v + theorems: + - equilateral_rot_2 + - equilateral_isosceles_3 + - equilateral_isosceles_2 + - equilateral_strict_swap_4 + - conga_isosceles + - equilateral_swap + - equilateral_strict_equilateral + - equilateral_strict_conga_3 + - equilateral_strict_conga_2 + - equilateral_strict_conga_1 + - equilateral_swap_rot + - equilateral_strict_swap_1 + - equilateral_rot + - isosceles_conga + - isosceles_foot__midpoint_conga + - equilateral_strict_neq + - equilateral_cong + - conga3_equilateral + - path: theories/Main/Highschool/SegmTrisect.v + theorems: + - SegmTrisectUniqueness + - SegmTrisectExistence + - SegmTrisectExistenceUniqueness + - SegmTrisectFirstThirdUniqueness + - SegmTrisectFirstThirdExistence + - path: theories/Main/Annexes/rhombus.v + theorems: + - PlgLeft + - PlgExABC2 + - RhombusExABC1 + - PlgEx + - PlgAABB + - RhombusUnicity + - RhombusEx + - PlgEquivDef + - RhombusExABC2 + - PlgExABC1 + - ColCongMid + - path: theories/Main/Annexes/suma.v + theorems: + - nsams__obtuse + - trisuma_perm_213 + - sams_lea_lta789_suma2__lta123 + - sams_lta2_suma2__lta456 + - sams_suma__lea456789 + - bet_per2__suma + - sams_lea_lta123_suma2__lta + - sams_lea123_suma2__lea + - sams_suma__lea123789 + - nbet_sams_suma__acute + - sams_lea2_suma2__conga123 + - out213__sams + - ex_trisuma + - sams_lea_lta456_suma2__lta123 + - sams_lea_lta456_suma2__lta + - sams_lea2_suma2__lea123 + - out546_suma__conga + - sams_sym + - sams_lea_lta123_suma2__lta456 + - bet_suma2__or_conga + - suma_middle_comm + - sams_assoc_1 + - suppa__sams + - trisuma_dec + - trisuma_perm_321 + - sams_lta2_suma2__lta + - suma_left_comm + - out546__sams + - acute_suma__nbet + - suma_dec + - trisuma_perm_132 + - sams_right_comm + - acute2_suma2__conga + - per2__sams + - suma_right_comm + - sams_assoc + - os_ts__sams + - trisuma2__conga + - conga3_trisuma__trisuma + - sams_assoc_2 + - bet_suma__per + - bet_per_suma__per123 + - col_suma__col + - sams2_suma2__conga456 + - out546__suma + - sams_suma__out213 + - bet_suma__suppa + - sams2_suma2__conga + - bet2_suma__suma + - trisuma_perm_312 + - ex_suma + - sams_lea2_suma2__conga456 + - sams_lea456_suma2__lea + - suma_sym + - inangle__suma + - conga_trisuma__trisuma + - suma2__conga + - sams_lta2_suma2__lta123 + - bet_per_suma__per456 + - suma2_obtuse2__conga + - bet_suma__sams + - acute_per__sams + - acute2__sams + - obtuse__nsams + - ts__suma + - out6_suma__suma + - sams123231 + - sams_left_comm + - sams_comm + - sams_dec + - suma_assoc_1 + - out213_suma__conga + - acute2_suma__nbet + - conga_sams_nos__nts + - suma2__or_conga_suppa + - suma_suppa__bet + - ncol_suma__ncol + - per2_suma__bet + - sams_distincts + - sams2_suma2__conga123 + - suma_distincts + - suma_comm + - sams_lea2__sams + - bet__suma + - acute__sams + - path: theories/Main/Tarski_dev/Ch02_cong.v + theorems: + - not_cong_4312 + - cong_trivial_identity + - cong_diff + - cong_transitivity + - cong_3_sym + - cong_reverse_identity + - construction_uniqueness + - cong_diff_2 + - cong3_transitivity + - eq_dec_points + - cong_right_commutativity + - five_segment_with_def + - l2_11 + - not_cong_1243 + - cong_diff_3 + - Cong_cases + - cong_diff_4 + - bet_cong3 + - cong_commutativity + - cong_left_commutativity + - not_cong_2134 + - cong_3_swap + - not_cong_3412 + - cong_3421 + - Cong_perm + - not_cong_2143 + - cong_3_swap_2 + - cong_4321 + - not_cong_4321 + - path: theories/Main/Meta_theory/Models/beeson_to_tarski.v + theorems: + - BetT_symmetry + - IT_chara + - pasch + - another_point + - bet_id + - cong_stability_eq_stability + - pasch_col_case + - IT_to_T + - five_segment + - segment_construction + - path: theories/Elements/OriginalProofs/proposition_09.v + theorems: + - proposition_09 + - path: theories/Algebraic/Counter_models/nD/counter_model_five_segment.v + theorems: + - cong_inner_transitivity + - five_segment + - inner_pasch + - col__colI + - bet_bet' + - row2_cong_nD + - cong_pseudo_reflexivity + - row_mx_behead + - to_nD_behead + - Col_Col + - euclid + - cong_identity + - to_nD_head + - bet_inner_transitivity + - bet'_abc + - segment_construction + - eq_head_behead + - path: theories/Algebraic/Counter_models/nD/counter_model_bet_inner_transitivity.v + theorems: + - coplanar_midpoint + - col__Col + - Midpoint_midpoint + - midpointBR + - midpointxx + - cong_midpoint + - onemx_neq0 + - bet'_midpoint + - midpointLR + - colI__Col + - ColD + - col__colI + - euclid + - midpointBL + - colI__Col_13 + - betR_midpoint + - ncol__ncolI + - bet_midpoint + - inner_pasch + - bet_inner_transitivity + - bet_symmetry + - Col__col + - Col_23 + - five_segment + - cong_identity + - cong_pseudo_reflexivity + - segment_construction + - path: theories/Algebraic/Counter_models/nD/independent_version_to_beeson.v + theorems: + - ITbetween_identity + - bet_stability + - ITsegment_construction + - altIT_to_IT + - ITinner_pasch + - ITfive_segment + - weak_Bstability + - cong_stability + - ITbetween_symmetry + - ITlower_dim + - path: theories/Coinc/CongR.v + theorems: + - eq_pseudo_refl + - identify_lengths_ok + - proper_3 + - collect_congs + - proper_2_aux + - proper_1 + - exists_witness_ok + - pick_lengths_ok_1 + - CTcong_right_comm + - ss_ok_empty + - pick_lengths_ok_2 + - positive_dec + - identify_lengths + - test_cong_ok + - proper_0 + - proper_2 + - path: theories/Elements/OriginalProofs/lemma_TTorder.v + theorems: + - lemma_TTorder + - path: theories/Main/Meta_theory/Models/tarski_to_gupta_inspired.v + theorems: + - T_to_GI + - cong_inner_transitivity' + - T2D_to_TG2D + - path: theories/Algebraic/Counter_models/nD/counter_model_cong_identity.v + theorems: + - bet_inner_transitivityP + - inner_pasch + - cong_identity + - segment_construction + - bet_symmetry + - cong_inner_transitivity + - upper_dim + - lower_dim + - path: theories/Elements/OriginalProofs/lemma_crossbar2.v + theorems: + - lemma_crossbar2 + - path: theories/Elements/OriginalProofs/lemma_9_5b.v + theorems: + - lemma_9_5b + - path: theories/Main/Meta_theory/Parallel_postulates/TCP_tarski.v + theorems: + - impossible_case_3 + - impossible_case_4_1 + - impossible_case_4 + - triangle_circumscription_implies_tarski_s_euclid + - impossible_case_2 + - impossible_case_1 + - triangle_circumscription_implies_tarski_s_euclid_aux1 + - triangle_circumscription_implies_tarski_s_euclid_aux + - path: theories/Elements/OriginalProofs/euclidean_tactics.v + theorems: + - nCol_not_Col + - not_nCol_Col + - Col_or_nCol + - Col_nCol_False + - eq_or_neq + - nCol_or_Col + - path: theories/Main/Meta_theory/Models/makarios_to_tarski.v + theorems: + - Tarski_follows_from_Makarios_Variant + - LmCoghGrab + - Tarski_follows_from_Makarios_Variant_with_decidable_point_equality' + - Mcong_reflexivity + - five_segment + - Mcong_trivial_identity + - Mcong_symmetry + - cong_pre_pseudo_reflexivity + - cong_pseudo_reflexivity + - between_trivial + - path: theories/Main/Annexes/midpoint_theorems.v + theorems: + - triangle_mid_par_flat + - triangle_mid_par_cong + - col123_124__col234 + - col_permut231 + - col_permut321 + - triangle_mid_par_flat_cong_2 + - triangle_mid_par + - col_permut213 + - triangle_mid_par_flat_cong_aux + - triangle_mid_par_strict_cong_aux + - triangle_mid_par_strict_cong_simp + - triangle_mid_par_strict_cong_1 + - triangle_mid_par_flat_cong + - triangle_mid_par_strict + - triangle_mid_par_strict_cong + - col_permut312 + - triangle_mid_par_cong_1 + - triangle_mid_par_flat_cong_1 + - triangle_par_mid + - triangle_mid_par_strict_cong_2 + - path: theories/Elements/OriginalProofs/lemma_3_7a.v + theorems: + - lemma_3_7a + - path: theories/Algebraic/Counter_models/nD/counter_model_cong_inner_transitivity.v + theorems: + - bet_symmetry + - cong_inner_transitivity + - bet_inner_transitivity + - segment_construction + - cong_axay + - continuity + - inner_pasch + - five_segment + - cong_pseudo_reflexivity + - path: theories/Main/Tarski_dev/Ch10_line_reflexivity_2.v + theorems: + - image_preserves_col + - image_spec_preserves_per + - cong_cop_per2_1 + - l10_16 + - cop_not_par_other_side + - cop__cong_on_bissect + - cong_cop_per2 + - image_preserves_bet + - cop_image_in2__col + - all_coplanar + - hilbert_s_version_of_pasch_aux + - intersection_with_image_gen + - image_preserves_per + - l10_10 + - perp2__col + - image_gen_preserves_ncol + - cong_cop_per2_gen + - l10_10_spec + - cong_cop_image__col + - cop_not_par_same_side + - l10_12 + - two_sides_cases + - per2__col + - not_par_two_sides + - hilbert_s_version_of_pasch + - image_gen_preserves_bet + - image_gen_preserves_col + - path: theories/Main/Tarski_dev/Ch13_2_length.v + theorems: + - ex_points_lg + - ex_eqL + - all_eql + - null_len + - is_len_cong_is_len + - ex_eql + - lg_cong + - lg_null_trivial + - not_cong_is_len1 + - lg_cong_lg + - lg_exists + - ex_point_lg + - not_cong_is_len + - ex_point_lg_bet + - eqL_equivalence + - lg_null_dec + - lg_sym + - ex_point_lg_out + - ex_lg + - path: theories/Coinc/Utils/sets.v + theorems: + - compare_spec + - eqListRefl + - eqST_dec + - OCPTlOK + - OCP + - lt_antiref + - lengthOne + - eqb_eq + - OCPALengthOK + - STadd_iff + - lt + - ltListIrrefl + - eq_equiv + - PermSorted + - lt_irrefl + - OCPPerm + - OCPSortedAux + - eqbListEqList + - leb_total + - lt + - eqListSortOCP + - compare_spec + - eqListOK + - ltTrans + - lt_trans + - eqbST_eqST + - CPLOCPTlOK + - lengthAtLeastOne + - eqListSym + - OCPSortedTl + - lt_trans + - ltListTrans + - eqListTrans + - STempty_b + - compareListSpec + - path: theories/Main/Tarski_dev/Ch16_coordinates.v + theorems: + - characterization_of_betweenness + - exists_coord + - point_of_coordinates_on_an_axis + - grid_exchange_axes + - characterization_of_collinearity + - coord_exchange_axes + - l16_9_2 + - eq_points_coordinates + - Cs_not_Col + - characterization_of_betweenness_aux + - length_eq_or_opp + - cong_3_3_cong_5 + - point_of_coordinates + - point_of_coordinates_origin + - square_distance_formula_aux + - exists_grid_spec + - bet_betCood + - Cd_Col + - exists_projp + - same_abscissa_col + - path: theories/Main/Annexes/Tagged_predicates.v + theorems: + - Par_tagged_Par + - Cong_Cong_tagged + - Perp_in_Perp_in_tagged + - Bet_tagged_Bet + - Diff_perm + - Per_Per_tagged + - Par_strict_Par_strict_tagged + - Plg_Plg_tagged + - Perp_tagged_Perp + - Cong_tagged_Cong + - Diff_Diff_tagged + - Bet_Bet_tagged + - Mid_Mid_tagged + - Perp_in_tagged_Perp_in + - Par_strict_tagged_Par_strict + - Per_tagged_Per + - NCol_NCol_tagged + - Col_Col_tagged + - Col_tagged_Col + - Perp_Perp_tagged + - Plg_tagged_Plg + - path: theories/Elements/OriginalProofs/lemma_supplements2.v + theorems: + - lemma_supplements2 + - path: theories/Algebraic/Counter_models/nD/independent_version_to_tarski.v + theorems: + - bet_axx + - Coplanar_dec + - Col_dec + - Col_TCol + - Cop_TCop + - altIT_to_T_PED + - altIT_euclidean_to_T_euclidean + - altIT_to_T + - NCol_TNCol + - path: theories/Algebraic/Counter_models/nD/counter_model_bet_symmetry.v + theorems: + - ColP + - bet_inner_transitivity + - five_segment + - CopP + - cong_pseudo_reflexivity + - cong_identity + - col__colI + - bet_symmetry + - inner_pasch + - euclid + - cong_inner_transitivity + - path: theories/Main/Tarski_dev/Ch13_6_Desargues_Hessenberg.v + theorems: + - l13_15 + - l13_19_par_aux + - l13_18 + - l13_15_par + - l13_19_par + - l13_19_aux + - l13_15_2 + - l13_18_2 + - l13_15_1 + - l13_15_2_aux + - l13_18_3 + - path: theories/Main/Tactics/CoincR_for_concy.v + theorems: + - collect_coincs_for_concy + - collect_wds_for_concy + - test_coinc_ok_for_concy + - path: theories/Elements/OriginalProofs/lemma_parallelflip.v + theorems: + - lemma_parallelflip + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_col.v + theorems: + - Tarski_is_a_Coinc_predicates_for_col + - diff_perm_1 + - Tarski_is_a_Arity_for_col + - col_3 + - col_perm_2 + - Tarski_is_a_Coinc_theory_for_col + - diff_perm_2 + - col_bd + - path: theories/Elements/OriginalProofs/proposition_44A.v + theorems: + - proposition_44A + - path: theories/Main/Tarski_dev/Ch13_5_Pappus_Pascal.v + theorems: + - cop_par__perp2 + - l13_10_aux3 + - l13_6_bis + - per13_preserves_bet_inv + - lcos3_lcos2 + - cop_per2__perp + - lcos_lcos_cop__col + - l13_10_aux1 + - l13_11 + - l13_14 + - l13_10_aux5 + - path: theories/Main/Meta_theory/Parallel_postulates/legendre.v + theorems: + - legendre_s_third_theorem_aux + - legendre_s_fourth_theorem + - legendre_aux + - legendre_s_fourth_theorem_aux + - legendre_s_second_theorem + - stronger_legendre_s_first_theorem + - legendre_aux1 + - legendre_s_first_theorem + - path: theories/Algebraic/Counter_models/Planar/counter_model_cong_pseudo_reflexivity.v + theorems: + - bet_inner_transitivity + - lower_dim + - bet_symmetry + - inner_pasch + - euclid + - cong_identity + - upper_dim + - five_segment + - cong_pseudo_reflexivity + - path: theories/Main/Meta_theory/Parallel_postulates/par_trans_NID.v + theorems: + - col2_dec + - playfair__playfair_ter + - not_ex_forall_not_7 + - playfair_quater__playfair + - par_trans__par_dec + - par_dec_NID + - playfair_ter__playfair + - path: theories/Algebraic/Counter_models/nD/counter_model_segment_construction.v + theorems: + - Midpoint_midpoint + - bet'__bet + - coplanar_1324 + - nCol__ncol + - cong_midpoint + - euclid + - col__ColT + - midpointBR + - five_segment + - Col_aaa + - bet_midpoint + - bet_symmetry + - midpointLR + - midpointBL + - col__colI + - nCol__nCol' + - cong_identity + - segment_construction + - cong_pseudo_reflexivity + - onemx_neq0 + - betR_midpoint + - bet_inner_transitivity + - ncol__nCol + - ColT__Col + - coplanar_midpoint + - midpointxx + - cong_inner_transitivity + - coplanar_3214 + - Col'__Col + - cong_abab + - inner_pasch + - path: theories/Elements/OriginalProofs/proposition_39.v + theorems: + - proposition_39 + - path: theories/Algebraic/Counter_models/nD/independence.v + theorems: + - decidability__stability + - big_andb_and + - tnth_nseq + - nth_nseq + - path: theories/Main/Meta_theory/Continuity/elementary_continuity_props.v + theorems: + - equivalent_variants_of_line_circle + - equivalent_variants_of_circle_circle + - circle_circle_bis__euclid_22 + - circle_circle__circle_circle_two + - circle_circle_bis__circle_circle_axiom + - euclid_22__circle_circle + - circle_circle_bis__one_point_line_circle + - circle_circle__circle_circle_bis + - path: theories/Main/Highschool/midpoint_thales.v + theorems: + - midpoint_thales_reci + - midpoint_thales + - path: theories/Elements/OriginalProofs/proposition_14.v + theorems: + - proposition_14 + - path: theories/Main/Tarski_dev/Ch03_bet.v + theorems: + - l3_9_4 + - bet_neq21__neq + - outer_transitivity_between + - l3_17 + - bet_dec_eq_dec_b + - cong_dec_eq_dec_b + - bet_neq12__neq + - between_equality + - not_bet_distincts + - between_exchange2 + - between_exchange4 + - l2_11_b + - Bet_perm + - between_symmetry + - between_trivial2 + - between_exchange3 + - outer_transitivity_between2 + - between_inner_transitivity + - bet_col + - between_trivial + - between_equality_2 + - bet_neq32__neq + - another_point + - BetSEq + - Bet_cases + - point_construction_different + - two_distinct_points + - path: theories/Elements/OriginalProofs/proposition_40.v + theorems: + - proposition_40 + - path: theories/Main/Meta_theory/Parallel_postulates/rah_thales_postulate.v + theorems: + - rah__thales_postulate + - path: theories/Main/Meta_theory/Dimension_axioms/upper_dim_3.v + theorems: + - median_planes_implies_upper_dim + - upper_dim_3_equivalent_axioms + - upper_dim_3_stab + - median_planes_aux + - orthonormal_family_axiom_implies_space_separation + - upper_dim_implies_orthonormal_family_axiom + - orthonormal_family_axiom_implies_orth_at2__col + - orthonormal_family_aux + - space_separation_implies_median_planes + - orthonormal_family_axiom_implies_not_two_sides_one_side + - plane_intersection_implies_space_separation + - space_separation_implies_plane_intersection + - path: theories/Main/Annexes/tangency.v + theorems: + - tangentat_perp + - tangency_chara2 + - tangent_neq + - intercc__neq + - tangency_chara + - interccat__ncol + - tangent_out + - tangentcc__neq + - onc2__oreq + - tangency_chara3 + - cop_onc2__oreq + - diam_not_tangent + - path: theories/Elements/OriginalProofs/proposition_26A.v + theorems: + - proposition_26A + - path: theories/Main/Meta_theory/Parallel_postulates/triangle_playfair_bis.v + theorems: + - legendre_aux1 + - triangle__playfair_bis + - legendre_aux2 + - legendre_aux + - path: theories/Elements/OriginalProofs/lemma_angledistinct.v + theorems: + - lemma_angledistinct + - path: theories/Elements/OriginalProofs/lemma_squaresequal.v + theorems: + - lemma_squaresequal + - path: theories/Main/Annexes/perp_bisect.v + theorems: + - cong_mid_perp_bisect + - perp_bisect_existence_cop + - perp_bisect_cong + - perp_bisect_equiv_def + - perp_bisect_sym_1 + - perp_bisect_cong_1 + - perp_bisect_cop2_existence + - cong_cop_perp_bisect + - perp_bisect_sym_2 + - cong_cop2_perp_bisect_col + - perp_bisect_cong2 + - perp_bisect_sym_3 + - perp_mid_perp_bisect + - perp_bisect_cong_2 + - path: theories/Main/Meta_theory/Parallel_postulates/rectangle_existence_rah.v + theorems: + - rectangle_existence__rah + - path: theories/Main/Meta_theory/Models/tarski_to_cong_theory.v + theorems: + - Tarski_is_a_Cong_theory + - path: theories/Main/Meta_theory/Continuity/completeness.v + theorems: + - line_completeness_aux + - ncompleteness_for_planes__upper_dim + - nupper_dim_3__completeness_for_3d_spaces + - extension_to_3d__upper_dim_3 + - line_completeness__completeness_for_planes + - line_extension_reverse_bet + - line_completeness__completeness_for_3d_spaces + - not_archimedes__line_completeness + - line_extension_stability + - extension_reverse_col + - pres_bet_line__col + - ncompleteness_for_3d_spaces__upper_dim_3 + - extension__line_extension + - nupper_dim__completeness_for_planes + - archimedes_aux + - line_extension_symmetry + - path: theories/Elements/OriginalProofs/lemma_parallelsymmetric.v + theorems: + - lemma_parallelsymmetric + - path: theories/Elements/OriginalProofs/proposition_45.v + theorems: + - proposition_45 + - path: theories/Main/Highschool/bisector.v + theorems: + - bisector_perp_equality + - not_col_bfoot_not_equality + - perp_equality_bisector + - not_col_efoot_not_equality + - bisector_existence + - equality_foot_out_out + - path: theories/Elements/OriginalProofs/lemma_Euclid4.v + theorems: + - lemma_Euclid4 + - path: theories/Main/Annexes/defect.v + theorems: + - defect_distincts + - t22_16_1bis + - ex_defect + - defect_perm_231 + - defect_perm_321 + - t22_16_1 + - defect2__conga + - defect_perm_213 + - rah_defect__out + - t22_16_2aux1 + - conga_defect__defect + - col_defect__out + - conga3_defect__defect + - defect_ncol_out__rah + - defect_perm_132 + - t22_16_2aux + - t22_16_2 + - path: theories/Main/Meta_theory/Continuity/grad.v + theorems: + - grad__col + - gradexp_clos_trans + - gradexp2__gradexp123 + - gradexp2__gradexp456 + - grad112__eq + - grad121__eq + - grad__ex_gradexp_le + - bet_cong2_grad__grad2 + - grad_neq__neq13 + - gradexp__grad + - grad2__grad123 + - gradexpinv_clos_trans + - grad2__grad456 + - reach__ex_gradexp_le + - gradexp__gradexpinv + - reach__ex_gradexp_lt + - grad__le + - grad__bet + - reach__grad_min + - path: theories/Elements/OriginalProofs/lemma_lessthancongruence.v + theorems: + - lemma_lessthancongruence + - path: theories/Main/Meta_theory/Parallel_postulates/triangle_existential_triangle.v + theorems: + - triangle__existential_triangle + - path: theories/Main/Tarski_dev/Ch04_col.v + theorems: + - col_permutation_4 + - l4_19 + - col_permutation_2 + - l4_16 + - l4_18 + - not_col_permutation_1 + - l4_14 + - not_col_permutation_4 + - not_col_permutation_2 + - col_permutation_1 + - l4_13 + - Col_cases + - NCol_cases + - NCol_perm + - col_trivial_1 + - not_col_distincts + - not_col_permutation_3 + - col_trivial_2 + - l4_17 + - Col_perm + - not_col_permutation_5 + - col_cong_3_cong_3_eq + - col_trivial_3 + - col_permutation_3 + - path: theories/Elements/euclid_to_tarski.v + theorems: + - nullsegment2 + - Tarski_follows_Euclid + - cong_sym + - Tcong_symmetric + - Bet_sym + - nullsegment1 + - lemma_congruenceflip + - cong_eq + - Tcong_reflexive + - TCong_neq_Cong + - path: theories/Algebraic/Counter_models/nD/stronger_pasch.v + theorems: + - addition_segments + - stronger_inner_pasch + - bet_outer_connectivity + - g2_24 + - path: theories/Elements/OriginalProofs/lemma_10_12.v + theorems: + - lemma_10_12 + - path: theories/Elements/OriginalProofs/lemma_angleorderrespectscongruence.v + theorems: + - lemma_angleorderrespectscongruence + - path: theories/Main/Meta_theory/Parallel_postulates/thales_converse_postulate_weak_triangle_circumscription_principle.v + theorems: + - thales_converse_postulate__weak_triangle_circumscription_principle + - path: theories/Elements/OriginalProofs/lemma_supplementsymmetric.v + theorems: + - lemma_supplementsymmetric + - path: theories/Elements/OriginalProofs/lemma_squareparallelogram.v + theorems: + - lemma_squareparallelogram + - path: theories/Main/Tactics/CoincR_for_cop.v + theorems: + - collect_coincs_for_cop + - st_ok_empty_for_cop + - test_coinc_ok_for_cop + - collect_wds_for_cop + - path: theories/Elements/OriginalProofs/lemma_collinear4.v + theorems: + - lemma_collinear4 + - path: theories/Elements/OriginalProofs/lemma_droppedperpendicularunique.v + theorems: + - lemma_droppedperpendicularunique + - path: theories/Elements/OriginalProofs/lemma_together.v + theorems: + - lemma_together + - path: theories/Main/Meta_theory/Continuity/aristotle.v + theorems: + - aristotle__obtuse_case_elimination + - greenberg__aristotle + - aristotle__greenberg + - aristotle__acute_or_right + - path: theories/Algebraic/Counter_models/nD/gupta_inspired_to_independent_version.v + theorems: + - GI_to_IT + - GI_euclidean_to_IT_euclidean + - exists_cong_per_f + - bet__col + - basisG + - GI2D_to_T2D + - path: theories/Elements/OriginalProofs/proposition_32.v + theorems: + - proposition_32 + - path: theories/Main/Meta_theory/Parallel_postulates/playfair_midpoint.v + theorems: + - playfair_s_postulate_implies_midpoint_converse_postulate + - path: theories/Elements/OriginalProofs/lemma_paralleldef2A.v + theorems: + - lemma_paralleldef2A + - path: theories/Main/Meta_theory/Continuity/dedekind_variant.v + theorems: + - dedekind_equiv + - path: theories/Elements/OriginalProofs/lemma_fiveline.v + theorems: + - lemma_fiveline + - path: theories/Elements/OriginalProofs/lemma_diagonalsbisect.v + theorems: + - lemma_diagonalsbisect + - path: theories/Algebraic/Counter_models/Planar/counter_model_bet_inner_transitivity.v + theorems: + - inner_pasch + - lower_dim + - cong_pseudo_reflexivity + - bet_symmetry + - five_segment + - cong_inner_transitivity + - col_alt_def + - cong_identity + - upper_dim + - bet_inner_transitivity + - path: theories/Elements/OriginalProofs/lemma_congruencesymmetric.v + theorems: + - lemma_congruencesymmetric + - path: theories/Elements/OriginalProofs/lemma_collinearorder.v + theorems: + - lemma_collinearorder + - path: theories/Elements/OriginalProofs/proposition_22.v + theorems: + - lemma_togethera + - proposition_22 + - path: theories/Main/Meta_theory/Parallel_postulates/original_spp_inverse_projection_postulate.v + theorems: + - original_spp__inverse_projection_postulate + - path: theories/Elements/OriginalProofs/lemma_paralleldef2B.v + theorems: + - lemma_paralleldef2B + - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_proclus.v + theorems: + - alternate_interior__proclus_aux + - alternate_interior__proclus + - path: theories/Elements/OriginalProofs/lemma_parallelNC.v + theorems: + - lemma_parallelNC + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_concyclic.v + theorems: + - concy_perm_1 + - concy_perm_2 + - Tarski_is_a_Arity_for_concy + - concyclic_gen_1123 + - concy_bd + - Tarski_is_a_Coinc_theory_for_concy + - not_col_perm_2 + - Tarski_is_a_Coinc_predicates_for_concy + - path: theories/Main/Tactics/CoincR_for_col.v + theorems: + - st_ok_empty_for_col + - collect_coincs_for_col + - collect_wds_for_col + - test_coinc_ok_for_col + - path: theories/Elements/OriginalProofs/lemma_angleordertransitive.v + theorems: + - lemma_angleordertransitive + - path: theories/Main/Highschool/Euler_line.v + theorems: + - Euler_line_special_case + - gravity_center_change_triangle + - concyclic_not_col_or_eq_aux + - Euler_line + - concyclic_not_col_or_eq + - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_2_par_par_perp_perp.v + theorems: + - par_perp_2_par_implies_par_perp_perp + - path: theories/Elements/OriginalProofs/lemma_rectanglerotate.v + theorems: + - lemma_rectanglerotate + - path: theories/Main/Meta_theory/Parallel_postulates/SPP_ID.v + theorems: + - strong_parallel_postulate_implies_inter_dec + - path: theories/Elements/OriginalProofs/lemma_26helper.v + theorems: + - lemma_26helper + - path: theories/Elements/OriginalProofs/lemma_supplementofright.v + theorems: + - lemma_supplementofright + - path: theories/Elements/OriginalProofs/lemma_samesidetransitive.v + theorems: + - lemma_samesidetransitive + - path: theories/Elements/OriginalProofs/lemma_lessthanadditive.v + theorems: + - lemma_lessthanadditive + - path: theories/Elements/OriginalProofs/proposition_39A.v + theorems: + - proposition_39A + - path: theories/Main/Meta_theory/Continuity/dedekind_completeness.v + theorems: + - dedekind_variant__completeness + - extension_grad + - extension_gradexp + - path: theories/Elements/OriginalProofs/lemma_partnotequalwhole.v + theorems: + - lemma_partnotequalwhole + - path: theories/Main/Meta_theory/Continuity/archimedes.v + theorems: + - t22_24 + - archi__obtuse_case_elimination + - t22_24_aux1 + - t22_19 + - t22_23 + - t22_18 + - t22_24_aux + - path: theories/Main/Meta_theory/Parallel_postulates/weak_triangle_circumscription_principle_bachmann_s_lotschnittaxiom.v + theorems: + - weak_triangle_circumscription_principle__bachmann_s_lotschnittaxiom + - path: theories/Elements/OriginalProofs/proposition_36.v + theorems: + - proposition_36 + - path: theories/Elements/OriginalProofs/lemma_3_7b.v + theorems: + - lemma_3_7b + - path: theories/Main/Meta_theory/Decidability/equivalence_between_decidability_properties_of_basic_relations.v + theorems: + - equivalence_between_decidability_properties_of_basic_relations + - eq_dec_bet_dec + - cong_dec_eq_dec + - eq_dec_cong_dec + - path: theories/Elements/OriginalProofs/lemma_oppositesidesymmetric.v + theorems: + - lemma_oppositesidesymmetric + - path: theories/Main/Highschool/sesamath_exercises.v + theorems: + - sesamath_4ieme_G2_ex39 + - sesamath_4ieme_G2_ex36 + - sesamath_4ieme_G2_ex38 + - sesamath_4ieme_G2_ex47 + - sesamath_4ieme_G2_ex41 + - sesamath_4ieme_G2_ex36_aux + - sesamath_4ieme_G2_ex35 + - path: theories/Elements/OriginalProofs/lemma_collinearright.v + theorems: + - lemma_collinearright + - path: theories/Elements/OriginalProofs/lemma_equalanglestransitive.v + theorems: + - lemma_equalanglestransitive + - path: theories/Algebraic/Counter_models/nD/counter_model_cong_pseudo_reflexivity.v + theorems: + - bet_symmetry + - inner_pasch + - euclid + - five_segment + - cong_pseudo_reflexivity + - cong_identity + - cong_inner_transitivity + - onemx_neq0 + - ColP + - path: theories/Elements/OriginalProofs/lemma_Pasch_outer2.v + theorems: + - lemma_Pasch_outer2 + - path: theories/Elements/OriginalProofs/lemma_differenceofparts.v + theorems: + - lemma_differenceofparts + - path: theories/Elements/OriginalProofs/lemma_Playfairhelper2.v + theorems: + - lemma_Playfairhelper2 + - path: theories/Elements/OriginalProofs/lemma_ABCequalsCBA.v + theorems: + - lemma_ABCequalsCBA + - path: theories/Algebraic/Counter_models/nD/counter_model_lower_dim.v + theorems: + - upper_dim + - nth_basis + - mulmx11_eq0 + - bet_cong__mid + - zeromxE + - path: theories/Elements/OriginalProofs/lemma_altitudebisectsbase.v + theorems: + - lemma_altitudebisectsbase + - path: theories/Main/Meta_theory/Parallel_postulates/midpoint_playfair.v + theorems: + - midpoint_converse_postulate_implies_playfair + - path: theories/Main/Meta_theory/Parallel_postulates/SPP_tarski.v + theorems: + - impossible_case_8 + - strong_parallel_postulate_implies_tarski_s_euclid_aux + - impossible_case_6 + - impossible_case_5 + - path: theories/Elements/OriginalProofs/lemma_angleaddition.v + theorems: + - lemma_angleaddition + - path: theories/Elements/OriginalProofs/proposition_28A.v + theorems: + - proposition_28A + - path: theories/Algebraic/Counter_models/Planar/counter_model_lower_dim.v + theorems: + - segment_construction + - point_equality_decidability + - cong_identity + - five_segment + - cong_pseudo_reflexivity + - inner_pasch + - cong_inner_transitivity + - upper_dim + - lower_dim + - path: theories/Elements/OriginalProofs/lemma_equaltorightisright.v + theorems: + - lemma_equaltorightisright + - path: theories/Elements/OriginalProofs/proposition_01.v + theorems: + - proposition_01 + - path: theories/Elements/OriginalProofs/lemma_tworays.v + theorems: + - lemma_tworays + - path: theories/Main/Meta_theory/Parallel_postulates/original_euclid_original_spp.v + theorems: + - original_euclid__original_spp + - path: theories/Main/Meta_theory/Continuity/first_order.v + theorems: + - dedekind__fod + - path: theories/Algebraic/Counter_models/Planar/gupta_inspired_to_independent_tarski.v + theorems: + - GI_euclidean_to_IT_euclidean + - GI2D_to_IT2D + - GI_to_IT + - path: theories/Elements/OriginalProofs/lemma_supplements.v + theorems: + - lemma_supplements + - path: theories/Main/Meta_theory/Parallel_postulates/par_trans_playfair.v + theorems: + - par_trans_implies_playfair + - path: theories/Elements/OriginalProofs/lemma_parallelcollinear2.v + theorems: + - lemma_parallelcollinear2 + - path: theories/Elements/OriginalProofs/proposition_03.v + theorems: + - proposition_03 + - path: theories/Elements/OriginalProofs/lemma_parallelbetween.v + theorems: + - lemma_parallelbetween + - path: theories/Main/Highschool/incenter.v + theorems: + - incenter_permut132 + - incenter_dec + - incenter_permut231 + - incenter_permut321 + - incenter_exists + - path: theories/Elements/OriginalProofs/lemma_raystrict.v + theorems: + - lemma_raystrict + - path: theories/Elements/OriginalProofs/lemma_NCdistinct.v + theorems: + - lemma_NCdistinct + - path: theories/Elements/OriginalProofs/lemma_tarskiparallelflip.v + theorems: + - lemma_tarskiparallelflip + - path: theories/Main/Meta_theory/Parallel_postulates/szmielew.v + theorems: + - szmielew_s_theorem + - aah__hpp + - path: theories/Coinc/Utils/general_tactics.v + theorems: + - ltac_something_eq + - ltac_something_hide + - path: theories/Main/Meta_theory/Continuity/first_order_dedekind_circle_circle.v + theorems: + - circle_circle_aux + - path: theories/Main/Highschool/exercises.v + theorems: + - Per_mid_rectangle + - path: theories/Elements/OriginalProofs/lemma_doublereverse.v + theorems: + - lemma_doublereverse + - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_legendre_s_parallel_postulate.v + theorems: + - bachmann_s_lotschnittaxiom__legendre_s_parallel_postulate + - path: theories/Main/Meta_theory/Parallel_postulates/weak_tarski_s_parallel_postulate_weak_inverse_projection_postulate.v + theorems: + - weak_tarski_s_parallel_postulate__weak_inverse_projection_postulate_aux + - path: theories/Main/Meta_theory/Parallel_postulates/posidonius_postulate_rah.v + theorems: + - posidonius_postulate__rah + - path: theories/Main/Meta_theory/Models/tarski_to_coinc_theory_for_cop.v + theorems: + - cop_3 + - cop_bd + - cop_perm_2 + - Tarski_is_a_Arity_for_cop + - cop_perm_1 + - not_col_perm_2 + - Tarski_is_a_Coinc_theory_for_cop + - path: theories/Elements/OriginalProofs/lemma_subtractequals.v + theorems: + - lemma_subtractequals + - path: theories/Elements/OriginalProofs/proposition_35.v + theorems: + - proposition_35 + - path: theories/Elements/OriginalProofs/proposition_33.v + theorems: + - proposition_33 + - path: theories/Elements/OriginalProofs/lemma_RTcongruence.v + theorems: + - lemma_RTcongruence + - path: theories/Elements/OriginalProofs/proposition_30A.v + theorems: + - proposition_30A + - path: theories/Main/Meta_theory/Parallel_postulates/proclus_SPP.v + theorems: + - proclus_s_postulate_implies_strong_parallel_postulate + - path: theories/Main/Meta_theory/Parallel_postulates/rectangle_principle_rectangle_existence.v + theorems: + - rectangle_principle__rectangle_existence + - path: theories/Elements/OriginalProofs/proposition_06.v + theorems: + - proposition_06 + - path: theories/Main/Meta_theory/Parallel_postulates/proclus_bis_proclus.v + theorems: + - proclus_bis__proclus + - path: theories/Coinc/Permutations.v + theorems: + - PermCoincOK + - PermWdOK + - path: theories/Elements/OriginalProofs/lemma_PGsymmetric.v + theorems: + - lemma_PGsymmetric + - path: theories/Elements/OriginalProofs/lemma_layoffunique.v + theorems: + - lemma_layoffunique + - path: theories/Elements/OriginalProofs/proposition_04.v + theorems: + - proposition_04 + - path: theories/Elements/OriginalProofs/proposition_28D.v + theorems: + - proposition_28D + - path: theories/Main/Meta_theory/Models/tarski_continuous_to_trc.v + theorems: + - TC_to_TRC + - path: theories/Elements/OriginalProofs/lemma_TGsymmetric.v + theorems: + - lemma_TGsymmetric + - path: theories/Main/Meta_theory/Parallel_postulates/consecutive_interior_angles_alternate_interior_angles.v + theorems: + - consecutive_interior__alternate_interior + - path: theories/Elements/OriginalProofs/lemma_diagonalsmeet.v + theorems: + - lemma_diagonalsmeet + - path: theories/Elements/OriginalProofs/lemma_samesidesymmetric.v + theorems: + - lemma_samesidesymmetric + - path: theories/Elements/OriginalProofs/lemma_betweennesspreserved.v + theorems: + - lemma_betweennesspreserved + - path: theories/Elements/OriginalProofs/lemma_equalitysymmetric.v + theorems: + - lemma_equalitysymmetric + - path: theories/Elements/OriginalProofs/lemma_ray2.v + theorems: + - lemma_ray2 + - path: theories/Elements/OriginalProofs/lemma_oppositesideflip.v + theorems: + - lemma_oppositesideflip + - path: theories/Elements/OriginalProofs/lemma_samesideflip.v + theorems: + - lemma_samesideflip + - path: theories/Elements/OriginalProofs/lemma_8_2.v + theorems: + - lemma_8_2 + - path: theories/Main/Meta_theory/Parallel_postulates/playfair_par_trans.v + theorems: + - playfair_implies_par_trans + - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_playfair_bis.v + theorems: + - alternate_interior__playfair_aux + - alternate_interior__playfair_bis + - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_perp_par_perp_2_par.v + theorems: + - par_perp_perp_implies_par_perp_2_par + - path: theories/Main/Tarski_dev/Ch15_pyth_rel.v + theorems: + - opp_same_square + - PythRel_uniqueness + - PythOK + - PythRel_exists + - path: theories/Elements/OriginalProofs/lemma_30helper.v + theorems: + - lemma_30helper + - path: theories/Algebraic/Counter_models/Planar/independent_tarski_to_gupta_inspired.v + theorems: + - IT_euclidean_to_GI_euclidean + - IT_to_GI + - IT2D_to_GI2D + - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_perp_TCP.v + theorems: + - inter_dec_plus_par_perp_perp_imply_triangle_circumscription + - path: theories/Main/Meta_theory/Parallel_postulates/weak_inverse_projection_postulate_bachmann_s_lotschnittaxiom.v + theorems: + - weak_inverse_projection_postulate__bachmann_s_lotschnittaxiom_aux + - weak_inverse_projection_postulate__bachmann_s_lotschnittaxiom + - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_triangle.v + theorems: + - alternate_interior__triangle + - path: theories/Elements/OriginalProofs/proposition_36A.v + theorems: + - proposition_36A + - path: theories/Elements/OriginalProofs/lemma_equalanglessymmetric.v + theorems: + - lemma_equalanglessymmetric + - path: theories/Main/Meta_theory/Parallel_postulates/thales_postulate_thales_converse_postulate.v + theorems: + - thales_postulate__thales_converse_postulate + - path: theories/Main/Highschool/varignon.v + theorems: + - varignon_aux + - varignon' + - varignon + - path: theories/Main/Meta_theory/Parallel_postulates/tarski_playfair.v + theorems: + - tarski_s_euclid_implies_playfair + - path: theories/Main/Meta_theory/Parallel_postulates/playfair_bis_playfair.v + theorems: + - playfair_bis__playfair + - path: theories/Main/Meta_theory/Models/tarski_to_col_theory.v + theorems: + - Tarski_is_a_Col_theory + - path: theories/Elements/OriginalProofs/proposition_25.v + theorems: + - proposition_25 + - path: theories/Main/Meta_theory/Continuity/dedekind_archimedes.v + theorems: + - archimedes_aux + - dedekind_variant__archimedes + - path: theories/Elements/OriginalProofs/lemma_crossbar.v + theorems: + - lemma_crossbar + - path: theories/Elements/OriginalProofs/proposition_11.v + theorems: + - proposition_11 + - path: theories/Main/Meta_theory/Parallel_postulates/existential_playfair_rah.v + theorems: + - existential_playfair__rah + - path: theories/Algebraic/Counter_models/nD/stability_properties.v + theorems: + - two_points + - point_equality_stability__betS_stability + - cong_stability__point_equality_stability + - point_equality_stability__cong_stability + - path: theories/Main/Meta_theory/Models/tarski_to_makarios.v + theorems: + - Makarios_Variant_follows_from_Tarski + - lower_dim_ex + - path: theories/Elements/OriginalProofs/lemma_ondiameter.v + theorems: + - lemma_ondiameter + - path: theories/Main/Meta_theory/Parallel_postulates/alternate_interior_angles_consecutive_interior_angles.v + theorems: + - alternate_interior__consecutive_interior + - path: theories/Elements/OriginalProofs/proposition_38.v + theorems: + - proposition_38 + - path: theories/Elements/OriginalProofs/proposition_26B.v + theorems: + - proposition_26B + - path: theories/Elements/OriginalProofs/proposition_19.v + theorems: + - proposition_19 + - path: theories/Elements/OriginalProofs/lemma_trichotomy1.v + theorems: + - lemma_trichotomy1 + - path: theories/Elements/OriginalProofs/lemma_rayimpliescollinear.v + theorems: + - lemma_rayimpliescollinear + - path: theories/Elements/OriginalProofs/lemma_collinearparallel.v + theorems: + - lemma_collinearparallel + - path: theories/Elements/OriginalProofs/lemma_twoperpsparallel.v + theorems: + - lemma_twoperpsparallel + - path: theories/Elements/OriginalProofs/lemma_35helper.v + theorems: + - lemma_35helper + - path: theories/Elements/OriginalProofs/lemma_extensionunique.v + theorems: + - lemma_extensionunique + - path: theories/Elements/OriginalProofs/lemma_9_5a.v + theorems: + - lemma_9_5a + - path: theories/Elements/OriginalProofs/lemma_trichotomy2.v + theorems: + - lemma_trichotomy2 + - path: theories/Elements/OriginalProofs/lemma_righttogether.v + theorems: + - lemma_righttogether + - path: theories/Main/Meta_theory/Parallel_postulates/euclid_5_original_euclid.v + theorems: + - euclid_5__original_euclid + - path: theories/Main/Meta_theory/Parallel_postulates/playfair_existential_playfair.v + theorems: + - playfair__existential_playfair + - path: theories/Elements/OriginalProofs/proposition_48A.v + theorems: + - proposition_48A + - path: theories/Elements/OriginalProofs/proposition_28B.v + theorems: + - proposition_28B + - path: theories/Elements/OriginalProofs/proposition_15.v + theorems: + - proposition_15b + - proposition_15a + - proposition_15 + - path: theories/Elements/OriginalProofs/proposition_41.v + theorems: + - proposition_41 + - path: theories/Elements/OriginalProofs/lemma_EFreflexive.v + theorems: + - lemma_EFreflexive + - path: theories/Elements/OriginalProofs/lemma_equalanglesNC.v + theorems: + - lemma_equalanglesNC + - path: theories/Elements/OriginalProofs/lemma_rectanglereverse.v + theorems: + - lemma_rectanglereverse + - path: theories/Elements/OriginalProofs/lemma_3_6a.v + theorems: + - lemma_3_6a + - path: theories/Elements/OriginalProofs/proposition_16.v + theorems: + - proposition_16 + - path: theories/Elements/OriginalProofs/proposition_35A.v + theorems: + - proposition_35A + - path: theories/Elements/OriginalProofs/proposition_43B.v + theorems: + - proposition_43B + - path: theories/Elements/OriginalProofs/lemma_equalanglesflip.v + theorems: + - lemma_equalanglesflip + - path: theories/Elements/OriginalProofs/lemma_rightreverse.v + theorems: + - lemma_rightreverse + - path: theories/Elements/OriginalProofs/lemma_notperp.v + theorems: + - lemma_notperp + - path: theories/Main/Meta_theory/Parallel_postulates/weak_inverse_projection_postulate_weak_tarski_s_parallel_postulate.v + theorems: + - weak_inverse_projection_postulate__weak_tarski_s_parallel_postulate + - path: theories/Elements/OriginalProofs/lemma_squarerectangle.v + theorems: + - lemma_squarerectangle + - path: theories/Elements/OriginalProofs/lemma_paste5.v + theorems: + - lemma_paste5 + - path: theories/Elements/OriginalProofs/lemma_angletrichotomy2.v + theorems: + - lemma_angletrichotomy2 + - path: theories/Main/Meta_theory/Continuity/cantor_variant.v + theorems: + - nested_bis__nested + - cantor__cantor_variant + - path: theories/Elements/OriginalProofs/lemma_NCorder.v + theorems: + - lemma_NCorder + - path: theories/Elements/OriginalProofs/lemma_PGrotate.v + theorems: + - lemma_PGrotate + - path: theories/Elements/OriginalProofs/lemma_equalangleshelper.v + theorems: + - lemma_equalangleshelper + - path: theories/Elements/OriginalProofs/lemma_triangletoparallelogram.v + theorems: + - lemma_triangletoparallelogram + - path: theories/Elements/OriginalProofs/proposition_06a.v + theorems: + - proposition_06a + - path: theories/Main/Meta_theory/Parallel_postulates/rah_posidonius_postulate.v + theorems: + - rah__posidonius + - path: theories/Elements/OriginalProofs/lemma_outerconnectivity.v + theorems: + - lemma_outerconnectivity + - path: theories/Elements/OriginalProofs/proposition_42.v + theorems: + - proposition_42 + - path: theories/Elements/OriginalProofs/lemma_congruenceflip.v + theorems: + - lemma_congruenceflip + - path: theories/Main/Meta_theory/Parallel_postulates/par_perp_perp_playfair.v + theorems: + - par_perp_perp_implies_playfair + - path: theories/Elements/OriginalProofs/lemma_localextension.v + theorems: + - lemma_localextension + - path: theories/Elements/OriginalProofs/lemma_rightangleNC.v + theorems: + - lemma_rightangleNC + - path: theories/Elements/OriginalProofs/lemma_layoff.v + theorems: + - lemma_layoff + - path: theories/Elements/OriginalProofs/lemma_ETreflexive.v + theorems: + - lemma_ETreflexive + - path: theories/Main/Meta_theory/Parallel_postulates/tarski_s_euclid_remove_degenerated_cases.v + theorems: + - tarski_s_euclid_remove_degenerated_cases + - path: theories/Elements/OriginalProofs/lemma_8_7.v + theorems: + - lemma_8_7 + - path: theories/Elements/OriginalProofs/lemma_twolines2.v + theorems: + - lemma_twolines2 + - path: theories/Elements/OriginalProofs/lemma_TTflip2.v + theorems: + - lemma_TTflip2 + - path: theories/Elements/OriginalProofs/lemma_PGflip.v + theorems: + - lemma_PGflip + - path: theories/Elements/OriginalProofs/proposition_05.v + theorems: + - proposition_05 + - path: theories/Elements/OriginalProofs/lemma_crossimpliesopposite.v + theorems: + - lemma_crossimpliesopposite + - path: theories/Elements/OriginalProofs/lemma_NChelper.v + theorems: + - lemma_NChelper + - path: theories/Main/Meta_theory/Parallel_postulates/existential_saccheri_rah.v + theorems: + - existential_saccheri__rah + - path: theories/Elements/OriginalProofs/lemma_ray4.v + theorems: + - lemma_ray4 + - path: theories/Elements/OriginalProofs/lemma_trapezoiddiagonals.v + theorems: + - lemma_trapezoiddiagonals + - path: theories/Elements/OriginalProofs/lemma_supplementinequality.v + theorems: + - lemma_supplementinequality + - path: theories/Elements/OriginalProofs/proposition_21.v + theorems: + - proposition_21 + - path: theories/Elements/OriginalProofs/lemma_collinear2.v + theorems: + - lemma_collinear2 + - path: theories/Elements/OriginalProofs/lemma_Playfairhelper.v + theorems: + - lemma_Playfairhelper + - path: theories/Elements/OriginalProofs/proposition_33B.v + theorems: + - proposition_33B + - path: theories/Elements/OriginalProofs/lemma_lessthanbetween.v + theorems: + - lemma_lessthanbetween + - path: theories/Elements/OriginalProofs/lemma_angletrichotomy.v + theorems: + - lemma_angletrichotomy + - path: theories/Elements/OriginalProofs/lemma_collinearparallel2.v + theorems: + - lemma_collinearparallel2 + - path: theories/Elements/OriginalProofs/proposition_20.v + theorems: + - proposition_20 + - path: theories/Elements/OriginalProofs/lemma_TTtransitive.v + theorems: + - lemma_TTtransitive + - path: theories/Elements/OriginalProofs/proposition_34.v + theorems: + - proposition_34 + - path: theories/Elements/OriginalProofs/lemma_equalanglesreflexive.v + theorems: + - lemma_equalanglesreflexive + - path: theories/Elements/OriginalProofs/lemma_midpointunique.v + theorems: + - lemma_midpointunique + - path: theories/Main/Meta_theory/Parallel_postulates/playfair_alternate_interior_angles.v + theorems: + - playfair__alternate_interior + - path: theories/Elements/OriginalProofs/lemma_squareflip.v + theorems: + - lemma_squareflip + - path: theories/Elements/OriginalProofs/lemma_interior5.v + theorems: + - lemma_interior5 + - path: theories/Main/Meta_theory/Parallel_postulates/universal_posidonius_postulate_par_perp_perp.v + theorems: + - universal_posidonius_postulate__perpendicular_transversal_postulate_aux + - universal_posidonius_postulate__perpendicular_transversal_postulate + - path: theories/Elements/OriginalProofs/proposition_31.v + theorems: + - proposition_31 + - path: theories/Main/Meta_theory/Parallel_postulates/similar_rah.v + theorems: + - similar__rah + - similar__rah_aux + - path: theories/Elements/OriginalProofs/proposition_23B.v + theorems: + - proposition_23B + - path: theories/Elements/OriginalProofs/proposition_27B.v + theorems: + - proposition_27B + - path: theories/Elements/OriginalProofs/lemma_sameside2.v + theorems: + - lemma_sameside2 + - path: theories/Elements/OriginalProofs/lemma_parallelcollinear.v + theorems: + - lemma_parallelcollinear + - path: theories/Elements/OriginalProofs/proposition_23.v + theorems: + - proposition_23 + - path: theories/Main/Meta_theory/Parallel_postulates/playfair_universal_posidonius_postulate.v + theorems: + - playfair__universal_posidonius_postulate + - path: theories/Elements/OriginalProofs/proposition_29C.v + theorems: + - proposition_29C + - path: theories/Elements/OriginalProofs/lemma_inequalitysymmetric.v + theorems: + - lemma_inequalitysymmetric + - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_weak_triangle_circumscription_principle.v + theorems: + - bachmann_s_lotschnittaxiom__weak_triangle_circumscription_principle + - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_variant.v + theorems: + - bachmann_s_lotschnittaxiom_aux + - path: theories/Elements/OriginalProofs/lemma_betweennotequal.v + theorems: + - lemma_betweennotequal + - path: theories/Elements/OriginalProofs/lemma_parallelPasch.v + theorems: + - lemma_parallelPasch + - path: theories/Elements/OriginalProofs/lemma_Playfair.v + theorems: + - lemma_Playfair + - path: theories/Elements/OriginalProofs/lemma_ray1.v + theorems: + - lemma_ray1 + - path: theories/Elements/OriginalProofs/lemma_crisscross.v + theorems: + - lemma_crisscross + - path: theories/Elements/OriginalProofs/proposition_18.v + theorems: + - proposition_18 + - path: theories/Elements/OriginalProofs/proposition_17.v + theorems: + - proposition_17 + - path: theories/Elements/OriginalProofs/proposition_46.v + theorems: + - proposition_46 + - path: theories/Main/Meta_theory/Parallel_postulates/proclus_aristotle.v + theorems: + - proclus__aristotle + - path: theories/Elements/OriginalProofs/proposition_27.v + theorems: + - proposition_27 + - path: theories/Main/Meta_theory/Parallel_postulates/existential_triangle_rah.v + theorems: + - existential_triangle__rah + - path: theories/Main/Meta_theory/Continuity/cantor_completeness.v + theorems: + - cantor__completeness + - path: theories/Elements/OriginalProofs/lemma_ray3.v + theorems: + - lemma_ray3 + - path: theories/Elements/OriginalProofs/proposition_13.v + theorems: + - proposition_13 + - path: theories/Main/Meta_theory/Parallel_postulates/inverse_projection_postulate_proclus_bis.v + theorems: + - inverse_projection_postulate__proclus_bis + - path: theories/Elements/OriginalProofs/lemma_squareunique.v + theorems: + - lemma_squareunique + - path: theories/Elements/OriginalProofs/lemma_3_5b.v + theorems: + - lemma_3_5b + - path: theories/Elements/OriginalProofs/lemma_angleorderrespectscongruence2.v + theorems: + - lemma_angleorderrespectscongruence2 + - path: theories/Elements/OriginalProofs/lemma_altitudeofrighttriangle.v + theorems: + - lemma_altitudeofrighttriangle + - path: theories/Elements/OriginalProofs/proposition_42B.v + theorems: + - proposition_42B + - path: theories/Elements/OriginalProofs/proposition_29.v + theorems: + - proposition_29 + - path: theories/Elements/OriginalProofs/proposition_47.v + theorems: + - proposition_47 + - path: theories/Elements/OriginalProofs/lemma_PGrectangle.v + theorems: + - lemma_PGrectangle + - path: theories/Elements/OriginalProofs/proposition_48.v + theorems: + - proposition_48 + - path: theories/Main/Meta_theory/Continuity/archimedes_cantor_dedekind.v + theorems: + - archimedes_cantor__dedekind_variant + - path: theories/Elements/OriginalProofs/proposition_47B.v + theorems: + - proposition_47B + - path: theories/Elements/OriginalProofs/lemma_extension.v + theorems: + - lemma_extension + - path: theories/Main/Meta_theory/Parallel_postulates/rah_triangle.v + theorems: + - rah__triangle + - path: theories/Main/Meta_theory/Parallel_postulates/rah_similar.v + theorems: + - rah__similar + - path: theories/Elements/OriginalProofs/proposition_11B.v + theorems: + - proposition_11B + - path: theories/Elements/OriginalProofs/lemma_lessthannotequal.v + theorems: + - lemma_lessthannotequal + - path: theories/Main/Meta_theory/Parallel_postulates/bachmann_s_lotschnittaxiom_weak_inverse_projection_postulate.v + theorems: + - bachmann_s_lotschnittaxiom__weak_inverse_projection_postulate + - path: theories/Elements/OriginalProofs/lemma_together2.v + theorems: + - lemma_together2 diff --git a/GeoCoq/train/local.meta.json b/GeoCoq/train/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..b2a33417d6668f823584bd6873d70237f31cdb61 --- /dev/null +++ b/GeoCoq/train/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8844a4b9a736403ca9f4b197831b3d01eca557324a6e7e5c571cc00d297617cf +size 390 diff --git a/GeoCoq/train/local_data_0000010000.json b/GeoCoq/train/local_data_0000010000.json new file mode 100644 index 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+dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 43 +datasets: +- project: /category-theory/ + files: + - path: Lib/MapDecide.v + theorems: + - map_expr_substitution_eq + - formula_denote_subst_formula + - path: Theory/Isomorphism.v + theorems: + - from_equiv_implies_iso_equiv + - path: Theory/Category.v + theorems: + - cod_id + - path: Instance/Lambda/Step.v + theorems: + - Value_cannot_start + - path: Instance/Lambda/Sem.v + theorems: + - RenSem_inil + - path: Instance/Two/Discrete.v + theorems: + - TwoDHom_Y_X_absurd + - path: Construction/Free/Quiver.v + theorems: + - tlistDA_morDA + - morDA_tlistDA + - QuiverHomomorphismEquivalence_Symmetric + - path: Structure/Cartesian.v + theorems: + - fork_comp + - exr_split + - exr_fork_assoc + - path: Construction/Free.v + theorems: + - morD_sound + - path: Structure/Monoidal/Semicartesian/Proofs.v + theorems: + - proj_right_natural + - path: Instance/Fun.v + theorems: + - nat_id_right + - whisker_right_id + - whisker_flip + - path: Theory/Metacategory/ArrowsOnly.v + theorems: + - mapsto_inv + - path: Construction/Comma/Adjunction.v + theorems: + - lawvere_from_morph_iso_functorial + - lawvere_to_morph_iso_functorial + - surjective_tripleF + - path: Instance/Lambda/Ren.v + theorems: + - RenExp_DropAll + - Keep_idRen + - path: Construction/Product.v + theorems: + - Product_Opposite + - path: Theory/Coq/Map.v + theorems: + - lookup_delete + - path: Theory/Natural/Transformation.v + theorems: + - fun_id_left + - path: Theory/Adjunction.v + theorems: + - Build_Adjunction' + - path: Structure/UniversalProperty/Universal/Arrow.v + theorems: + - fmap_respects' + - path: Tools/Abstraction.v + theorems: + - ccc_curry + - path: Lib/FMapExt.v + theorems: + - filter_empty + - filter_idempotent + - path: Structure/Group/Proofs.v + theorems: + - left_inverse_unique + - path: Structure/Monoidal/Proofs.v + theorems: + - bimap_triangle_left + - bimap_triangle_right + - path: Instance/Lambda/Multi.v + theorems: + - values_final + - path: Theory/Metacategory/DecideExample.v + theorems: + - formula_backward_example + - path: Structure/Cocartesian.v + theorems: + - left_comp + - path: Instance/Lambda/Norm.v + theorems: + - step_preserves_SN' + - path: Instance/Lambda.v + theorems: + - composition_assoc_sym + - path: Theory/Morphisms.v + theorems: + - epi_compose + - path: Functor/Bifunctor.v + theorems: + - bimap_id_left_right + - path: Instance/Comp.v + theorems: + - from_free_unique diff --git a/category-theory/eval/local.meta.json b/category-theory/eval/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..693a4c7b44fb27b461f215964745a859b3bd9531 --- /dev/null +++ b/category-theory/eval/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2ae9f0783bf50ffdb7d6be44de5cc618366edf8abadd4bc6862ae41a5feac384 +size 386 diff --git a/category-theory/eval/local_data_0000000208.json b/category-theory/eval/local_data_0000000208.json new file mode 100644 index 0000000000000000000000000000000000000000..a98c20a3c207a67582de370f3c974675956b3f45 --- /dev/null +++ b/category-theory/eval/local_data_0000000208.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2ef7aff604ab3706149378f57f2913181690b8ecad533c784a55b7c3b5778148 +size 295969 diff --git a/category-theory/eval/local_lemma_0000000000.json b/category-theory/eval/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/category-theory/eval/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/category-theory/test/category-theory_test.yaml b/category-theory/test/category-theory_test.yaml new file mode 100644 index 0000000000000000000000000000000000000000..916014bbf31ceefe5a63259a7224d6f595674e16 --- /dev/null +++ b/category-theory/test/category-theory_test.yaml @@ -0,0 +1,224 @@ +name: category-theory_test +num_files: 56 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 101 +datasets: +- project: /category-theory/ + files: + - path: Tools/Abstraction.v + theorems: + - forall_distribute + - ccc_apply_pair + - path: Instance/Fun.v + theorems: + - "nat_\u03C1" + - "nat_\u03BB" + - path: Structure/Cartesian.v + theorems: + - second_id + - second_split + - exl_second + - split_first + - path: Instance/Lambda/Full.v + theorems: + - Plug_injective + - path: Construction/Free/Quiver.v + theorems: + - tlist'_equiv_lengths + - transport_comp + - tlist_quiver_equiv + - Forgetful + - transport_quiver_cod + - path: Structure/Cartesian/Closed.v + theorems: + - uncurry_comp + - uncurry_inj + - eval_first + - path: Structure/Monoidal/Braided.v + theorems: + - Yang_Baxter_equation + - path: Instance/Lambda.v + theorems: + - composition_identity_left + - path: Lib/MapDecide.v + theorems: + - term_denote_subst_term + - formula_sound + - path: Structure/Bicartesian.v + theorems: + - fork_merge + - path: Functor/Diagonal.v + theorems: + - Diagonal_Product_Two + - path: Instance/Roof.v + theorems: + - RPos_RNeg_absurd + - RPos_RZero_absurd + - path: Lib/FMapExt.v + theorems: + - add_mapsto_iffT + - for_all_remove_inv_2 + - add_equal_iff + - path: Theory/Coq/Map.v + theorems: + - pneq_dec + - size_delete + - delete_comm + - delete_idem + - path: Theory/Adjunction.v + theorems: + - from_adj_unit + - counit_fmap_unit + - adj_univ + - left_adjoint_iso + - path: Structure/UniversalProperty.v + theorems: + - preyoneda + - univ_property_unique_up_to_unique_iso + - path: Theory/Metacategory/ArrowsOnly.v + theorems: + - Two_2_object + - composition_right + - path: Solver/Normal.v + theorems: + - exprAD_sound' + - ex_normalize + - from_morphism_to_morphism_r + - path: Theory/Coq/Monad/Proofs.v + theorems: + - arrow_IsMonad + - path: Instance/Lambda/Multi.v + theorems: + - multistep_Fst1 + - path: Structure/Binoidal.v + theorems: + - inj_left_right + - path: Solver/Decide.v + theorems: + - expr_sound + - ex_categorical + - path: Structure/Cocartesian.v + theorems: + - left_fork + - left_right + - path: Lib/NETList.v + theorems: + - netlist_app_length + - netlist_concat_app + - path: Structure/Monoidal/Semicartesian/Proofs.v + theorems: + - proj_left_left + - proj_right_tensor_id + - path: Theory/Functor.v + theorems: + - ToAFunctor_FromAFunctor + - transport_relation_exchange + - transport_adjunction + - proper_transport_r + - fun_equiv_comp_assoc + - fun_strict_equiv_id_left + - path: Instance/Coq/Monad.v + theorems: + - join_fmap_fmap_x + - join_fmap_ret_x + - path: Theory/Morphisms.v + theorems: + - id_monic + - id_epic + - id_invol + - path: Theory/Coq/Tuple/Proofs.v + theorems: + - unsplit + - path: Theory/Isomorphism.v + theorems: + - to_equiv_implies_iso_equiv + - path: Construction/Free.v + theorems: + - composition_tapp + - path: Instance/Shapes.v + theorems: + - sized_resized + - sized + - path: Instance/Lambda/Step.v + theorems: + - Step_deterministic + - path: Instance/Lambda/Ren.v + theorems: + - RenExp_ValueP + - RenVar_idRen + - path: Functor/Bifunctor.v + theorems: + - bimap_id_right_left + - path: Instance/Comp.v + theorems: + - Project + - path: Theory/Natural/Transformation.v + theorems: + - fun_id_right + - path: Theory/Coq/List/Proofs.v + theorems: + - Permutation_insert + - Permutation_Forall + - path: Monad/Adjunction.v + theorems: + - Adjunction_Nat_Monad + - path: Construction/Comma/Adjunction.v + theorems: + - lawvere_eqv_counit_fmap_unit + - lawvere_to_functorial + - path: Adjunction/Hom.v + theorems: + - hom_unit_naturality_consequence + - path: Instance/Lambda/Norm.v + theorems: + - Exp_SN + - path: Instance/Fun/Cartesian.v + theorems: + - ump_product_auto3 + - ump_product_auto1 + - ump_product' + - path: Lib/Datatypes.v + theorems: + - match_last + - last_rcons + - map_inj + - path: Instance/Lambda/Sem.v + theorems: + - SubSem_idSub + - path: Functor/Construction/Product/Monoidal.v + theorems: + - ProductFunctor_LaxMonoidal_ap_functor_nat + - path: Theory/Metacategory/DecideExample.v + theorems: + - problem3 + - path: Functor/Hom.v + theorems: + - Curried_Hom + - path: Instance/Parallel.v + theorems: + - ParHom_Id_false_absurd + - path: Structure/Monoidal/Cartesian/Proofs.v + theorems: + - proj_left_braid + - path: Solver/Reify.v + theorems: + - test_lookup + - path: Construction/Cayley.v + theorems: + - Cayley_Right + - path: Theory/Coq/Applicative/Proofs.v + theorems: + - pure_comp + - path: Construction/Comma.v + theorems: + - comma_proj_com_iso + - path: Structure/Terminal.v + theorems: + - one_comp + - path: Structure/Monoidal/Proofs.v + theorems: + - tensor_id_left_inj diff --git a/category-theory/test/local.meta.json b/category-theory/test/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..1c68a7dd0260f3f971ce7d6eab266cc798b20791 --- /dev/null +++ b/category-theory/test/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:444a8740f43110b5ed1c444d76ceb519b247f1926371ab33ca53a2927f6171f7 +size 386 diff --git a/category-theory/test/local_data_0000000610.json b/category-theory/test/local_data_0000000610.json new file mode 100644 index 0000000000000000000000000000000000000000..e7558b05627a48ff5fca5ad4f3236c910de410f0 --- /dev/null +++ b/category-theory/test/local_data_0000000610.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2744d6246e2c3783bd255150c4a47724877c018f384569a94aa274c9d087ad8a +size 939929 diff --git a/category-theory/test/local_lemma_0000000000.json b/category-theory/test/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/category-theory/test/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/category-theory/train/category-theory_train.yaml b/category-theory/train/category-theory_train.yaml new file mode 100644 index 0000000000000000000000000000000000000000..458028f430ecf51be0a0cac7493e7dee8dc3bd74 --- /dev/null +++ b/category-theory/train/category-theory_train.yaml @@ -0,0 +1,798 @@ +name: category-theory_train +num_files: 107 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 573 +datasets: +- project: /category-theory/ + files: + - path: Structure/Cartesian.v + theorems: + - exr_first + - swap_second + - swap_inj_l + - exl_fork_comp + - first_comp + - exl_split + - exr_fork + - fork_exl_exr + - exl'_fork + - second_fork + - swap_fork + - first_id + - exr_second + - split_comp + - first_split + - swap_inj_r + - split_id + - exl_fork + - fork'_natural + - swap_first + - split_second + - exr'_fork + - first_second + - fork_inv + - swap_invol + - exr_fork_comp + - exl_fork_assoc + - exl_first + - second_comp + - split_fork + - first_fork + - path: Lib/FMapExt.v + theorems: + - filter_not_in + - update_empty_r + - MapsTo_singleton + - update_add + - for_all_remove_inv + - Oeq_neq_sym + - filter_add_false + - for_all_empty + - update_empty_l + - take_first_None + - for_all_impl + - remove_add + - add_associative + - in_mapsto_iff + - mapsto_dec + - update_find_l + - filter_add_false_r + - filter_add_true_r + - filter_add_true + - for_all_add_true + - in_mapsto_iffT + - for_all_remove + - filter_Empty + - update_find_r + - not_in_mapsto_iff + - Proper_Oeq_negb + - path: Instance/Parallel.v + theorems: + - ParHom_inv_t + - ParHom_Y_X_absurd + - ParHom_inv + - path: Structure/Monoidal/Proofs.v + theorems: + - tensor_id_right_inj + - unit_identity + - id_unit_left + - triangle_identity_right + - bimap_id_unit_left + - id_unit_right + - inverse_pentagon_identity + - triangle_identity_left + - bimap_unit_right_id + - inverse_triangle_identity + - path: Structure/Cone.v + theorems: + - Cocone + - path: Tools/Abstraction.v + theorems: + - ccc_terminal + - convert_fork + - ccc_apply + - ccc_id + - distribute_forall + - path: Structure/Cocartesian.v + theorems: + - cover_inl + - cover_inr + - inr_right + - inr_left + - inl_right + - right_id + - paws_invol + - right_cover + - left_cover + - cover_right + - inr_merge + - paws_left + - left_id + - right_comp + - merge_comp + - cover_left + - merge_inl_inr + - cover_id + - right_fork + - inl_left + - paws_fork + - cover_comp + - paws_right + - inl_merge + - merge_inv + - path: Construction/Free/Quiver.v + theorems: + - QuiverComp + - transport_comp_mid + - tlist'_quiver_equiv_transitive + - transport_quiver_dom + - tlist'_quiver_equiv_symmetric + - QuiverHomomorphismEquivalence_Transitive + - QuiverComp + - tlist'_quiver_equiv_reflexive + - InducedFunctor + - transport_id + - inductive_implies_fixpoint + - fixpoint_implies_inductive + - QuiverHomomorphismEquivalence_Reflexive + - QuiverHomomorphismOfFunctor + - UniversalArrowQuiverCat + - UnitQuiverCatAdjunction + - transport_r_comp + - path: Structure/Monoidal/Closed.v + theorems: + - curry_inj + - curry_uncurry + - curry_eval + - uncurry_curry + - uncurry_inj + - path: Theory/Coq/Map.v + theorems: + - fromList_toList + - compress_delete + - lookup_insert + - insert_delete + - MapsTo_lookup + - toList_fromList + - compress_equiv + - size_compress + - path: Functor/Structure/Monoidal.v + theorems: + - ap_iso_to_from + - ap_iso_from_to + - path: Instance/Lambda/Norm.v + theorems: + - SubExp_SN + - step_preserves_SN + - step_preserves_halting + - value_halts + - strong_normalization + - multistep_preserves_SN + - normal_forms_unique + - multistep_preserves_SN' + - path: Structure/Monoidal/Cartesian/Proofs.v + theorems: + - proj_right_braid + - eliminate_left_diagonal + - proj_right_left_diagonal + - proj_left_id_diagonal + - proj_right_id_diagonal + - unit_right_eliminate + - unit_left_eliminate + - proj_left_right_diagonal + - eliminate_right_diagonal + - path: Lib/MapDecide.v + theorems: + - formula_substitution_eq + - map_expr_size_subst_all_map_expr + - find_mapsto_iff_ex + - formula_size_subst_formula + - term_substitution_eq + - term_denote_substitution + - formula_size_subst_all_formula + - map_expr_denote_subst_map_expr + - map_contains_MapsTo + - term_beq_sound + - map_expr_size_subst_map_expr + - terms_not_conflicted + - path: Instance/Shapes.v + theorems: + - Trie_case0 + - trie_vec + - vec_Trie_left + - resized_sized + - size_unsize + - vec_trie + - vec_Trie_right + - Trie_map_flatten + - resized + - map_id + - map_concat + - Trie_right + - Trie_left + - map_map + - map_append + - path: Construction/Subcategory.v + theorems: + - Full_Implies_Full_Functor + - path: Instance/Lambda/Full.v + theorems: + - AppR_LAM + - Ctxt_id_left + - Ctxt_id_right + - strong_progress + - Ctxt_comp_assoc + - AppL_LAM + - Redex_deterministic + - Plug_functional + - path: Adjunction/Hom.v + theorems: + - hom_fmap_counit_unit + - hom_counit_fmap_unit + - hom_counit_naturality_consequence + - path: Instance/Sets/Par.v + theorems: + - from_to_impossible + - to_from_impossible + - from_to + - to_from + - path: Theory/Adjunction.v + theorems: + - fmap_counit_unit + - to_adj_unit + - to_adj_comp_law + - right_adjoint_iso + - unit_comp + - to_adj_counit + - fmap_from_adj_unit + - from_adj_respects + - from_adj_comp_law + - counit_comp + - from_adj_counit + - adj_univ_impl + - fmap_to_adj_counit + - adj_monic + - to_adj_respects + - path: Solver/Normal.v + theorems: + - exprAD_sound + - from_morphism_to_morphism + - from_morphism_app_r + - append_assoc + - from_morphism_app + - exprSD_enough + - path: Theory/Metacategory/ArrowsOnly.v + theorems: + - composition_identity_left + - Two_2_morphism + - composable_pairs_step_disjoint + - Two_2_object + - composition_respects + - composition_associative + - peano_rect' + - composition_identity_right + - composable_pairs_step_find + - composable_pairs_succ + - composition_left + - peano_rect' + - find_add_inv + - path: Structure/Monoidal/Semicartesian.v + theorems: + - eliminate_comp + - path: Functor/Construction/Product/Monoidal.v + theorems: + - ProductFunctor_fst_LaxMonoidal_ap_functor_nat + - ProductFunctor_Monoidal_proj2_ap_functor_iso + - ProductFunctor_LaxMonoidal_proj2_ap_functor_nat + - ProductFunctor_proj_LaxMonoidal + - ProductFunctor_snd_LaxMonoidal_ap_functor_nat + - ProductFunctor_LaxMonoidal_proj1_ap_functor_nat + - path: Natural/Transformation/Opposite.v + theorems: + - Opposite_Transform_invol + - path: Structure/BiCCC.v + theorems: + - unmerge_uncurry + - uncurry_merge + - path: Instance/Two/Discrete.v + theorems: + - TwoDHom_inv_t + - TwoDHom_inv + - TwoDHom_X_Y_absurd + - path: Lib/NETList.v + theorems: + - netlist_find_sublist_app + - netlist_rev_length + - netlist_concat_tadd + - netlist_rev_unit + - netlist_rev_involutive + - netlist_find_sublist_nat_ex1 + - netlist_app_assoc + - netlist_cons_uncons + - netlist_app_comm_cons + - netlist_rev_app_distr + - path: Theory/Coq/Monad/Proofs.v + theorems: + - Identity_IsMonad + - Compose_IsMonad + - path: Theory/Coq/List/Proofs.v + theorems: + - olast_spec + - olast_last + - sortBy_sorted + - olast_cons + - Forall_insert + - Permutation_sortBy + - StronglySorted_insert + - Forall_append + - StronglySorted_cat + - Forall_ordered + - path: Theory/Isomorphism.v + theorems: + - iso_sym_right_inverse + - iso_sym_left_inverse + - to_equiv_implies_iso_equiv + - to_equiv_implies_iso_equiv + - path: Structure/Monoidal/Relevance.v + theorems: + - braid2_natural + - path: Theory/Coq/Maybe/Proofs.v + theorems: + - ap_endo_nothing + - fmap_endo_nothing + - alt_endo_nothing + - bind_endo_just + - fmap_endo_just + - Maybe_choose_spec + - bind_endo_nothing + - alt_endo_just + - ap_endo_just + - path: Lib/TList.v + theorems: + - tlist_concat_tnil + - tlist_rev_length + - tlist_app_assoc + - tlist_app_cons + - tlist_app_length + - tlist_app_tnil_r + - tlist_rev_involutive + - tlist_rev_unit + - tlist_rev_app_distr + - tlist_concat_app + - tlist_app_comm_cons + - tlist_concat_tcons + - tlist_app_tnil_l + - path: Lib/Datatypes.v + theorems: + - Forall_app + - let_projT1 + - last_app_cons + - length_remove + - last_Forall + - Acc_symprod2 + - let_snd + - list_rect2 + - last_cons + - let_projT2 + - let_fst + - wf_symprod2 + - path: Structure/Cartesian/Closed.v + theorems: + - curry_comp + - curry_inj + - uncurry_comp_r + - curry_unfork + - curry_id + - uncurry_curry + - curry_uncurry + - curry_fork + - eval_curry + - curry_eval + - curry_comp_l + - path: Theory/Morphisms.v + theorems: + - sections_are_monic + - flip_Retraction + - retractions_are_epic + - flip_invol + - id_idem + - flip_Section + - monic_compose + - path: Instance/Lambda/Sem.v + theorems: + - SubSem_ScR + - RenSem_idRen + - SemExp_wk + - SubSem_ScS + - SemVar_SubSem + - SemExp_RenSem + - ScR_idRen + - SubSem_RcS + - SubSem_inil + - SemExp_SubSem + - RenSem_RcR + - RenSem_skip1 + - SemVar_RenSem + - path: Theory/Universal/Arrow.v + theorems: + - AdjunctionFromUniversalArrows + - ump_universal_arrows + - universal_arrow_from_UMP + - LeftAdjointFunctorFromUniversalArrows + - path: Instance/Lambda/Multi.v + theorems: + - multi_trans + - multi_R + - multistep_Snd1 + - multistep_Pair + - multistep_Pair2 + - multistep_AppR + - multistep_Pair1 + - path: Theory/Functor.v + theorems: + - fun_equiv_id_left + - fun_equiv_id_right + - fun_equiv_fmap_from + - fun_strict_equiv_comp_assoc + - transport_functorial_dom + - proper_transport_r + - transport_trans + - FullyFaithful + - transport_f_equal + - transport_functorial_cod + - FromAFunctor_ToAFunctor + - proper_transport_r + - proper_transport + - fun_equiv_to_fmap + - fun_strict_equiv_id_right + - fobj_Compose + - transport_r_trans + - path: Instance/Comp.v + theorems: + - Bool + - Product + - component_compose + - component_compose + - component_compose + - component_compose + - GroupEq + - induced_hom + - path: Functor/Structure/Cartesian/Closed.v + theorems: + - exp_out_inj + - exp_out_in + - exp_in_inj + - fmap_eval + - exp_in_out + - path: Theory/Coq/Tuple/Proofs.v + theorems: + - fst_snd + - uncurry_works + - path: Solver/Reify.v + theorems: + - test_reifyExpr + - test_addToObjList + - test_allObjs + - test_addToCatList + - test_catLists + - test_reifyTerm + - test_toList + - ex_find_vars + - ex_reify_and_change + - test_lookupObj + - test_listSize + - test_updateCat + - test_lookupCat + - test_lookupArr + - test_reifyObj + - test_addToList + - test_addToArrList + - test_allVars + - path: Instance/Cat.v + theorems: + - Cat_Iso_from_Full + - path: Structure/UniversalProperty/Universal/Arrow.v + theorems: + - UniversalArrowIsUniversalProperty + - path: Instance/Sets.v + theorems: + - surjectivity_is_epic + - bijective_is_iso + - injectivity_is_monic + - setoid_morphism_compose_respects + - path: Theory/Coq/Functor/Proofs.v + theorems: + - compose_fmap + - path: Theory/Natural/Transformation.v + theorems: + - fun_id_right_and_sym + - fun_id_right_sym_and + - fun_comp_assoc + - fun_id_left_sym + - fun_id_right_sym + - fun_comp_assoc_and_sym + - Build_Transform' + - nat_compose_respects + - fun_comp_assoc_sym + - fun_comp_assoc_sym_and + - fun_id_left_and_sym + - fun_id_left_sym_and + - path: Instance/Roof.v + theorems: + - RZero_RPos_id + - RNeg_RZero_absurd + - RNeg_RNeg_id + - RoofHom_inv + - RNeg_RPos_absurd + - RoofHom_inv_t + - path: Instance/Lambda.v + theorems: + - composition_assoc + - composition_identity_right + - SemExp_identity + - SemExp_composition + - path: Instance/Two.v + theorems: + - TwoHom_inv_t + - TwoHom_Y_X_absurd + - TwoHom_inv + - path: Instance/Fun.v + theorems: + - "nat_\u03B1" + - "nat_\u03B1_whisker_assoc" + - whisker_right_dist + - whisker_left_dist + - "nat_\u03B1_whisker_right_comp" + - "nat_\u03B1_whisker_left_comp" + - nat_id_left + - whisker_left_id + - nat_comp_assoc + - "nat_\u03B1_nat_\u03B1" + - Functor_Setoid_Nat_Iso + - whisker_left_right + - path: Instance/Cones/Comma.v + theorems: + - Cones_Comma + - Cones_to_Comma + - Cones_from_Comma + - path: Construction/Product.v + theorems: + - fst_comp + - snd_comp + - path: Theory/Coq/Applicative/Proofs.v + theorems: + - Compose_IsApplicative + - compose_ap + - Identity_IsApplicative + - ap_comp + - arrow_IsApplicative + - path: Instance/One/Diagonal.v + theorems: + - Diagonal_Unique + - path: Structure/Monoidal/Naturality.v + theorems: + - monoidal_naturality + - path: Instance/Lambda/Step.v + theorems: + - AppR_LAM + - strong_progress + - AppL_LAM + - value_is_nf + - Value_irreducible + - path: Functor/Bifunctor.v + theorems: + - bimap_id_id + - bimap_comp + - bimap_fmap + - bimap_comp_id_left + - bimap_comp_id_right + - path: Instance/Fun/Cartesian.v + theorems: + - ump_product_auto4 + - ump_product_auto4 + - fmap_respects' + - ump_product_auto2 + - path: Structure/Group/Proofs.v + theorems: + - mappend_inverse + - mempty_left_diagonal + - mempty_right_diagonal + - path: Structure/Initial.v + theorems: + - zero_comp + - path: Adjunction/Natural/Transformation/Opposite.v + theorems: + - Opposite_Adjunction_Transform_invol + - path: Instance/Lambda/Ren.v + theorems: + - DropAll_nil_idRen + - RenExp_idRen + - RenExp_preserves_size + - NoRen_idRen + - Keep_RcR + - RcR_idRen_right + - RcR_idRen_left + - RenVar_RcR + - RcR_assoc + - RenVar_skip1 + - RenExp_RcR + - path: Structure/UniversalProperty.v + theorems: + - univ_property_respects_iso + - univ_property_unique + - representability_by_yoneda + - path: Theory/Metacategory/DecideExample.v + theorems: + - map_decide_test + - problem4 + - problem5 + - problem100 + - path: Construction/Comma/Adjunction.v + theorems: + - lawvere_iso_to_from + - Left_Functoriality + - lawvere_from_to + - "\u03BC_\u03BA_\u03B8" + - "\u03C6_\u03C8_equiv" + - lawvere_iso_from + - lawvere_iso_from_to + - "\u03C8_\u03C6_equiv" + - "\u03B7_\u03B8_\u03BA" + - "expand_\u03C6_\u03C8" + - lawvere_iso_to + - Adjunction_Comma + - lawvere_eqv_fmap_counit_unit + - surjective_tripleG + - lawvere_to_from + - Right_Functoriality + - "expand_\u03C8_\u03C6" + - lawvere_from_functorial + - path: Instance/Lambda/Log.v + theorems: + - ExpP_P + - ExpR_R + - path: Functor/Hom.v + theorems: + - Yoneda_Embedding' + - Curried_Hom + - path: Theory/Category.v + theorems: + - dom_id + - cod_comp + - dom_comp + - Build_Category'_Coq + - path: Construction/Free.v + theorems: + - morDA_tlistDA + - tlistDA_morDA + - composition_tnil + - path: Construction/Enriched.v + theorems: + - Functor_is_Enriched_over_Set + - Category_is_Enriched_over_Set + - path: Instance/Lambda/Value.v + theorems: + - ValueP_dec + - ValueP_irrelevance + - path: Monad/Kleisli.v + theorems: + - monad_id_right + - monad_comp_assoc + - monad_id_left + - path: Structure/UniversalProperty/Limit.v + theorems: + - LimitIsUniversalProperty + - ump_limit_construct_proper + - cone_coherence_auto1 + - cone_equiv_to_morphism_equiv + - ump_limit_construct_recover + - ump_limit_construct_recover' + - ump_limit_construct + - path: Construction/Cayley.v + theorems: + - Cayley_Left + - path: Functor/Diagonal.v + theorems: + - Diagonal_Product_Two + - path: Functor/Structure/Cartesian.v + theorems: + - prod_out_in + - prod_out_inj + - prod_in_inj + - prod_in_out + - path: Construction/Opposite.v + theorems: + - op_invol + - path: Theory/Kan/Extension.v + theorems: + - left_adjoint_impl + - left_adjoints_preserve + - path: Adjunction/Opposite.v + theorems: + - Opposite_Adjunction_invol + - path: Lib/IList.v + theorems: + - ith + - ith_exact + - path: Structure/Limit/Cartesian.v + theorems: + - Cartesian_Limit + - path: Structure/Monoid.v + theorems: + - uncurry_exl_fork_exr + - mappend_assoc_sym + - path: Instance/Lambda/Sound.v + theorems: + - soundness + - path: Theory/Coq/List.v + theorems: + - oends + - path: Structure/Cone/Const.v + theorems: + - Cone_Comma + - Cone_Natural_Transform + - path: Functor/Structure/Monoidal/Pure.v + theorems: + - pure_natural + - fmap_pure + - path: Structure/Monoidal/Symmetric.v + theorems: + - bimap_braid + - braid_bimap_braid + - hexagon_rotated + - path: Instance/Poset.v + theorems: + - eq_equiv + - path: Monad/Monoid.v + theorems: + - Monoid_Monad + - path: Solver/Decide.v + theorems: + - expr_tauto + - path: Functor/Traversable/Product.v + theorems: + - ProductFunctor_Traversable_ap_functor_nat + - path: Instance/Coq/Monad.v + theorems: + - join_fmap_join_x + - join_ret_x + - path: Structure/UniversalProperty/Cartesian.v + theorems: + - CartesianProductIsUniversalProperty + - path: Instance/Lambda/Exp.v + theorems: + - Exp_size_preserved + - path: Functor/Opposite.v + theorems: + - Opposite_Functor_invol + - path: Construction/Product/Comma.v + theorems: + - Comma_Product + - path: Structure/Monoidal/Semicartesian/Proofs.v + theorems: + - proj_right_right + - proj_left_tensor_id + - proj_left_natural + - path: Construction/Comma.v + theorems: + - comma_proj_mor_iso + - path: Structure/Limit/Kan/Extension.v + theorems: + - Kan_Limit + - path: Structure/Monoidal/Internal/Product.v + theorems: + - exl_swap + - exr_swap + - path: Structure/Limit/Terminal.v + theorems: + - Terminal_Limit + - path: Structure/Monoidal/Semicocartesian.v + theorems: + - generate_comp + - path: Monad/Adjunction.v + theorems: + - Adjunction_Monad diff --git a/category-theory/train/local.meta.json b/category-theory/train/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..06c2995356de6d26fb3b9be91d97f3d9023ee4cf --- /dev/null +++ b/category-theory/train/local.meta.json @@ -0,0 +1,3 @@ +version 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b/coq/train/local_data_0000205604.json new file mode 100644 index 0000000000000000000000000000000000000000..602a9587fac98d09ff09458a78a3ce34a8b8167e --- /dev/null +++ b/coq/train/local_data_0000205604.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:acd2743f849f3929925c73928f05d60e1ca7d66aa8b98f841badf6382a3d0956 +size 63625027 diff --git a/coq/train/local_lemma_0000000000.json b/coq/train/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/coq/train/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/crossoint.py b/crossoint.py new file mode 100644 index 0000000000000000000000000000000000000000..ad9fcdfdb42cc1bc7aeb159cf6ac494a7bd90bd2 --- /dev/null +++ b/crossoint.py @@ -0,0 +1,103 @@ +import mlcroissant as mlc, validate + +# Define dataset metadata +metadata = mlc.nodes.Metadata( + name="Your Dataset Name", # Replace with your dataset name + description="Description of your dataset", # Replace with your dataset description + license="https://creativecommons.org/licenses/by/4.0/", # Replace with your dataset license + url="https://example.com/your-dataset-url", # Replace with your dataset URL +) + +# Define dataset distribution (assuming your JSON file is stored locally) +distribution = mlc.nodes.Distribution( + name="your_data.json", # Replace with your JSON file name + content_url="path/to/your/data.json", # Replace with the actual path to your JSON file + encoding_format="application/json", +) +metadata.distribution = [distribution] + +# Define RecordSet to represent the structure of your training data +record_set = mlc.nodes.RecordSet( + name="training_data", + description="Records representing the training data", + field=[ + mlc.nodes.Field( + name="proof_id", + data_type="sc:Text", + description="Unique identifier for the proof", + source=mlc.nodes.Source(extract={"json_path": "proof_id"}), + ), + mlc.nodes.Field( + name="all_useful_defns_theorems", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="List of all useful definitions and theorems", + source=mlc.nodes.Source(extract={"json_path": "all_useful_defns_theorems"}), + ), + mlc.nodes.Field( + name="goal_description", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="Description of the goal", + source=mlc.nodes.Source(extract={"json_path": "goal_description"}), + ), + mlc.nodes.Field( + name="start_goals", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="List of start goals", + source=mlc.nodes.Source(extract={"json_path": "start_goals"}), + ), + mlc.nodes.Field( + name="end_goals", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="List of end goals", + source=mlc.nodes.Source(extract={"json_path": "end_goals"}), + ), + mlc.nodes.Field( + name="proof_steps", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="List of proof steps", + source=mlc.nodesaste(extract={"json_path": "proof_steps"}), + ), + mlc.nodes.Field( + name="simplified_goals", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="List of simplified goals", + source=mlc.nodes.Source(extract={"json_path": "simplified_goals"}), + ), + mlc.nodes.Field( + name="addition_state_info", + data_type="sc:Text", # Consider using a more specific type if appropriate + description="Additional state information", + source=mlc.nodes.Source(extract={"json_path": "addition_state_info"}), + ), + mlc.nodes.Field( + name="file_path", + data_type="sc:Text", + description="File path", + source=mlc.nodes.Source(extract={"json_path": "file_path"}), + ), + mlc.nodes.Field( + name="project_id", + data_type="sc:Text", + description="Project identifier", + source=mlc.nodes.Source(extract={"json_path": "project_id"}), + ), + mlc.nodes.Field( + name="theorem_name", + data_type="sc:Text", + description="Name of the theorem", + source=mlc.nodes.Source(extract={"json_path": "theorem_name"}), + ), + + ], +) +metadata.record_set = [record_set] + + + +# Print or save the Croissant metadata +print(metadata.to_json()) + +# Optional: Save to a file +# with open("croissant_metadata.json", "w") as f: +# f.write(metadata.to_json()) +validate(metadata.to_json()) \ No newline at end of file diff --git a/lean/eval/leandojo-eval.yaml b/lean/eval/leandojo-eval.yaml new file mode 100644 index 0000000000000000000000000000000000000000..acb6baa4c3bcf79aa1a6d42b6f353ee7b82bfa76 --- /dev/null +++ b/lean/eval/leandojo-eval.yaml @@ -0,0 +1,2504 @@ +name: leandojo-eval +num_files: 729 +language: LEAN4 +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 1035 +datasets: +- project: /mathlib4 + files: + - path: Mathlib/Data/Set/Pointwise/Basic.lean + theorems: + - Set.div_subset_range + - Set.pow_mem_pow + - path: Mathlib/Analysis/SpecialFunctions/Pow/Real.lean + theorems: + - Complex.abs_cpow_eq_rpow_re_of_nonneg + - Real.rpow_div_two_eq_sqrt + - path: Mathlib/Order/Filter/AtTopBot.lean + theorems: + - Filter.exists_eventually_atTop + - Filter.tendsto_const_mul_atTop_of_pos + - path: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean + theorems: + - CategoryTheory.ShortComplex.HasRightHomology.hasKernel + - path: Mathlib/RingTheory/WittVector/Verschiebung.lean + theorems: + - WittVector.verschiebungFun_coeff + - path: Mathlib/Order/WellFounded.lean + theorems: + - WellFounded.eq_strictMono_iff_eq_range_aux + - path: Mathlib/Algebra/Order/Group/MinMax.lean + theorems: + - max_one_div_max_inv_one_eq_self + - path: Mathlib/Data/List/Intervals.lean + theorems: + - List.Ico.filter_lt_of_succ_bot + - List.Ico.length + - path: Mathlib/Data/Seq/Computation.lean + theorems: + - Computation.results_of_terminates' + - path: Mathlib/Probability/ProbabilityMassFunction/Basic.lean + theorems: + - PMF.toMeasure_injective + - path: Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean + theorems: + - expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul + - expNegInvGlue.pos_of_pos + - path: Mathlib/SetTheory/Game/PGame.lean + theorems: + - SetTheory.PGame.add_le_add_right' + - SetTheory.PGame.isOption_neg + - SetTheory.PGame.moveLeft_neg_symm + - SetTheory.PGame.bddAbove_range_of_small + - path: Mathlib/Data/List/Range.lean + theorems: + - List.unzip_enum_eq_prod + - List.pairwise_lt_finRange + - List.chain'_range_succ + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean + theorems: + - "CliffordAlgebra.\u03B9_range_map_reverse" + - CliffordAlgebra.submodule_comap_pow_reverse + - path: Mathlib/Data/Nat/Order/Basic.lean + theorems: + - Nat.div_eq_self + - Nat.add_sub_one_le_mul + - path: Mathlib/Topology/Compactness/Compact.lean + theorems: + - IsCompact.nhdsSet_prod_eq_biSup + - IsCompact.union + - IsCompact.inter_right + - path: .lake/packages/std/Std/Data/List/Lemmas.lean + theorems: + - List.diff_cons_right + - List.length_insert_of_not_mem + - List.forall_mem_append + - List.map_subset + - List.take_take + - List.exists_of_replaceF + - List.drop_sizeOf_le + - List.suffix_of_suffix_length_le + - List.append_ne_nil_of_left_ne_nil + - List.findIdx?_eq_some_iff + - List.erase_of_not_mem + - List.drop_eq_nil_iff_le + - List.take_eq_take + - path: Mathlib/NumberTheory/ADEInequality.lean + theorems: + - ADEInequality.admissible_of_one_lt_sumInv_aux' + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean + theorems: + - InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero + - EuclideanGeometry.angle_eq_arcsin_of_angle_eq_pi_div_two + - InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero + - InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero + - path: Mathlib/FieldTheory/SeparableDegree.lean + theorems: + - IntermediateField.finSepDegree_bot + - path: Mathlib/NumberTheory/FLT/Four.lean + theorems: + - Fermat42.exists_pos_odd_minimal + - path: Mathlib/Data/Matrix/Basis.lean + theorems: + - Matrix.StdBasisMatrix.mul_of_ne + - path: Mathlib/Data/Matrix/Block.lean + theorems: + - Matrix.toBlock_diagonal_disjoint + - path: Mathlib/Data/Polynomial/AlgebraMap.lean + theorems: + - Polynomial.aeval_apply_smul_mem_of_le_comap' + - path: Mathlib/Algebra/Order/Sub/Canonical.lean + theorems: + - tsub_tsub_eq_add_tsub_of_le + - path: Mathlib/Data/Nat/Parity.lean + theorems: + - Nat.even_add + - Odd.ne_two_of_dvd_nat + - pow_eq_neg_one_iff + - path: Mathlib/ModelTheory/Basic.lean + theorems: + - FirstOrder.Language.Embedding.coe_injective + - path: Mathlib/Topology/Covering.lean + theorems: + - isCoveringMap_iff_isCoveringMapOn_univ + - path: Mathlib/Data/Int/Cast/Lemmas.lean + theorems: + - Int.cast_nonneg + - Int.cast_le_neg_one_of_neg + - path: Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean + theorems: + - NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra + - path: Mathlib/Data/Polynomial/FieldDivision.lean + theorems: + - Polynomial.mod_eq_self_iff + - Polynomial.rootSet_prod + - Polynomial.monic_normalize + - Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivative + - path: Mathlib/MeasureTheory/Measure/WithDensity.lean + theorems: + - MeasureTheory.withDensity_eq_zero + - MeasureTheory.withDensity_add_measure + - MeasureTheory.restrict_withDensity + - path: Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean + theorems: + - HasSum.sigma + - path: Mathlib/Data/Set/Lattice.lean + theorems: + - Set.biUnion_iUnion + - Set.iUnion_of_singleton_coe + - path: Mathlib/GroupTheory/Perm/Fin.lean + theorems: + - Fin.cycleRange_apply + - path: Mathlib/Data/Set/Basic.lean + theorems: + - Set.ite_right + - Set.ite_compl + - path: Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean + theorems: + - CategoryTheory.Limits.HasZeroObject.zeroIsoIsTerminal_hom + - CategoryTheory.Limits.HasZeroObject.zeroIsoIsInitial_inv + - "CategoryTheory.Limits.image.\u03B9_zero'" + - "CategoryTheory.Limits.image_\u03B9_comp_eq_zero" + - CategoryTheory.Limits.HasZeroObject.zeroIsoTerminal_inv + - path: Mathlib/Analysis/Seminorm.lean + theorems: + - Seminorm.smul_inf + - Seminorm.bddBelow_range_add + - Seminorm.comp_smul + - absorbent_ball_zero + - path: Mathlib/Analysis/Calculus/FDeriv/Basic.lean + theorems: + - DifferentiableWithinAt.hasFDerivWithinAt + - fderivWithin_zero_of_isolated + - HasFDerivWithinAt.lim + - path: Mathlib/Data/Complex/ExponentialBounds.lean + theorems: + - Real.exp_neg_one_gt_d9 + - path: Mathlib/SetTheory/Cardinal/Ordinal.lean + theorems: + - Cardinal.ciSup_mul_ciSup + - Cardinal.mk_cardinal + - path: Mathlib/Geometry/Euclidean/Circumcenter.lean + theorems: + - Affine.Simplex.dist_circumcenter_sq_eq_sq_sub_circumradius + - path: Mathlib/Order/Filter/Archimedean.lean + theorems: + - Filter.Tendsto.atBot_zsmul_const + - Filter.Tendsto.atTop_mul_const' + - Filter.Eventually.int_cast_atBot + - path: Mathlib/Data/Polynomial/Eval.lean + theorems: + - "Polynomial.eval\u2082_monomial" + - Polynomial.coeff_comp_degree_mul_degree + - Polynomial.eval_ofNat + - path: Mathlib/Data/PNat/Factors.lean + theorems: + - PNat.factorMultiset_gcd + - path: Mathlib/MeasureTheory/MeasurableSpace/Defs.lean + theorems: + - MeasurableSpace.generateFrom_insert_univ + - MeasurableSet.univ + - path: Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean + theorems: + - CategoryTheory.Limits.KernelFork.app_one + - "CategoryTheory.Limits.CokernelCofork.IsColimit.isIso_\u03C0" + - CategoryTheory.Limits.cokernel.map_desc + - path: Mathlib/SetTheory/Surreal/Dyadic.lean + theorems: + - Surreal.dyadicMap_apply_pow' + - path: Mathlib/Order/LocallyFinite.lean + theorems: + - Multiset.mem_Ioo + - path: Mathlib/Condensed/Equivalence.lean + theorems: + - Condensed.StoneanProfinite.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily + - path: Mathlib/Analysis/NormedSpace/lpSpace.lean + theorems: + - lp.norm_le_of_tendsto + - "Mem\u2113p.infty_mul" + - path: Mathlib/GroupTheory/Subgroup/Pointwise.lean + theorems: + - Subgroup.sup_eq_closure_mul + - Subgroup.conj_smul_le_of_le + - path: Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean + theorems: + - SimpleGraph.ComponentCompl.exists_adj_boundary_pair + - path: Mathlib/Algebra/Module/Zlattice.lean + theorems: + - Zspan.repr_floor_apply + - path: Mathlib/Analysis/InnerProductSpace/Symmetric.lean + theorems: + - LinearMap.IsSymmetric.conj_inner_sym + - path: Mathlib/Tactic/Abel.lean + theorems: + - Mathlib.Tactic.Abel.term_add_termg + - path: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean + theorems: + - PartialHomeomorph.contDiffOn_extend_coord_change + - path: Mathlib/Data/Fintype/Basic.lean + theorems: + - Set.toFinset_image + - path: Mathlib/Order/Filter/Basic.lean + theorems: + - Filter.frequently_and_distrib_left + - Filter.iInf_principal_finite + - Filter.iSup_join + - Filter.sup_join + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean + theorems: + - Orientation.two_zsmul_oangle_smul_right_of_ne_zero + - Orientation.oangle_self + - path: Mathlib/Algebra/BigOperators/Intervals.lean + theorems: + - Finset.prod_Ioc_mul_eq_prod_Icc + - path: Mathlib/Topology/Compactification/OnePoint.lean + theorems: + - OnePoint.ne_infty_iff_exists + - OnePoint.continuousAt_infty + - OnePoint.inseparable_iff + - path: Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean + theorems: + - UniformOnFun.topologicalSpace_eq + - UniformOnFun.tendsto_iff_tendstoUniformlyOn + - path: Mathlib/LinearAlgebra/Matrix/Determinant.lean + theorems: + - Matrix.det_eq_of_forall_col_eq_smul_add_pred + - path: Mathlib/Topology/Algebra/Group/Basic.lean + theorems: + - QuotientGroup.isOpenMap_coe + - topologicalGroup_iInf + - tendsto_inv_nhdsWithin_Iic + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean + theorems: + - CategoryTheory.Limits.biprod.mapBiprod_hom_desc + - path: Mathlib/Data/Set/Pairwise/Basic.lean + theorems: + - Set.pairwise_insert + - path: Mathlib/Topology/MetricSpace/Perfect.lean + theorems: + - Perfect.small_diam_aux + - path: Mathlib/Topology/PartialHomeomorph.lean + theorems: + - PartialHomeomorph.subtypeRestr_symm_eqOn + - PartialHomeomorph.ofSet_trans' + - PartialHomeomorph.IsImage.interior + - path: Mathlib/Analysis/InnerProductSpace/Basic.lean + theorems: + - inner_add_right + - Orthonormal.sum_inner_products_le + - path: Mathlib/Order/SuccPred/Limit.lean + theorems: + - Order.mem_range_pred_of_not_isPredLimit + - path: Mathlib/Data/Polynomial/Module/Basic.lean + theorems: + - PolynomialModule.comp_smul + - path: Mathlib/Data/Int/Bitwise.lean + theorems: + - Int.bit_val + - Int.testBit_bit_zero + - path: Mathlib/Probability/Kernel/Disintegration.lean + theorems: + - ProbabilityTheory.set_integral_condKernel_univ_right + - path: Mathlib/FieldTheory/RatFunc.lean + theorems: + - RatFunc.num_mul_eq_mul_denom_iff + - RatFunc.induction_on + - RatFunc.ofFractionRing_sub + - path: Mathlib/Data/Finset/Card.lean + theorems: + - Finset.le_card_of_inj_on_range + - path: Mathlib/Topology/Order/LowerUpperTopology.lean + theorems: + - Topology.IsLower.isClosed_upperClosure + - path: Mathlib/Analysis/Calculus/SmoothSeries.lean + theorems: + - differentiable_tsum + - path: Mathlib/GroupTheory/Index.lean + theorems: + - Subgroup.relindex_self + - Subgroup.relindex_bot_right + - path: Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean + theorems: + - UpperHalfPlane.isometry_pos_mul + - path: Mathlib/Data/Polynomial/EraseLead.lean + theorems: + - Polynomial.eraseLead_C_mul_X_pow + - path: Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean + theorems: + - Asymptotics.isLittleO_pow_pow_atTop_of_lt + - path: Mathlib/Data/Finsupp/Basic.lean + theorems: + - Finsupp.equivMapDomain_trans + - path: Mathlib/Order/SupIndep.lean + theorems: + - Finset.supIndep_univ_bool + - CompleteLattice.independent_iff_supIndep_univ + - path: Mathlib/RingTheory/PowerBasis.lean + theorems: + - PowerBasis.natDegree_minpoly + - path: Mathlib/CategoryTheory/Subobject/Basic.lean + theorems: + - CategoryTheory.Subobject.ofMkLEMk_refl + - path: Mathlib/Order/SupClosed.lean + theorems: + - infClosed_empty + - path: Mathlib/Algebra/Ring/Hom/Defs.lean + theorems: + - RingHom.map_ite_one_zero + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean + theorems: + - InnerProductGeometry.cos_angle_mul_norm_mul_norm + - path: Mathlib/MeasureTheory/PiSystem.lean + theorems: + - generateFrom_piiUnionInter_le + - MeasurableSpace.DynkinSystem.ext + - path: Mathlib/RingTheory/Localization/InvSubmonoid.lean + theorems: + - IsLocalization.finiteType_of_monoid_fg + - path: Mathlib/Analysis/Calculus/ContDiff/Bounds.lean + theorems: + - norm_iteratedFDeriv_clm_apply + - norm_iteratedFDeriv_clm_apply_const + - path: Mathlib/LinearAlgebra/Matrix/Adjugate.lean + theorems: + - Matrix.adjugate_diagonal + - Matrix.adjugate_transpose + - path: Mathlib/RingTheory/WittVector/Teichmuller.lean + theorems: + - WittVector.map_teichmullerFun + - path: Mathlib/Geometry/Manifold/Instances/Sphere.lean + theorems: + - contMDiff_coe_sphere + - stereographic_apply_neg + - path: Mathlib/CategoryTheory/Abelian/RightDerived.lean + theorems: + - CategoryTheory.InjectiveResolution.toRightDerivedZero'_comp_iCycles + - path: Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean + theorems: + - MvQPF.wrepr_equiv + - path: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean + theorems: + - CategoryTheory.Limits.biprod.braid_natural + - path: Mathlib/RingTheory/DedekindDomain/Ideal.lean + theorems: + - FractionalIdeal.coe_ideal_span_singleton_inv_mul + - multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity + - Ideal.prime_span_singleton_iff + - path: Mathlib/Analysis/Convex/StrictConvexSpace.lean + theorems: + - combo_mem_ball_of_ne + - path: Mathlib/Deprecated/Subgroup.lean + theorems: + - IsGroupHom.injective_of_trivial_ker + - Group.normalClosure_subset + - path: Mathlib/LinearAlgebra/Lagrange.lean + theorems: + - Lagrange.eval_basis_self + - Lagrange.eval_nodal_at_node + - path: Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean + theorems: + - NonUnitalAlgHom.coe_range + - path: Mathlib/Data/Fintype/Perm.lean + theorems: + - mem_perms_of_finset_iff + - path: Mathlib/GroupTheory/Subgroup/Basic.lean + theorems: + - Subgroup.eq_bot_of_subsingleton + - Subgroup.Normal.mem_comm + - CommGroup.center_eq_top + - path: Mathlib/Algebra/Module/LocalizedModule.lean + theorems: + - IsLocalizedModule.iso_symm_apply_aux + - IsLocalizedModule.mk'_sub + - path: Mathlib/GroupTheory/Coset.lean + theorems: + - Subgroup.card_comap_dvd_of_injective + - path: Mathlib/Data/Real/NNReal.lean + theorems: + - NNReal.mul_eq_mul_left + - Real.toNNReal_pos + - NNReal.sSup_of_not_bddAbove + - path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean + theorems: + - CochainComplex.HomComplex.Cochain.leftUnshift_smul + - CochainComplex.HomComplex.Cochain.shift_v' + - path: Mathlib/Algebra/EuclideanDomain/Defs.lean + theorems: + - EuclideanDomain.lt_one + - path: Mathlib/Combinatorics/Hall/Basic.lean + theorems: + - Fintype.all_card_le_filter_rel_iff_exists_injective + - path: Mathlib/Analysis/InnerProductSpace/Rayleigh.lean + theorems: + - IsSelfAdjoint.hasEigenvector_of_isMaxOn + - path: Mathlib/Order/Bounds/Basic.lean + theorems: + - IsGreatest.insert + - path: Mathlib/Tactic/Ring/RingNF.lean + theorems: + - Mathlib.Tactic.RingNF.mul_neg + - path: Mathlib/Data/Matrix/Basic.lean + theorems: + - Matrix.dotProduct_pUnit + - Matrix.mulVec_zero + - Matrix.mul_neg + - path: Mathlib/GroupTheory/Sylow.lean + theorems: + - IsPGroup.sylow_mem_fixedPoints_iff + - path: Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean + theorems: + - gramSchmidt_orthonormal' + - gramSchmidt_triangular + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean + theorems: + - ContinuousLinearMap.le_of_opNorm_le_of_le + - path: Mathlib/Algebra/GroupWithZero/Basic.lean + theorems: + - "mul_eq_right\u2080" + - path: Mathlib/Data/Finsupp/Multiset.lean + theorems: + - Finsupp.toMultiset_sup + - path: Mathlib/Data/Polynomial/RingDivision.lean + theorems: + - Polynomial.roots_C + - Polynomial.root_mul + - Polynomial.not_dvd_of_natDegree_lt + - Polynomial.zero_of_eval_zero + - Polynomial.roots_pow + - path: Mathlib/CategoryTheory/Limits/Shapes/Images.lean + theorems: + - CategoryTheory.Limits.image.factorThruImage_preComp + - path: Mathlib/MeasureTheory/Integral/IntegrableOn.lean + theorems: + - MeasureTheory.hasFiniteIntegral_restrict_of_bounded + - integrableOn_Ico_iff_integrableOn_Ioo + - path: Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean + theorems: + - Gamma_is_cong_sub + - Gamma1_in_Gamma0 + - path: Mathlib/Order/Monotone/Odd.lean + theorems: + - monotone_of_odd_of_monotoneOn_nonneg + - path: Mathlib/Data/Nat/Log.lean + theorems: + - Nat.log_le_clog + - path: Mathlib/Data/Set/Sigma.lean + theorems: + - Set.range_sigmaMk + - path: Mathlib/Algebra/Order/Ring/Abs.lean + theorems: + - abs_lt_of_sq_lt_sq + - path: Mathlib/Algebra/Order/Field/Basic.lean + theorems: + - one_le_one_div + - inv_le_one + - inv_le_inv + - path: Mathlib/MeasureTheory/OuterMeasure/Basic.lean + theorems: + - MeasureTheory.OuterMeasure.restrict_mono + - MeasureTheory.OuterMeasure.sUnion_null_iff + - MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt + - MeasureTheory.OuterMeasure.ext_nonempty + - MeasureTheory.OuterMeasure.iSup_apply + - MeasureTheory.OuterMeasure.union_null + - path: Mathlib/Combinatorics/SimpleGraph/Prod.lean + theorems: + - SimpleGraph.Connected.boxProd + - path: Mathlib/Logic/Equiv/PartialEquiv.lean + theorems: + - PartialEquiv.symm_symm + - PartialEquiv.prod_symm + - PartialEquiv.IsImage.leftInvOn_piecewise + - path: Mathlib/Topology/MetricSpace/IsometricSMul.lean + theorems: + - nndist_div_left + - path: Mathlib/Topology/Category/Profinite/Nobeling.lean + theorems: + - Profinite.NobelingProof.contained_eq_proj + - path: .lake/packages/std/Std/Data/AssocList.lean + theorems: + - Std.AssocList.length_toList + - path: Mathlib/Analysis/Calculus/ContDiff/Basic.lean + theorems: + - ContinuousLinearEquiv.iteratedFDerivWithin_comp_right + - ContDiff.pow + - contDiffOn_top_iff_derivWithin + - path: Mathlib/InformationTheory/Hamming.lean + theorems: + - hammingDist_lt_one + - path: Mathlib/Data/Nat/Count.lean + theorems: + - Nat.count_lt_count_succ_iff + - path: Mathlib/Analysis/BoundedVariation.lean + theorems: + - LocallyBoundedVariationOn.ae_differentiableWithinAt + - variationOnFromTo.eq_left_iff + - path: Mathlib/Data/List/NodupEquivFin.lean + theorems: + - List.sublist_iff_exists_fin_orderEmbedding_get_eq + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean + theorems: + - Real.Angle.neg_coe_abs_toReal_of_sign_nonpos + - path: Mathlib/Probability/Kernel/MeasurableIntegral.lean + theorems: + - MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'' + - Measurable.lintegral_kernel_prod_right' + - path: Mathlib/NumberTheory/Padics/PadicVal.lean + theorems: + - one_le_padicValNat_of_dvd + - path: Mathlib/Topology/Connected/PathConnected.lean + theorems: + - PathConnectedSpace.exists_path_through_family' + - path: Mathlib/Algebra/BigOperators/Finprod.lean + theorems: + - finprod_mem_insert' + - finprod_mem_eq_one_of_infinite + - finprod_eq_dif + - path: Mathlib/MeasureTheory/Measure/Restrict.lean + theorems: + - MeasureTheory.Measure.restrict_toMeasurable + - MeasureTheory.Measure.restrict_sUnion_congr + - path: Mathlib/Topology/LocallyConstant/Basic.lean + theorems: + - LocallyConstant.piecewise_apply_right + - path: Mathlib/CategoryTheory/MorphismProperty.lean + theorems: + - CategoryTheory.MorphismProperty.RespectsIso.isoClosure_eq + - path: Mathlib/Logic/Function/Basic.lean + theorems: + - Function.surjective_of_right_cancellable_Prop + - path: Mathlib/SetTheory/Ordinal/NaturalOps.lean + theorems: + - Ordinal.nadd_nat + - path: Mathlib/MeasureTheory/Integral/Bochner.lean + theorems: + - MeasureTheory.SimpleFunc.integral_eq_sum_filter + - MeasureTheory.integral_undef + - MeasureTheory.ae_le_trim_of_stronglyMeasurable + - path: Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean + theorems: + - MeasureTheory.LpAddConst_of_one_le + - path: Mathlib/NumberTheory/Padics/PadicNumbers.lean + theorems: + - padicNormE.eq_of_norm_add_lt_right + - Padic.valuation_one + - PadicSeq.norm_nonarchimedean_aux + - Padic.padicNormE_lim_le + - path: Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean + theorems: + - CategoryTheory.tensorLeftHomEquiv_whiskerLeft_comp_evaluation + - path: Mathlib/Data/Fin/Basic.lean + theorems: + - Fin.val_bit1 + - Fin.le_pred_iff + - Fin.rev_castPred + - path: Mathlib/RingTheory/Polynomial/Bernstein.lean + theorems: + - bernsteinPolynomial.sum_mul_smul + - path: Mathlib/Analysis/Convex/Body.lean + theorems: + - ConvexBody.hausdorffEdist_ne_top + - path: Mathlib/Analysis/Matrix.lean + theorems: + - Matrix.linfty_opNorm_def + - path: Mathlib/Algebra/Homology/Single.lean + theorems: + - "ChainComplex.toSingle\u2080Equiv_symm_apply_f_zero" + - path: Mathlib/Data/MvPolynomial/Basic.lean + theorems: + - MvPolynomial.prod_X_pow_eq_monomial + - MvPolynomial.induction_on_monomial + - "MvPolynomial.eval_eval\u2082" + - path: Mathlib/Analysis/Asymptotics/Asymptotics.lean + theorems: + - Asymptotics.isLittleO_pow_sub_sub + - Asymptotics.isLittleO_insert + - path: Mathlib/Analysis/NormedSpace/Pointwise.lean + theorems: + - cthickening_cthickening + - path: Mathlib/LinearAlgebra/BilinearForm/Basic.lean + theorems: + - BilinForm.flip_flip_aux + - path: Mathlib/Topology/MetricSpace/Contracting.lean + theorems: + - ContractingWith.isFixedPt_fixedPoint_iterate + - path: Mathlib/Data/Int/CharZero.lean + theorems: + - Int.cast_eq_one + - path: Mathlib/Topology/IndicatorConstPointwise.lean + theorems: + - tendsto_indicator_const_apply_iff_eventually + - path: Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean + theorems: + - Polynomial.IsWeaklyEisensteinAt.map + - path: Mathlib/Topology/Algebra/Constructions.lean + theorems: + - Units.topology_eq_inf + - path: Mathlib/RingTheory/TensorProduct/Basic.lean + theorems: + - Algebra.TensorProduct.map_comp_includeLeft + - path: Mathlib/Data/QPF/Univariate/Basic.lean + theorems: + - QPF.Cofix.dest_corec + - QPF.Fix.rec_eq + - QPF.Fix.ind + - path: Mathlib/Algebra/Group/Hom/Defs.lean + theorems: + - map_comp_div' + - path: Mathlib/Topology/Instances/NNReal.lean + theorems: + - NNReal.summable_nat_add_iff + - NNReal.tendsto_tsum_compl_atTop_zero + - path: Mathlib/Algebra/GroupPower/Ring.lean + theorems: + - add_sq + - path: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean + theorems: + - Measurable.ennreal_tsum + - path: Mathlib/Analysis/Convex/Side.lean + theorems: + - AffineSubspace.sSameSide_vadd_right_iff + - AffineSubspace.wOppSide_iff_exists_right + - path: Mathlib/Data/NNRat/Lemmas.lean + theorems: + - Rat.toNNRat_div + - Rat.toNNRat_inv + - path: Mathlib/Analysis/Normed/Group/Basic.lean + theorems: + - dist_prod_prod_le_of_le + - path: Mathlib/Algebra/Homology/Exact.lean + theorems: + - CategoryTheory.exact_of_image_eq_kernel + - path: Mathlib/CategoryTheory/Preadditive/Generator.lean + theorems: + - CategoryTheory.isCoseparator_iff_faithful_preadditiveYoneda + - path: Mathlib/Data/List/Sigma.lean + theorems: + - List.mem_keys_kinsert + - path: Mathlib/Analysis/BoxIntegral/Partition/Split.lean + theorems: + - BoxIntegral.Box.coe_splitUpper + - BoxIntegral.Prepartition.IsPartition.exists_splitMany_le + - BoxIntegral.Box.splitLower_eq_self + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean + theorems: + - Orientation.tan_oangle_add_right_smul_rotation_pi_div_two + - Orientation.oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two + - path: Mathlib/MeasureTheory/Integral/Lebesgue.lean + theorems: + - MeasureTheory.lintegral_iSup + - ENNReal.count_const_le_le_of_tsum_le + - MeasureTheory.lintegral_lintegral_mul + - path: Mathlib/GroupTheory/PushoutI.lean + theorems: + - Monoid.PushoutI.NormalWord.eq_one_of_smul_normalized + - path: Mathlib/Order/LiminfLimsup.lean + theorems: + - Filter.blimsup_mono + - Filter.isCobounded_principal + - Filter.limsup_congr + - path: Mathlib/Data/Fin/OrderHom.lean + theorems: + - Fin.predAbove_rev_left + - path: Mathlib/Algebra/Algebra/Spectrum.lean + theorems: + - spectrum.add_mem_add_iff + - spectrum.unit_smul_eq_smul + - spectrum.map_star + - path: Mathlib/Data/Polynomial/Derivative.lean + theorems: + - Polynomial.derivative_C_mul_X_sq + - path: Mathlib/CategoryTheory/Subobject/Comma.lean + theorems: + - CategoryTheory.CostructuredArrow.unop_left_comp_underlyingIso_hom_unop + - path: Mathlib/MeasureTheory/Measure/Dirac.lean + theorems: + - MeasureTheory.Measure.tsum_indicator_apply_singleton + - path: Mathlib/Order/CompactlyGenerated/Basic.lean + theorems: + - DirectedOn.disjoint_sSup_left + - path: Mathlib/Data/Polynomial/Inductions.lean + theorems: + - Polynomial.divX_X_pow + - path: Mathlib/Data/Real/Basic.lean + theorems: + - Real.cauchy_sub + - path: .lake/packages/std/Std/Data/Nat/Gcd.lean + theorems: + - Nat.Coprime.gcd_mul_left_cancel_right + - path: Mathlib/Topology/NoetherianSpace.lean + theorems: + - TopologicalSpace.NoetherianSpace.exists_finset_irreducible + - path: Mathlib/NumberTheory/Dioph.lean + theorems: + - Dioph.ext + - path: Mathlib/LinearAlgebra/Matrix/ToLin.lean + theorems: + - Matrix.toLin'_mul_apply + - path: .lake/packages/std/Std/Data/Nat/Lemmas.lean + theorems: + - Nat.compare_eq_eq + - path: Mathlib/Analysis/Convex/Gauge.lean + theorems: + - gauge_neg + - continuousAt_gauge + - path: Mathlib/Data/Multiset/Sum.lean + theorems: + - Multiset.mem_disjSum + - path: Mathlib/Order/ConditionallyCompleteLattice/Basic.lean + theorems: + - WithTop.iInf_empty + - ciSup_unique + - path: Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean + theorems: + - CochainComplex.mappingCone.inr_snd + - CochainComplex.mappingCone.inl_v_descCochain_v + - path: Mathlib/Data/Set/Intervals/OrdConnected.lean + theorems: + - OrderEmbedding.image_Ioo + - path: Mathlib/Data/Set/Card.lean + theorems: + - Set.encard_union_le + - Set.Nat.encard_range + - Set.one_lt_ncard_iff + - Set.eq_of_subset_of_ncard_le + - path: Mathlib/Data/Part.lean + theorems: + - Part.bind_some_eq_map + - Part.mem_restrict + - Part.sdiff_get_eq + - Part.bind_none + - path: Mathlib/ModelTheory/Types.lean + theorems: + - FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_univ_iff + - path: Mathlib/Data/List/Rotate.lean + theorems: + - List.rotate'_cons_succ + - List.rotate'_length_mul + - List.rotate_rotate + - path: Mathlib/Analysis/Convex/StoneSeparation.lean + theorems: + - exists_convex_convex_compl_subset + - path: Mathlib/LinearAlgebra/Orientation.lean + theorems: + - Basis.orientation_unitsSMul + - path: Mathlib/GroupTheory/CoprodI.lean + theorems: + - Monoid.CoprodI.lift_word_prod_nontrivial_of_not_empty + - path: Mathlib/Algebra/GCDMonoid/Basic.lean + theorems: + - gcd_mul_dvd_mul_gcd + - path: Mathlib/Data/Nat/Bitwise.lean + theorems: + - Nat.xor_self + - path: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean + theorems: + - MeasureTheory.StronglyMeasurable.norm_approxBounded_le + - MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae + - MeasureTheory.StronglyMeasurable.piecewise + - List.aestronglyMeasurable_prod' + - path: Mathlib/Algebra/Order/Monoid/Lemmas.lean + theorems: + - lt_one_of_mul_lt_right + - path: Mathlib/Algebra/Order/Monovary.lean + theorems: + - monovaryOn_iff_mul_rearrangement + - path: Mathlib/Order/UpperLower/Basic.lean + theorems: + - isLowerSet_iff_Iic_subset + - UpperSet.codisjoint_coe + - path: Mathlib/LinearAlgebra/Vandermonde.lean + theorems: + - Matrix.det_vandermonde + - path: Mathlib/Combinatorics/SimpleGraph/Subgraph.lean + theorems: + - SimpleGraph.Subgraph.deleteVerts_le + - SimpleGraph.Subgraph.coe_degree + - path: Mathlib/Algebra/Group/Basic.lean + theorems: + - div_mul_mul_cancel + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean + theorems: + - LinearIsometry.norm_toContinuousLinearMap_comp + - path: Mathlib/Data/Set/Intervals/Group.lean + theorems: + - Set.pairwise_disjoint_Ioo_add_int_cast + - path: .lake/packages/std/Std/Data/RBMap/WF.lean + theorems: + - Std.RBNode.RedRed.balance2 + - path: Mathlib/CategoryTheory/Comma/Presheaf.lean + theorems: + - "CategoryTheory.OverPresheafAux.MakesOverArrow.map\u2081" + - CategoryTheory.OverPresheafAux.counitForward_val_snd + - path: Mathlib/Topology/VectorBundle/Basic.lean + theorems: + - VectorBundleCore.coordChange_linear_comp + - path: Mathlib/Topology/NhdsSet.lean + theorems: + - nhdsSet_diagonal + - path: Mathlib/Data/Nat/Pairing.lean + theorems: + - Nat.unpair_left_le + - path: Mathlib/RingTheory/MvPolynomial/Homogeneous.lean + theorems: + - MvPolynomial.homogeneousComponent_zero + - path: Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean + theorems: + - smoothSheafCommRing.isUnit_stalk_iff + - path: Mathlib/Topology/Algebra/UniformGroup.lean + theorems: + - UniformGroup.uniformContinuous_iff_open_ker + - path: Mathlib/Algebra/Lie/Killing.lean + theorems: + - LieAlgebra.IsKilling.ker_restrictBilinear_eq_bot_of_isCartanSubalgebra + - path: Mathlib/Geometry/Euclidean/Sphere/Power.lean + theorems: + - EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi + - path: Mathlib/Geometry/Manifold/ChartedSpace.lean + theorems: + - TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq + - path: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean + theorems: + - MeasureTheory.integrable_of_intervalIntegral_norm_bounded + - path: Mathlib/GroupTheory/Complement.lean + theorems: + - Subgroup.IsComplement.equiv_snd_eq_iff_rightCosetEquivalence + - path: Mathlib/RingTheory/Localization/Basic.lean + theorems: + - IsLocalization.map_eq_zero_iff + - IsLocalization.algebraMap_eq_map_map_submonoid + - path: Mathlib/Data/MvPolynomial/Degrees.lean + theorems: + - MvPolynomial.degrees_rename + - MvPolynomial.totalDegree_zero + - MvPolynomial.degrees_monomial_eq + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean + theorems: + - "CliffordAlgebra.foldl_prod_map_\u03B9" + - path: Mathlib/Topology/MetricSpace/Bounded.lean + theorems: + - Metric.dist_le_diam_of_mem' + - path: Mathlib/Analysis/NormedSpace/Banach.lean + theorems: + - LinearMap.continuous_of_seq_closed_graph + - ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl + - path: Mathlib/Data/List/Basic.lean + theorems: + - List.head_mem + - List.getLastI_eq_getLast? + - List.get_set_of_ne + - List.map_bijective_iff + - List.disjoint_map + - List.zipWith_flip + - path: Mathlib/GroupTheory/FiniteAbelian.lean + theorems: + - AddCommGroup.equiv_directSum_zmod_of_finite + - path: Mathlib/Data/Bool/Count.lean + theorems: + - List.Chain'.length_div_two_le_count_bool + - path: Mathlib/Analysis/SpecialFunctions/Gaussian.lean + theorems: + - GaussianFourier.integrable_cexp_neg_mul_sq_norm_add + - norm_cexp_neg_mul_sq + - integrable_mul_cexp_neg_mul_sq + - path: Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean + theorems: + - SimpleGraph.incMatrix_apply_eq_zero_iff + - SimpleGraph.incMatrix_apply' + - path: Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean + theorems: + - ModularCyclotomicCharacter.id + - path: Mathlib/Data/Polynomial/Degree/Definitions.lean + theorems: + - Polynomial.eq_X_add_C_of_degree_eq_one + - Polynomial.ne_zero_of_natDegree_gt + - Polynomial.degree_linear + - path: Mathlib/Analysis/InnerProductSpace/Adjoint.lean + theorems: + - LinearIsometryEquiv.adjoint_eq_symm + - path: Mathlib/Topology/ContinuousFunction/Bounded.lean + theorems: + - BoundedContinuousFunction.add_norm_nonneg + - BoundedContinuousFunction.abs_diff_coe_le_dist + - path: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean + theorems: + - CategoryTheory.Limits.pullback_symmetry_hom_of_epi_eq + - path: Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean + theorems: + - CircleDeg1Lift.translationNumber_one + - CircleDeg1Lift.translationNumber_mul_of_commute + - path: Mathlib/Combinatorics/SimpleGraph/Basic.lean + theorems: + - SimpleGraph.not_mem_neighborSet_self + - SimpleGraph.mem_incidence_iff_neighbor + - SimpleGraph.disjoint_edgeSet + - path: Mathlib/Analysis/Convex/Combination.lean + theorems: + - Set.Finite.convexHull_eq_image + - path: Mathlib/LinearAlgebra/Matrix/Circulant.lean + theorems: + - Matrix.circulant_mul_comm + - path: Mathlib/NumberTheory/Divisors.lean + theorems: + - Nat.image_div_divisors_eq_divisors + - Nat.swap_mem_divisorsAntidiagonal + - path: Mathlib/Topology/MetricSpace/Basic.lean + theorems: + - dist_pos + - MetricSpace.replaceUniformity_eq + - path: Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean + theorems: + - CategoryTheory.solutionSetCondition_of_isRightAdjoint + - path: Mathlib/Topology/UniformSpace/Completion.lean + theorems: + - CauchyFilter.uniformInducing_pureCauchy + - path: Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean + theorems: + - Set.self_mem_ordConnectedComponent + - path: Mathlib/Data/Nat/ModEq.lean + theorems: + - Nat.add_mod_add_of_le_add_mod + - Nat.odd_mod_four_iff + - path: Mathlib/Topology/UniformSpace/Equicontinuity.lean + theorems: + - Filter.HasBasis.uniformEquicontinuousOn_iff_left + - path: Mathlib/Algebra/GroupWithZero/NeZero.lean + theorems: + - inv_mul_cancel + - path: Mathlib/Data/Fin/Tuple/Curry.lean + theorems: + - Function.FromTypes.curry_uncurry + - path: Mathlib/Order/JordanHolder.lean + theorems: + - JordanHolderLattice.second_iso_of_eq + - path: Mathlib/Data/Vector/Snoc.lean + theorems: + - Vector.map_snoc + - path: Mathlib/Topology/ContinuousOn.lean + theorems: + - continuous_mulIndicator + - nhdsWithin_pi_eq' + - path: Mathlib/NumberTheory/SumPrimeReciprocals.lean + theorems: + - one_half_le_sum_primes_ge_one_div + - path: Mathlib/Data/Polynomial/Basic.lean + theorems: + - Polynomial.coeff_update_apply + - Polynomial.support_monomial + - Polynomial.toFinsupp_add + - path: Mathlib/Analysis/Calculus/LineDeriv/Basic.lean + theorems: + - hasLineDerivAt_iff_tendsto_slope_zero + - HasLineDerivAt.of_comp + - path: Mathlib/Topology/Baire/Lemmas.lean + theorems: + - "IsG\u03B4.dense_biUnion_interior_of_closed" + - "dense_biInter_of_G\u03B4" + - path: Mathlib/LinearAlgebra/Span.lean + theorems: + - Submodule.iSup_eq_span + - Submodule.mem_sSup + - Submodule.sup_span + - path: Mathlib/Algebra/BigOperators/Fin.lean + theorems: + - Fin.partialProd_succ' + - path: Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean + theorems: + - CategoryTheory.ProjectiveResolution.iso_inv_naturality + - path: Mathlib/Algebra/Order/Ring/Defs.lean + theorems: + - le_mul_of_le_one_right + - one_le_bit1 + - path: Mathlib/RingTheory/Valuation/Basic.lean + theorems: + - Valuation.one_lt_val_iff + - Valuation.map_sub_le + - path: Mathlib/Order/GaloisConnection.lean + theorems: + - "GaloisConnection.l_iSup\u2082" + - path: Mathlib/GroupTheory/Transfer.lean + theorems: + - MonoidHom.transferSylow_eq_pow_aux + - path: Mathlib/LinearAlgebra/Basis.lean + theorems: + - Basis.constr_apply + - path: Mathlib/Computability/Primrec.lean + theorems: + - Nat.Primrec.sub + - Primrec.bind_decode_iff + - Primrec.const + - path: Mathlib/Analysis/Subadditive.lean + theorems: + - Subadditive.apply_mul_add_le + - path: Mathlib/RingTheory/UniqueFactorizationDomain.lean + theorems: + - Associates.map_subtype_coe_factors' + - UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors + - Associates.FactorSet.prod_eq_zero_iff + - UniqueFactorizationMonoid.dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors + - path: Mathlib/Data/Seq/WSeq.lean + theorems: + - Stream'.WSeq.LiftRel.refl + - path: Mathlib/Combinatorics/Composition.lean + theorems: + - List.map_length_splitWrtCompositionAux + - CompositionAsSet.mem_boundaries_iff_exists_blocks_sum_take_eq + - path: Mathlib/Order/WellFoundedSet.lean + theorems: + - Set.PartiallyWellOrderedOn.wellFoundedOn + - path: Mathlib/Analysis/Calculus/FDeriv/Analytic.lean + theorems: + - ContinuousMultilinearMap.cPolynomialAt + - path: Mathlib/LinearAlgebra/BilinearForm/Properties.lean + theorems: + - BilinForm.dualBasis_dualBasis + - path: Mathlib/Data/Set/Defs.lean + theorems: + - Set.mem_univ_pi + - path: Mathlib/Data/Rat/Defs.lean + theorems: + - Rat.coe_nat_eq_divInt + - Rat.zero_add + - path: .lake/packages/std/Std/Data/Array/Lemmas.lean + theorems: + - Array.get?_push_lt + - Array.contains_def + - path: Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean + theorems: + - FreeAbelianGroup.support_nsmul + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean + theorems: + - CliffordAlgebra.changeForm_changeForm + - CliffordAlgebra.changeForm.associated_neg_proof + - path: Mathlib/Data/List/OfFn.lean + theorems: + - List.last_ofFn + - path: Mathlib/Algebra/Ring/Defs.lean + theorems: + - ite_sub_ite + - path: Mathlib/Analysis/Convex/Between.lean + theorems: + - wbtw_iff_right_eq_or_left_mem_image_Ici + - path: Mathlib/Analysis/Convex/Star.lean + theorems: + - StarConvex.preimage_add_left + - path: Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean + theorems: + - IsCoercive.antilipschitz + - IsCoercive.bounded_below + - path: Mathlib/Logic/Basic.lean + theorems: + - xor_false + - path: Mathlib/Analysis/Calculus/LHopital.lean + theorems: + - HasDerivAt.lhopital_zero_atTop_on_Ioi + - deriv.lhopital_zero_nhds + - path: Mathlib/Data/Polynomial/Reverse.lean + theorems: + - Polynomial.reverse_neg + - Polynomial.reverse_mul + - path: Mathlib/Algebra/Homology/HomotopyCofiber.lean + theorems: + - "HomologicalComplex.cylinder.map_\u03B9\u2080_eq_map_\u03B9\u2081" + - path: Mathlib/Probability/Independence/Kernel.lean + theorems: + - ProbabilityTheory.kernel.iIndep_comap_mem_iff + - path: Mathlib/Data/Real/Pointwise.lean + theorems: + - Real.sSup_smul_of_nonpos + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean + theorems: + - Orientation.det_rotation + - path: Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean + theorems: + - affineSpan_le_toAffineSubspace_span + - vectorSpan_eq_span_vsub_set_left + - AffineSubspace.spanPoints_subset_coe_of_subset_coe + - path: Mathlib/MeasureTheory/Group/Convolution.lean + theorems: + - MeasureTheory.Measure.mconv_add + - path: Mathlib/Algebra/BigOperators/Basic.lean + theorems: + - Multiset.prod_sum + - path: Mathlib/LinearAlgebra/LinearIndependent.lean + theorems: + - LinearIndependent.maximal_iff + - LinearIndependent.disjoint_span_image + - path: Mathlib/RingTheory/Polynomial/GaussLemma.lean + theorems: + - Polynomial.Monic.irreducible_iff_irreducible_map_fraction_map + - path: Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean + theorems: + - CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_snd_snd + - CategoryTheory.Limits.diagonalObjPullbackFstIso_hom_snd + - path: Mathlib/Data/Polynomial/Expand.lean + theorems: + - Polynomial.contract_expand + - Polynomial.natDegree_expand + - path: Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean + theorems: + - TopCat.pullback_fst_image_snd_preimage + - path: Mathlib/Data/PFunctor/Univariate/Basic.lean + theorems: + - PFunctor.liftp_iff' + - PFunctor.liftr_iff + - path: .lake/packages/std/Std/Data/Int/DivMod.lean + theorems: + - Int.fdiv_self + - path: Mathlib/Data/String/Lemmas.lean + theorems: + - String.leftpad_suffix + - String.leftpad_length + - path: Mathlib/RingTheory/MvPowerSeries/Basic.lean + theorems: + - MvPowerSeries.monomial_mul_monomial + - MvPowerSeries.algebraMap_apply + - MvPowerSeries.commute_monomial + - path: Mathlib/Data/Nat/Size.lean + theorems: + - Nat.size_pow + - path: Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean + theorems: + - "AlgebraicTopology.DoldKan.MorphComponents.preComp_\u03C6" + - path: Mathlib/Data/Finset/Fold.lean + theorems: + - Finset.fold_image + - path: Mathlib/Combinatorics/Partition.lean + theorems: + - Nat.Partition.partition_one_parts + - path: Mathlib/Combinatorics/Derangements/Basic.lean + theorems: + - derangements.Equiv.RemoveNone.fiber_none + - path: Mathlib/Computability/TuringMachine.lean + theorems: + - Turing.TM2to1.addBottom_modifyNth + - "Turing.eval_maximal\u2081" + - path: Mathlib/Logic/Equiv/Basic.lean + theorems: + - Equiv.sumCongr_refl + - path: Mathlib/CategoryTheory/Category/Basic.lean + theorems: + - CategoryTheory.eq_whisker + - path: Mathlib/CategoryTheory/EqToHom.lean + theorems: + - CategoryTheory.eqToIso_map_trans + - path: Mathlib/Data/Real/GoldenRatio.lean + theorems: + - geom_gold_isSol_fibRec + - goldConj_neg + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean + theorems: + - Complex.cos_pi_div_two + - Real.sin_lt_sin_of_lt_of_le_pi_div_two + - path: Mathlib/MeasureTheory/Measure/EverywherePos.lean + theorems: + - MeasureTheory.Measure.measure_eq_zero_of_subset_diff_everywherePosSubset + - path: Mathlib/GroupTheory/OrderOfElement.lean + theorems: + - Commute.orderOf_mul_dvd_lcm + - isOfFinOrder_iff_zpow_eq_one + - orderOf_eq_prime + - path: Mathlib/Algebra/Order/Monoid/WithTop.lean + theorems: + - WithTop.map_add + - path: Mathlib/Combinatorics/Additive/SalemSpencer.lean + theorems: + - MulSalemSpencer.mul_right + - path: Mathlib/Probability/ProbabilityMassFunction/Constructions.lean + theorems: + - PMF.map_comp + - path: Mathlib/Data/Polynomial/Coeff.lean + theorems: + - Polynomial.coeff_mul_C + - path: Mathlib/Data/Complex/Exponential.lean + theorems: + - Real.exp_approx_start + - Real.quadratic_le_exp_of_nonneg + - Complex.sin_zero + - Complex.cos_two_mul + - path: Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean + theorems: + - Path.Homotopy.continuous_transReflReparamAux + - path: Mathlib/RingTheory/WittVector/Truncated.lean + theorems: + - TruncatedWittVector.card + - path: Mathlib/Analysis/ODE/Gronwall.lean + theorems: + - ODE_solution_unique_of_eventually + - path: Mathlib/Topology/Connected/Basic.lean + theorems: + - "PreconnectedSpace.induction\u2082'" + - IsPreconnected.prod + - IsPreconnected.union' + - connectedComponentIn_eq + - path: Mathlib/Data/Option/NAry.lean + theorems: + - "Option.map\u2082_assoc" + - "Option.map\u2082_map_left_anticomm" + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean + theorems: + - Real.surjOn_tan + - path: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean + theorems: + - NNReal.rpow_neg_one + - ENNReal.top_rpow_of_neg + - path: Mathlib/Probability/Kernel/Basic.lean + theorems: + - ProbabilityTheory.kernel.sum_add + - ProbabilityTheory.kernel.integral_piecewise + - ProbabilityTheory.kernel.integral_deterministic + - ProbabilityTheory.kernel.restrict_apply' + - path: Mathlib/RingTheory/Ideal/Operations.lean + theorems: + - Submodule.mem_smul_span + - Ideal.IsPrime.multiset_prod_le + - path: Mathlib/RingTheory/Etale.lean + theorems: + - Algebra.FormallyUnramified.localization_map + - path: Mathlib/RingTheory/Ideal/QuotientOperations.lean + theorems: + - Ideal.mem_quotient_iff_mem + - Ideal.fst_comp_quotientInfEquivQuotientProd + - path: Mathlib/CategoryTheory/EffectiveEpi/Preserves.lean + theorems: + - CategoryTheory.effectiveEpiFamilyStructOfEquivalence_aux + - path: Mathlib/Logic/Relation.lean + theorems: + - Relation.TransGen.head_induction_on + - Relation.map_map + - Relation.reflTransGen_eq_reflGen + - Relation.ReflTransGen.trans_induction_on + - path: Mathlib/RingTheory/Polynomial/Vieta.lean + theorems: + - MvPolynomial.prod_C_add_X_eq_sum_esymm + - path: Mathlib/Algebra/Periodic.lean + theorems: + - Function.Periodic.const_inv_smul + - path: Mathlib/Algebra/Lie/Solvable.lean + theorems: + - LieAlgebra.derivedSeriesOfIdeal_le + - path: Mathlib/Algebra/BigOperators/Module.lean + theorems: + - Finset.sum_range_by_parts + - path: Mathlib/CategoryTheory/Sites/Sheaf.lean + theorems: + - CategoryTheory.Presheaf.isSheaf_iff_multiequalizer + - path: Mathlib/SetTheory/Cardinal/Basic.lean + theorems: + - Cardinal.one_lt_iff_nontrivial + - Cardinal.self_le_power + - Cardinal.mk_subset_ge_of_subset_image_lift + - path: Mathlib/Topology/Basic.lean + theorems: + - closure_diff_frontier + - interior_eq_empty_iff_dense_compl + - path: Mathlib/Data/Finsupp/ToDFinsupp.lean + theorems: + - finsuppLequivDFinsupp_apply_apply + - path: Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean + theorems: + - one_lt_mul_iff + - path: Mathlib/NumberTheory/RamificationInertia.lean + theorems: + - Ideal.ramificationIdx_lt + - path: Mathlib/Analysis/Analytic/Meromorphic.lean + theorems: + - MeromorphicAt.order_eq_top_iff + - path: Mathlib/Algebra/Order/ToIntervalMod.lean + theorems: + - toIcoMod_sub_zsmul + - toIocMod_eq_iff + - path: Mathlib/Computability/Language.lean + theorems: + - Language.mul_self_kstar_comm + - path: Mathlib/MeasureTheory/MeasurableSpace/Basic.lean + theorems: + - MeasurableSpace.comap_compl + - MeasurableEmbedding.iff_comap_eq + - measurableSet_tendsto + - path: Mathlib/Analysis/Convex/GaugeRescale.lean + theorems: + - image_gaugeRescaleHomeomorph_interior + - path: Mathlib/LinearAlgebra/Basis/VectorSpace.lean + theorems: + - LinearMap.exists_rightInverse_of_surjective + - Submodule.exists_le_ker_of_lt_top + - path: Mathlib/AlgebraicTopology/DoldKan/PInfty.lean + theorems: + - AlgebraicTopology.DoldKan.PInfty_comp_QInfty + - AlgebraicTopology.DoldKan.QInfty_f_comp_PInfty_f + - AlgebraicTopology.DoldKan.QInfty_idem + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean + theorems: + - MeasureTheory.condexp_condexp_of_le + - path: Mathlib/MeasureTheory/Integral/FundThmCalculus.lean + theorems: + - intervalIntegral.integral_comp_smul_deriv'' + - path: Mathlib/Algebra/ModEq.lean + theorems: + - AddCommGroup.intCast_modEq_intCast' + - path: Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean + theorems: + - CategoryTheory.Limits.Multiequalizer.hom_ext + - path: Mathlib/Analysis/SpecialFunctions/Log/Basic.lean + theorems: + - Mathlib.Meta.Positivity.log_pos_of_isRat_neg + - Real.tendsto_log_atTop + - path: Mathlib/Data/QPF/Multivariate/Basic.lean + theorems: + - MvQPF.liftR_iff + - path: Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean + theorems: + - HomotopyCategory.distinguished_iff_iso_trianglehOfDegreewiseSplit + - path: Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean + theorems: + - midpoint_sub_add + - path: Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean + theorems: + - Module.End.isRoot_of_hasEigenvalue + - path: Mathlib/Data/Finmap.lean + theorems: + - Finmap.mem_singleton + - path: Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean + theorems: + - Real.Gamma_one + - Complex.Gamma_ofReal + - path: Mathlib/Data/Rat/Floor.lean + theorems: + - Rat.floor_def' + - path: Mathlib/Algebra/Homology/Homotopy.lean + theorems: + - Homotopy.comp_nullHomotopicMap + - path: Mathlib/LinearAlgebra/Matrix/Symmetric.lean + theorems: + - Matrix.IsSymm.fromBlocks + - path: Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean + theorems: + - "jacobiTheta\u2082_add_left" + - path: Mathlib/Data/PFunctor/Univariate/M.lean + theorems: + - PFunctor.M.cases_mk + - path: Mathlib/Geometry/RingedSpace/PresheafedSpace.lean + theorems: + - AlgebraicGeometry.PresheafedSpace.Hom.ext + - path: Mathlib/Topology/Algebra/Order/Compact.lean + theorems: + - IsCompact.exists_isLeast + - path: Mathlib/Topology/UniformSpace/UniformConvergence.lean + theorems: + - TendstoUniformly.prod_map + - path: Mathlib/Data/Set/Pointwise/ListOfFn.lean + theorems: + - Set.mem_pow + - path: Mathlib/RingTheory/Derivation/Basic.lean + theorems: + - Derivation.eqOn_adjoin + - path: Mathlib/RingTheory/FractionalIdeal/Basic.lean + theorems: + - FractionalIdeal.coeIdeal_mul + - path: Mathlib/CategoryTheory/Abelian/LeftDerived.lean + theorems: + - CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_hom_naturality + - path: Mathlib/Combinatorics/Young/YoungDiagram.lean + theorems: + - YoungDiagram.exists_not_mem_row + - path: Mathlib/CategoryTheory/Monoidal/Preadditive.lean + theorems: + - CategoryTheory.leftDistributor_assoc + - path: Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean + theorems: + - AkraBazziRecurrence.GrowsPolynomially.neg_iff + - path: Mathlib/LinearAlgebra/Finsupp.lean + theorems: + - Finsupp.totalOn_range + - path: Mathlib/Data/IsROrC/Basic.lean + theorems: + - IsROrC.exists_norm_eq_mul_self + - IsROrC.div_re + - IsROrC.nonneg_iff + - IsROrC.conj_smul + - IsROrC.one_re + - path: Mathlib/Data/Nat/Defs.lean + theorems: + - Nat.mul_div_cancel_left' + - Nat.div_le_div_right + - Nat.div_lt_one_iff + - path: Mathlib/Order/CompleteBooleanAlgebra.lean + theorems: + - "disjoint_iSup\u2082_iff" + - path: Mathlib/Algebra/CharP/MixedCharZero.lean + theorems: + - MixedCharZero.reduce_to_maximal_ideal + - path: Mathlib/Data/Set/Pointwise/BigOperators.lean + theorems: + - Set.multiset_prod_subset_multiset_prod + - path: Mathlib/Probability/StrongLaw.lean + theorems: + - ProbabilityTheory.abs_truncation_le_abs_self + - path: Mathlib/Combinatorics/SetFamily/LYM.lean + theorems: + - Finset.mem_falling + - path: Mathlib/NumberTheory/Rayleigh.lean + theorems: + - compl_beattySeq + - path: Mathlib/Init/Data/Int/CompLemmas.lean + theorems: + - Int.natAbs_bit0_step + - path: Mathlib/Algebra/Star/StarAlgHom.lean + theorems: + - NonUnitalStarAlgHom.mk_coe + - path: Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean + theorems: + - AlgebraicClosure.Step.isIntegral + - path: Mathlib/NumberTheory/NumberField/Embeddings.lean + theorems: + - NumberField.InfinitePlace.isUnramified_mk_iff_forall_isConj + - NumberField.InfinitePlace.card_eq_card_isUnramifiedIn + - NumberField.InfinitePlace.isReal_of_mk_isReal + - NumberField.InfinitePlace.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq + - path: Mathlib/Topology/Maps.lean + theorems: + - OpenEmbedding.open_iff_image_open + - path: Mathlib/Topology/Order/Basic.lean + theorems: + - eventually_abs_sub_lt + - path: Mathlib/Data/Finset/Image.lean + theorems: + - Finset.image_biUnion + - path: Mathlib/Topology/Constructions.lean + theorems: + - continuous_sigma_iff + - path: Mathlib/RingTheory/Adjoin/Basic.lean + theorems: + - Algebra.adjoin_adjoin_coe_preimage + - path: Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean + theorems: + - CategoryTheory.Subgroupoid.generated_le_generatedNormal + - path: Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean + theorems: + - Finset.sups_aux + - Finset.truncatedInf_union + - path: Mathlib/Topology/MetricSpace/Thickening.lean + theorems: + - Metric.ediam_cthickening_le + - path: Mathlib/LinearAlgebra/Eigenspace/Basic.lean + theorems: + - Module.End.iSup_generalizedEigenspace_inf_le_add + - Module.End.hasGeneralizedEigenvalue_of_hasEigenvalue + - path: Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean + theorems: + - AlgebraicTopology.alternatingFaceMapComplex_obj_d + - path: Mathlib/Topology/Order/MonotoneConvergence.lean + theorems: + - tendsto_atBot_isLUB + - path: Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean + theorems: + - MultilinearMap.bound_of_shell + - path: Mathlib/MeasureTheory/Integral/SetToL1.lean + theorems: + - MeasureTheory.tendsto_setToFun_of_L1 + - path: Mathlib/Data/Matrix/Notation.lean + theorems: + - Matrix.tail_transpose + - path: Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean + theorems: + - Basis.addHaar_eq_iff + - path: Mathlib/Data/Finset/LocallyFinite/Basic.lean + theorems: + - Finset.Ioc_subset_Ioo_right + - Finset.map_add_left_Ioc + - Finset.Ioc_insert_left + - path: Mathlib/Data/Sign.lean + theorems: + - SignType.pos_iff + - SignType.mul_assoc + - abs_mul_sign + - SignType.le_trans + - path: Mathlib/Data/Real/Sqrt.lean + theorems: + - Real.sqrt_one_add_le + - path: Mathlib/GroupTheory/FixedPointFree.lean + theorems: + - MonoidHom.FixedPointFree.prod_pow_eq_one + - path: Mathlib/CategoryTheory/Whiskering.lean + theorems: + - CategoryTheory.Functor.pentagon + - path: Mathlib/NumberTheory/NumberField/CanonicalEmbedding.lean + theorems: + - NumberField.canonicalEmbedding.nnnorm_eq + - NumberField.mixedEmbedding.finrank + - path: Mathlib/Data/Nat/Sqrt.lean + theorems: + - Nat.sqrt_le_add + - Nat.le_three_of_sqrt_eq_one + - path: Mathlib/Topology/MetricSpace/Isometry.lean + theorems: + - IsometryEquiv.ediam_preimage + - path: Mathlib/Topology/Homotopy/Contractible.lean + theorems: + - ContinuousMap.nullhomotopic_of_constant + - path: Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean + theorems: + - "SimplicialObject.Splitting.cofan_inj_\u03C0Summand_eq_id" + - path: Mathlib/GroupTheory/GroupAction/Hom.lean + theorems: + - DistribMulActionHom.ext_ring + - path: Mathlib/Algebra/MonoidAlgebra/Support.lean + theorems: + - MonoidAlgebra.mem_span_support + - path: Mathlib/SetTheory/Lists.lean + theorems: + - Lists'.to_ofList + - Lists'.ofList_subset + - path: Mathlib/Data/Nat/Prime.lean + theorems: + - Nat.prime_mul_iff + - Nat.minFac_le_div + - path: Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean + theorems: + - Euclidean.closure_ball + - path: Mathlib/Topology/MetricSpace/PseudoMetric.lean + theorems: + - edist_le_ofReal + - nndist_dist + - dist_dist_dist_le_right + - Metric.eventually_isCompact_closedBall + - path: Mathlib/Tactic/NormNum/OfScientific.lean + theorems: + - Mathlib.Meta.NormNum.isRat_ofScientific_of_true + - path: Mathlib/SetTheory/Ordinal/Arithmetic.lean + theorems: + - Ordinal.bsup_eq_blsub_iff_succ + - Ordinal.IsNormal.blsub_eq + - Ordinal.enum_succ_eq_top + - Ordinal.sup_typein_succ + - path: Mathlib/RingTheory/RootsOfUnity/Complex.lean + theorems: + - Complex.isPrimitiveRoot_exp + - path: Mathlib/Analysis/InnerProductSpace/Projection.lean + theorems: + - smul_orthogonalProjection_singleton + - path: Mathlib/Analysis/ConstantSpeed.lean + theorems: + - hasConstantSpeedOnWith_of_subsingleton + - path: Mathlib/CategoryTheory/Monoidal/Category.lean + theorems: + - CategoryTheory.MonoidalCategory.comp_tensor_id + - CategoryTheory.MonoidalCategory.associator_naturality_right + - path: Mathlib/Tactic/NormNum/LegendreSymbol.lean + theorems: + - "Mathlib.Meta.NormNum.jacobiSymNat.qr\u2083" + - path: Mathlib/LinearAlgebra/RootSystem/Basic.lean + theorems: + - RootPairing.isReduced_iff + - path: Mathlib/NumberTheory/ZetaFunction.lean + theorems: + - riemannZeta_one_sub + - path: Mathlib/Analysis/Convex/Normed.lean + theorems: + - Convex.thickening + - path: Mathlib/Algebra/Free.lean + theorems: + - FreeMagma.length_toFreeSemigroup + - path: Mathlib/Algebra/Ring/CentroidHom.lean + theorems: + - CentroidHom.centroid_eq_centralizer_mulLeftRight + - path: Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean + theorems: + - AffineMap.image_vsub_image + - path: Mathlib/Topology/Partial.lean + theorems: + - pcontinuous_iff' + - path: Mathlib/MeasureTheory/Constructions/Cylinders.lean + theorems: + - MeasureTheory.disjoint_cylinder_iff + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean + theorems: + - WeierstrassCurve.Jacobian.equation_zero' + - WeierstrassCurve.Jacobian.equation_some + - path: Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean + theorems: + - GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_eq_none_iff + - path: Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean + theorems: + - MeasureTheory.ProbabilityMeasure.continuous_map + - MeasureTheory.ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure + - path: Mathlib/LinearAlgebra/Matrix/SchurComplement.lean + theorems: + - Matrix.det_one_sub_mul_comm + - path: Mathlib/Order/Directed.lean + theorems: + - directedOn_image + - path: .lake/packages/std/Std/Data/List/Pairwise.lean + theorems: + - List.Pairwise.imp_of_mem + - path: Mathlib/LinearAlgebra/Matrix/ZPow.lean + theorems: + - Matrix.pow_inv_comm' + - path: Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean + theorems: + - mfderiv_sub + - path: Mathlib/Data/Multiset/Basic.lean + theorems: + - Multiset.filter_attach' + - Multiset.union_le + - Multiset.one_le_count_iff_mem + - Multiset.pairwise_coe_iff + - Multiset.lt_replicate_succ + - path: Mathlib/LinearAlgebra/Matrix/Transvection.lean + theorems: + - Matrix.transvection_mul_apply_of_ne + - path: Mathlib/Geometry/Manifold/ContMDiff/Basic.lean + theorems: + - contMDiff_inclusion + - path: Mathlib/Algebra/GeomSum.lean + theorems: + - geom_sum_pos' + - path: Mathlib/Data/Vector/Basic.lean + theorems: + - Vector.reverse_get_zero + - path: Mathlib/LinearAlgebra/Matrix/PosDef.lean + theorems: + - Matrix.PosSemidef.one + - path: Mathlib/MeasureTheory/Measure/MeasureSpace.lean + theorems: + - "MeasureTheory.measure_sUnion\u2080" + - MeasureTheory.Measure.measure_le_sInf + - MeasureTheory.sum_measure_preimage_singleton + - path: Mathlib/MeasureTheory/Measure/Hausdorff.lean + theorems: + - MeasureTheory.Measure.mkMetric_le_liminf_sum + - path: Mathlib/MeasureTheory/Measure/GiryMonad.lean + theorems: + - MeasureTheory.Measure.join_map_join + - MeasureTheory.Measure.measurable_of_measurable_coe + - path: Mathlib/Data/Nat/Factorial/Cast.lean + theorems: + - Nat.cast_descFactorial_two + - path: Mathlib/MeasureTheory/Measure/Typeclasses.lean + theorems: + - "MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion\u2080" + - path: Mathlib/Data/Nat/Factorial/DoubleFactorial.lean + theorems: + - Nat.doubleFactorial_two_mul + - path: Mathlib/Data/Finset/Basic.lean + theorems: + - Finset.singleton_inter_of_not_mem + - Finset.filter_inter + - Finset.erase_insert + - path: Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean + theorems: + - AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage + - path: .lake/packages/std/Std/Data/Rat/Lemmas.lean + theorems: + - Rat.mkRat_mul_mkRat + - Rat.maybeNormalize_eq + - path: Mathlib/LinearAlgebra/Determinant.lean + theorems: + - Basis.det_unitsSMul + - path: Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean + theorems: + - "AlgebraicTopology.DoldKan.h\u03C3'_eq" + - path: Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean + theorems: + - Complex.cpow_neg + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean + theorems: + - Real.arccos_neg_one + - path: Mathlib/GroupTheory/Submonoid/Operations.lean + theorems: + - Submonoid.mrange_inl_sup_mrange_inr + - Submonoid.equivMapOfInjective_coe_mulEquiv + - path: Mathlib/Analysis/InnerProductSpace/LinearPMap.lean + theorems: + - LinearPMap.adjoint_apply_of_not_dense + - path: Mathlib/Algebra/Group/Units.lean + theorems: + - Units.isUnit_units_mul + - path: Mathlib/RingTheory/Polynomial/IntegralNormalization.lean + theorems: + - Polynomial.integralNormalization_support + - path: Mathlib/MeasureTheory/Function/LocallyIntegrable.lean + theorems: + - MeasureTheory.IntegrableOn.continuousOn_mul_of_subset + - path: Mathlib/Data/Finset/NAry.lean + theorems: + - "Finset.image\u2082_left_comm" + - path: Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean + theorems: + - hasFDerivAt_of_restrictScalars + - path: Mathlib/Algebra/Order/Chebyshev.lean + theorems: + - sum_div_card_sq_le_sum_sq_div_card + - path: Mathlib/NumberTheory/Padics/RingHoms.lean + theorems: + - PadicInt.denseRange_int_cast + - path: Mathlib/Analysis/Calculus/Deriv/Add.lean + theorems: + - deriv_sub_const + - path: Mathlib/LinearAlgebra/Dual.lean + theorems: + - TensorProduct.dualDistribInvOfBasis_apply + - Subspace.finrank_add_finrank_dualCoannihilator_eq + - LinearMap.range_dualMap_le_dualAnnihilator_ker + - Submodule.iSup_dualAnnihilator_le_iInf + - path: Mathlib/Analysis/NormedSpace/MStructure.lean + theorems: + - IsLprojection.compl_mul + - path: .lake/packages/std/Std/Data/PairingHeap.lean + theorems: + - Std.PairingHeapImp.Heap.noSibling_merge + - path: Mathlib/RingTheory/RootsOfUnity/Basic.lean + theorems: + - IsPrimitiveRoot.of_map_of_injective + - path: Mathlib/Data/Finset/Pointwise.lean + theorems: + - Finset.preimage_mul_right_one + - Finset.op_smul_finset_smul_eq_smul_smul_finset + - Finset.inv_smul_mem_iff + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean + theorems: + - Matrix.Represents.one + - path: Mathlib/RingTheory/Noetherian.lean + theorems: + - isNoetherian_of_fg_of_noetherian + - path: Mathlib/Probability/Independence/ZeroOne.lean + theorems: + - ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self + - ProbabilityTheory.kernel.measure_eq_zero_or_one_of_indepSet_self + - path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean + theorems: + - AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.addMem' + - path: Mathlib/Topology/Homeomorph.lean + theorems: + - Homeomorph.symm_map_nhds_eq + - path: Mathlib/ModelTheory/Substructures.lean + theorems: + - FirstOrder.Language.Substructure.comap_id + - path: Mathlib/Topology/AlexandrovDiscrete.lean + theorems: + - isOpen_iff_forall_specializes + - path: Mathlib/CategoryTheory/Products/Bifunctor.lean + theorems: + - CategoryTheory.Bifunctor.diagonal' + - path: Mathlib/Data/Nat/Digits.lean + theorems: + - Nat.toDigitsCore_lens_eq + - path: Mathlib/Combinatorics/SimpleGraph/Connectivity.lean + theorems: + - SimpleGraph.Walk.edges_copy + - SimpleGraph.Walk.support_concat + - path: Mathlib/RingTheory/Polynomial/Dickson.lean + theorems: + - Polynomial.map_dickson + - path: Mathlib/ModelTheory/Semantics.lean + theorems: + - FirstOrder.Language.BoundedFormula.realize_toPrenex + - path: Mathlib/Analysis/SpecificLimits/Normed.lean + theorems: + - isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt + - path: Mathlib/Analysis/Normed/Group/Pointwise.lean + theorems: + - singleton_div_closedBall + - path: Mathlib/Analysis/MeanInequalitiesPow.lean + theorems: + - ENNReal.rpow_add_rpow_le_add + - path: Mathlib/Computability/TMToPartrec.lean + theorems: + - Turing.PartrecToTM2.codeSupp_case + - Turing.PartrecToTM2.trNormal_supports + - Turing.PartrecToTM2.K'.elim_update_main + - Turing.PartrecToTM2.trNormal_respects + - path: Mathlib/RingTheory/Coprime/Basic.lean + theorems: + - IsCoprime.of_mul_right_right + - path: Mathlib/Data/Sym/Basic.lean + theorems: + - Sym.map_id' + - path: Mathlib/Geometry/Manifold/LocalDiffeomorph.lean + theorems: + - IsLocalDiffeomorphAt.contMDiffAt + - path: Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean + theorems: + - MeasureTheory.snorm_le_snorm_mul_snorm_of_nnnorm + - path: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean + theorems: + - MeasureTheory.Measure.inv_rnDeriv_aux + - path: Mathlib/Order/CompleteLattice.lean + theorems: + - iSup_split_single + - path: Mathlib/Topology/UniformSpace/Basic.lean + theorems: + - uniformity_lift_le_swap + - UniformSpace.toTopologicalSpace_iInf + - Disjoint.exists_uniform_thickening + - path: Mathlib/Data/Nat/Interval.lean + theorems: + - Nat.Ico_insert_succ_left + - path: Mathlib/Algebra/Order/Rearrangement.lean + theorems: + - Monovary.sum_smul_comp_perm_eq_sum_smul_iff + - path: Mathlib/Data/List/AList.lean + theorems: + - AList.lookup_insert + - path: Mathlib/LinearAlgebra/SesquilinearForm.lean + theorems: + - LinearMap.IsRefl.nondegenerate_of_separatingRight + - path: Mathlib/NumberTheory/DiophantineApproximation.lean + theorems: + - Real.exists_rat_eq_convergent + - path: Mathlib/RingTheory/Jacobson.lean + theorems: + - Ideal.Polynomial.jacobson_bot_of_integral_localization + - path: Mathlib/Data/Vector3.lean + theorems: + - Vector3.append_left + - path: Mathlib/Topology/Homotopy/Product.lean + theorems: + - Path.Homotopic.comp_pi_eq_pi_comp + - path: Mathlib/AlgebraicGeometry/Gluing.lean + theorems: + - "AlgebraicGeometry.Scheme.GlueData.\u03B9_isoCarrier_inv" + - path: Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean + theorems: + - CategoryTheory.Preadditive.epi_iff_surjective' + - path: Mathlib/Data/Polynomial/Splits.lean + theorems: + - Polynomial.splits_of_splits_mul' + - path: Mathlib/Data/ENNReal/Real.lean + theorems: + - ENNReal.ofReal_sub + - ENNReal.mul_iInf + - path: Mathlib/Order/BooleanAlgebra.lean + theorems: + - inf_sdiff_eq_bot_iff + - path: Mathlib/Data/List/Zip.lean + theorems: + - List.revzip_map_fst + - List.map_prod_right_eq_zip + - path: Mathlib/Analysis/Normed/Group/Hom.lean + theorems: + - NormedAddGroupHom.mem_ker + - path: Mathlib/RingTheory/NonUnitalSubring/Basic.lean + theorems: + - "NonUnitalSubring.closure_induction\u2082" + - path: Mathlib/MeasureTheory/Group/Measure.lean + theorems: + - MeasureTheory.Measure.isHaarMeasure_map + - path: Mathlib/CategoryTheory/Pi/Basic.lean + theorems: + - CategoryTheory.isIso_pi_iff + - path: Mathlib/Logic/Equiv/List.lean + theorems: + - Denumerable.list_ofNat_succ + - path: Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean + theorems: + - AlgebraicGeometry.locallyOfFiniteType_eq + - path: Mathlib/MeasureTheory/Covering/Vitali.lean + theorems: + - Vitali.exists_disjoint_subfamily_covering_enlargment_closedBall + - path: Mathlib/CategoryTheory/Abelian/FunctorCategory.lean + theorems: + - CategoryTheory.Abelian.FunctorCategory.coimageImageComparison_app' + - path: Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean + theorems: + - CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_leftAdd + - path: Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean + theorems: + - Complex.abs_mul_cos_add_sin_mul_I + - path: Mathlib/RingTheory/Localization/Module.lean + theorems: + - LinearIndependent.of_isLocalizedModule + - path: Mathlib/Data/Real/EReal.lean + theorems: + - EReal.exists_between_coe_real + - EReal.sub_lt_sub_of_lt_of_le + - path: Mathlib/Topology/Gluing.lean + theorems: + - "TopCat.GlueData.eqvGen_of_\u03C0_eq" + - path: Mathlib/MeasureTheory/Measure/Sub.lean + theorems: + - MeasureTheory.Measure.sub_apply + - path: Mathlib/MeasureTheory/Covering/VitaliFamily.lean + theorems: + - VitaliFamily.mem_filterAt_iff + - path: Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean + theorems: + - Matrix.nondegenerate_of_det_ne_zero + - path: Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean + theorems: + - MulChar.pow_apply' + - path: Mathlib/Order/ModularLattice.lean + theorems: + - wellFounded_lt_exact_sequence + - path: Mathlib/Data/Sym/Sym2.lean + theorems: + - Sym2.other_spec + - path: Mathlib/RingTheory/DedekindDomain/Different.lean + theorems: + - Submodule.le_traceDual + - aeval_derivative_mem_differentIdeal + - path: Mathlib/Analysis/NormedSpace/CompactOperator.lean + theorems: + - IsCompactOperator.continuous + - path: Mathlib/Algebra/Order/CompleteField.lean + theorems: + - LinearOrderedField.cutMap_bddAbove + - path: Mathlib/Analysis/Convolution.lean + theorems: + - BddAbove.continuous_convolution_left_of_integrable + - HasCompactSupport.convolution_integrand_bound_left + - path: Mathlib/MeasureTheory/Measure/Stieltjes.lean + theorems: + - StieltjesFunction.measure_Ioo + - path: Mathlib/Topology/Algebra/InfiniteSum/Basic.lean + theorems: + - Set.Finite.summable + - path: Mathlib/Order/Filter/Cofinite.lean + theorems: + - "Filter.coprod\u1D62_cofinite" + - path: Mathlib/CategoryTheory/Monoidal/End.lean + theorems: + - "CategoryTheory.\u03BC_inv_naturality\u2097" + - path: Mathlib/Probability/Kernel/MeasureCompProd.lean + theorems: + - MeasureTheory.Measure.dirac_compProd_apply + - path: Mathlib/Data/List/Sublists.lean + theorems: + - List.sublists'_eq_sublists + - path: Mathlib/Algebra/Divisibility/Units.lean + theorems: + - isRelPrime_mul_unit_right_right + - path: Mathlib/Data/Int/Units.lean + theorems: + - Int.isUnit_iff + - path: Mathlib/Data/Num/Lemmas.lean + theorems: + - PosNum.to_nat_inj + - path: Mathlib/RingTheory/PrincipalIdealDomain.lean + theorems: + - gcd_isUnit_iff + - path: Mathlib/GroupTheory/Perm/Cycle/Type.lean + theorems: + - Equiv.Perm.cycleType_eq + - path: Mathlib/Data/Int/Parity.lean + theorems: + - Int.add_one_ediv_two_mul_two_of_odd + - Int.ne_of_odd_add + - path: Mathlib/Combinatorics/Additive/Behrend.lean + theorems: + - Behrend.map_succ + - path: Mathlib/Data/Set/Intervals/Monoid.lean + theorems: + - Set.image_const_add_Ioi + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean + theorems: + - MeasureTheory.snorm_condexpL2_le + - path: Mathlib/LinearAlgebra/AffineSpace/Combination.lean + theorems: + - Finset.affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one + - path: Mathlib/Data/Nat/Choose/Basic.lean + theorems: + - Nat.choose_one_right + - path: Mathlib/Order/Disjoint.lean + theorems: + - Disjoint.left_le_of_le_sup_left + - path: Mathlib/Algebra/Category/Ring/Constructions.lean + theorems: + - CommRingCat.pushoutCocone_inl + - path: Mathlib/Data/List/Indexes.lean + theorems: + - List.foldlIdx_eq_foldlIdxSpec + - path: Mathlib/Analysis/InnerProductSpace/Orientation.lean + theorems: + - Orientation.abs_volumeForm_apply_of_pairwise_orthogonal + - path: Mathlib/GroupTheory/GroupAction/Basic.lean + theorems: + - MulAction.disjoint_image_image_iff + - path: Mathlib/Analysis/LocallyConvex/WithSeminorms.lean + theorems: + - WithSeminorms.partial_sups + - path: Mathlib/Algebra/GroupPower/Order.lean + theorems: + - zero_pow_le_one + - path: Mathlib/Data/Ordmap/Ordset.lean + theorems: + - Ordnode.balanceL_eq_balance + - Ordnode.Raised.add_left + - path: Mathlib/Data/List/Nodup.lean + theorems: + - List.Nodup.erase_get + - path: Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean + theorems: + - integrableOn_exp_Iic + - path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean + theorems: + - ApproximatesLinearOn.open_image + - path: Mathlib/RingTheory/Polynomial/Basic.lean + theorems: + - MvPolynomial.prime_rename_iff + - path: Mathlib/Topology/Order/NhdsSet.lean + theorems: + - nhdsSet_Ici + - path: Mathlib/NumberTheory/Multiplicity.lean + theorems: + - padicValNat.pow_add_pow + - path: Mathlib/Analysis/Calculus/Deriv/Basic.lean + theorems: + - DifferentiableAt.derivWithin + - path: Mathlib/RingTheory/RootsOfUnity/Minpoly.lean + theorems: + - IsPrimitiveRoot.minpoly_eq_pow_coprime + - path: Mathlib/Data/List/MinMax.lean + theorems: + - List.getElem_le_maximum_of_length_pos + - path: Mathlib/Analysis/Calculus/ContDiff/Defs.lean + theorems: + - iteratedFDerivWithin_succ_apply_right + - contDiffWithinAt_succ_iff_hasFDerivWithinAt' + - path: Mathlib/SetTheory/Cardinal/Cofinality.lean + theorems: + - Cardinal.nfpBFamily_lt_ord_of_isRegular + - path: Mathlib/Geometry/Manifold/MFDeriv/Basic.lean + theorems: + - mfderiv_comp_of_eq + - path: Mathlib/Algebra/Quaternion.lean + theorems: + - Quaternion.self_mul_star + - path: Mathlib/Logic/Denumerable.lean + theorems: + - Nat.Subtype.ofNat_surjective_aux + - path: Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean + theorems: + - QuadraticForm.associated_baseChange + - path: Mathlib/Init/Data/Nat/Bitwise.lean + theorems: + - Nat.bodd_add + - path: Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean + theorems: + - hasDerivAt_sqrt_mul_log + - path: Mathlib/RingTheory/PowerSeries/Order.lean + theorems: + - PowerSeries.coeff_of_lt_order + - path: Mathlib/MeasureTheory/Function/LpSpace.lean + theorems: + - MeasureTheory.Lp.dist_def + - path: Mathlib/Geometry/Manifold/VectorBundle/Basic.lean + theorems: + - Bundle.contMDiffAt_totalSpace + - path: Mathlib/RepresentationTheory/Rep.lean + theorems: + - Rep.ihom_ev_app_hom + - path: Mathlib/Data/Int/GCD.lean + theorems: + - Int.ne_zero_of_gcd + - path: Mathlib/CategoryTheory/Yoneda.lean + theorems: + - CategoryTheory.isIso_of_yoneda_map_bijective + - path: Mathlib/CategoryTheory/Shift/Pullback.lean + theorems: + - CategoryTheory.pullbackShiftFunctorZero_hom_app + - path: Mathlib/Computability/PartrecCode.lean + theorems: + - Nat.Partrec.Code.eval_prec_zero + - path: Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean + theorems: + - CategoryTheory.ShortComplex.LeftHomologyMapData.cyclesMap_eq + - CategoryTheory.ShortComplex.f'_cyclesMap' + - path: Mathlib/RingTheory/WittVector/Identities.lean + theorems: + - WittVector.coeff_p_pow_eq_zero + - path: Mathlib/Topology/Instances/ENNReal.lean + theorems: + - ENNReal.tendsto_nhds_top_iff_nnreal + - path: Mathlib/Computability/Halting.lean + theorems: + - ComputablePred.computable_iff_re_compl_re + - path: Mathlib/Data/Fin/Tuple/Basic.lean + theorems: + - Fin.snoc_init_self + - Fin.snoc_zero + - path: Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean + theorems: + - "AlgebraicGeometry.\u0393Spec.right_triangle" + - path: Mathlib/GroupTheory/Perm/Cycle/Basic.lean + theorems: + - Equiv.Perm.isCycleOn_singleton + - path: Mathlib/RingTheory/Ideal/LocalRing.lean + theorems: + - LocalRing.isLocalRingHom_residue + - path: Mathlib/Data/PFunctor/Multivariate/Basic.lean + theorems: + - MvPFunctor.comp.get_mk + - path: .lake/packages/std/Std/Data/Array/Monadic.lean + theorems: + - Array.SatisfiesM_anyM + - path: Mathlib/CategoryTheory/SmallObject/Construction.lean + theorems: + - "CategoryTheory.SmallObject.\u03B9FunctorObj_\u03C0FunctorObj" + - path: Mathlib/GroupTheory/Subsemigroup/Center.lean + theorems: + - Set.neg_mem_center + - path: Mathlib/Topology/Category/TopCat/Opens.lean + theorems: + - TopologicalSpace.Opens.map_eq + - path: Mathlib/Data/Set/Prod.lean + theorems: + - Set.univ_prod_univ + - path: Mathlib/LinearAlgebra/Semisimple.lean + theorems: + - Module.End.isSemisimple_neg + - path: Mathlib/SetTheory/Ordinal/Exponential.lean + theorems: + - Ordinal.log_opow_mul_add + - path: Mathlib/Topology/MetricSpace/Lipschitz.lean + theorems: + - LipschitzWith.dist + - path: Mathlib/RingTheory/FractionalIdeal/Operations.lean + theorems: + - FractionalIdeal.spanSingleton_eq_spanSingleton + - path: Mathlib/LinearAlgebra/PiTensorProduct.lean + theorems: + - PiTensorProduct.induction_on' + - path: Mathlib/Data/DFinsupp/Basic.lean + theorems: + - DFinsupp.extendWith_zero + - path: Mathlib/Deprecated/Submonoid.lean + theorems: + - Submonoid.isSubmonoid + - path: Mathlib/LinearAlgebra/LinearPMap.lean + theorems: + - LinearPMap.mem_graph_snd_inj + - path: Mathlib/Topology/Algebra/GroupCompletion.lean + theorems: + - AddMonoidHom.completion_add + - path: Mathlib/Topology/Compactness/LocallyCompact.lean + theorems: + - exists_mem_nhdsSet_isCompact_mapsTo + - path: Mathlib/Algebra/Lie/IdealOperations.lean + theorems: + - LieSubmodule.lie_sup + - path: Mathlib/CategoryTheory/Limits/MonoCoprod.lean + theorems: + - CategoryTheory.Limits.MonoCoprod.monoCoprod_of_preservesCoprod_of_reflectsMono + - path: Mathlib/Topology/Algebra/Ring/Ideal.lean + theorems: + - QuotientRing.quotientMap_coe_coe + - path: Mathlib/LinearAlgebra/AffineSpace/Ordered.lean + theorems: + - left_lt_lineMap_iff_lt + - path: Mathlib/Analysis/MellinInversion.lean + theorems: + - rexp_neg_image_aux + - path: Mathlib/Algebra/Lie/Nilpotent.lean + theorems: + - LieSubmodule.lowerCentralSeries_eq_lcs_comap + - LieModule.isTrivial_of_nilpotencyLength_le_one + - path: Mathlib/LinearAlgebra/Prod.lean + theorems: + - Submodule.prod_comap_inl + - LinearMap.tailing_sup_tunnel_succ_le_tunnel + - path: Mathlib/FieldTheory/Adjoin.lean + theorems: + - IntermediateField.adjoin_simple_toSubalgebra_of_integral + - path: Mathlib/Analysis/Convex/Strict.lean + theorems: + - DirectedOn.strictConvex_sUnion + - path: Mathlib/Data/List/BigOperators/Basic.lean + theorems: + - List.prod_drop_succ + - path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean + theorems: + - ProjectiveSpectrum.vanishingIdeal_iUnion + - path: Mathlib/Data/Multiset/Powerset.lean + theorems: + - Multiset.powersetCard_zero_left + - path: Mathlib/Algebra/Ring/BooleanRing.lean + theorems: + - ofBoolAlg_symmDiff + - path: Mathlib/MeasureTheory/Measure/VectorMeasure.lean + theorems: + - MeasureTheory.VectorMeasure.nonpos_of_restrict_le_zero + - path: Mathlib/Analysis/Convex/Segment.lean + theorems: + - openSegment_eq_Ioo' + - path: Mathlib/Control/Fold.lean + theorems: + - Traversable.toList_spec + - path: Mathlib/Analysis/SpecialFunctions/Log/Base.lean + theorems: + - Real.logb_inv + - path: Mathlib/FieldTheory/Finite/Basic.lean + theorems: + - ZMod.pow_card_sub_one_eq_one + - path: Mathlib/Probability/Martingale/Upcrossing.lean + theorems: + - MeasureTheory.upperCrossingTime_le + - path: Mathlib/Topology/Separation.lean + theorems: + - Filter.coclosedCompact_eq_cocompact + - path: Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean + theorems: + - ContinuousMap.Homotopy.apply_zero_path + - path: Mathlib/Order/OmegaCompletePartialOrder.lean + theorems: + - OmegaCompletePartialOrder.ContinuousHom.map_continuous' + - path: Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean + theorems: + - Algebra.TensorProduct.rTensor_ker + - path: Mathlib/Init/Data/Bool/Lemmas.lean + theorems: + - Bool.and_eq_false_eq_eq_false_or_eq_false + - path: Mathlib/Order/Filter/NAry.lean + theorems: + - "Filter.map_map\u2082" + - path: Mathlib/LinearAlgebra/Matrix/BilinearForm.lean + theorems: + - Matrix.toBilin_comp + - path: Mathlib/Order/InitialSeg.lean + theorems: + - InitialSeg.acc + - path: Mathlib/Analysis/Calculus/TangentCone.lean + theorems: + - uniqueDiffWithinAt_of_mem_nhds + - path: Mathlib/Algebra/Star/Subalgebra.lean + theorems: + - StarSubalgebra.sInf_toSubalgebra + - path: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean + theorems: + - MeasureTheory.Measure.rnDeriv_smul_left' + - path: Mathlib/Data/Finsupp/Defs.lean + theorems: + - Finsupp.support_single_disjoint + - Finsupp.erase_single_ne + - path: Mathlib/LinearAlgebra/AffineSpace/Basis.lean + theorems: + - AffineBasis.affineCombination_coord_eq_self + - path: Mathlib/Order/CompleteLatticeIntervals.lean + theorems: + - Set.Iic.coe_iInf + - path: Mathlib/Algebra/Order/Group/PosPart.lean + theorems: + - leOnePart_one + - path: Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean + theorems: + - HomologicalComplex.d_pOpcycles + - path: Mathlib/RingTheory/ClassGroup.lean + theorems: + - ClassGroup.mk_eq_one_of_coe_ideal + - path: Mathlib/Analysis/InnerProductSpace/PiL2.lean + theorems: + - OrthonormalBasis.sum_repr_symm + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean + theorems: + - "Wbtw.oangle\u2083\u2081\u2082_eq_zero" + - path: Mathlib/Order/Heyting/Basic.lean + theorems: + - Disjoint.sup_sdiff_cancel_left + - sup_sdiff_distrib + - path: Mathlib/Algebra/Lie/Matrix.lean + theorems: + - Matrix.lieConj_symm_apply + - path: Mathlib/RingTheory/Discriminant.lean + theorems: + - Algebra.discr_of_matrix_mulVec + - path: Mathlib/Probability/Kernel/Condexp.lean + theorems: + - MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_id + - path: Mathlib/Data/ENNReal/Inv.lean + theorems: + - ENNReal.mul_lt_of_lt_div + - path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean + theorems: + - CochainComplex.HomComplex.Cochain.ofHom_injective + - path: Mathlib/RingTheory/Localization/Away/Basic.lean + theorems: + - selfZPow_pow_sub + - path: Mathlib/Order/Hom/Lattice.lean + theorems: + - LatticeHom.withTopWithBot_id + - path: Mathlib/RingTheory/JacobsonIdeal.lean + theorems: + - Ideal.exists_mul_sub_mem_of_sub_one_mem_jacobson + - path: Mathlib/Data/Nat/GCD/Basic.lean + theorems: + - Nat.gcd_add_self_right + - path: .lake/packages/std/Std/Data/String/Lemmas.lean + theorems: + - String.Iterator.forward_eq_nextn + - path: Mathlib/Algebra/Order/Ring/Lemmas.lean + theorems: + - mul_eq_mul_iff_eq_and_eq_of_pos' + - path: Mathlib/Data/List/Perm.lean + theorems: + - List.length_permutations'Aux + - path: Mathlib/RingTheory/FreeCommRing.lean + theorems: + - FreeCommRing.isSupported_of + - path: Mathlib/Data/ZMod/Basic.lean + theorems: + - ZMod.nontrivial_iff + - ZMod.nat_cast_self' + - path: Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean + theorems: + - ModuleCat.MonoidalCategory.associator_naturality + - path: Mathlib/Data/Set/Intervals/WithBotTop.lean + theorems: + - WithTop.preimage_coe_Ioc + - WithTop.preimage_coe_Iio_top + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean + theorems: + - CliffordAlgebra.induction + - path: Mathlib/Order/Filter/Lift.lean + theorems: + - Filter.lift_iInf_of_map_univ + - path: Mathlib/Topology/UrysohnsBounded.lean + theorems: + - exists_bounded_mem_Icc_of_closed_of_le + - path: Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean + theorems: + - GaussianInt.toComplex_re + - path: Mathlib/Algebra/GroupPower/Basic.lean + theorems: + - pow_bit0' + - path: Mathlib/Algebra/Lie/Submodule.lean + theorems: + - LieSubmodule.iInf_coe + - path: Mathlib/RingTheory/ChainOfDivisors.lean + theorems: + - coe_factor_orderIso_map_eq_one_iff + - path: Mathlib/MeasureTheory/Decomposition/Jordan.lean + theorems: + - MeasureTheory.SignedMeasure.absolutelyContinuous_ennreal_iff + - path: Mathlib/Analysis/Complex/Liouville.lean + theorems: + - Complex.liouville_theorem_aux + - path: Mathlib/Analysis/InnerProductSpace/Positive.lean + theorems: + - ContinuousLinearMap.isPositive_zero + - path: Mathlib/Data/Set/Image.lean + theorems: + - Set.range_subtype_map + - path: Mathlib/Combinatorics/Additive/ETransform.lean + theorems: + - Finset.MulETransform.card + - path: Mathlib/Algebra/GroupWithZero/Units/Basic.lean + theorems: + - Units.mk0_one + - path: Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean + theorems: + - "AlgebraicTopology.DoldKan.\u03C3_comp_PInfty" + - path: Mathlib/Algebra/Order/Sub/Defs.lean + theorems: + - AddLECancellable.lt_add_of_tsub_lt_right + - path: Mathlib/Topology/Category/TopCat/Limits/Products.lean + theorems: + - TopCat.prodIsoProd_hom_apply + - path: Mathlib/RingTheory/IntegralDomain.lean + theorems: + - sum_hom_units_eq_zero + - path: Mathlib/Algebra/Order/Group/Defs.lean + theorems: + - div_le_div_iff_right + - path: Mathlib/Computability/Partrec.lean + theorems: + - Nat.rfindOpt_mono + - path: Mathlib/RingTheory/Subsemiring/Basic.lean + theorems: + - Subsemiring.mem_sSup_of_directedOn + - path: Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean + theorems: + - quadraticChar_eq_pow_of_char_ne_two' + - path: Mathlib/Probability/IdentDistrib.lean + theorems: + - ProbabilityTheory.IdentDistrib.ae_snd + - path: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean + theorems: + - elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal + - path: Mathlib/Topology/MetricSpace/Infsep.lean + theorems: + - Set.infsep_pair_le_toReal_inf + - path: Mathlib/RingTheory/WittVector/Basic.lean + theorems: + - WittVector.ghostFun_sub + - path: Mathlib/Data/Nat/Choose/Multinomial.lean + theorems: + - Nat.multinomial_nil + - path: Mathlib/RingTheory/Filtration.lean + theorems: + - Ideal.isIdempotentElem_iff_eq_bot_or_top_of_localRing + - path: Mathlib/FieldTheory/Laurent.lean + theorems: + - RatFunc.laurentAux_algebraMap + - path: Mathlib/Probability/Kernel/CondDistrib.lean + theorems: + - MeasureTheory.Integrable.integral_norm_condDistrib_map + - path: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean + theorems: + - measurableSet_integrable + - path: Mathlib/Data/Finset/Sigma.lean + theorems: + - Finset.sigma_nonempty + - path: Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean + theorems: + - "CategoryTheory.Limits.\u03B9_colimitConstInitial_hom" + - path: Mathlib/Topology/MetricSpace/HausdorffDistance.lean + theorems: + - Metric.infDist_image + - path: Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean + theorems: + - "CategoryTheory.Pretriangulated.Triangle.mor\u2083_eq_zero_iff_mono\u2081" + - path: Mathlib/LinearAlgebra/Contraction.lean + theorems: + - comp_dualTensorHom + - path: Mathlib/Order/Filter/ModEq.lean + theorems: + - Nat.frequently_modEq + - path: Mathlib/Data/Set/Pointwise/Interval.lean + theorems: + - Set.preimage_neg_Ioc + - path: Mathlib/Data/PNat/Basic.lean + theorems: + - PNat.sub_coe + - path: Mathlib/MeasureTheory/Measure/Haar/Basic.lean + theorems: + - MeasureTheory.Measure.haar.index_elim + - path: Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean + theorems: + - Matrix.det_smul_inv_vecMul_eq_cramer_transpose + - path: Mathlib/NumberTheory/LucasLehmer.lean + theorems: + - "LucasLehmer.X.\u03C9_mul_\u03C9b" + - path: Mathlib/Analysis/BoxIntegral/Partition/Basic.lean + theorems: + - BoxIntegral.Prepartition.ofWithBot_le + - path: Mathlib/RepresentationTheory/Basic.lean + theorems: + - Representation.asAlgebraHom_single + - path: Mathlib/LinearAlgebra/Dimension/DivisionRing.lean + theorems: + - rank_dual_eq_card_dual_of_aleph0_le_rank' + - path: Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean + theorems: + - CategoryTheory.Limits.parallelPair_functor_obj + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean + theorems: + - WeierstrassCurve.Affine.CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal' + - path: Mathlib/RingTheory/PowerSeries/Trunc.lean + theorems: + - PowerSeries.trunc_coe_eq_self + - path: Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean + theorems: + - Real.negMulLog_one + - path: Mathlib/Data/Polynomial/Degree/TrailingDegree.lean + theorems: + - Polynomial.nextCoeffUp_C_eq_zero + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean + theorems: + - MeasureTheory.snorm_one_condexp_le_snorm + - path: Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean + theorems: + - "TopCat.Presheaf.SheafConditionEqualizerProducts.res_\u03C0" diff --git a/lean/eval/local.meta.json b/lean/eval/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..aebd9954261314692889f0761f42a83db4dcb4f7 --- /dev/null +++ b/lean/eval/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0bcd139a3115ff33c2c81f7d15f7838e969d0ac8406dbb2453e4ddbd361734d0 +size 368 diff --git a/lean/eval/local_data_0000001000.json b/lean/eval/local_data_0000001000.json new file mode 100644 index 0000000000000000000000000000000000000000..1b11c49ff479cd4442867a885301292d2395a026 --- /dev/null +++ b/lean/eval/local_data_0000001000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:88ce0992f0095cb3ec51c0ebc21f381000be535703e24a648f5addaf939ee020 +size 1625727 diff --git a/lean/eval/local_data_0000002000.json b/lean/eval/local_data_0000002000.json new file mode 100644 index 0000000000000000000000000000000000000000..868f8bb390885d3550ed3562b78cab0ac1eeb2a1 --- /dev/null +++ b/lean/eval/local_data_0000002000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6cf7e6ab334cab26c2d59ae893ce7f87216ccc5c9687328446c02f08e6de7959 +size 1566080 diff --git a/lean/eval/local_data_0000003000.json b/lean/eval/local_data_0000003000.json new file mode 100644 index 0000000000000000000000000000000000000000..82dd73cd98fff047c6f2f113942744ba8d8ccd52 --- /dev/null +++ b/lean/eval/local_data_0000003000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0d0c626dec3f2c00d765ccdb4fda457c6ff1a9d567a5421590809807bf601cda +size 1667420 diff --git a/lean/eval/local_data_0000004000.json b/lean/eval/local_data_0000004000.json new file mode 100644 index 0000000000000000000000000000000000000000..af04ebf02283fccc3f448e37bc8c7aa36ae1cd32 --- /dev/null +++ b/lean/eval/local_data_0000004000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f29b1cfb5e4e270827ca418ffda8be3165c7d2e98e030228688c6e81c57b755f +size 1573087 diff --git a/lean/eval/local_data_0000004220.json b/lean/eval/local_data_0000004220.json new file mode 100644 index 0000000000000000000000000000000000000000..c708ed66b168b3e3085e439b20292b6d4ee06921 --- /dev/null +++ b/lean/eval/local_data_0000004220.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d57c8d0a71e69839668b1b84056fbe990a948fe77e3fb96876d5f2d0b9c24e2d +size 578788 diff --git a/lean/eval/local_lemma_0000000000.json b/lean/eval/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/lean/eval/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/lean/test/leandojo-test.yaml b/lean/test/leandojo-test.yaml new file mode 100644 index 0000000000000000000000000000000000000000..153c960047202afab672d15edc3641318625df45 --- /dev/null +++ b/lean/test/leandojo-test.yaml @@ -0,0 +1,2421 @@ +name: leandojo-eval +num_files: 709 +language: LEAN4 +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 991 +datasets: +- project: /mathlib4 + files: + - path: Mathlib/Analysis/SpecificLimits/Normed.lean + theorems: + - tendsto_pow_const_mul_const_pow_of_abs_lt_one + - dist_partial_sum_le_of_le_geometric + - path: Mathlib/SetTheory/Cardinal/Cofinality.lean + theorems: + - Ordinal.cof_eq_one_iff_is_succ + - Ordinal.aleph0_le_cof + - path: Mathlib/GroupTheory/GroupAction/Quotient.lean + theorems: + - MulAction.Quotient.mk_smul_out' + - Subgroup.normalCore_eq_ker + - path: Mathlib/Data/Complex/Abs.lean + theorems: + - Complex.lim_im + - Complex.abs_le_abs_re_add_abs_im + - path: Mathlib/Tactic/ComputeDegree.lean + theorems: + - Mathlib.Tactic.ComputeDegree.coeff_int_cast_ite + - path: Mathlib/RingTheory/ChainOfDivisors.lean + theorems: + - DivisorChain.second_of_chain_is_irreducible + - path: Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean + theorems: + - mul_right_mem_nonZeroDivisors_eq_zero_iff + - path: Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean + theorems: + - Set.smul_mem_centralizer + - path: Mathlib/Dynamics/PeriodicPts.lean + theorems: + - MulAction.pow_add_period_smul + - path: Mathlib/Algebra/Order/ToIntervalMod.lean + theorems: + - toIcoMod_eq_iff + - toIcoMod_add_zsmul + - toIcoMod_apply_left + - toIcoMod_le_toIocMod + - iUnion_Ioc_int_cast + - toIcoMod_zsmul_add' + - path: Mathlib/Analysis/InnerProductSpace/Projection.lean + theorems: + - sub_orthogonalProjection_mem_orthogonal + - exists_norm_eq_iInf_of_complete_convex + - path: Mathlib/Data/Real/Sign.lean + theorems: + - Real.sign_zero + - path: Mathlib/LinearAlgebra/Reflection.lean + theorems: + - Module.preReflection_preReflection + - path: .lake/packages/std/Std/Data/String/Lemmas.lean + theorems: + - Substring.ValidFor.all + - String.data_take + - String.Iterator.ValidFor.nextn + - path: Mathlib/Order/Iterate.lean + theorems: + - Monotone.seq_pos_lt_seq_of_lt_of_le + - path: Mathlib/Data/Polynomial/Eval.lean + theorems: + - "Polynomial.eval\u2082_at_zero" + - "Polynomial.eval\u2082_mul'" + - path: Mathlib/Algebra/Group/Commute/Defs.lean + theorems: + - mul_pow_sub_one + - path: Mathlib/Data/Complex/Cardinality.lean + theorems: + - mk_complex + - mk_univ_complex + - path: Mathlib/FieldTheory/AbelRuffini.lean + theorems: + - gal_isSolvable_tower + - path: Mathlib/Computability/TuringMachine.lean + theorems: + - Turing.TM1to1.tr_supports + - Turing.TM0.univ_supports + - Turing.BlankExtends.below_of_le + - path: Mathlib/Data/Ordmap/Ordset.lean + theorems: + - Ordnode.pos_size_of_mem + - Ordnode.node4L_size + - Ordnode.size_dual + - path: Mathlib/SetTheory/Ordinal/FixedPoint.lean + theorems: + - Ordinal.apply_lt_nfpBFamily + - path: Mathlib/RingTheory/WittVector/Identities.lean + theorems: + - WittVector.coeff_p_pow + - path: Mathlib/Data/Real/EReal.lean + theorems: + - EReal.add_lt_add_left_coe + - path: Mathlib/Data/List/Basic.lean + theorems: + - List.map_bind + - "List.map\u2082Left_eq_map\u2082Left'" + - List.nthLe_cons + - Function.LeftInverse.list_map + - path: Mathlib/RingTheory/Algebraic.lean + theorems: + - IsAlgebraic.algHom + - path: Mathlib/Algebra/Group/Defs.lean + theorems: + - inv_mul_cancel_comm_assoc + - path: Mathlib/RingTheory/HahnSeries/Multiplication.lean + theorems: + - HahnSeries.single_mul_coeff_add + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean + theorems: + - EulerSine.tendsto_integral_cos_pow_mul_div + - path: Mathlib/Topology/Algebra/Group/Basic.lean + theorems: + - nhds_translation_mul_inv + - Subgroup.coe_topologicalClosure_bot + - path: Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean + theorems: + - groupCohomology.map_one_snd_of_isTwoCocycle + - path: Mathlib/Analysis/Normed/Group/Basic.lean + theorems: + - MonoidHomClass.lipschitz_of_bound + - NormedCommGroup.uniformity_basis_dist + - path: Mathlib/Order/GaloisConnection.lean + theorems: + - OrderIso.bddBelow_preimage + - path: Mathlib/Data/Num/Lemmas.lean + theorems: + - Num.add_toZNum + - path: Mathlib/MeasureTheory/Function/SimpleFuncDense.lean + theorems: + - MeasureTheory.SimpleFunc.nearestPtInd_succ + - MeasureTheory.SimpleFunc.tendsto_nearestPt + - path: Mathlib/Probability/Kernel/IntegralCompProd.lean + theorems: + - ProbabilityTheory.set_integral_compProd_univ_left + - path: Mathlib/FieldTheory/RatFunc.lean + theorems: + - RatFunc.ofFractionRing_mk' + - RatFunc.num_eq_zero_iff + - path: Mathlib/Algebra/AddConstMap/Basic.lean + theorems: + - AddConstMapClass.map_add_zsmul + - path: Mathlib/MeasureTheory/Measure/AEDisjoint.lean + theorems: + - Disjoint.aedisjoint + - path: Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean + theorems: + - "TensorAlgebra.ringQuot_mkAlgHom_freeAlgebra_\u03B9_eq_\u03B9" + - path: Mathlib/Analysis/Convex/Between.lean + theorems: + - Wbtw.left_ne_right_of_ne_right + - Wbtw.left_ne_right_of_ne_left + - path: Mathlib/Data/Finset/Sups.lean + theorems: + - Finset.sized_compls + - Finset.powerset_infs_powerset_self + - Finset.forall_disjSups_iff + - path: Mathlib/Algebra/Module/PID.lean + theorems: + - Submodule.isSemisimple_torsionBy_of_irreducible + - path: Mathlib/RingTheory/Polynomial/Dickson.lean + theorems: + - Polynomial.dickson_add_two + - path: Mathlib/MeasureTheory/Function/LpSpace.lean + theorems: + - ContinuousLinearMap.add_compLpL + - MeasureTheory.indicatorConstLp_empty + - MeasureTheory.dist_indicatorConstLp_eq_norm + - "MeasureTheory.Lp.cauchy_complete_\u2112p" + - ContinuousLinearMap.smul_compLp + - path: .lake/packages/std/Std/Data/List/Perm.lean + theorems: + - List.nil_subperm + - List.Perm.pmap + - List.Perm.symm + - path: Mathlib/Data/Real/NNReal.lean + theorems: + - NNReal.iSup_div + - Real.toNNReal_sum_of_nonneg + - path: Mathlib/Topology/UniformSpace/UniformEmbedding.lean + theorems: + - uniformEmbedding_iff' + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean + theorems: + - Matrix.charmatrix_apply_ne + - path: Mathlib/Data/PFunctor/Multivariate/M.lean + theorems: + - MvPFunctor.M.dest'_eq_dest' + - path: Mathlib/Analysis/Calculus/ContDiff/Defs.lean + theorems: + - hasFTaylorSeriesUpToOn_top_iff + - HasFTaylorSeriesUpTo.continuous + - ftaylorSeriesWithin_univ + - ContDiffWithinAt.continuousWithinAt + - path: Mathlib/FieldTheory/Adjoin.lean + theorems: + - IntermediateField.adjoin_eq_top_of_adjoin_eq_top + - IntermediateField.exists_finset_of_mem_supr'' + - path: Mathlib/Analysis/SpecialFunctions/Stirling.lean + theorems: + - Stirling.log_stirlingSeq_bounded_by_constant + - Stirling.stirlingSeq_zero + - path: Mathlib/Algebra/Homology/Exact.lean + theorems: + - CategoryTheory.kernel_comp_cokernel + - CategoryTheory.exact_zero_left_of_mono + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean + theorems: + - Matrix.coeff_det_one_add_X_smul_one + - path: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean + theorems: + - Real.borel_eq_generateFrom_Ioi_rat + - borel_eq_generateFrom_Ico + - borel_eq_generateFrom_Ioc + - path: Mathlib/Order/Filter/Bases.lean + theorems: + - Filter.generate_eq_generate_inter + - Filter.IsBasis.mem_filter_iff + - path: Mathlib/Data/Real/Hyperreal.lean + theorems: + - Hyperreal.infinitesimal_pos_iff_infinitePos_inv + - path: Mathlib/Probability/CondCount.lean + theorems: + - ProbabilityTheory.condCount_singleton + - path: Mathlib/Analysis/InnerProductSpace/Basic.lean + theorems: + - norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero + - inner_eq_norm_mul_iff_div + - InnerProductSpace.Core.inner_self_ofReal_re + - OrthogonalFamily.summable_iff_norm_sq_summable + - DFinsupp.inner_sum + - path: Mathlib/Analysis/InnerProductSpace/Calculus.lean + theorems: + - HasDerivWithinAt.norm_sq + - path: Mathlib/Algebra/BigOperators/Basic.lean + theorems: + - Finset.mulIndicator_biUnion + - Multiset.add_eq_union_left_of_le + - path: Mathlib/Analysis/Complex/AbelLimit.lean + theorems: + - Real.tendsto_tsum_powerSeries_nhdsWithin_lt + - Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet + - path: Mathlib/Order/JordanHolder.lean + theorems: + - CompositionSeries.Equivalent.symm + - CompositionSeries.mem_eraseTop + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean + theorems: + - Collinear.two_zsmul_oangle_eq_right + - "EuclideanGeometry.oangle_swap\u2081\u2082_sign" + - path: Mathlib/Data/MvPolynomial/Degrees.lean + theorems: + - MvPolynomial.degreeOf_lt_iff + - path: Mathlib/Data/Set/Pointwise/Interval.lean + theorems: + - Set.image_sub_const_Ici + - Set.image_neg_Ioo + - Set.image_mul_right_Icc + - path: Mathlib/Algebra/IsPrimePow.lean + theorems: + - isPrimePow_nat_iff + - path: Mathlib/Data/Nat/Factorial/Cast.lean + theorems: + - Nat.cast_descFactorial + - path: Mathlib/Data/List/BigOperators/Lemmas.lean + theorems: + - unop_map_list_prod + - path: Mathlib/NumberTheory/VonMangoldt.lean + theorems: + - ArithmeticFunction.vonMangoldt_ne_zero_iff + - path: Mathlib/Topology/Sober.lean + theorems: + - genericPoint_specializes + - path: .lake/packages/std/Std/Data/Rat/Lemmas.lean + theorems: + - Rat.neg_divInt + - Rat.neg_mkRat + - path: Mathlib/RingTheory/Polynomial/Pochhammer.lean + theorems: + - descPochhammer_zero_eval_zero + - path: Mathlib/Topology/ContinuousOn.lean + theorems: + - continuousWithinAt_update_same + - nhdsWithin_pi_univ_eq + - path: Mathlib/MeasureTheory/Measure/Portmanteau.lean + theorems: + - MeasureTheory.le_liminf_measure_open_of_forall_tendsto_measure + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean + theorems: + - EuclideanGeometry.angle_eq_zero_of_angle_eq_pi_left + - path: Mathlib/Data/Matrix/Basis.lean + theorems: + - Matrix.StdBasisMatrix.mul_right_apply_of_ne + - Matrix.mem_range_scalar_iff_commute_stdBasisMatrix + - path: Mathlib/LinearAlgebra/SesquilinearForm.lean + theorems: + - LinearMap.IsRefl.domRestrict + - LinearMap.IsAdjointPair.sub + - path: Mathlib/Topology/MetricSpace/Antilipschitz.lean + theorems: + - antilipschitzWith_iff_le_mul_nndist + - AntilipschitzWith.comp + - path: Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean + theorems: + - pow_div_pow_eventuallyEq_atTop + - path: Mathlib/Algebra/FreeAlgebra.lean + theorems: + - FreeAlgebra.induction + - path: Mathlib/Probability/Martingale/Upcrossing.lean + theorems: + - MeasureTheory.upcrossingsBefore_bot + - path: Mathlib/Data/Set/UnionLift.lean + theorems: + - Set.iUnionLift_const + - path: Mathlib/NumberTheory/Divisors.lean + theorems: + - Nat.properDivisors_eq_empty + - Nat.sup_divisors_id + - path: Mathlib/ModelTheory/Substructures.lean + theorems: + - FirstOrder.Language.Substructure.range_subtype + - FirstOrder.Language.Substructure.closure_withConstants_eq + - path: Mathlib/RingTheory/UniqueFactorizationDomain.lean + theorems: + - Associates.le_of_count_ne_zero + - UniqueFactorizationMonoid.prime_of_normalized_factor + - path: Mathlib/Probability/Distributions/Geometric.lean + theorems: + - ProbabilityTheory.geometricPMFRealSum + - path: Mathlib/Analysis/LocallyConvex/WithSeminorms.lean + theorems: + - SeminormFamily.basisSets_add + - WithSeminorms.hasBasis + - path: Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean + theorems: + - SemiNormedGroupCat.ExplicitCoker.map_desc + - SemiNormedGroupCat.explicitCokernelDesc_normNoninc + - SemiNormedGroupCat.explicitCokernelIso_hom_desc + - "SemiNormedGroupCat.explicitCokernel\u03C0_apply_dom_eq_zero" + - path: Mathlib/Data/Seq/Computation.lean + theorems: + - Computation.LiftRel.equiv + - path: Mathlib/MeasureTheory/Covering/Besicovitch.lean + theorems: + - Besicovitch.ae_tendsto_measure_inter_div + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean + theorems: + - Real.cos_pi + - Real.cos_pi_div_four + - Real.sin_pi_div_two_sub + - Real.sin_eq_sqrt_one_sub_cos_sq + - path: Mathlib/Computability/AkraBazzi/AkraBazzi.lean + theorems: + - AkraBazziRecurrence.eventually_r_lt_n + - AkraBazziRecurrence.base_nonempty + - path: Mathlib/Topology/MetricSpace/Thickening.lean + theorems: + - Metric.cthickening_eq_biUnion_closedBall + - Metric.mem_thickening_iff + - path: Mathlib/Analysis/Calculus/Gradient/Basic.lean + theorems: + - HasGradientWithinAt.congr + - path: Mathlib/Analysis/InnerProductSpace/PiL2.lean + theorems: + - OrthonormalBasis.orthogonalProjection_eq_sum + - path: Mathlib/Topology/Instances/NNReal.lean + theorems: + - NNReal.tendsto_atTop_zero_of_summable + - path: Mathlib/Data/Finsupp/Basic.lean + theorems: + - Finsupp.embDomain_eq_mapDomain + - Finsupp.comapDomain_zero + - path: Mathlib/Logic/Relation.lean + theorems: + - Relation.comp_assoc + - path: Mathlib/Order/Filter/AtTopBot.lean + theorems: + - Filter.HasAntitoneBasis.subbasis_with_rel + - Filter.tendsto_const_mul_atBot_of_neg + - path: Mathlib/Topology/UniformSpace/Compact.lean + theorems: + - nhdsSet_diagonal_eq_uniformity + - path: Mathlib/Algebra/Lie/Semisimple.lean + theorems: + - LieAlgebra.subsingleton_of_semisimple_lie_abelian + - path: Mathlib/Analysis/Convex/Segment.lean + theorems: + - mem_segment_translate + - mem_segment_sub_add + - path: Mathlib/Algebra/Order/Group/Defs.lean + theorems: + - inv_mul_lt_one_iff + - path: Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean + theorems: + - BoxIntegral.TaggedPrepartition.IsSubordinate.disjUnion + - path: Mathlib/GroupTheory/Submonoid/Membership.lean + theorems: + - MulMemClass.mul_mem_add_closure + - Submonoid.mem_closure_singleton + - path: Mathlib/CategoryTheory/Subobject/Limits.lean + theorems: + - CategoryTheory.Limits.factorThruKernelSubobject_comp_kernelSubobjectIso + - CategoryTheory.Limits.le_kernelSubobject + - path: Mathlib/Topology/Gluing.lean + theorems: + - TopCat.GlueData.rel_equiv + - path: Mathlib/CategoryTheory/Abelian/Generator.lean + theorems: + - CategoryTheory.Abelian.has_injective_coseparator + - path: Mathlib/Analysis/Calculus/Deriv/Inv.lean + theorems: + - HasDerivWithinAt.div + - path: Mathlib/Topology/CompactOpen.lean + theorems: + - ContinuousMap.compactOpen_eq_iInf_induced + - path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean + theorems: + - CochainComplex.HomComplex.Cochain.shift_zero + - path: Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean + theorems: + - MeasureTheory.ProbabilityMeasure.apply_mono + - path: Mathlib/LinearAlgebra/DFinsupp.lean + theorems: + - Submodule.mem_iSup_finset_iff_exists_sum + - Submodule.biSup_eq_range_dfinsupp_lsum + - path: Mathlib/MeasureTheory/Measure/Restrict.lean + theorems: + - mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem + - MeasureTheory.Measure.restrict_eq_self + - MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const + - path: Mathlib/Data/Sym/Basic.lean + theorems: + - Sym.cons_equiv_eq_equiv_cons + - path: Mathlib/Analysis/Convex/Function.lean + theorems: + - StrictConcaveOn.translate_left + - path: Mathlib/Data/Finset/Basic.lean + theorems: + - Finset.singleton_inter_of_mem + - Finset.biUnion_singleton_eq_self + - Finset.left_eq_union + - Finset.inter_erase + - Finset.insert_union_comm + - Finset.sdiff_eq_self + - path: Mathlib/MeasureTheory/Integral/IntegrableOn.lean + theorems: + - ContinuousOn.aemeasurable + - MeasurableEmbedding.integrableAtFilter_map_iff + - path: Mathlib/CategoryTheory/Monoidal/Category.lean + theorems: + - CategoryTheory.MonoidalCategory.associator_naturality_left + - CategoryTheory.MonoidalCategory.whiskerLeft_eqToHom + - CategoryTheory.MonoidalCategory.id_whiskerLeft + - CategoryTheory.MonoidalCategory.whiskerRight_tensor_symm + - CategoryTheory.MonoidalCategory.triangle_assoc_comp_left_inv + - path: Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean + theorems: + - MeasureTheory.levyProkhorovDist_triangle + - path: Mathlib/Data/DFinsupp/Basic.lean + theorems: + - DFinsupp.erase_def + - DFinsupp.add_closure_iUnion_range_single + - DFinsupp.support_neg + - DFinsupp.filter_def + - DFinsupp.single_eq_of_ne + - path: Mathlib/Analysis/Convex/Jensen.lean + theorems: + - ConcaveOn.min_le_of_mem_Icc + - path: Mathlib/Topology/EMetricSpace/Basic.lean + theorems: + - edist_congr_right + - EMetric.mk_uniformity_basis_le + - path: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean + theorems: + - MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto + - MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded + - path: Mathlib/Algebra/Order/Monovary.lean + theorems: + - "antivaryOn_inv\u2080" + - antivary_iff_forall_mul_nonpos + - antivaryOn_iff_forall_mul_nonpos + - path: Mathlib/Combinatorics/SetFamily/Kleitman.lean + theorems: + - Finset.card_biUnion_le_of_intersecting + - path: Mathlib/Data/List/Nodup.lean + theorems: + - List.Nodup.pairwise_coe + - List.inj_on_of_nodup_map + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean + theorems: + - LinearMap.bound_of_shell + - path: Mathlib/Data/Sym/Sym2.lean + theorems: + - Sym2.diag_injective + - Sym2.mem_map + - path: Mathlib/Topology/ContinuousFunction/NonUnitalFunctionalCalculus.lean + theorems: + - "cfc\u2099Hom_comp" + - path: Mathlib/RingTheory/PowerSeries/WellKnown.lean + theorems: + - PowerSeries.map_exp + - path: Mathlib/Algebra/Lie/Nilpotent.lean + theorems: + - LieModule.isNilpotent_iff_exists_ucs_eq_top + - path: Mathlib/Algebra/Lie/IdealOperations.lean + theorems: + - LieSubmodule.lie_le_right + - path: Mathlib/CategoryTheory/Quotient.lean + theorems: + - CategoryTheory.Quotient.compClosure_eq_self + - path: Mathlib/Algebra/Module/LinearMap/Basic.lean + theorems: + - LinearMap.map_smul_inv + - path: Mathlib/Analysis/InnerProductSpace/l2Space.lean + theorems: + - lp.inner_single_right + - lp.inner_single_left + - Orthonormal.isHilbertSum + - path: Mathlib/Data/Finset/PiAntidiagonal.lean + theorems: + - Finset.mem_piAntidiagonal + - path: Mathlib/AlgebraicGeometry/Gluing.lean + theorems: + - AlgebraicGeometry.Scheme.OpenCover.hom_ext + - path: Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean + theorems: + - CliffordAlgebra.ofEven_comp_toEven + - path: Mathlib/Data/Fin/Basic.lean + theorems: + - Fin.nontrivial_iff_two_le + - Fin.liftFun_iff_succ + - Fin.sub_one_lt_iff + - Fin.coe_orderIso_apply + - path: Mathlib/SetTheory/Cardinal/ENat.lean + theorems: + - Cardinal.nat_lt_ofENat + - Cardinal.nat_eq_ofENat + - path: Mathlib/Topology/Algebra/Module/Basic.lean + theorems: + - Submodule.ClosedComplemented.exists_submodule_equiv_prod + - ContinuousLinearMap.map_neg + - path: Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean + theorems: + - MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp + - "MeasureTheory.SimpleFunc.measure_lt_top_of_mem\u2112p_indicator" + - path: Mathlib/Data/Polynomial/Reverse.lean + theorems: + - Polynomial.coeff_one_reverse + - path: Mathlib/Data/Int/GCD.lean + theorems: + - Int.lcm_zero_right + - path: Mathlib/Analysis/Convex/Topology.lean + theorems: + - Convex.openSegment_interior_closure_subset_interior + - path: Mathlib/RingTheory/Etale.lean + theorems: + - Algebra.FormallyEtale.iff_unramified_and_smooth + - path: Mathlib/Algebra/DirectLimit.lean + theorems: + - AddCommGroup.DirectLimit.congr_symm_apply_of + - path: Mathlib/Topology/Constructions.lean + theorems: + - isOpen_sigma_fst_preimage + - isClosedMap_sigmaMk + - path: Mathlib/Order/LiminfLimsup.lean + theorems: + - Filter.blimsup_eq_limsup_subtype + - Filter.bliminf_false + - Filter.frequently_lt_of_lt_limsup + - path: Mathlib/GroupTheory/Exponent.lean + theorems: + - Monoid.exponent_eq_iSup_orderOf' + - path: Mathlib/Data/ZMod/Basic.lean + theorems: + - ZMod.int_cast_mod + - ZMod.val_mul_of_lt + - path: Mathlib/Algebra/Order/Pointwise.lean + theorems: + - LinearOrderedField.smul_Iic + - path: Mathlib/Analysis/Complex/Basic.lean + theorems: + - Complex.normSq_eq_norm_sq + - path: Mathlib/Data/Nat/Totient.lean + theorems: + - Nat.totient_prime + - path: Mathlib/SetTheory/Game/PGame.lean + theorems: + - SetTheory.PGame.leftMoves_add_cases + - SetTheory.PGame.le_neg_iff + - path: Mathlib/MeasureTheory/Function/AEEqFun.lean + theorems: + - "MeasureTheory.AEEqFun.coeFn_comp\u2082" + - path: Mathlib/Algebra/GCDMonoid/Multiset.lean + theorems: + - Multiset.gcd_eq_zero_iff + - path: Mathlib/Analysis/BoundedVariation.lean + theorems: + - LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_real + - path: Mathlib/LinearAlgebra/Finsupp.lean + theorems: + - Finsupp.lcongr_single + - Finsupp.total_comp_lmapDomain + - path: Mathlib/Analysis/SpecialFunctions/Log/Basic.lean + theorems: + - Real.continuousAt_log_iff + - path: Mathlib/Topology/ContinuousFunction/Weierstrass.lean + theorems: + - continuousMap_mem_polynomialFunctions_closure + - path: Mathlib/Analysis/InnerProductSpace/Orthogonal.lean + theorems: + - Submodule.sub_mem_orthogonal_of_inner_left + - path: Mathlib/GroupTheory/Perm/Cycle/Basic.lean + theorems: + - Equiv.Perm.SameCycle.conj + - Equiv.Perm.IsCycle.eq_on_support_inter_nonempty_congr + - path: Mathlib/Algebra/Module/LocalizedModule.lean + theorems: + - IsLocalizedModule.mk'_neg + - path: .lake/packages/std/Std/Data/Array/Lemmas.lean + theorems: + - Array.size_zipWith + - Array.get_extract_loop_lt + - Array.data_swap + - path: Mathlib/MeasureTheory/Measure/MeasureSpace.lean + theorems: + - MeasureTheory.Measure.AbsolutelyContinuous.smul + - MeasureTheory.ae_map_mem_range + - path: Mathlib/Probability/Independence/Basic.lean + theorems: + - ProbabilityTheory.iIndep_iff + - ProbabilityTheory.iIndepSet_iff + - path: Mathlib/Data/List/Forall2.lean + theorems: + - "List.forall\u2082_cons" + - path: Mathlib/Algebra/Order/Group/PosPart.lean + theorems: + - leOnePart_le_one + - mabs_div_eq_leOnePart_sq + - one_lt_ltOnePart + - path: Mathlib/Topology/MetricSpace/PiNat.lean + theorems: + - PiNat.mem_cylinder_iff_le_firstDiff + - path: Mathlib/Analysis/Analytic/Inverse.lean + theorems: + - FormalMultilinearSeries.comp_rightInv_aux1 + - path: Mathlib/Topology/Connected/Basic.lean + theorems: + - IsConnected.union + - isPreconnected_of_forall_constant + - QuotientMap.image_connectedComponent + - path: Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean + theorems: + - MeasureTheory.ae_ae_add_linearMap_mem_iff + - path: Mathlib/Data/Multiset/Bind.lean + theorems: + - Multiset.bind_congr + - path: Mathlib/Algebra/Algebra/Subalgebra/Basic.lean + theorems: + - AlgHom.coe_range + - path: Mathlib/Analysis/Convex/StrictConvexBetween.lean + theorems: + - Sbtw.dist_lt_max_dist + - Wbtw.dist_le_max_dist + - path: Mathlib/LinearAlgebra/Matrix/PosDef.lean + theorems: + - Matrix.PosSemidef.conjTranspose_mul_mul_same + - path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean + theorems: + - CochainComplex.HomComplex.Cochain.comp_assoc_of_first_is_zero_cochain + - path: Mathlib/Order/Filter/Basic.lean + theorems: + - Filter.not_eventually + - Filter.comap_neBot_iff + - Filter.iInf_neBot_of_directed + - Filter.map_const + - Filter.prod_map_seq_comm + - Filter.EventuallyLE.congr + - path: Mathlib/MeasureTheory/Function/SimpleFunc.lean + theorems: + - MeasureTheory.SimpleFunc.FinMeasSupp.add + - MeasureTheory.SimpleFunc.approx_apply + - path: Mathlib/Analysis/InnerProductSpace/TwoDim.lean + theorems: + - Orientation.norm_kahler + - Orientation.inner_mul_areaForm_sub' + - Orientation.kahler_mul + - path: Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean + theorems: + - Real.volume_eq_stieltjes_id + - path: Mathlib/Combinatorics/SimpleGraph/Connectivity.lean + theorems: + - SimpleGraph.reachable_delete_edges_iff_exists_walk + - path: Mathlib/RingTheory/Ideal/Operations.lean + theorems: + - Ideal.iInf_span_singleton + - Ideal.radical_eq_iff + - Ideal.sup_mul_eq_of_coprime_left + - path: Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean + theorems: + - CategoryTheory.ShortComplex.LeftHomologyData.liftCycles_comp_cyclesIso_hom + - "CategoryTheory.ShortComplex.toCycles_comp_leftHomology\u03C0" + - path: Mathlib/CategoryTheory/Bicategory/Adjunction.lean + theorems: + - CategoryTheory.Bicategory.right_triangle_of_left_triangle + - path: Mathlib/GroupTheory/Subsemigroup/Membership.lean + theorems: + - Subsemigroup.mem_sSup_of_mem + - path: Mathlib/Algebra/Order/Hom/Monoid.lean + theorems: + - OrderMonoidWithZeroHom.toOrderMonoidHom_injective + - path: Mathlib/MeasureTheory/Integral/Average.lean + theorems: + - MeasureTheory.measure_setLaverage_le_pos + - MeasureTheory.laverage_eq + - path: Mathlib/Algebra/Order/Floor.lean + theorems: + - Nat.one_le_ceil_iff + - Int.fract_natCast + - path: Mathlib/Data/Set/Intervals/Basic.lean + theorems: + - Set.right_mem_Iic + - path: Mathlib/Topology/Algebra/Polynomial.lean + theorems: + - "Polynomial.continuous_eval\u2082" + - path: Mathlib/Data/Complex/Exponential.lean + theorems: + - Real.cosh_sub + - Real.exp_1_approx_succ_eq + - Complex.cos_sq + - path: Mathlib/Data/PFunctor/Multivariate/Basic.lean + theorems: + - MvPFunctor.const.get_map + - path: Mathlib/Logic/Embedding/Basic.lean + theorems: + - Equiv.embeddingCongr_apply_trans + - path: Mathlib/Algebra/Order/Module/Defs.lean + theorems: + - neg_of_smul_neg_right + - smul_pos_iff_of_neg_left + - PosSMulReflectLE.lift + - path: Mathlib/SetTheory/Ordinal/Notation.lean + theorems: + - ONote.repr_sub + - path: Mathlib/MeasureTheory/MeasurableSpace/Defs.lean + theorems: + - MeasurableSpace.measurableSpace_iSup_eq + - MeasurableSpace.measurableSet_bot_iff + - path: Mathlib/Combinatorics/SetFamily/Shadow.lean + theorems: + - Set.Sized.upShadow + - path: Mathlib/GroupTheory/Submonoid/Operations.lean + theorems: + - Submonoid.eq_bot_of_subsingleton + - path: Mathlib/RingTheory/IsAdjoinRoot.lean + theorems: + - IsAdjoinRootMonic.modByMonicHom_root_pow + - path: Mathlib/LinearAlgebra/Matrix/Spectrum.lean + theorems: + - Matrix.IsHermitian.eigenvectorMatrix_apply + - path: Mathlib/Algebra/GradedMonoid.lean + theorems: + - SetLike.pow_mem_graded + - path: Mathlib/Data/ENNReal/Real.lean + theorems: + - ENNReal.ofReal_lt_one + - path: Mathlib/Data/Nat/Log.lean + theorems: + - Nat.clog_eq_one + - path: Mathlib/LinearAlgebra/Dual.lean + theorems: + - Module.forall_dual_apply_eq_zero_iff + - Subspace.dualRestrict_comp_dualLift + - Subspace.map_dualCoannihilator + - Submodule.exists_dual_map_eq_bot_of_lt_top + - path: Mathlib/Algebra/Homology/HomologicalBicomplex.lean + theorems: + - "HomologicalComplex\u2082.d_comm" + - path: Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean + theorems: + - MeasureTheory.snorm'_measure_zero_of_neg + - MeasureTheory.snorm_zero' + - MeasureTheory.snormEssSup_mono_measure + - "MeasureTheory.mem\u2112p_const" + - path: Mathlib/Analysis/Calculus/LineDeriv/Basic.lean + theorems: + - HasLineDerivWithinAt.congr_mono + - HasFDerivWithinAt.hasLineDerivWithinAt + - path: Mathlib/AlgebraicTopology/SimplexCategory.lean + theorems: + - SimplexCategory.factorThruImage_eq + - SimplexCategory.iso_eq_iso_refl + - path: Mathlib/Data/Nat/Factorization/Root.lean + theorems: + - Nat.floorRoot_ne_zero + - path: Mathlib/GroupTheory/SpecificGroups/Dihedral.lean + theorems: + - DihedralGroup.orderOf_r_one + - DihedralGroup.r_one_pow + - path: Mathlib/Topology/Compactness/SigmaCompact.lean + theorems: + - CompactExhaustion.mem_diff_shiftr_find + - path: Mathlib/Algebra/Function/Support.lean + theorems: + - Function.mulSupport_extend_one + - Function.mulSupport_binop_subset + - Function.range_subset_insert_image_mulSupport + - path: Mathlib/Data/Set/Card.lean + theorems: + - Set.ncard_singleton_inter + - Set.one_le_encard_iff_nonempty + - Set.ncard_pair + - Set.ncard_insert_le + - Set.encard_eq_three + - Set.fiber_ncard_ne_zero_iff_mem_image + - path: Mathlib/RingTheory/WittVector/InitTail.lean + theorems: + - WittVector.init_init + - path: Mathlib/GroupTheory/HNNExtension.lean + theorems: + - HNNExtension.lift_t + - path: Mathlib/Data/List/BigOperators/Basic.lean + theorems: + - List.prod_isUnit_iff + - Commute.list_prod_right + - List.prod_map_neg + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean + theorems: + - WeierstrassCurve.Jacobian.equiv_of_Z_eq_zero + - path: Mathlib/Data/Int/Lemmas.lean + theorems: + - Int.natAbs_inj_of_nonneg_of_nonneg + - path: Mathlib/Data/Fintype/Basic.lean + theorems: + - Fintype.choose_subtype_eq + - path: Mathlib/RingTheory/Valuation/Basic.lean + theorems: + - Valuation.map_one_sub_of_lt + - path: Mathlib/Algebra/EuclideanDomain/Defs.lean + theorems: + - EuclideanDomain.div_add_mod' + - path: Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean + theorems: + - Real.BohrMollerup.logGammaSeq_add_one + - path: Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean + theorems: + - CategoryTheory.Localization.essSurj_mapArrow_of_hasLeftCalculusofFractions + - CategoryTheory.MorphismProperty.LeftFraction.map_ofInv_hom_id + - path: Mathlib/Analysis/PSeries.lean + theorems: + - Real.not_summable_indicator_one_div_natCast + - Finset.le_sum_condensed' + - Real.summable_one_div_nat_pow + - path: Mathlib/Algebra/Lie/Submodule.lean + theorems: + - LieSubmodule.coe_toSubmodule_mk + - LieHom.mem_idealRange + - path: Mathlib/MeasureTheory/MeasurableSpace/Basic.lean + theorems: + - measurableSet_range_inl + - path: Mathlib/Data/List/Rotate.lean + theorems: + - List.rotate_eq_nil_iff + - List.get_rotate + - path: Mathlib/NumberTheory/Cyclotomic/Basic.lean + theorems: + - IsCyclotomicExtension.iff_union_of_dvd + - IsCyclotomicExtension.of_union_of_dvd + - IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic + - path: Mathlib/Data/List/Lattice.lean + theorems: + - List.nil_bagInter + - path: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean + theorems: + - MeasureTheory.lintegral_fn_integral_sub + - path: Mathlib/Topology/Order/Basic.lean + theorems: + - interior_Ico + - nhds_eq_iInf_abs_sub + - path: Mathlib/Computability/Halting.lean + theorems: + - Partrec.merge' + - path: Mathlib/Topology/Separation.lean + theorems: + - isClosed_setOf_specializes + - compl_singleton_mem_nhdsSet_iff + - Set.EqOn.of_subset_closure + - path: Mathlib/Algebra/Homology/TotalComplexSymmetry.lean + theorems: + - "HomologicalComplex\u2082.totalFlipIso_hom_f_D\u2082" + - path: Mathlib/Order/SuccPred/Limit.lean + theorems: + - Order.not_isSuccLimit_of_noMax + - path: Mathlib/RingTheory/PolynomialAlgebra.lean + theorems: + - matPolyEquiv_map_C + - path: Mathlib/RingTheory/RootsOfUnity/Minpoly.lean + theorems: + - IsPrimitiveRoot.minpoly_dvd_expand + - path: Mathlib/CategoryTheory/Closed/Ideal.lean + theorems: + - CategoryTheory.bijection_natural + - path: Mathlib/Algebra/Lie/Weights/Basic.lean + theorems: + - LieModule.exists_weightSpace_le_ker_of_isNoetherian + - path: Mathlib/Analysis/LocallyConvex/AbsConvex.lean + theorems: + - gaugeSeminormFamily_ball + - path: Mathlib/Data/Nat/Sqrt.lean + theorems: + - Nat.sqrt_lt_self + - path: Mathlib/NumberTheory/Pell.lean + theorems: + - Pell.exists_of_not_isSquare + - path: Mathlib/Topology/ContinuousFunction/Algebra.lean + theorems: + - algebraMap_apply + - path: Mathlib/Algebra/Order/Rearrangement.lean + theorems: + - Monovary.sum_comp_perm_smul_lt_sum_smul_iff + - path: Mathlib/LinearAlgebra/AffineSpace/Matrix.lean + theorems: + - AffineBasis.det_smul_coords_eq_cramer_coords + - path: Mathlib/Topology/UniformSpace/Basic.lean + theorems: + - UniformSpace.to_nhds_mono + - uniformity_le_symm + - path: Mathlib/Data/List/Sigma.lean + theorems: + - List.keys_kreplace + - List.dlookup_kinsert + - path: Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean + theorems: + - GeneralizedContinuedFraction.convergents'_of_int + - path: Mathlib/RingTheory/TensorProduct/Basic.lean + theorems: + - LinearMap.baseChange_mul + - Algebra.TensorProduct.lift_comp_includeRight + - Algebra.TensorProduct.lift_includeLeft_includeRight + - path: Mathlib/Algebra/CharP/Basic.lean + theorems: + - Ring.two_ne_zero + - path: Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean + theorems: + - UpperHalfPlane.im_inv_neg_coe_pos + - path: Mathlib/Data/Option/Basic.lean + theorems: + - Option.guard_eq_some' + - path: Mathlib/RingTheory/Ideal/Norm.lean + theorems: + - Ideal.absNorm_eq_zero_iff + - path: Mathlib/MeasureTheory/Measure/Haar/Unique.lean + theorems: + - MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure + - MeasureTheory.Measure.isHaarMeasure_eq_of_isProbabilityMeasure + - path: Mathlib/Analysis/Analytic/Composition.lean + theorems: + - FormalMultilinearSeries.comp_removeZero + - path: Mathlib/Logic/Hydra.lean + theorems: + - Relation.cutExpand_fibration + - Relation.cutExpand_add_left + - path: Mathlib/LinearAlgebra/Matrix/ToLin.lean + theorems: + - LinearMap.toMatrixAlgEquiv_apply + - LinearMap.toMatrixAlgEquiv_mul + - LinearMap.toMatrix_basis_equiv + - path: Mathlib/NumberTheory/Rayleigh.lean + theorems: + - Beatty.hit_or_miss + - path: Mathlib/Analysis/Convolution.lean + theorems: + - ConvolutionExistsAt.distrib_add + - path: Mathlib/Data/Finset/LocallyFinite/Basic.lean + theorems: + - Finset.map_add_left_Ioo + - Finset.Icc_erase_right + - path: Mathlib/Algebra/Group/Ext.lean + theorems: + - CommGroup.toGroup_injective + - path: Mathlib/LinearAlgebra/AffineSpace/Combination.lean + theorems: + - Finset.affineCombinationSingleWeights_apply_self + - path: Mathlib/GroupTheory/Perm/Fin.lean + theorems: + - Fin.cycleRange_symm_succ + - path: Mathlib/Topology/Algebra/UniformGroup.lean + theorems: + - UniformContinuous.pow_const + - path: Mathlib/Topology/MetricSpace/Bounded.lean + theorems: + - Metric.ediam_univ_eq_top_iff_noncompact + - Metric.diam_triple + - path: Mathlib/Algebra/Order/Field/Basic.lean + theorems: + - div_mul_le_div_mul_of_div_le_div + - le_div_iff_of_neg' + - div_pos_iff + - path: Mathlib/MeasureTheory/Function/UniformIntegrable.lean + theorems: + - MeasureTheory.tendsto_Lp_of_tendsto_ae + - path: .lake/packages/std/Std/Data/HashMap/WF.lean + theorems: + - Std.HashMap.Imp.Buckets.exists_of_update + - path: Mathlib/RingTheory/Polynomial/ScaleRoots.lean + theorems: + - Polynomial.support_scaleRoots_eq + - path: Mathlib/CategoryTheory/Limits/MonoCoprod.lean + theorems: + - CategoryTheory.Limits.MonoCoprod.mono_binaryCofanSum_inr' + - path: Mathlib/Algebra/Module/Injective.lean + theorems: + - Module.Baer.injective + - path: Mathlib/LinearAlgebra/RootSystem/Basic.lean + theorems: + - RootPairing.isCrystallographic_iff + - path: Mathlib/RepresentationTheory/Action/Monoidal.lean + theorems: + - Action.rightUnitor_hom_hom + - path: Mathlib/Data/Part.lean + theorems: + - Part.toOption_isNone + - path: Mathlib/Algebra/BigOperators/Intervals.lean + theorems: + - Finset.prod_Ico_add' + - Finset.prod_Ico_mul_eq_prod_Icc + - Finset.prod_Ico_reflect + - path: Mathlib/Algebra/Lie/Weights/Chain.lean + theorems: + - LieModule.weightSpaceChain_neg + - path: Mathlib/CategoryTheory/Limits/Types.lean + theorems: + - CategoryTheory.Limits.Types.Image.lift_fac + - path: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean + theorems: + - CategoryTheory.ShortComplex.RightHomologyMapData.opcyclesMap_eq + - CategoryTheory.ShortComplex.opcyclesMap'_comp + - path: Mathlib/Order/Filter/Lift.lean + theorems: + - Filter.prod_lift'_lift' + - path: Mathlib/AlgebraicGeometry/Morphisms/Basic.lean + theorems: + - AlgebraicGeometry.targetAffineLocallyOfOpenCover + - AlgebraicGeometry.AffineTargetMorphismProperty.isLocalOfOpenCoverImply + - path: Mathlib/RingTheory/Norm.lean + theorems: + - Algebra.norm_eq_prod_embeddings_gen + - path: Mathlib/Data/PNat/Prime.lean + theorems: + - PNat.gcd_eq_right_iff_dvd + - path: Mathlib/SetTheory/Ordinal/Arithmetic.lean + theorems: + - Ordinal.bsup_not_succ_of_ne_bsup + - Ordinal.IsNormal.eq_iff_zero_and_succ + - Acc.rank_lt_of_rel + - Ordinal.div_nonempty + - path: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean + theorems: + - intervalIntegral.integral_interval_sub_interval_comm + - intervalIntegral.smul_integral_comp_add_mul + - intervalIntegral.integral_cases + - intervalIntegral.integral_congr + - IntervalIntegrable.comp_add_right + - path: Mathlib/RingTheory/DedekindDomain/Ideal.lean + theorems: + - FractionalIdeal.mul_inv_cancel + - irreducible_pow_sup_of_le + - path: Mathlib/Analysis/SpecialFunctions/Pow/Real.lean + theorems: + - Real.exists_rat_pow_btwn + - Real.one_le_rpow + - path: Mathlib/RingTheory/Subring/Basic.lean + theorems: + - coe_int_mem + - path: Mathlib/LinearAlgebra/Matrix/Basis.lean + theorems: + - Basis.coePiBasisFun.toMatrix_eq_transpose + - path: Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean + theorems: + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.map_epi_on_summand_id" + - path: Mathlib/Topology/Baire/Lemmas.lean + theorems: + - dense_biInter_of_isOpen + - path: .lake/packages/std/Std/Data/Range/Lemmas.lean + theorems: + - Std.Range.forIn_eq_forIn_range' + - path: Mathlib/Logic/Nontrivial/Basic.lean + theorems: + - Function.Injective.exists_ne + - path: Mathlib/MeasureTheory/OuterMeasure/Basic.lean + theorems: + - MeasureTheory.OuterMeasure.restrict_apply + - MeasureTheory.OuterMeasure.le_trim + - path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean + theorems: + - ProjectiveSpectrum.basicOpen_mul_le_left + - path: Mathlib/Algebra/Parity.lean + theorems: + - range_two_mul + - path: Mathlib/Logic/Lemmas.lean + theorems: + - dite_dite_distrib_right + - path: Mathlib/Data/Matroid/Dual.lean + theorems: + - Matroid.dual_dual + - path: Mathlib/Data/Polynomial/Smeval.lean + theorems: + - Polynomial.smeval_C + - path: Mathlib/Algebra/ModEq.lean + theorems: + - AddCommGroup.modEq_refl + - path: Mathlib/Logic/Function/Conjugate.lean + theorems: + - Function.Semiconj.inverses_right + - path: Mathlib/Topology/Algebra/InfiniteSum/Ring.lean + theorems: + - summable_mul_right_iff + - path: Mathlib/LinearAlgebra/Alternating/Basic.lean + theorems: + - AlternatingMap.curryLeft_same + - path: Mathlib/RingTheory/FractionalIdeal/Operations.lean + theorems: + - FractionalIdeal.map_symm_map + - path: Mathlib/Logic/Function/Basic.lean + theorems: + - Function.sometimes_spec + - Function.extend_def + - path: Mathlib/Algebra/Homology/Homotopy.lean + theorems: + - Homotopy.nullHomotopicMap'_f + - path: Mathlib/Analysis/Complex/PhragmenLindelof.lean + theorems: + - PhragmenLindelof.right_half_plane_of_tendsto_zero_on_real + - PhragmenLindelof.horizontal_strip + - path: Mathlib/CategoryTheory/Comma/Presheaf.lean + theorems: + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2081_fst" + - path: Mathlib/LinearAlgebra/BilinearForm/Basic.lean + theorems: + - BilinForm.coe_injective + - path: Mathlib/Data/Vector.lean + theorems: + - Vector.toList_drop + - path: Mathlib/Data/Int/CardIntervalMod.lean + theorems: + - Nat.count_modEq_card + - Int.Ioc_filter_modEq_card + - path: Mathlib/SetTheory/Cardinal/Basic.lean + theorems: + - Cardinal.power_mul + - Cardinal.lift_le_aleph0 + - Cardinal.lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le + - path: Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean + theorems: + - CategoryTheory.Limits.prod.comp_lift + - path: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean + theorems: + - PartialHomeomorph.extend_image_nhd_mem_nhds_of_boundaryless + - path: Mathlib/LinearAlgebra/Pi.lean + theorems: + - LinearMap.pi_zero + - LinearMap.iInf_ker_proj + - path: Mathlib/Data/Polynomial/Basic.lean + theorems: + - Polynomial.toFinsupp_sub + - path: Mathlib/RingTheory/Ideal/Basic.lean + theorems: + - Ideal.mem_span_singleton + - Ideal.eq_bot_or_top + - path: Mathlib/Topology/Semicontinuous.lean + theorems: + - lowerSemicontinuousWithinAt_tsum + - lowerSemicontinuousAt_tsum + - upperSemicontinuousAt_sum + - path: Mathlib/Data/Multiset/Basic.lean + theorems: + - Multiset.Rel.add + - Multiset.cons_subset + - Multiset.erase_add_right_pos + - path: Mathlib/MeasureTheory/Measure/VectorMeasure.lean + theorems: + - MeasureTheory.Measure.toENNRealVectorMeasure_ennrealToMeasure + - MeasureTheory.VectorMeasure.restrict_add + - MeasureTheory.Measure.toSignedMeasure_congr + - MeasureTheory.VectorMeasure.hasSum_of_disjoint_iUnion + - path: Mathlib/Algebra/Order/Sub/Defs.lean + theorems: + - AddLECancellable.tsub_eq_of_eq_add_rev + - path: Mathlib/Analysis/NormedSpace/Spectrum.lean + theorems: + - spectrum.differentiableOn_inverse_one_sub_smul + - path: Mathlib/Computability/Partrec.lean + theorems: + - Computable.ofOption + - path: Mathlib/CategoryTheory/Limits/Shapes/Types.lean + theorems: + - CategoryTheory.Limits.Types.equalizerIso_hom_comp_subtype + - path: Mathlib/Control/EquivFunctor.lean + theorems: + - EquivFunctor.mapEquiv.injective + - path: Mathlib/Dynamics/Ergodic/MeasurePreserving.lean + theorems: + - MeasureTheory.MeasurePreserving.restrict_image_emb + - MeasureTheory.MeasurePreserving.restrict_preimage_emb + - path: Mathlib/Analysis/Calculus/Implicit.lean + theorems: + - ImplicitFunctionData.map_nhds_eq + - path: Mathlib/Data/Finmap.lean + theorems: + - Finmap.keys_union + - Finmap.keysLookupEquiv_symm_apply_keys + - path: Mathlib/RingTheory/Ideal/Cotangent.lean + theorems: + - Ideal.to_quotient_square_range + - path: Mathlib/Topology/Algebra/Order/Field.lean + theorems: + - Filter.Tendsto.div_atTop + - path: Mathlib/NumberTheory/Liouville/LiouvilleWith.lean + theorems: + - LiouvilleWith.int_add_iff + - path: Mathlib/AlgebraicGeometry/Properties.lean + theorems: + - AlgebraicGeometry.basicOpen_eq_bot_iff + - path: Mathlib/Order/ModularLattice.lean + theorems: + - eq_of_le_of_inf_le_of_sup_le + - path: Mathlib/Combinatorics/SimpleGraph/Maps.lean + theorems: + - SimpleGraph.map_le_iff_le_comap + - path: Mathlib/Data/Set/Intervals/Instances.lean + theorems: + - Set.Ioc.coe_eq_one + - Set.Icc.one_sub_nonneg + - path: Mathlib/Data/Matroid/Basic.lean + theorems: + - Matroid.base_compl_iff_mem_maximals_disjoint_base + - path: Mathlib/Algebra/Group/Basic.lean + theorems: + - conj_eq_one_iff + - path: Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean + theorems: + - AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_zero_or_pos_or_neg + - path: Mathlib/Analysis/Convex/Intrinsic.lean + theorems: + - intrinsicClosure_empty + - path: Mathlib/GroupTheory/Complement.lean + theorems: + - Subgroup.coe_transferFunction + - Subgroup.IsComplement.equiv_mul_left + - path: Mathlib/Analysis/Normed/Group/HomCompletion.lean + theorems: + - NormedAddGroupHom.ker_le_ker_completion + - path: Mathlib/MeasureTheory/Constructions/Polish.lean + theorems: + - MeasurableSet.isClopenable + - path: Mathlib/Topology/Bases.lean + theorems: + - Pairwise.countable_of_isOpen_disjoint + - path: Mathlib/Data/List/Infix.lean + theorems: + - List.map_reverse_inits + - path: Mathlib/Algebra/Module/Zlattice.lean + theorems: + - Zlattice.FG + - path: Mathlib/SetTheory/ZFC/Basic.lean + theorems: + - Class.coe_sUnion + - ZFSet.toSet_sep + - ZFSet.toSet_subset_iff + - path: Mathlib/Analysis/Asymptotics/Theta.lean + theorems: + - Asymptotics.isTheta_inv + - path: Mathlib/Data/PFunctor/Univariate/M.lean + theorems: + - PFunctor.M.ext_aux + - path: Mathlib/Order/ConditionallyCompleteLattice/Basic.lean + theorems: + - WithTop.coe_iSup + - path: Mathlib/NumberTheory/NumberField/CanonicalEmbedding.lean + theorems: + - NumberField.mixedEmbedding.convexBodySum_volume + - path: Mathlib/Algebra/GCDMonoid/Basic.lean + theorems: + - normalize_coe_units + - lcm_mul_right + - Associates.dvd_out_iff + - path: Mathlib/Data/Set/Image.lean + theorems: + - Set.forall_mem_range + - Set.image_union_image_compl_eq_range + - Set.image_preimage_eq_of_subset + - path: Mathlib/FieldTheory/ChevalleyWarning.lean + theorems: + - char_dvd_card_solutions_of_fintype_sum_lt + - path: .lake/packages/std/Std/Data/List/Count.lean + theorems: + - List.count_le_count_map + - path: Mathlib/Data/Fintype/BigOperators.lean + theorems: + - Finset.prod_toFinset_eq_subtype + - path: Mathlib/MeasureTheory/Integral/IntervalAverage.lean + theorems: + - interval_average_symm + - path: Mathlib/Geometry/Euclidean/Basic.lean + theorems: + - EuclideanGeometry.dist_smul_vadd_eq_dist + - EuclideanGeometry.dist_left_midpoint_eq_dist_right_midpoint + - path: Mathlib/RingTheory/RingHomProperties.lean + theorems: + - RingHom.RespectsIso.cancel_right_isIso + - path: Mathlib/Data/Set/Prod.lean + theorems: + - Set.range_pair_subset + - path: Mathlib/LinearAlgebra/Matrix/Block.lean + theorems: + - Matrix.blockTriangular_stdBasisMatrix + - path: Mathlib/Topology/Instances/Sign.lean + theorems: + - continuousAt_sign_of_pos + - path: Mathlib/Data/DFinsupp/Order.lean + theorems: + - DFinsupp.support_tsub + - path: Mathlib/Topology/MetricSpace/Basic.lean + theorems: + - MetricSpace.replaceBornology_eq + - path: Mathlib/Data/Nat/Digits.lean + theorems: + - Nat.ofDigits_one_cons + - path: Mathlib/NumberTheory/FLT/Four.lean + theorems: + - Fermat42.comm + - path: Mathlib/Data/Nat/GCD/Basic.lean + theorems: + - Nat.coprime_pow_left_iff + - path: Mathlib/Logic/Equiv/PartialEquiv.lean + theorems: + - PartialEquiv.refl_restr_target + - path: Mathlib/Analysis/Complex/AbsMax.lean + theorems: + - Complex.exists_mem_frontier_isMaxOn_norm + - path: Mathlib/Computability/Reduce.lean + theorems: + - ComputablePred.computable_of_manyOneReducible + - path: Mathlib/NumberTheory/NumberField/Embeddings.lean + theorems: + - NumberField.InfinitePlace.exists_smul_eq_of_comap_eq + - path: Mathlib/CategoryTheory/Bicategory/Basic.lean + theorems: + - CategoryTheory.Bicategory.leftUnitor_inv_naturality + - path: Mathlib/Analysis/Calculus/Deriv/Mul.lean + theorems: + - HasDerivAtFilter.const_smul + - path: Mathlib/CategoryTheory/Monoidal/Preadditive.lean + theorems: + - "CategoryTheory.leftDistributor_ext\u2082_right" + - path: Mathlib/Algebra/ContinuedFractions/Translations.lean + theorems: + - GeneralizedContinuedFraction.first_denominator_eq + - path: Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean + theorems: + - PartialHomeomorph.MDifferentiable.symm_comp_deriv + - path: Mathlib/GroupTheory/Index.lean + theorems: + - Subgroup.inf_relindex_left + - path: Mathlib/Data/Finset/Lattice.lean + theorems: + - Finset.card_le_of_interleaved + - Finset.sup_le_iff + - path: Mathlib/RingTheory/Ideal/Over.lean + theorems: + - Ideal.eq_bot_of_comap_eq_bot + - path: Mathlib/RingTheory/Derivation/Basic.lean + theorems: + - Derivation.leibniz_of_mul_eq_one + - path: Mathlib/Data/List/Chain.lean + theorems: + - List.Chain'.infix + - path: Mathlib/Data/Real/Irrational.lean + theorems: + - irrational_iff_ne_rational + - path: Mathlib/Data/List/Zip.lean + theorems: + - List.unzip_zip_left + - path: Mathlib/MeasureTheory/Group/Prod.lean + theorems: + - MeasureTheory.absolutelyContinuous_map_mul_right + - path: Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean + theorems: + - CategoryTheory.Equivalence.precoherent_eq + - path: Mathlib/MeasureTheory/Function/L1Space.lean + theorems: + - MeasureTheory.L1.dist_def + - path: Mathlib/Analysis/SpecialFunctions/Integrals.lean + theorems: + - integral_sin_pow_aux + - path: Mathlib/Data/Polynomial/Laurent.lean + theorems: + - LaurentPolynomial.T_sub + - LaurentPolynomial.invert_C + - path: Mathlib/Topology/Algebra/Order/Compact.lean + theorems: + - cocompact_le_atBot_atTop + - path: Mathlib/LinearAlgebra/QuadraticForm/Prod.lean + theorems: + - QuadraticForm.posDef_prod_iff + - path: Mathlib/MeasureTheory/Integral/Lebesgue.lean + theorems: + - NNReal.count_const_le_le_of_tsum_le + - MeasureTheory.lintegral_indicator_le + - path: Mathlib/CategoryTheory/Adjunction/Opposites.lean + theorems: + - CategoryTheory.Adjunction.homEquiv_leftAdjointUniq_hom_app + - path: Mathlib/Topology/UniformSpace/CompactConvergence.lean + theorems: + - ContinuousMap.mem_compactConvergence_entourage_iff + - Filter.HasBasis.compactConvergenceUniformity + - path: Mathlib/Algebra/Order/Sub/Canonical.lean + theorems: + - AddLECancellable.tsub_lt_iff_left + - path: Mathlib/MeasureTheory/Group/Integral.lean + theorems: + - MeasureTheory.Integrable.comp_div_right + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean + theorems: + - SzemerediRegularity.eps_pos + - path: Mathlib/Topology/MetricSpace/PseudoMetric.lean + theorems: + - Real.Ioo_eq_ball + - swap_dist + - path: Mathlib/Order/Lattice.lean + theorems: + - sup_right_comm + - inf_eq_right + - path: Mathlib/Algebra/Module/Submodule/Bilinear.lean + theorems: + - "Submodule.map\u2082_flip" + - "Submodule.map\u2082_eq_span_image2" + - path: Mathlib/Order/WithBot.lean + theorems: + - WithBot.some_le_some + - path: Mathlib/Geometry/Manifold/InteriorBoundary.lean + theorems: + - ModelWithCorners.boundary_eq_complement_interior + - path: Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean + theorems: + - natDegree_minpolyDiv_succ + - path: Mathlib/LinearAlgebra/Ray.lean + theorems: + - eq_zero_of_sameRay_self_neg + - path: Mathlib/Data/Finset/Pointwise.lean + theorems: + - "Finset.smul_univ\u2080" + - Finset.preimage_mul_left_singleton + - path: Mathlib/Data/Setoid/Partition.lean + theorems: + - Setoid.classes_ker_subset_fiber_set + - path: Mathlib/Topology/Connected/PathConnected.lean + theorems: + - Path.map_trans + - path: Mathlib/Algebra/Module/Submodule/Map.lean + theorems: + - LinearMap.map_codRestrict + - path: Mathlib/NumberTheory/RamificationInertia.lean + theorems: + - Ideal.powQuotSuccInclusion_injective + - path: Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean + theorems: + - CategoryTheory.ShortComplex.quasiIso_iff_isIso_descOpcycles + - CategoryTheory.ShortComplex.quasiIso_opMap_iff + - path: Mathlib/Data/Nat/Fib/Zeckendorf.lean + theorems: + - Nat.zeckendorf_succ + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean + theorems: + - EuclideanGeometry.dist_div_sin_oangle_right_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_sub_right_of_oangle_eq_pi_div_two + - path: Mathlib/LinearAlgebra/Contraction.lean + theorems: + - dualTensorHomEquivOfBasis_apply + - homTensorHomEquiv_toLinearMap + - path: Mathlib/GroupTheory/Subgroup/Basic.lean + theorems: + - Subgroup.Normal.comap + - mul_mem_cancel_right + - Subgroup.map_eq_comap_of_inverse + - path: Mathlib/Algebra/Periodic.lean + theorems: + - Function.Antiperiodic.neg_eq + - path: Mathlib/Data/Polynomial/Splits.lean + theorems: + - Polynomial.splits_mul + - path: Mathlib/Data/Finset/Card.lean + theorems: + - Finset.injOn_of_card_image_eq + - path: Mathlib/Tactic/Ring/Basic.lean + theorems: + - Mathlib.Tactic.Ring.div_pf + - Mathlib.Tactic.Ring.mul_one + - path: Mathlib/Data/Set/Pairwise/Lattice.lean + theorems: + - Set.pairwise_sUnion + - path: Mathlib/AlgebraicGeometry/StructureSheaf.lean + theorems: + - AlgebraicGeometry.StructureSheaf.comap_const + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean + theorems: + - Orientation.oangle_sign_sub_smul_right + - Orientation.oangle_eq_zero_iff_sameRay + - path: Mathlib/LinearAlgebra/LinearIndependent.lean + theorems: + - linearIndependent_sum + - LinearIndependent.map' + - linearIndependent_iff + - path: Mathlib/RingTheory/Polynomial/Basic.lean + theorems: + - Polynomial.exists_irreducible_of_natDegree_pos + - path: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean + theorems: + - MeasureTheory.Measure.mutuallySingular_singularPart + - path: Mathlib/CategoryTheory/Limits/Shapes/Images.lean + theorems: + - "CategoryTheory.Limits.image.map_\u03B9" + - path: Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean + theorems: + - Complex.Gamma_eq_GammaAux + - path: Mathlib/Analysis/Complex/CauchyIntegral.lean + theorems: + - Complex.circleIntegral_div_sub_of_differentiable_on_off_countable + - Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable + - path: Mathlib/Algebra/Lie/Basic.lean + theorems: + - LieHom.ext_iff + - path: Mathlib/LinearAlgebra/PiTensorProduct.lean + theorems: + - PiTensorProduct.tprodCoeff_eq_smul_tprod + - path: Mathlib/MeasureTheory/Measure/Tilted.lean + theorems: + - MeasureTheory.tilted_neg_same + - path: Mathlib/Data/Holor.lean + theorems: + - Holor.cprankMax_upper_bound + - path: Mathlib/Data/Polynomial/Degree/Definitions.lean + theorems: + - Polynomial.degree_X_sub_C + - Polynomial.leadingCoeff_quadratic + - Polynomial.X_pow_add_C_ne_one + - path: Mathlib/Analysis/Calculus/LHopital.lean + theorems: + - deriv.lhopital_zero_atBot_on_Iio + - path: Mathlib/MeasureTheory/Measure/Stieltjes.lean + theorems: + - StieltjesFunction.iInf_Ioi_eq + - StieltjesFunction.measure_singleton + - StieltjesFunction.measure_Ioc + - path: Mathlib/Analysis/Normed/Group/Pointwise.lean + theorems: + - IsCompact.closedBall_mul + - div_ball_one + - path: Mathlib/MeasureTheory/Group/Arithmetic.lean + theorems: + - List.aemeasurable_prod + - path: Mathlib/Algebra/Ring/Prod.lean + theorems: + - RingHom.prod_unique + - path: Mathlib/Algebra/Order/Archimedean.lean + theorems: + - existsUnique_sub_zsmul_mem_Ico + - path: Mathlib/Logic/Unique.lean + theorems: + - unique_subtype_iff_exists_unique + - path: Mathlib/GroupTheory/SpecificGroups/Cyclic.lean + theorems: + - IsCyclic.ext + - path: Mathlib/LinearAlgebra/QuadraticForm/Basic.lean + theorems: + - QuadraticForm.associated_left_inverse + - path: Mathlib/Algebra/Order/CompleteField.lean + theorems: + - LinearOrderedField.le_inducedMap_mul_self_of_mem_cutMap + - path: Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean + theorems: + - Matrix.nonsing_inv_mul_cancel_right + - path: Mathlib/MeasureTheory/Constructions/Prod/Basic.lean + theorems: + - MeasureTheory.Measure.measure_prod_null + - MeasureTheory.Measure.prod_restrict + - path: Mathlib/Computability/TMToPartrec.lean + theorems: + - Turing.PartrecToTM2.move_ok + - path: Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean + theorems: + - CochainComplex.mappingCone.liftCochain_descCochain + - CochainComplex.mappingCone.ext_to + - path: Mathlib/MeasureTheory/Measure/FiniteMeasure.lean + theorems: + - MeasureTheory.FiniteMeasure.eq_of_forall_apply_eq + - path: Mathlib/Data/Analysis/Topology.lean + theorems: + - Ctop.ofEquiv_val + - path: Mathlib/NumberTheory/Modular.lean + theorems: + - ModularGroup.re_T_zpow_smul + - path: Mathlib/RingTheory/Polynomial/Content.lean + theorems: + - Polynomial.primPart_mul + - Polynomial.normUnit_content + - path: Mathlib/Order/Sublattice.lean + theorems: + - Sublattice.map_symm_eq_iff_eq_map + - path: Mathlib/Analysis/NormedSpace/PiLp.lean + theorems: + - PiLp.norm_equiv_symm_const' + - path: Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean + theorems: + - MeasureTheory.measure_eq_trim + - path: Mathlib/Probability/Process/Stopping.lean + theorems: + - MeasureTheory.IsStoppingTime.measurableSet_lt + - MeasureTheory.stronglyMeasurable_stoppedValue_of_le + - path: .lake/packages/std/Std/Data/Int/Order.lean + theorems: + - Int.natAbs_add_le + - path: Mathlib/FieldTheory/Finite/Trace.lean + theorems: + - FiniteField.trace_to_zmod_nondegenerate + - path: Mathlib/Probability/Independence/Kernel.lean + theorems: + - ProbabilityTheory.kernel.iIndepFun.indepFun_mul_left + - path: Mathlib/NumberTheory/ClassNumber/Finite.lean + theorems: + - ClassGroup.mkMMem_surjective + - path: Mathlib/Algebra/GroupWithZero/Hom.lean + theorems: + - NeZero.of_map + - path: Mathlib/LinearAlgebra/Quotient.lean + theorems: + - Submodule.unique_quotient_iff_eq_top + - path: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean + theorems: + - NNReal.orderIsoRpow_symm_eq + - ENNReal.top_rpow_of_pos + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean + theorems: + - InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero + - path: Mathlib/GroupTheory/Perm/Cycle/Factors.lean + theorems: + - Equiv.Perm.cycleFactorsFinset_injective + - Equiv.Perm.IsCycle.cycleOf + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean + theorems: + - CliffordAlgebra.evenOdd_comap_reverse + - path: Mathlib/AlgebraicGeometry/Spec.lean + theorems: + - AlgebraicGeometry.Spec.sheafedSpaceMap_comp + - path: Mathlib/Data/Nat/PartENat.lean + theorems: + - PartENat.withTopEquiv_ofNat + - PartENat.get_le_get + - path: Mathlib/NumberTheory/LSeries/Convergence.lean + theorems: + - LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' + - path: Mathlib/Data/Nat/Order/Lemmas.lean + theorems: + - Nat.dvd_right_iff_eq + - Nat.div_eq_zero_iff + - path: Mathlib/Analysis/Convex/Side.lean + theorems: + - AffineSubspace.sSameSide_iff_exists_left + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean + theorems: + - Real.Angle.cos_toReal + - path: Mathlib/Data/Erased.lean + theorems: + - Erased.out_mk + - path: Mathlib/Topology/EMetricSpace/Lipschitz.lean + theorems: + - continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith + - path: Mathlib/Algebra/Order/CauSeq/BigOperators.lean + theorems: + - IsCauSeq.of_abv_le + - IsCauSeq.series_ratio_test + - path: Mathlib/CategoryTheory/Functor/FullyFaithful.lean + theorems: + - CategoryTheory.preimage_comp + - path: Mathlib/Analysis/Calculus/Deriv/Comp.lean + theorems: + - HasDerivAt.iterate + - path: Mathlib/Algebra/Quandle.lean + theorems: + - Rack.self_distrib_inv + - path: Mathlib/ModelTheory/Algebra/Field/CharP.lean + theorems: + - FirstOrder.Field.realize_eqZero + - path: Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean + theorems: + - Polynomial.cyclotomic_coeff_zero + - path: Mathlib/Analysis/ODE/Gronwall.lean + theorems: + - le_gronwallBound_of_liminf_deriv_right_le + - path: Mathlib/Analysis/SpecialFunctions/Exp.lean + theorems: + - Real.comap_exp_nhds_zero + - path: Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean + theorems: + - PrimeSpectrum.basicOpen_mul_le_left + - PrimeSpectrum.vanishingIdeal_strict_anti_mono_iff + - path: Mathlib/LinearAlgebra/Matrix/Transvection.lean + theorems: + - Matrix.det_transvection_of_ne + - path: Mathlib/Data/Matrix/Basic.lean + theorems: + - Matrix.comp_equiv_symm_dotProduct + - path: Mathlib/NumberTheory/Dioph.lean + theorems: + - Dioph.mod_dioph + - path: Mathlib/Data/Set/Lattice.lean + theorems: + - Set.iInter_eq_empty_iff + - Set.biInter_iUnion + - Set.iUnion_setOf + - path: Mathlib/RingTheory/Nullstellensatz.lean + theorems: + - MvPolynomial.vanishingIdeal_pointToPoint + - path: Mathlib/Geometry/Manifold/IntegralCurve.lean + theorems: + - isIntegralCurve_iff_isIntegralCurveAt + - path: Mathlib/LinearAlgebra/Lagrange.lean + theorems: + - Lagrange.natDegree_basisDivisor_self + - path: Mathlib/ModelTheory/Algebra/Ring/Basic.lean + theorems: + - FirstOrder.Ring.realize_one + - path: .lake/packages/std/Std/Data/Int/DivMod.lean + theorems: + - Int.negSucc_emod + - path: Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean + theorems: + - IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_eq + - path: Mathlib/Analysis/Fourier/FourierTransformDeriv.lean + theorems: + - Real.hasDerivAt_fourierChar + - path: Mathlib/Data/Multiset/LocallyFinite.lean + theorems: + - Multiset.Ico_filter_lt_of_le_right + - path: Mathlib/Topology/Algebra/InfiniteSum/Basic.lean + theorems: + - tsum_subtype_eq_of_support_subset + - hasSum_extend_zero + - path: Mathlib/Topology/Homotopy/Product.lean + theorems: + - Path.Homotopic.prod_projLeft_projRight + - path: Mathlib/Algebra/Homology/Additive.lean + theorems: + - HomologicalComplex.singleMapHomologicalComplex_hom_app_ne + - path: Mathlib/Algebra/Group/Units.lean + theorems: + - eq_divp_iff_mul_eq + - IsUnit.div_left_inj + - path: .lake/packages/std/Std/Data/List/Lemmas.lean + theorems: + - List.drop_take + - List.diff_cons + - List.Sublist.eq_of_length + - List.range'_eq_nil + - path: Mathlib/Data/Nat/Order/Basic.lean + theorems: + - Nat.findGreatest_spec + - path: Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean + theorems: + - "ExteriorAlgebra.\u03B9Multi_span_fixedDegree" + - path: Mathlib/LinearAlgebra/CrossProduct.lean + theorems: + - triple_product_eq_det + - path: Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean + theorems: + - CategoryTheory.Limits.isIso_of_source_target_iso_zero + - path: Mathlib/Algebra/Divisibility/Units.lean + theorems: + - isRelPrime_mul_unit_left_right + - path: Mathlib/Analysis/Analytic/Basic.lean + theorems: + - HasFPowerSeriesOnBall.eventually_eq_zero + - HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' + - path: Mathlib/GroupTheory/FreeAbelianGroup.lean + theorems: + - FreeAbelianGroup.lift.map_hom + - path: Mathlib/CategoryTheory/Limits/Presheaf.lean + theorems: + - CategoryTheory.final_toCostructuredArrow_comp_pre + - path: Mathlib/Combinatorics/Young/SemistandardTableau.lean + theorems: + - SemistandardYoungTableau.ext + - path: Mathlib/Data/Nat/Pow.lean + theorems: + - Nat.one_lt_two_pow' + - path: Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean + theorems: + - "CategoryTheory.Limits.\u03B9_colimitLimitIso_limit_\u03C0" + - path: Mathlib/Data/Stream/Init.lean + theorems: + - Stream'.mem_interleave_left + - path: Mathlib/Data/Nat/ModEq.lean + theorems: + - Nat.odd_mul_odd + - path: Mathlib/Data/Nat/Factorial/Basic.lean + theorems: + - Nat.factorial_mul_ascFactorial + - path: Mathlib/CategoryTheory/Limits/HasLimits.lean + theorems: + - CategoryTheory.Limits.colimit.pre_post + - CategoryTheory.Limits.colimit.post_post + - path: Mathlib/RingTheory/DedekindDomain/Different.lean + theorems: + - Submodule.traceDual_top' + - path: Mathlib/Topology/Order/MonotoneConvergence.lean + theorems: + - tendsto_atTop_ciInf + - path: Mathlib/NumberTheory/SumFourSquares.lean + theorems: + - Int.exists_sq_add_sq_add_one_eq_mul + - path: Mathlib/NumberTheory/Padics/RingHoms.lean + theorems: + - PadicInt.ext_of_toZModPow + - path: Mathlib/RingTheory/Noetherian.lean + theorems: + - isNoetherian_iff_wellFounded + - path: Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean + theorems: + - Polynomial.cyclotomic_pos_and_nonneg + - path: Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean + theorems: + - "Complex.Gamma\u211D_mul_Gamma\u211D_add_one" + - path: Mathlib/AlgebraicGeometry/Pullbacks.lean + theorems: + - AlgebraicGeometry.Scheme.Pullback.cocycle_snd_fst_snd + - path: Mathlib/Topology/MetricSpace/Isometry.lean + theorems: + - IsometryEquiv.preimage_sphere + - IsometryEquiv.completeSpace_iff + - path: Mathlib/Probability/Kernel/MeasureCompProd.lean + theorems: + - MeasureTheory.Measure.snd_dirac_unit_compProd_const + - path: Mathlib/Tactic/NormNum/Pow.lean + theorems: + - Mathlib.Meta.NormNum.isNat_zpow_pos + - path: Mathlib/Data/Polynomial/Derivative.lean + theorems: + - Polynomial.iterate_derivative_X_pow_eq_smul + - path: Mathlib/RingTheory/FinitePresentation.lean + theorems: + - Algebra.FiniteType.of_finitePresentation + - path: Mathlib/CategoryTheory/ComposableArrows.lean + theorems: + - CategoryTheory.ComposableArrows.hom_ext_succ + - path: Mathlib/Algebra/GroupWithZero/Units/Basic.lean + theorems: + - mul_div_cancel_of_imp + - path: Mathlib/LinearAlgebra/Basic.lean + theorems: + - LinearEquiv.ofBijective_symm_apply_apply + - Submodule.map_comap_eq + - path: Mathlib/Algebra/CubicDiscriminant.lean + theorems: + - Cubic.natDegree_of_b_eq_zero + - path: Mathlib/Init/Data/Nat/Lemmas.lean + theorems: + - Nat.bit1_lt_bit0 + - path: Mathlib/Order/CompleteSublattice.lean + theorems: + - CompleteSublattice.coe_sInf' + - path: Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean + theorems: + - CategoryTheory.Limits.Concrete.pullbackMk_surjective + - path: Mathlib/Analysis/NormedSpace/Star/Multiplier.lean + theorems: + - DoubleCentralizer.norm_fst + - DoubleCentralizer.norm_fst_eq_snd + - path: Mathlib/Data/Nat/MaxPowDiv.lean + theorems: + - Nat.maxPowDiv.go_eq + - path: Mathlib/GroupTheory/Perm/Basic.lean + theorems: + - Equiv.mul_swap_eq_swap_mul + - path: Mathlib/Topology/UniformSpace/UniformConvergence.lean + theorems: + - tendstoLocallyUniformlyOn_univ + - path: Mathlib/NumberTheory/SmoothNumbers.lean + theorems: + - Nat.mem_smoothNumbers' + - path: Mathlib/Tactic/NormNum/Result.lean + theorems: + - Mathlib.Meta.NormNum.IsRat.to_isNat + - path: Mathlib/Algebra/BigOperators/Associated.lean + theorems: + - Associates.prod_mk + - path: Mathlib/CategoryTheory/Sites/IsSheafFor.lean + theorems: + - CategoryTheory.Presieve.isSheafFor_arrows_iff_pullbacks + - CategoryTheory.Presieve.FamilyOfElements.Compatible.pullback + - path: Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean + theorems: + - SimpleGraph.adjMatrix_pow_apply_eq_card_walk + - path: Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean + theorems: + - continuousAt_const_cpow + - path: Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean + theorems: + - "CochainComplex.mappingConeCompTriangleh_comm\u2081" + - path: Mathlib/Algebra/Tropical/BigOperators.lean + theorems: + - List.trop_sum + - path: Mathlib/Analysis/Seminorm.lean + theorems: + - Seminorm.preimage_metric_ball + - path: Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean + theorems: + - Algebra.adjoin.powerBasis'_gen + - path: Mathlib/Combinatorics/SetFamily/Compression/UV.lean + theorems: + - UV.shadow_compression_subset_compression_shadow + - UV.compression_self + - path: Mathlib/NumberTheory/Zsqrtd/Basic.lean + theorems: + - Zsqrtd.nonnegg_comm + - path: Mathlib/Data/Rat/Cast/Order.lean + theorems: + - Rat.cast_abs + - Rat.cast_pos_of_pos + - path: Mathlib/Analysis/NormedSpace/FiniteDimension.lean + theorems: + - Basis.opNorm_le + - path: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean + theorems: + - MeasureTheory.Measure.rnDeriv_eq_zero_of_mutuallySingular + - path: Mathlib/Data/IsROrC/Basic.lean + theorems: + - IsROrC.re_eq_add_conj + - path: Mathlib/GroupTheory/DoubleCoset.lean + theorems: + - Doset.eq + - path: Mathlib/LinearAlgebra/Dimension/Free.lean + theorems: + - nonempty_linearEquiv_of_lift_rank_eq + - path: Mathlib/CategoryTheory/Category/Basic.lean + theorems: + - CategoryTheory.whisker_eq + - path: Mathlib/Algebra/Module/Submodule/LinearMap.lean + theorems: + - Submodule.subtype_comp_inclusion + - path: Mathlib/Algebra/Algebra/Operations.lean + theorems: + - Submodule.le_one_toAddSubmonoid + - path: Mathlib/Topology/MetricSpace/Dilation.lean + theorems: + - Dilation.ediam_image + - path: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean + theorems: + - Measurable.stronglyMeasurable + - path: Mathlib/LinearAlgebra/Dimension/Finite.lean + theorems: + - CompleteLattice.Independent.subtype_ne_bot_le_rank + - path: Mathlib/Analysis/BoxIntegral/Box/Basic.lean + theorems: + - BoxIntegral.Box.mk'_eq_coe + - path: Mathlib/Algebra/Ring/Semiconj.lean + theorems: + - SemiconjBy.sub_right + - path: Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean + theorems: + - MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable + - path: Mathlib/Data/Vector/Basic.lean + theorems: + - Vector.get_tail + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean + theorems: + - WeierstrassCurve.Affine.CoordinateRing.exists_smul_basis_eq + - path: Mathlib/AlgebraicTopology/DoldKan/Projections.lean + theorems: + - AlgebraicTopology.DoldKan.Q_idem + - path: Mathlib/Analysis/BoxIntegral/Partition/Split.lean + theorems: + - BoxIntegral.Prepartition.not_disjoint_imp_le_of_subset_of_mem_splitMany + - path: Mathlib/Algebra/Category/GroupCat/FilteredColimits.lean + theorems: + - GroupCat.FilteredColimits.colimitInvAux_eq_of_rel + - path: Mathlib/Order/CompleteLattice.lean + theorems: + - iSup_option_elim + - "iInf\u2082_comm" + - path: Mathlib/GroupTheory/Perm/Cycle/Type.lean + theorems: + - Equiv.Perm.cycleType_eq' + - Equiv.Perm.IsThreeCycle.isCycle + - path: Mathlib/RingTheory/Polynomial/Nilpotent.lean + theorems: + - Polynomial.isNilpotent_X_mul_iff + - path: Mathlib/GroupTheory/GroupAction/Hom.lean + theorems: + - MulSemiringActionHom.id_comp + - path: Mathlib/RingTheory/PrincipalIdealDomain.lean + theorems: + - IsBezout.gcd_dvd_right + - path: Mathlib/Topology/DenseEmbedding.lean + theorems: + - DenseInducing.interior_compact_eq_empty + - path: Mathlib/Algebra/Field/Basic.lean + theorems: + - Commute.inv_sub_inv + - one_div_mul_add_mul_one_div_eq_one_div_add_one_div + - path: Mathlib/Geometry/Manifold/MFDeriv/Basic.lean + theorems: + - mfderiv_congr_point + - path: Mathlib/Topology/MetricSpace/PartitionOfUnity.lean + theorems: + - EMetric.exists_continuous_real_forall_closedBall_subset + - path: Mathlib/Data/Finset/Image.lean + theorems: + - Finset.map_subset_map + - path: Mathlib/Topology/MetricSpace/Infsep.lean + theorems: + - Set.infsep_pair_eq_toReal + - path: Mathlib/CategoryTheory/Sites/Sheaf.lean + theorems: + - CategoryTheory.Presheaf.isSheaf_iff_isSheaf_comp + - path: Mathlib/Data/Complex/Basic.lean + theorems: + - Complex.normSq_sub + - path: Mathlib/RingTheory/Discriminant.lean + theorems: + - Algebra.discr_not_zero_of_basis + - Algebra.discr_mul_isIntegral_mem_adjoin + - path: Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean + theorems: + - "CategoryTheory.ShortComplex.SnakeInput.naturality_\u03B4" + - path: Mathlib/Data/Set/MemPartition.lean + theorems: + - memPartitionSet_eq_iff + - path: Mathlib/Topology/Algebra/Module/FiniteDimension.lean + theorems: + - LinearMap.continuous_of_isClosed_ker + - path: Mathlib/Data/Nat/PrimeFin.lean + theorems: + - Nat.Prime.primeFactors + - path: Mathlib/Order/SymmDiff.lean + theorems: + - bihimp_of_le + - Disjoint.symmDiff_eq_sup + - symmDiff_left_comm + - path: Mathlib/Algebra/Homology/HomotopyCofiber.lean + theorems: + - HomologicalComplex.homotopyCofiber.sndX_inrX + - "HomologicalComplex.cylinder.\u03B9\u2081_\u03C0" + - path: Mathlib/SetTheory/Cardinal/Ordinal.lean + theorems: + - Cardinal.mk_finsupp_of_infinite + - path: Mathlib/Topology/Basic.lean + theorems: + - nhds_basis_opens' + - path: Mathlib/FieldTheory/SeparableDegree.lean + theorems: + - Polynomial.IsSeparableContraction.natSepDegree_eq + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean + theorems: + - "CliffordAlgebra.lift_comp_\u03B9" + - path: Mathlib/ModelTheory/DirectLimit.lean + theorems: + - FirstOrder.Language.DirectLimit.equiv_iff + - path: Mathlib/CategoryTheory/Comma/Arrow.lean + theorems: + - CategoryTheory.Arrow.isIso_of_isIso_left_of_isIso_right + - path: Mathlib/GroupTheory/SemidirectProduct.lean + theorems: + - SemidirectProduct.lift_inr + - path: Mathlib/Topology/SeparatedMap.lean + theorems: + - isLocallyInjective_iff_isOpen_diagonal + - path: Mathlib/Data/Nat/Choose/Multinomial.lean + theorems: + - Nat.binomial_eq + - path: Mathlib/Probability/Kernel/CondDistrib.lean + theorems: + - ProbabilityTheory.condDistrib_apply_of_ne_zero + - path: Mathlib/Probability/Kernel/Disintegration/Density.lean + theorems: + - ProbabilityTheory.kernel.set_lintegral_density + - ProbabilityTheory.kernel.density_mono_set + - path: Mathlib/Data/Multiset/Fintype.lean + theorems: + - Multiset.toEmbedding_coeEquiv_trans + - path: Mathlib/Tactic/PushNeg.lean + theorems: + - Mathlib.Tactic.PushNeg.not_nonempty_eq + - path: Mathlib/Data/Finset/NAry.lean + theorems: + - "Finset.image_image\u2082_distrib" + - "Finset.forall_image\u2082_iff" + - path: Mathlib/CategoryTheory/Limits/Opposites.lean + theorems: + - CategoryTheory.Limits.PullbackCone.op_inl + - path: Mathlib/Algebra/Lie/Killing.lean + theorems: + - LieIdeal.coe_killingCompl_top + - path: Mathlib/Combinatorics/Colex.lean + theorems: + - Finset.Colex.toColex_sdiff_lt_toColex_sdiff' + - path: Mathlib/Analysis/NormedSpace/Pointwise.lean + theorems: + - closedBall_add_closedBall + - path: Mathlib/Combinatorics/Composition.lean + theorems: + - CompositionAsSet.boundary_length + - path: Mathlib/Tactic/LinearCombination.lean + theorems: + - Mathlib.Tactic.LinearCombination.eq_of_add_pow + - path: Mathlib/SetTheory/Cardinal/ToNat.lean + theorems: + - Cardinal.mk_toNat_of_infinite + - path: .lake/packages/std/Std/Data/RBMap/WF.lean + theorems: + - Std.RBNode.balance2_eq + - path: Mathlib/MeasureTheory/Integral/SetIntegral.lean + theorems: + - "integral_withDensity_eq_integral_smul\u2080" + - path: Mathlib/Algebra/BigOperators/Multiset/Basic.lean + theorems: + - Multiset.prod_induction_nonempty + - Multiset.sum_int_mod + - Multiset.one_le_prod_of_one_le + - path: Mathlib/RingTheory/JacobsonIdeal.lean + theorems: + - Ideal.eq_jacobson_iff_sInf_maximal + - path: Mathlib/Data/Rat/Star.lean + theorems: + - NNRat.addSubmonoid_closure_range_pow + - path: Mathlib/Topology/Compactness/Lindelof.lean + theorems: + - lindelofSpace_of_countable_subfamily_closed + - path: Mathlib/Data/Finset/Fold.lean + theorems: + - Finset.fold_union_inter + - path: Mathlib/LinearAlgebra/FiniteDimensional.lean + theorems: + - LinearMap.finrank_range_add_finrank_ker + - path: Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean + theorems: + - Finset.card_mul_mul_le_card_div_mul_card_div + - path: Mathlib/ModelTheory/Semantics.lean + theorems: + - FirstOrder.Language.BoundedFormula.realize_exs + - path: Mathlib/Algebra/EuclideanDomain/Basic.lean + theorems: + - EuclideanDomain.eq_div_of_mul_eq_right + - path: Mathlib/Computability/DFA.lean + theorems: + - DFA.pumping_lemma + - path: Mathlib/CategoryTheory/Localization/Equivalence.lean + theorems: + - CategoryTheory.Functor.IsLocalization.of_equivalence_source + - path: Mathlib/Data/Real/Pointwise.lean + theorems: + - Real.sSup_smul_of_nonneg + - path: Mathlib/SetTheory/Ordinal/Basic.lean + theorems: + - Ordinal.lift_down' + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean + theorems: + - Real.arccos_of_le_neg_one + - path: Mathlib/Analysis/Calculus/FDeriv/Mul.lean + theorems: + - HasFDerivWithinAt.mul_const + - hasStrictFDerivAt_finset_prod + - path: Mathlib/MeasureTheory/Integral/DominatedConvergence.lean + theorems: + - MeasureTheory.hasSum_integral_of_summable_integral_norm + - path: Mathlib/RingTheory/Jacobson.lean + theorems: + - Ideal.Polynomial.isJacobson_polynomial_of_isJacobson + - path: Mathlib/Analysis/Calculus/BumpFunction/Normed.lean + theorems: + - ContDiffBump.measure_closedBall_div_le_integral + - path: Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean + theorems: + - Complex.isTheta_exp_arg_mul_im + - path: Mathlib/Algebra/Module/Torsion.lean + theorems: + - Module.isTorsionBySet_singleton_iff + - path: Mathlib/Algebra/Order/Group/Abs.lean + theorems: + - mabs_div_comm + - path: Mathlib/Data/Finsupp/Defs.lean + theorems: + - Finsupp.update_self + - path: Mathlib/Analysis/Calculus/FDeriv/Basic.lean + theorems: + - hasFDerivWithinAt_diff_singleton + - path: Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean + theorems: + - jacobiSym.eq_zero_iff_not_coprime + - path: Mathlib/SetTheory/Ordinal/Exponential.lean + theorems: + - Ordinal.opow_dvd_opow + - path: Mathlib/RingTheory/Trace.lean + theorems: + - Algebra.trace_eq_zero_of_not_exists_basis + - path: Mathlib/Analysis/NormedSpace/lpSpace.lean + theorems: + - LipschitzWith.uniformly_bounded + - path: Mathlib/Data/Nat/Fib/Basic.lean + theorems: + - Nat.fib_two_mul_add_two + - path: .lake/packages/std/Std/Data/Nat/Gcd.lean + theorems: + - Nat.gcd_mul_lcm + - Nat.lcm_comm + - path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean + theorems: + - HasStrictFDerivAt.to_localInverse + - path: Mathlib/Order/Monotone/Union.lean + theorems: + - StrictMonoOn.union + - path: Mathlib/RingTheory/Flat/Basic.lean + theorems: + - Module.Flat.of_retract + - path: Mathlib/Algebra/Algebra/Tower.lean + theorems: + - IsScalarTower.algebraMap_apply + - path: Mathlib/Topology/PartialHomeomorph.lean + theorems: + - PartialHomeomorph.prod_eq_prod_of_nonempty + - path: Mathlib/Geometry/Manifold/ContMDiff/Basic.lean + theorems: + - ContMDiff.comp + - path: Mathlib/Data/Sign.lean + theorems: + - SignType.nonpos_iff_ne_one + - path: Mathlib/Data/Set/Subset.lean + theorems: + - Set.image_val_sUnion + - path: Mathlib/LinearAlgebra/Basis.lean + theorems: + - Basis.ext' + - Basis.repr_self_apply + - path: Mathlib/FieldTheory/PurelyInseparable.lean + theorems: + - isPurelyInseparable_of_finSepDegree_eq_one + - path: Mathlib/Data/List/Join.lean + theorems: + - List.length_join + - path: Mathlib/Data/Matrix/Rank.lean + theorems: + - Matrix.rank_diagonal + - path: Mathlib/Combinatorics/Schnirelmann.lean + theorems: + - exists_of_schnirelmannDensity_eq_zero + - path: Mathlib/RingTheory/IntegralClosure.lean + theorems: + - IsIntegral.fg_adjoin_singleton + - path: Mathlib/Data/Finset/Sigma.lean + theorems: + - Finset.card_sigmaLift + - path: Mathlib/Data/Option/NAry.lean + theorems: + - "Option.map_map\u2082" + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean + theorems: + - MeasureTheory.AEStronglyMeasurable'.continuous_comp + - path: Mathlib/LinearAlgebra/QuotientPi.lean + theorems: + - Submodule.piQuotientLift_single + - path: Mathlib/Order/Monotone/Basic.lean + theorems: + - monotone_dual_iff + - path: Mathlib/Order/PropInstances.lean + theorems: + - Pi.isCompl_iff + - path: Mathlib/Data/Set/Basic.lean + theorems: + - Set.mem_dite_empty_right + - path: Mathlib/RingTheory/Finiteness.lean + theorems: + - Submodule.fg_unit + - path: Mathlib/Topology/MetricSpace/Lipschitz.lean + theorems: + - lipschitzWith_iff_dist_le_mul + - path: Mathlib/Data/Complex/Module.lean + theorems: + - rank_real_of_complex + - path: Mathlib/Data/Rat/Defs.lean + theorems: + - Rat.mul_add + - path: Mathlib/Topology/LocallyFinite.lean + theorems: + - LocallyFinite.closure + - path: Mathlib/Algebra/Order/Monoid/MinMax.lean + theorems: + - fn_min_mul_fn_max + - path: Mathlib/Data/Int/Div.lean + theorems: + - Int.ofNat_add_negSucc_of_ge + - path: Mathlib/Data/List/NodupEquivFin.lean + theorems: + - List.sublist_of_orderEmbedding_get?_eq + - path: Mathlib/MeasureTheory/Function/ContinuousMapDense.lean + theorems: + - MeasureTheory.Lp.boundedContinuousFunction_dense + - path: Mathlib/Logic/Equiv/Functor.lean + theorems: + - Functor.mapEquiv_refl + - path: Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean + theorems: + - SimpleGraph.lapMatrix_toLin'_apply_eq_zero_iff_forall_reachable + - path: Mathlib/Tactic/NormNum/DivMod.lean + theorems: + - Mathlib.Meta.NormNum.isInt_dvd_false + - path: Mathlib/Topology/Algebra/Order/LiminfLimsup.lean + theorems: + - iUnion_Ici_eq_Ioi_of_lt_of_tendsto + - path: Mathlib/NumberTheory/DirichletCharacter/Basic.lean + theorems: + - DirichletCharacter.changeLevel_trans + - path: Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean + theorems: + - GeneralizedContinuedFraction.exists_rat_eq_nth_numerator + - path: Mathlib/Analysis/Normed/Group/AddCircle.lean + theorems: + - AddCircle.norm_coe_eq_abs_iff + - path: .lake/packages/std/Std/Data/AssocList.lean + theorems: + - Std.AssocList.contains_eq + - path: Mathlib/Data/PFun.lean + theorems: + - PFun.coe_preimage + - path: Mathlib/Data/PNat/Interval.lean + theorems: + - PNat.card_Ico + - path: Mathlib/Data/Nat/Choose/Cast.lean + theorems: + - Nat.cast_choose + - path: Mathlib/FieldTheory/Separable.lean + theorems: + - Polynomial.rootMultiplicity_le_one_of_separable + - path: Mathlib/Data/Fin/Tuple/Basic.lean + theorems: + - Fin.insertNth_apply_succAbove + - path: Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean + theorems: + - ContinuousMultilinearMap.uncurry0_curry0 + - path: Mathlib/Algebra/Order/Ring/Lemmas.lean + theorems: + - exists_square_le' + - path: Mathlib/Data/Nat/Defs.lean + theorems: + - Nat.pred_sub + - path: Mathlib/Analysis/Calculus/TangentCone.lean + theorems: + - UniqueDiffWithinAt.prod + - path: Mathlib/Analysis/Normed/Field/Basic.lean + theorems: + - antilipschitzWith_mul_right + - path: Mathlib/Combinatorics/SetFamily/FourFunctions.lean + theorems: + - collapse_modular + - path: Mathlib/Init/Logic.lean + theorems: + - let_eq + - path: Mathlib/Data/Finset/NoncommProd.lean + theorems: + - Multiset.noncommFoldr_eq_foldr + - path: Mathlib/Probability/Kernel/Composition.lean + theorems: + - ProbabilityTheory.kernel.snd_swapRight + - path: Mathlib/Data/Seq/Seq.lean + theorems: + - Stream'.Seq.dropn_tail + - path: Mathlib/Order/LocallyFinite.lean + theorems: + - Finset.map_subtype_embedding_Ici + - path: .lake/packages/std/Std/Data/Int/Gcd.lean + theorems: + - Int.gcd_neg + - path: Mathlib/Algebra/Order/Ring/WithTop.lean + theorems: + - WithTop.distrib' + - path: Mathlib/RingTheory/Kaehler.lean + theorems: + - Derivation.liftKaehlerDifferential_unique + - path: Mathlib/CategoryTheory/Limits/Fubini.lean + theorems: + - "CategoryTheory.Limits.colimitFlipCompColimIsoColimitCompColim_\u03B9_\u03B9\ + _hom" + - path: Mathlib/MeasureTheory/Covering/VitaliFamily.lean + theorems: + - VitaliFamily.tendsto_filterAt_iff + - path: Mathlib/Combinatorics/SetFamily/Shatter.lean + theorems: + - Finset.shatters_empty + - path: Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean + theorems: + - "CategoryTheory.Limits.Cocone.ofCofork_\u03B9" + - path: Mathlib/Order/Filter/Pi.lean + theorems: + - Filter.pi_mem_pi + - path: Mathlib/Topology/MetricSpace/HausdorffDistance.lean + theorems: + - Metric.hausdorffDist_triangle' + - path: Mathlib/Topology/Instances/ENNReal.lean + theorems: + - ENNReal.continuousOn_sub_left + - path: .lake/packages/std/Std/Data/RBMap/Lemmas.lean + theorems: + - Std.RBNode.Stream.next?_toList + - path: Mathlib/Analysis/Convex/Star.lean + theorems: + - StarConvex.sub diff --git a/lean/test/local.meta.json b/lean/test/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..a85f9565794a21c68cef905f05e6c473e1e74b2b --- /dev/null +++ b/lean/test/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:32394c9bb741775c35e8360ecfd3e689d0a350c7dc46afc7730c5be1555c8ac2 +size 368 diff --git a/lean/test/local_data_0000001000.json b/lean/test/local_data_0000001000.json new file mode 100644 index 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b/lean/test/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/lean/test/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/lean/train/leandojo-train.yaml b/lean/train/leandojo-train.yaml new file mode 100644 index 0000000000000000000000000000000000000000..c236eaa022916c6f83d826ddcb4c1e7c9823ef69 --- /dev/null +++ b/lean/train/leandojo-train.yaml @@ -0,0 +1,62729 @@ +name: leandojo-eval +num_files: 3284 +language: LEAN4 +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 56140 +datasets: +- project: /mathlib4 + files: + - path: Mathlib/CategoryTheory/Filtered/Basic.lean + theorems: + - CategoryTheory.IsFilteredOrEmpty.of_right_adjoint + - CategoryTheory.IsCofiltered.of_cone_nonempty + - CategoryTheory.IsFiltered.sup_objs_exists + - CategoryTheory.IsCofiltered.inf_objs_exists + - CategoryTheory.IsFiltered.cocone_nonempty + - CategoryTheory.IsFiltered.span + - CategoryTheory.IsCofilteredOrEmpty.of_left_adjoint + - "CategoryTheory.IsFiltered.coeq\u2083_condition\u2082" + - CategoryTheory.IsFiltered.sup_exists + - CategoryTheory.IsCofiltered.cone_nonempty + - "CategoryTheory.IsFiltered.coeq\u2083_condition\u2081" + - CategoryTheory.IsCofiltered.cospan + - CategoryTheory.IsFiltered.bowtie + - CategoryTheory.IsFiltered.of_cocone_nonempty + - CategoryTheory.Functor.ranges_directed + - CategoryTheory.IsFiltered.tulip + - CategoryTheory.IsCofiltered.inf_exists + - path: Mathlib/MeasureTheory/Function/L1Space.lean + theorems: + - MeasureTheory.HasFiniteIntegral.neg + - MeasureTheory.integrable_withDensity_iff_integrable_coe_smul + - MeasureTheory.Integrable.re + - MeasureTheory.Integrable.measure_ge_lt_top + - MeasureTheory.Integrable.norm_toL1 + - MeasureTheory.Integrable.essSup_smul + - MeasureTheory.Integrable.sup + - MeasureTheory.integrable_finset_sum + - MeasureTheory.L1.norm_sub_eq_lintegral + - MeasureTheory.Integrable.div_const + - MeasureTheory.integrable_smul_const + - MeasureTheory.hasFiniteIntegral_iff_ofReal + - MeasureTheory.HasFiniteIntegral.norm + - MeasureTheory.integrable_zero + - MeasureTheory.HasFiniteIntegral.min_zero + - MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm + - MeasureTheory.L1.norm_def + - MeasureTheory.AEEqFun.integrable_mk + - MeasureTheory.L1.ofReal_norm_sub_eq_lintegral + - MeasurableEmbedding.integrable_map_iff + - MeasureTheory.integrable_finset_sum_measure + - MeasureTheory.Integrable.edist_toL1_zero + - MeasureTheory.HasFiniteIntegral.max_zero + - MeasureTheory.AEEqFun.Integrable.add + - MeasureTheory.Integrable.add' + - MeasureTheory.Integrable.bdd_mul' + - MeasureTheory.lintegral_nnnorm_zero + - MeasureTheory.Integrable.measure_norm_ge_lt_top + - MeasureTheory.AEEqFun.integrable_coeFn + - MeasureTheory.integrable_map_measure + - MeasureTheory.L1.integrable_coeFn + - MeasureTheory.lintegral_edist_triangle + - MeasureTheory.isFiniteMeasure_withDensity_ofReal + - MeasureTheory.integrable_withDensity_iff + - MeasureTheory.integrable_const_iff + - MeasureTheory.hasFiniteIntegral_of_dominated_convergence + - MeasureTheory.hasFiniteIntegral_iff_ofNNReal + - MeasureTheory.Integrable.to_average + - MeasureTheory.Integrable.left_of_add_measure + - MeasureTheory.MeasurePreserving.integrable_comp + - MeasureTheory.Integrable.of_measure_le_smul + - MeasureTheory.lintegral_edist_lt_top + - MeasureTheory.ofReal_toReal_ae_eq + - MeasureTheory.Integrable.add_measure + - MeasureTheory.Integrable.bdd_mul + - "MeasureTheory.Mem\u2112p.integrable_norm_rpow" + - MeasureTheory.Integrable.sub + - MeasureTheory.Integrable.re_im_iff + - MeasureTheory.Integrable.real_toNNReal + - MeasureTheory.Integrable.smul_measure + - MeasureTheory.Integrable.smul_essSup + - MeasureTheory.hasFiniteIntegral_zero + - MeasureTheory.lintegral_norm_eq_lintegral_edist + - MeasureTheory.Integrable.smul_of_top_left + - "MeasureTheory.Mem\u2112p.integrable_norm_rpow'" + - MeasureTheory.Integrable.ofReal + - MeasureTheory.Integrable.abs + - MeasureTheory.HasFiniteIntegral.smul_measure + - MeasureTheory.integrable_smul_measure + - "MeasureTheory.mem\u2112p_one_iff_integrable" + - MeasureTheory.hasFiniteIntegral_iff_edist + - MeasureTheory.integrable_of_norm_sub_le + - MeasureTheory.HasFiniteIntegral.smul + - "MeasureTheory.integrable_withDensity_iff_integrable_coe_smul\u2080" + - MeasureTheory.Integrable.right_of_add_measure + - MeasureTheory.coe_toNNReal_ae_eq + - MeasureTheory.L1.ofReal_norm_eq_lintegral + - MeasureTheory.Integrable.smul_of_top_right + - MeasureTheory.hasFiniteIntegral_smul_iff + - MeasureTheory.hasFiniteIntegral_const_iff + - MeasureTheory.AEEqFun.integrable_iff_mem_L1 + - MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top + - MeasureTheory.Integrable.trim + - MeasureTheory.Integrable.nnnorm_toL1 + - MeasureTheory.Integrable.edist_toL1_toL1 + - MeasureTheory.Integrable.toL1_coeFn + - MeasureTheory.integrable_average + - MeasureTheory.integrable_norm_iff + - MeasureTheory.hasFiniteIntegral_zero_measure + - MeasureTheory.Integrable.inf + - MeasureTheory.integrable_map_equiv + - MeasureTheory.integrable_of_integrable_trim + - MeasureTheory.integrable_of_forall_fin_meas_le + - MeasureTheory.HasFiniteIntegral.restrict + - MeasureTheory.HasFiniteIntegral.add_measure + - MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz + - MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence + - "MeasureTheory.mem_\u21121_toReal_of_lintegral_ne_top" + - MeasureTheory.integrable_inv_smul_measure + - MeasureTheory.all_ae_ofReal_f_le_bound + - MeasureTheory.HasFiniteIntegral.mono + - MeasureTheory.Integrable.im + - MeasureTheory.lintegral_nnnorm_neg + - MeasureTheory.L1.edist_def + - MeasureTheory.integrable_withDensity_iff_integrable_smul' + - MeasureTheory.lintegral_nnnorm_eq_lintegral_edist + - MeasureTheory.hasFiniteIntegral_iff_norm + - MeasureTheory.Integrable.measure_le_lt_top + - path: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean + theorems: + - aestronglyMeasurable_union_iff + - Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact + - stronglyMeasurable_of_tendsto + - MeasureTheory.StronglyMeasurable.measurableSet_le + - MeasureTheory.AEStronglyMeasurable.piecewise + - Multiset.stronglyMeasurable_prod' + - MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite + - MeasureTheory.StronglyMeasurable.measurableSet_mulSupport + - MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite + - MeasureTheory.StronglyMeasurable.mono + - Embedding.comp_stronglyMeasurable_iff + - MeasureTheory.finStronglyMeasurable_zero + - Measurable.add_stronglyMeasurable + - MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable + - MeasureTheory.Subsingleton.stronglyMeasurable + - MeasureTheory.FinStronglyMeasurable.sup + - MeasureTheory.StronglyMeasurable.isSeparable_range + - HasCompactSupport.stronglyMeasurable_of_prod + - aestronglyMeasurable_of_tendsto_ae + - Measurable.stronglyMeasurable_add + - MeasureTheory.AEStronglyMeasurable.aestronglyMeasurable_uIoc_iff + - MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le + - stronglyMeasurable_const_smul_iff + - exists_stronglyMeasurable_limit_of_tendsto_ae + - Multiset.stronglyMeasurable_prod + - MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite + - MeasureTheory.StronglyMeasurable.prod_mk + - MeasurableEmbedding.stronglyMeasurable_extend + - Multiset.aestronglyMeasurable_prod' + - MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable + - MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_eq_fun + - MeasureTheory.Subsingleton.stronglyMeasurable' + - MeasureTheory.AEStronglyMeasurable.isSeparable_ae_range + - MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff + - aestronglyMeasurable_indicator_iff + - aestronglyMeasurable_withDensity_iff + - MeasureTheory.FinStronglyMeasurable.inf + - stronglyMeasurable_iff_measurable_separable + - MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_lt + - Continuous.stronglyMeasurable + - MeasureTheory.FinStronglyMeasurable.mul + - aestronglyMeasurable_const_smul_iff + - MeasureTheory.StronglyMeasurable.measurableSet_eq_fun + - MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set + - List.stronglyMeasurable_prod' + - MeasureTheory.StronglyMeasurable.finStronglyMeasurable + - MeasureTheory.measurable_uncurry_of_continuous_of_measurable + - MeasureTheory.AEStronglyMeasurable.sum_measure + - MeasureTheory.FinStronglyMeasurable.neg + - aestronglyMeasurable_iff_aemeasurable_separable + - Multiset.aestronglyMeasurable_prod + - MeasureTheory.StronglyMeasurable.tendsto_approxBounded_of_norm_le + - List.stronglyMeasurable_prod + - stronglyMeasurable_bot_iff + - Finset.stronglyMeasurable_prod + - List.aestronglyMeasurable_prod + - MeasureTheory.aestronglyMeasurable_zero_measure + - Finset.aestronglyMeasurable_prod + - MeasurableEmbedding.aestronglyMeasurable_map_iff + - stronglyMeasurable_of_stronglyMeasurable_union_cover + - MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on + - Measurable.sub_stronglyMeasurable + - aestronglyMeasurable_add_measure_iff + - MeasureTheory.StronglyMeasurable.measurableSet_lt + - MeasureTheory.stronglyMeasurable_const' + - MeasureTheory.FinStronglyMeasurable.const_smul + - Embedding.aestronglyMeasurable_comp_iff + - MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le + - path: Mathlib/GroupTheory/Perm/Cycle/Type.lean + theorems: + - Equiv.Perm.partition_eq_of_isConj + - Equiv.Perm.card_compl_support_modEq + - Equiv.Perm.IsThreeCycle.sign + - Equiv.Perm.swap_mul_swap_same_mem_closure_three_cycles + - Equiv.Perm.card_cycleType_eq_one + - Equiv.Perm.subgroup_eq_top_of_swap_mem + - Equiv.Perm.cycleType_inv + - Equiv.Perm.IsThreeCycle.inv_iff + - Equiv.Perm.sum_cycleType + - Equiv.Perm.isConj_iff_cycleType_eq + - Equiv.Perm.sign_of_cycleType + - Equiv.Perm.cycleType_le_of_mem_cycleFactorsFinset + - Equiv.Perm.lcm_cycleType + - Equiv.Perm.IsThreeCycle.card_support + - Equiv.Perm.cycleType_conj + - Equiv.Perm.orderOf_cycleOf_dvd_orderOf + - Equiv.Perm.Disjoint.cycleType + - Equiv.Perm.cycleType_extendDomain + - Equiv.Perm.cycleType_of_card_le_mem_cycleType_add_two + - Equiv.Perm.exists_fixed_point_of_prime' + - exists_prime_orderOf_dvd_card + - Equiv.Perm.cycleType_mul_inv_mem_cycleFactorsFinset_eq_sub + - Equiv.Perm.IsSwap.mul_mem_closure_three_cycles + - Equiv.Perm.isCycle_of_prime_order + - exists_prime_addOrderOf_dvd_card + - Equiv.Perm.two_dvd_card_support + - Equiv.Perm.isThreeCycle_swap_mul_swap_same + - Equiv.Perm.card_fixedPoints_modEq + - Equiv.Perm.two_le_of_mem_cycleType + - Equiv.Perm.cycleType_eq_zero + - Equiv.Perm.filter_parts_partition_eq_cycleType + - Equiv.Perm.isCycle_of_prime_order' + - Equiv.Perm.VectorsProdEqOne.one_eq + - Equiv.Perm.exists_fixed_point_of_prime + - Equiv.Perm.card_cycleType_eq_zero + - Equiv.Perm.isCycle_of_prime_order'' + - Equiv.Perm.IsThreeCycle.isThreeCycle_sq + - Equiv.Perm.sign_of_cycleType' + - Equiv.Perm.dvd_of_mem_cycleType + - Equiv.Perm.cycleType_prime_order + - Equiv.Perm.IsThreeCycle.orderOf + - Equiv.Perm.IsThreeCycle.inv + - card_support_eq_three_iff + - Equiv.Perm.isConj_of_cycleType_eq + - Equiv.Perm.mem_cycleType_iff + - path: Mathlib/Data/List/Basic.lean + theorems: + - List.indexOf_append_of_mem + - List.indexOf_eq_length + - List.scanr_cons + - List.cons_get_drop_succ + - List.foldr_map' + - List.reverse_replicate + - List.nthLe_succ_scanl + - List.nthLe_of_eq + - List.indexOf_get + - List.map_foldl_erase + - List.zipRight_eq_zipRight' + - List.getLast?_isNone + - List.zipRight'_nil_left + - Function.Involutive.exists_mem_and_apply_eq_iff + - List.length_injective_iff + - List.foldl_ext + - List.reverseRecOn_concat + - List.length_erase_add_one + - List.zipLeft_eq_zipLeft' + - List.modifyHead_modifyHead + - List.zipLeft_nil_right + - List.get?_succ_scanl + - List.forall_map_iff + - List.of_mem_filter + - List.map_injective_iff + - List.takeD_left' + - List.takeD_left + - List.cons_head?_tail + - List.foldl_eq_of_comm' + - List.sizeOf_dropSlice_lt + - List.indexOf_cons_eq + - List.getLast_reverse + - List.filter_true + - List.replicate_right_inj' + - List.bidirectionalRec_cons_append + - List.nthLe_tail + - "List.map\u2082Right_nil_left" + - List.takeWhile_takeWhile + - List.map_fst_add_enumFrom_eq_enumFrom + - List.get_length_sub_one + - List.foldl_assoc + - List.cons_eq_append_iff + - List.foldrM_eq_foldr + - List.scanl_cons + - List.append_cons_inj_of_not_mem + - List.map_eq_foldr + - List.get?_zero_scanl + - List.sublist_replicate_iff + - List.nthLe_reverse' + - List.map_diff + - List.tail_append_of_ne_nil + - List.replicate_add + - List.map_replicate + - "List.map\u2082Right_eq_zipWith" + - List.append_eq_cons_iff + - List.span_eq_take_drop + - List.nthLe_drop' + - List.Forall.imp + - List.subset_singleton_iff + - List.filter_comm + - List.nil_zipWith + - List.mem_pair + - List.modifyNthTail_modifyNthTail_same + - List.nthLe_length_sub_one + - List.take_one_drop_eq_of_lt_length + - List.isEmpty_iff_eq_nil + - List.splitOnP.go_ne_nil + - "List.get_reverse_aux\u2081" + - List.splitOnP_cons + - List.getLast_pmap + - List.modifyNthTail_modifyNthTail_le + - List.takeWhile_eq_nil_iff + - List.map_reverseAux + - List.get_enumFrom + - List.indexOf_cons_self + - List.map_comp_map + - List.pmap_map + - List.mem_filter_of_mem + - List.foldr_join + - List.map_subset_iff + - List.map_congr + - List.get_enum + - Function.Injective.list_map + - List.enumFrom_map + - "List.map\u2082Left_nil_right" + - List.foldl_eq_foldr' + - List.foldl_map' + - List.map_eq_map_iff + - List.enum_get? + - List.head!_mem_self + - List.dropWhile_eq_nil_iff + - List.lookmap_map_eq + - List.dropWhile_nthLe_zero_not + - List.foldl_eq_of_comm_of_assoc + - List.pmap_append + - List.splitAt_eq_take_drop + - List.length_pmap + - "List.map\u2082Left_eq_zipWith" + - List.dropSlice_eq + - List.takeWhile_eq_self_iff + - List.head!_append + - List.tail_replicate + - List.mem_enumFrom + - List.foldl_eq_foldr + - List.getLast_congr + - List.nthLe_drop + - List.mem_getLast?_eq_getLast + - List.takeWhile_idem + - List.lookmap.go_append + - List.nthLe_enumFrom + - List.getLast_cons + - List.modifyLast_append_one + - List.foldr_eta + - List.takeWhile_cons_of_neg + - List.eraseIdx_eq_take_drop_succ + - List.foldr_fixed' + - List.pmap_eq_map + - List.injective_foldl_comp + - "List.foldr_hom\u2082" + - List.sizeOf_lt_sizeOf_of_mem + - List.getLast?_append_of_ne_nil + - List.eq_nil_of_map_eq_nil + - List.takeI_left' + - List.reverse_cons' + - List.intercalate_splitOn + - List.length_concat' + - List.lookmap_congr + - List.map_pmap + - List.foldr_eq_of_comm' + - List.zipRight_nil_left + - List.enumFrom_eq_nil + - List.erase_get + - "List.get_reverse_aux\u2082" + - List.foldl_join + - List.foldl_assoc_comm_cons + - List.nthLe_cons_length + - List.removeNth_eq_nthTail + - List.eq_replicate_length + - List.reduceOption_length_le + - List.reduceOption_concat + - List.enumFrom_cons' + - List.lookmap_cons_none + - List.eq_cons_of_length_one + - List.nthLe_zero + - List.pmap_congr + - List.mem_attach + - List.foldlM_eq_foldl + - List.nthLe_take' + - List.get_pmap + - List.or_exists_of_exists_mem_cons + - List.attach_map_coe' + - List.reduceOption_length_eq_iff + - List.get?_pmap + - List.zipWith_nil + - List.concat_eq_reverse_cons + - List.splitOnP_eq_single + - List.length_eraseP_add_one + - List.getLast_append' + - List.foldr_concat + - List.head?_map + - List.map_id'' + - Decidable.List.eq_or_ne_mem_of_mem + - List.reduceOption_concat_of_some + - List.join_replicate_nil + - List.length_scanl + - List.mem_pure + - List.splitOnP_spec + - List.dropLast_append_getLast + - List.modifyLast_append + - List.modifyLast.go_append_one + - List.span.loop_eq_take_drop + - List.nthLe_set_of_ne + - "List.foldl_hom\u2082" + - List.reduceOption_get?_iff + - "List.map\u2082Right_eq_map\u2082Right'" + - List.map_concat + - List.filter_eq_foldr + - List.takeWhile_cons_of_pos + - List.getLast?_isSome + - List.enumFrom_get? + - List.mem_map_swap + - List.getLast_replicate_succ + - List.monotone_filter_right + - List.head?_append + - List.map_tail + - List.indexOf_append_of_not_mem + - List.enum_append + - List.head!_eq_head? + - List.takeI_left + - List.foldl1_eq_foldr1 + - List.get_attach + - List.pmap_eq_map_attach + - List.mem_pmap + - List.map_leftInverse_iff + - List.filter_false + - List.monotone_filter_left + - List.mem_map_of_involutive + - List.getLast?_append + - List.find?_mem + - List.dropLast_append_getLast? + - List.modifyNth_nil + - "List.map\u2082Right'_nil_left" + - List.foldr_ext + - List.map_surjective_iff + - List.reverse_foldl + - List.splitOn_intercalate + - List.lookmap_cons_some + - List.sublist_singleton + - List.pmap_eq_nil + - List.map_filter_eq_foldr + - List.ilast'_mem + - List.foldl_fixed' + - List.mem_takeWhile_imp + - List.foldl_op_eq_op_foldr_assoc + - List.filter_attach' + - List.indexOf_inj + - List.bind_ret_eq_map + - List.zipLeft'_nil_right + - List.map_erase + - List.map_join + - List.getLast?_append_cons + - List.getLast_append_singleton + - List.get?_injective + - List.nthLe_enum + - List.splitOnP_first + - List.reduceOption_map + - List.foldl_concat + - List.disjoint_pmap + - List.indexOf_le_length + - List.reduceOption_length_lt_iff + - List.tail_append_singleton_of_ne_nil + - List.modifyNth_eq_set + - List.eq_cons_of_mem_head? + - List.head!_mem_head? + - List.reduceOption_cons_of_some + - List.splitOnP.go_acc + - List.enumFrom_append + - "List.map\u2082Left'_nil_right" + - List.indexOf_get? + - List.reduceOption_cons_of_none + - List.length_lookmap + - List.nthLe_cons_aux + - List.forall_iff_forall_mem + - List.indexOf_cons_ne + - List.drop_tail + - List.eq_of_mem_map_const + - List.doubleton_eq + - List.mem_getLast?_cons + - List.get_eq_iff + - List.get_zero_scanl + - List.getLast_eq_nthLe + - List.getLast_map + - List.erase_diff_erase_sublist_of_sublist + - List.reduceOption_mem_iff + - List.map_eq_replicate_iff + - List.reduceOption_singleton + - List.lookmap_id' + - path: Mathlib/Data/Finset/Option.lean + theorems: + - Finset.eraseNone_insertNone + - Finset.eraseNone_map_some + - Finset.map_some_eraseNone + - Finset.forall_mem_eraseNone + - Finset.mem_eraseNone + - Finset.some_mem_insertNone + - Finset.eraseNone_empty + - Finset.forall_mem_insertNone + - Finset.image_some_eraseNone + - Finset.card_insertNone + - Finset.eraseNone_image_some + - Finset.eraseNone_eq_biUnion + - Finset.insertNone_eraseNone + - Finset.eraseNone_inter + - Finset.mem_insertNone + - Finset.eraseNone_none + - Finset.none_mem_insertNone + - Option.card_toFinset + - Finset.eraseNone_union + - Option.mem_toFinset + - path: Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean + theorems: + - HomogeneousLocalization.add_val + - HomogeneousLocalization.eq_num_div_den + - HomogeneousLocalization.pow_val + - HomogeneousLocalization.ext_iff_val + - HomogeneousLocalization.mul_val + - HomogeneousLocalization.isUnit_iff_isUnit_val + - HomogeneousLocalization.intCast_val + - HomogeneousLocalization.natCast_val + - HomogeneousLocalization.smul_val + - HomogeneousLocalization.neg_val + - HomogeneousLocalization.NumDenSameDeg.ext + - HomogeneousLocalization.sub_val + - path: Mathlib/Data/Vector/Basic.lean + theorems: + - Vector.get_set_eq_if + - Vector.eq_cons_iff + - Vector.get_ofFn + - Vector.toList_map + - Vector.append_nil + - Vector.reverse_reverse + - Vector.toList_ofFn + - Vector.nodup_iff_injective_get + - Vector.ne_cons_iff + - Vector.toList_singleton + - Vector.mOfFn_pure + - Vector.comp_traverse + - Vector.not_empty_toList + - Vector.nth_eq_nthLe + - Vector.scanl_singleton + - Vector.get_replicate + - Vector.tail_map + - Vector.insertNth_comm + - Vector.get_set_of_ne + - Vector.map_id + - Vector.ofFn_get + - Vector.scanl_cons + - Vector.prod_set' + - Vector.traverse_eq_map_id + - Vector.removeNth_insertNth' + - Vector.get_cons_succ + - Vector.id_traverse + - Vector.tail_ofFn + - Vector.traverse_def + - Vector.prod_set + - Vector.head_map + - Vector.get_append_cons_zero + - Vector.get_map + - "Vector.get_map\u2082" + - Vector.ext + - Vector.head_ofFn + - Vector.scanl_head + - Vector.naturality + - Vector.scanl_get + - Vector.get_tail_succ + - Vector.get_set_same + - Vector.get_cons_zero + - path: Mathlib/NumberTheory/NumberField/ClassNumber.lean + theorems: + - Rat.classNumber_eq + - RingOfIntegers.isPrincipalIdealRing_of_abs_discr_lt + - NumberField.exists_ideal_in_class_of_norm_le + - path: Mathlib/Topology/Algebra/Module/Simple.lean + theorems: + - LinearMap.isClosed_or_dense_ker + - path: Mathlib/CategoryTheory/Closed/Cartesian.lean + theorems: + - CategoryTheory.CartesianClosed.curry_id_eq_coev + - CategoryTheory.CartesianClosed.uncurry_id_eq_ev + - CategoryTheory.uncurry_pre + - CategoryTheory.pre_map + - CategoryTheory.initial_mono + - CategoryTheory.strict_initial + - CategoryTheory.pre_id + - path: Mathlib/Data/Finset/Basic.lean + theorems: + - Finset.coe_symmDiff + - Finset.pairwise_cons' + - Finset.erase_right_comm + - Finset.bind_toFinset + - Finset.Nonempty.exists_cons_eq + - Finset.erase_sdiff_erase + - Finset.forall_mem_cons + - Directed.exists_mem_subset_of_finset_subset_biUnion + - Finset.insert_sdiff_of_not_mem + - Finset.not_mem_sdiff_of_mem_right + - Finset.erase_biUnion + - Finset.disjoint_filter + - Finset.ssubset_iff_exists_cons_subset + - List.perm_of_nodup_nodup_toFinset_eq + - Finset.eq_singleton_or_nontrivial + - Finset.disjoint_erase_insert + - Finset.subset_singleton_iff + - Finset.ssubset_iff_exists_subset_erase + - Finset.insert_sdiff_insert' + - Finset.union_sdiff_symm + - Finset.not_mem_sdiff_of_not_mem_left + - Finset.not_subset + - Finset.insert_union_distrib + - Finset.val_toFinset + - Finset.piecewise_coe + - Multiset.toFinset_union + - Finset.biUnion_biUnion + - Finset.forall_mem_insert + - Finset.disjUnion_singleton + - List.toFinset_eq_empty_iff + - Finset.inter_subset_inter + - Finset.disjiUnion_filter_eq_of_maps_to + - Finset.sdiff_erase_self + - Finset.inter_biUnion + - DirectedOn.exists_mem_subset_of_finset_subset_biUnion + - Finset.erase_subset_iff_of_mem + - Finset.erase_injOn' + - Finset.disjoint_union_right + - Finset.erase_sdiff_distrib + - Finset.cons_induction + - Finset.coe_insert + - Finset.sdiff_insert_of_not_mem + - Finset.filter_False + - Finset.piecewise_eq_of_not_mem + - Finset.update_piecewise_of_mem + - Finset.erase_insert_of_ne + - Finset.disjoint_of_erase_left + - Finset.piecewise_cases + - Finset.Nontrivial.ne_singleton + - Multiset.toFinset_nonempty + - Finset.erase_union_distrib + - Finset.empty_inter + - Finset.eq_of_veq + - Finset.disjoint_erase_comm + - Finset.filter_comm + - Finset.coe_disjiUnion + - Finset.inter_singleton_of_mem + - Finset.subset_insert_iff_of_not_mem + - Finset.coe_eq_pair + - Finset.eq_of_mem_of_not_mem_erase + - Finset.disjoint_of_erase_right + - Finset.ssubset_singleton_iff + - Finset.sdiff_singleton_eq_self + - Finset.singleton_subset_singleton + - Finset.insert_val' + - Finset.coe_erase + - Finset.disjoint_union_left + - Finset.erase_insert_eq_erase + - Finset.inter_empty + - Finset.erase_eq_iff_eq_insert + - List.toFinset_surj_on + - Finset.disjoint_coe + - Finset.disjoint_insert_right + - Finset.inter_insert_of_not_mem + - Finset.coe_empty + - Finset.union_empty + - Finset.insert_erase + - Finset.piecewise_empty + - Set.pairwiseDisjoint_filter + - Finset.not_mem_empty + - Finset.sdiff_singleton_eq_erase + - Finset.inter_filter + - Finset.coe_biUnion + - Finset.singleton_iff_unique_mem + - Multiset.toFinset_bind_dedup + - Finset.filter_eq + - Finset.disjoint_insert_erase + - Multiset.toFinset_subset + - Finset.range_filter_eq + - Finset.sdiff_insert + - Multiset.toFinset_filter + - Finset.disjoint_right + - Finset.insert_sdiff_cancel + - Finset.singleton_biUnion + - Finset.coe_cons + - Finset.mem_disjiUnion + - Finset.piecewise_eq_of_mem + - Finset.inter_singleton_of_not_mem + - Finset.piecewise_same + - Finset.piecewise_insert + - Finset.erase_inter_comm + - Finset.erase_nonempty + - Finset.union_insert + - Finset.right_eq_union + - Finset.biUnion_congr + - Finset.filter_cons + - Finset.biUnion_nonempty + - Finset.disjoint_insert_left + - Finset.disjoint_biUnion_right + - Finset.coe_singleton + - Finset.forall_mem_not_eq + - Finset.erase_eq_empty_iff + - Finset.mem_erase_of_ne_of_mem + - List.toFinset_filter + - Finset.union_right_comm + - Multiset.toFinset_add + - Finset.filter_inter_distrib + - Finset.insert_subset_insert_iff + - Finset.filter_insert + - Finset.insert_sdiff_of_mem + - Finset.biUnion_subset_biUnion_of_subset_left + - Finset.not_disjoint_iff + - Finset.induction_on_union + - Finset.subset_inter + - coe_notMemRangeEquiv_symm + - Finset.empty_union + - Finset.cons_subset_cons + - Finset.coe_ssubset + - Finset.inter_insert_of_mem + - Finset.mem_biUnion + - Finset.disjUnion_eq_union + - Finset.Nonempty.cons_induction + - Finset.coe_subset_singleton + - Finset.range_nontrivial + - Finset.exists_mem_empty_iff + - Finset.filter_nonempty_iff + - Finset.filter_eq_self + - Finset.isEmpty_coe_sort + - Finset.erase_idem + - Finset.filter_eq_empty_iff + - Finset.piecewise_singleton + - Finset.disjoint_iff_ne + - Finset.coe_eq_singleton + - List.toFinset_range + - Finset.disjoint_or_nonempty_inter + - Finset.erase_cons_of_ne + - Multiset.toFinset_inter + - Finset.mem_disjUnion + - Finset.not_nontrivial_singleton + - Finset.inter_sdiff + - Finset.insert_subset_iff + - Finset.filter_not + - List.toFinset_append + - Finset.inter_right_comm + - Finset.Insert.comm + - Finset.update_piecewise_of_not_mem + - Finset.erase_sdiff_comm + - List.toFinset_eq + - Finset.filter_True + - Finset.nonempty_mk + - Finset.range_eq_empty_iff + - Finset.union_inter_cancel_left + - Finset.exists_mem_insert + - Finset.disjiUnion_empty + - Finset.filter_inter_filter_neg_eq + - Finset.piecewise_mem_set_pi + - Finset.insert_idem + - Finset.disjiUnion_disjiUnion + - List.toFinset_inter + - Finset.inter_sdiff_self + - Finset.eq_empty_iff_forall_not_mem + - Finset.filter_biUnion + - Finset.update_piecewise + - Finset.erase_inj + - Finset.mem_symmDiff + - Finset.inter_assoc + - Multiset.toFinset_singleton + - Finset.piecewise_insert_self + - Finset.biUnion_mono + - Finset.sdiff_erase + - Finset.sdiff_eq_filter + - Finset.filter_mem_eq_inter + - Finset.toList_toFinset + - Finset.filter_ne + - Finset.attach_nonempty_iff + - Finset.eq_singleton_iff_nonempty_unique_mem + - Finset.nodup_toList + - Finset.erase_cons + - Finset.sizeOf_lt_sizeOf_of_mem + - Finset.toList_cons + - Finset.disjoint_singleton_left + - Finset.sdiff_insert_insert_of_mem_of_not_mem + - Finset.disjoint_filter_filter' + - Finset.cons_sdiff_cons + - Multiset.toFinset_eq_singleton_iff + - List.toFinset_nonempty_iff + - Finset.val_eq_singleton_iff + - Finset.piecewise_le_piecewise' + - Finset.filter_congr_decidable + - Finset.mem_insert_coe + - Finset.biUnion_inter + - Finset.ne_insert_of_not_mem + - Finset.coe_disjUnion + - Finset.insert_inter_of_mem + - Finset.nonempty_iff_eq_singleton_default + - Multiset.toFinset_ssubset + - Multiset.toFinset_nsmul + - Finset.filter_eq' + - Finset.insert_inter_of_not_mem + - Finset.singleton_subset_coe + - Finset.not_mem_union + - Finset.filter_filter + - Finset.insert_val_of_not_mem + - Finset.toList_eq_singleton_iff + - Finset.filter_singleton + - Finset.coe_eq_empty + - Finset.inter_union_self + - Finset.inter_left_comm + - Finset.filter_ne' + - Finset.not_nontrivial_empty + - Finset.disjoint_singleton + - List.toFinset_replicate_of_ne_zero + - Finset.Nontrivial.sdiff_singleton_nonempty + - Finset.biUnion_filter_eq_of_maps_to + - Finset.singleton_subset_set_iff + - Finset.insert_inter_distrib + - List.toFinset_union + - Finset.erase_singleton + - List.toFinset_eq_iff_perm_dedup + - Finset.attach_eq_empty_iff + - Finset.filter_union_right + - Finset.erase_union_of_mem + - Finset.coe_pair + - Finset.filter_erase + - Finset.filter_union + - Finset.filter_and + - Finset.Nontrivial.erase_nonempty + - Finset.union_left_comm + - Finset.inter_comm + - Finset.disjoint_biUnion_left + - Finset.subset_union_elim + - Finset.not_disjoint_iff_nonempty_inter + - Multiset.toFinset_range + - Finset.subset_insert_iff + - List.toFinset.ext_iff + - Finset.erase_inter + - Multiset.disjoint_toFinset + - Finset.cons_eq_insert + - Finset.filter_or + - Finset.insert_union + - Finset.forall_mem_not_eq' + - Finset.union_inter_cancel_right + - Finset.sdiff_union_erase_cancel + - List.toFinset_cons + - Multiset.toFinset_dedup + - Multiset.Nodup.toFinset_inj + - Finset.biUnion_insert + - Finset.biUnion_subset + - Finset.Nontrivial.exists_cons_eq + - Finset.union_erase_of_mem + - Finset.filter_const + - Finset.piecewise_insert_of_ne + - Finset.eq_singleton_iff_unique_mem + - path: Mathlib/Data/Nat/Totient.lean + theorems: + - Nat.totient_pos + - Nat.totient_mul_prod_primeFactors + - Nat.totient_eq_one_iff + - Nat.totient_one + - Nat.Ico_filter_coprime_le + - Nat.totient_mul_of_prime_of_not_dvd + - Nat.totient_mul_of_prime_of_dvd + - Nat.totient_lt + - Nat.card_units_zmod_lt_sub_one + - Nat.totient_even + - Nat.totient_super_multiplicative + - Nat.totient_eq_card_lt_and_coprime + - Nat.totient_gcd_mul_totient_mul + - Nat.totient_dvd_of_dvd + - Nat.totient_eq_iff_prime + - Nat.totient_div_of_dvd + - Nat.filter_coprime_Ico_eq_totient + - Nat.totient_eq_prod_factorization + - Nat.totient_prime_pow_succ + - Nat.totient_prime_pow + - Nat.totient_eq_div_primeFactors_mul + - Nat.prime_iff_card_units + - Nat.totient_eq_mul_prod_factors + - Nat.sum_totient + - ZMod.card_units_eq_totient + - Nat.sum_totient' + - Nat.totient_mul + - path: Mathlib/Algebra/Group/Basic.lean + theorems: + - eq_one_div_of_mul_eq_one_right + - div_mul_eq_div_mul_one_div + - eq_of_div_eq_one + - eq_of_inv_mul_eq_one + - mul_mul_div_cancel + - mul_div_left_comm + - div_eq_inv_self + - mul_inv + - mul_eq_one_iff_eq_inv + - eq_inv_mul_iff_mul_eq + - inv_div' + - div_eq_div_mul_div + - mul_div_cancel_left + - inv_mul_eq_one + - eq_div_of_mul_eq' + - div_mul_cancel_left + - eq_iff_eq_of_mul_eq_mul + - mul_inv_eq_iff_eq_mul + - mul_div_cancel_right + - div_eq_of_eq_mul'' + - mul_mul_inv_cancel'_right + - inv_eq_one_div + - eq_div_of_mul_eq'' + - inv_div_left + - one_div_mul_one_div + - div_right_injective + - mul_inv_eq_of_eq_mul + - pow_natAbs_eq_one + - mul_rotate' + - mul_div_mul_right_eq_div + - div_mul_eq_mul_div + - mul_div_div_cancel + - eq_mul_of_inv_mul_eq + - eq_one_iff_eq_one_of_mul_eq_one + - div_div_div_eq + - div_div_eq_mul_div + - div_div_cancel_left + - div_eq_mul_one_div + - eq_of_one_div_eq_one_div + - mul_div_mul_left_eq_div + - div_mul_cancel + - div_mul_div_cancel' + - inv_div + - eq_mul_inv_of_mul_eq + - div_eq_one + - mul_div_right_comm + - eq_div_iff_mul_eq'' + - comp_mul_right + - div_mul_eq_div_div + - mul_eq_one_iff_inv_eq + - comp_mul_left + - mul_eq_of_eq_div + - multiplicative_of_symmetric_of_isTotal + - div_div_div_comm + - mul_mul_mul_comm + - dite_pow + - multiplicative_of_isTotal + - ite_mul_one + - eq_mul_inv_iff_mul_eq + - bit1_zero + - div_mul_div_cancel'' + - inv_inv_div_inv + - mul_inv_eq_one + - div_eq_inv_mul + - div_eq_div_iff_mul_eq_mul + - mul_div_mul_comm + - eq_mul_of_div_eq' + - eq_of_mul_inv_eq_one + - ne_iff_ne_of_mul_eq_mul + - div_mul_div_comm + - one_div_div + - mul_right_eq_self + - div_eq_of_eq_mul' + - eq_one_div_of_mul_eq_one_left + - inv_div_inv + - div_inv_eq_mul + - mul_eq_of_eq_mul_inv + - div_right_comm + - eq_mul_of_div_eq + - mul_div_cancel + - pow_dite + - div_eq_self + - mul_div_assoc + - div_div_div_cancel_left + - one_div_mul_one_div_rev + - inv_mul_eq_iff_eq_mul + - inv_mul' + - eq_mul_of_mul_inv_eq + - mul_eq_of_eq_div' + - div_mul_eq_div_div_swap + - eq_div_iff_mul_eq' + - div_one + - div_eq_iff_eq_mul' + - mul_eq_of_eq_inv_mul + - inv_mul_eq_div + - div_div_div_cancel_right' + - div_div + - div_mul_comm + - div_mul + - eq_inv_mul_of_mul_eq + - mul_rotate + - eq_iff_eq_of_div_eq_div + - one_div_one_div + - mul_comm_div + - div_div_self' + - mul_div + - div_eq_div_iff_comm + - inv_mul_eq_of_eq_mul + - ite_one_mul + - div_left_injective + - bit1_add' + - div_eq_iff_eq_mul + - div_self' + - div_left_inj + - mul_one_div + - div_eq_div_iff_div_eq_div + - div_mul_cancel_right + - mul_left_eq_self + - path: Mathlib/LinearAlgebra/Basic.lean + theorems: + - LinearMap.range_comp_of_range_eq_top + - LinearEquiv.conj_id + - LinearMap.range_eq_map + - LinearMap.range_domRestrict_le_range + - LinearMap.mem_submoduleImage + - Submodule.equivSubtypeMap_symm_apply + - Submodule.map_subtype_top + - LinearMap.comap_le_comap_iff + - Submodule.comap_subtype_eq_top + - LinearMap.ker_eq_bot_of_cancel + - IsLinearMap.isLinearMap_add + - LinearMap.range_le_iff_comap + - LinearEquiv.eq_bot_of_equiv + - LinearMap.funLeft_surjective_of_injective + - LinearMap.range_le_ker_iff + - LinearMap.funLeft_injective_of_surjective + - LinearEquiv.conj_trans + - LinearMap.range_codRestrict + - Submodule.range_subtype + - LinearMap.range_rangeRestrict + - LinearMap.range_smul' + - LinearEquiv.neg_apply + - Submodule.map_comap_eq_self + - LinearMap.range_zero + - LinearEquiv.arrowCongr_comp + - LinearEquiv.ofSubmodule'_toLinearMap + - LinearMap.mem_submoduleImage_of_le + - Submodule.range_inclusion + - Submodule.map_subtype_le + - LinearMap.ker_le_iff + - LinearMap.range_eq_top + - LinearMap.submoduleImage_apply_of_le + - LinearMap.range_smul + - LinearMap.range_le_bot_iff + - IsLinearMap.isLinearMap_sub + - LinearMap.eqOn_sup + - LinearEquiv.ofEq_rfl + - LinearMap.range_neg + - Submodule.map_subtype_range_inclusion + - LinearMap.eqLocus_eq_top + - LinearMap.surjective_rangeRestrict + - LinearMap.range_eq_bot + - path: Mathlib/Probability/StrongLaw.lean + theorems: + - ProbabilityTheory.moment_truncation_eq_intervalIntegral + - ProbabilityTheory.strong_law_aux2 + - ProbabilityTheory.integral_truncation_eq_intervalIntegral + - ProbabilityTheory.integral_truncation_le_integral_of_nonneg + - ProbabilityTheory.abs_truncation_le_bound + - ProbabilityTheory.sum_variance_truncation_le + - ProbabilityTheory.strong_law_aux5 + - ProbabilityTheory.tsum_prob_mem_Ioi_lt_top + - ProbabilityTheory.strong_law_aux6 + - ProbabilityTheory.strong_law_ae + - ProbabilityTheory.strong_law_aux1 + - ProbabilityTheory.strong_law_aux7 + - MeasureTheory.AEStronglyMeasurable.integrable_truncation + - ProbabilityTheory.truncation_eq_of_nonneg + - ProbabilityTheory.strong_law_aux4 + - ProbabilityTheory.strong_law_aux3 + - ProbabilityTheory.sum_prob_mem_Ioc_le + - MeasureTheory.AEStronglyMeasurable.truncation + - ProbabilityTheory.integral_truncation_eq_intervalIntegral_of_nonneg + - ProbabilityTheory.strong_law_ae_of_measurable + - ProbabilityTheory.strong_law_Lp + - ProbabilityTheory.tendsto_integral_truncation + - ProbabilityTheory.strong_law_ae_real + - ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg + - ProbabilityTheory.strong_law_ae_simpleFunc_comp + - ProbabilityTheory.truncation_zero + - ProbabilityTheory.truncation_eq_self + - path: Mathlib/Data/Matrix/Basic.lean + theorems: + - Matrix.two_mul_expl + - Matrix.diagonal_zero + - Matrix.submatrix_diagonal + - Matrix.mul_submatrix_one + - Matrix.map_id + - Matrix.transpose_zero + - Matrix.mulVec_mulVec + - Matrix.add_vecMul + - Matrix.conjTranspose_natCast_smul + - Matrix.vecMul_add + - Matrix.sub_mul + - Matrix.star_dotProduct_star + - Matrix.map_map + - Matrix.one_eq_pi_single + - Matrix.sub_mulVec + - Matrix.sub_dotProduct + - RingHom.map_mulVec + - RingHom.map_dotProduct + - Matrix.dotProduct_mulVec + - Matrix.single_vecMul_diagonal + - Matrix.op_smul_eq_mul_diagonal + - Matrix.zero_dotProduct + - Matrix.conjTranspose_inv_intCast_smul + - Matrix.transpose_transpose + - Matrix.ext_iff + - Matrix.diagonal_sub + - Matrix.transpose_add + - Matrix.one_mulVec + - Matrix.diagonal_mul_diagonal + - Matrix.mulVec_transpose + - Matrix.conjTranspose_sub + - Matrix.one_dotProduct_one + - Matrix.bit1_apply_eq + - Matrix.smul_dotProduct + - Matrix.diagonal_dotProduct + - Matrix.dotProduct_single + - Matrix.diagonal_mulVec_single + - Matrix.dotProduct_comm + - Matrix.diagonal_add + - Matrix.mulVec_smul + - Matrix.neg_mulVec + - Matrix.mul_assoc + - Matrix.conjTranspose_zero + - Matrix.transpose_eq_one + - Matrix.transpose_smul + - Matrix.transpose_sub + - Matrix.diagonal_map + - Matrix.smul_one_eq_diagonal + - Matrix.diagonal_smul + - Matrix.diagonal_neg + - Matrix.smul_mulVec_assoc + - Matrix.vec_one_mul + - Matrix.diagonal_apply_ne + - Matrix.conjTranspose_neg + - Matrix.map_algebraMap + - Matrix.conjTranspose_eq_zero + - RingHom.map_matrix_mul + - Matrix.diagonal_conjTranspose + - Matrix.diagonal_injective + - Matrix.conjTranspose_one + - Matrix.add_mulVec + - Matrix.add_mul + - Matrix.vecMul_smul + - Matrix.map_zero + - Matrix.zero_mul + - Matrix.neg_vecMul + - Matrix.neg_mul + - Matrix.star_dotProduct + - Matrix.smul_mul + - Matrix.dotProduct_zero + - Matrix.smul_eq_diagonal_mul + - Matrix.bit1_apply + - Matrix.smul_eq_mul_diagonal + - Matrix.bit1_apply_ne + - Matrix.single_dotProduct + - Matrix.mul_smul + - Matrix.map_mul + - Matrix.conjTranspose_zsmul + - Matrix.diagonal_transpose + - Matrix.vecMul_vecMul + - Matrix.algebraMap_matrix_apply + - Matrix.comp_equiv_dotProduct_comp_equiv + - Matrix.dotProduct_add + - Matrix.mulVec_sub + - Matrix.op_smul_one_eq_diagonal + - Matrix.scalar_commute_iff + - Matrix.vecMul_one + - Matrix.mul_sub + - Matrix.dotProduct_neg + - Matrix.mulVec_neg + - Matrix.zero_vecMul + - Matrix.transpose_map + - Matrix.dotProduct_smul + - Matrix.mulVec_add + - Matrix.conjTranspose_conjTranspose + - Matrix.diagonal_eq_diagonal_iff + - Matrix.one_mul + - Matrix.dotProduct_star + - Matrix.vecMul_transpose + - Matrix.conjTranspose_inv_natCast_smul + - Matrix.vecMul_zero + - Matrix.conjTranspose_smul_non_comm + - Matrix.conjTranspose_nsmul + - Matrix.mulVec_one + - Matrix.vecMul_sub + - Matrix.dotProduct_sub + - Matrix.one_submatrix_mul + - Matrix.mulVec_smul_assoc + - Matrix.dotProduct_assoc + - Matrix.conjTranspose_add + - Matrix.sum_elim_dotProduct_sum_elim + - Matrix.vecMul_mulVec + - Matrix.conjTranspose_ratCast_smul + - Matrix.conjTranspose_rat_smul + - Matrix.transpose_neg + - Matrix.add_dotProduct + - Matrix.map_one + - Matrix.conjTranspose_mul + - Matrix.mulVec_vecMul + - Matrix.mul_add + - Matrix.dotProduct_diagonal' + - Matrix.submatrix_mul_transpose_submatrix + - Matrix.neg_dotProduct + - RingHom.map_vecMul + - Matrix.mul_diagonal + - Matrix.mul_zero + - Matrix.conjTranspose_intCast_smul + - Matrix.mul_one + - Matrix.zero_mulVec + - Matrix.dotProduct_comp_equiv_symm + - Matrix.transpose_one + - Matrix.dotProduct_diagonal + - Matrix.sub_vecMul + - Matrix.transpose_mul + - Matrix.diagonal_apply_eq + - Matrix.mul_mul_apply + - Matrix.conjTranspose_eq_one + - Matrix.vecMul_neg + - path: Mathlib/Analysis/SpecialFunctions/Pow/Real.lean + theorems: + - Real.exists_rat_pow_btwn_rat + - Real.one_lt_rpow_iff_of_pos + - Real.rpow_inv_le_iff_of_pos + - Real.rpow_rpow_inv + - Complex.norm_prime_cpow_le_one_half + - Real.rpow_one + - Real.zero_rpow + - Real.abs_rpow_le_abs_rpow + - Real.exp_mul + - Real.le_rpow_inv_iff_of_pos + - Real.rpow_nat_cast + - Complex.cpow_inv_two_im_eq_sqrt + - Real.inv_rpow + - Real.rpow_sub_one + - Real.rpow_sum_of_nonneg + - Complex.norm_natCast_cpow_le_norm_natCast_cpow_of_pos + - Real.one_lt_rpow_of_pos_of_lt_one_of_neg + - Complex.abs_cpow_eq_rpow_re_of_pos + - Real.rpow_sub_nat' + - Real.exists_rat_pow_btwn_rat_aux + - Real.rpow_lt_rpow_of_neg + - Real.rpow_lt_one_iff_of_pos + - Real.rpow_sub_one' + - Mathlib.Meta.NormNum.isRat_rpow_pos + - Real.rpow_def_of_nonneg + - Real.rpow_add_int + - Real.rpow_le_rpow_of_exponent_nonpos + - Real.rpow_int_cast + - Real.rpow_le_one + - Real.rpow_of_add_eq + - Real.rpow_def_of_nonpos + - Real.rpow_lt_one_iff' + - Complex.cpow_ofReal_re + - Real.rpow_lt_rpow_of_exponent_gt + - Real.rpow_lt_rpow + - Complex.abs_cpow_of_imp + - Complex.cpow_inv_two_im_eq_neg_sqrt + - Real.rpow_add_nat + - Real.one_lt_rpow_iff + - Real.one_le_rpow_of_pos_of_le_one_of_nonpos + - Real.le_rpow_iff_log_le + - Real.rpow_le_one_of_one_le_of_nonpos + - Mathlib.Meta.NormNum.isNat_rpow_pos + - Real.exp_one_pow + - Real.rpow_sub + - Real.mul_log_eq_log_iff + - Real.rpow_lt_one_of_one_lt_of_neg + - Real.abs_log_mul_self_rpow_lt + - Real.mul_rpow + - Mathlib.Meta.NormNum.isNat_rpow_neg + - Real.rpow_lt_rpow_of_exponent_neg + - Real.le_rpow_inv_iff_of_neg + - Complex.abs_cpow_inv_nat + - Real.rpow_eq_zero + - Real.rpow_add_nat' + - Real.rpow_neg_one + - Real.rpow_le_rpow_left_iff + - Real.zero_rpow_eq_iff + - Complex.cpow_inv_two_re + - Real.norm_rpow_of_nonneg + - Real.eq_rpow_inv + - Real.strictAnti_rpow_of_base_lt_one + - Real.rpow_one_add' + - Real.log_rpow + - Real.sqrt_eq_rpow + - Real.abs_rpow_le_exp_log_mul + - Complex.cpow_ofReal + - Real.le_rpow_of_log_le + - Real.exp_one_rpow + - Mathlib.Meta.NormNum.isRat_rpow_neg + - Complex.ofReal_cpow + - Complex.abs_cpow_le + - Real.rpow_sub_int' + - Real.eq_zero_rpow_iff + - Complex.norm_natCast_cpow_of_pos + - Real.rpow_le_rpow_of_exponent_ge' + - Real.rpow_le_rpow_of_exponent_ge + - Real.rpow_zero_pos + - Real.rpow_natCast_mul + - Real.lt_rpow_iff_log_lt + - Real.zero_rpow_nonneg + - Real.rpow_lt_rpow_left_iff_of_base_lt_one + - Real.rpow_mul_natCast + - Real.one_lt_rpow + - Real.rpow_add_one + - Real.zero_rpow_le_one + - Real.rpow_left_injOn + - Real.rpow_def_of_neg + - Real.rpow_intCast_mul + - Complex.norm_natCast_cpow_of_re_ne_zero + - Complex.abs_cpow_inv_two_im + - Real.rpow_inv_lt_iff_of_neg + - Real.rpow_le_rpow_of_exponent_le + - Real.rpow_le_rpow_left_iff_of_base_lt_one + - Real.rpow_eq_zero_iff_of_nonneg + - Real.rpow_def_of_pos + - Real.rpow_two + - Real.rpow_lt_rpow_left_iff + - Real.rpow_add + - Real.strictMono_rpow_of_base_gt_one + - Complex.norm_natCast_cpow_le_norm_natCast_cpow_iff + - Real.one_rpow + - Real.rpow_le_rpow + - Real.rpow_add' + - Mathlib.Meta.NormNum.isInt_rpow_neg + - Real.rpow_inv_le_iff_of_neg + - Real.rpow_sub_nat + - Real.rpow_neg + - Real.rpow_add_int' + - Real.rpow_mul + - Real.log_le_rpow_div + - Complex.abs_cpow_of_ne_zero + - Real.rpow_zero + - Complex.cpow_ofReal_im + - Real.div_rpow + - Complex.ofReal_cpow_of_nonpos + - Real.rpow_nonneg + - Real.monotone_rpow_of_base_ge_one + - Real.le_rpow_add + - Real.rpow_lt_one + - Real.rpow_one_sub' + - Real.lt_rpow_of_log_lt + - Real.rpow_add_one' + - Real.rpow_le_one_iff_of_pos + - Real.rpow_pos_of_pos + - Real.rpow_add_of_nonneg + - Real.lt_rpow_inv_iff_of_neg + - Complex.one_sub_prime_cpow_ne_zero + - Real.rpow_lt_rpow_of_exponent_lt + - Real.rpow_le_rpow_of_nonpos + - Real.rpow_inv_rpow + - Real.rpow_inv_eq + - Complex.abs_cpow_real + - Real.pow_rpow_inv_natCast + - Real.abs_rpow_of_nonneg + - Real.rpow_lt_one_iff + - Real.antitone_rpow_of_base_le_one + - Real.rpow_inv_natCast_pow + - Real.rpow_sub_int + - Mathlib.Meta.NormNum.isInt_rpow_pos + - Complex.norm_log_natCast_le_rpow_div + - Real.rpow_mul_intCast + - Real.rpow_sub' + - path: Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean + theorems: + - LinearMap.rTensor_range + - lTensor.inverse_apply + - LinearMap.exact_subtype_mkQ + - lTensor.inverse_of_rightInverse_apply + - lTensor.inverse_of_rightInverse_comp_lTensor + - Ideal.map_includeLeft_eq + - le_comap_range_rTensor + - rTensor.inverse_of_rightInverse_apply + - le_comap_range_lTensor + - lTensor.inverse_comp_lTensor + - TensorProduct.map_surjective + - LinearMap.rTensor_surjective + - rTensor_exact + - rTensor.inverse_comp_rTensor + - rTensor.inverse_apply + - rTensor.inverse_of_rightInverse_comp_rTensor + - Ideal.map_includeRight_eq + - TensorProduct.map_ker + - lTensor_exact + - LinearMap.lTensor_range + - Algebra.TensorProduct.map_ker + - Algebra.TensorProduct.lTensor_ker + - rTensor_mkQ + - lTensor_mkQ + - LinearMap.lTensor_surjective + - path: Mathlib/Order/Filter/AtTopBot.lean + theorems: + - Filter.subseq_tendsto_of_neBot + - Filter.tendsto_const_mul_atBot_of_pos + - Filter.tendsto_mul_const_atTop_iff + - Filter.frequently_atTop' + - Filter.tendsto_comp_val_Iio_atBot + - Filter.tendsto_const_mul_pow_atBot_iff + - Filter.Tendsto.prod_atTop + - Filter.Tendsto.eventually_forall_le_atBot + - OrderIso.map_atTop + - Filter.eventually_atBot_prod_self + - Filter.tendsto_atTop_add_left_of_le' + - Filter.tendsto_mul_const_atBot_iff + - Filter.prod_map_atTop_eq + - Filter.tendsto_atTop + - Filter.tendsto_atTop_principal + - Filter.tendsto_mul_const_atBot_of_neg + - Filter.exists_le_of_tendsto_atTop + - Filter.tendsto_div_const_atTop_iff_pos + - Filter.exists_lt_of_tendsto_atTop + - Filter.map_add_atTop_eq_nat + - Filter.tendsto_atTop_finset_of_monotone + - Filter.Tendsto.prod_map_prod_atBot + - Filter.Tendsto.atTop_mul_atBot + - Filter.disjoint_atBot_atTop + - Filter.atTop_Ioi_eq + - Filter.Tendsto.atTop_mul_atTop + - Filter.Tendsto.atTop_div_const + - Filter.atTop_eq_generate_of_not_bddAbove + - Filter.inf_map_atTop_neBot_iff + - Filter.not_tendsto_pow_atTop_atBot + - Filter.exists_seq_monotone_tendsto_atTop_atTop + - Filter.map_div_atTop_eq_nat + - Filter.tendsto_comp_val_Ici_atTop + - Filter.tendsto_atTop_add_right_of_le' + - Filter.Tendsto.atBot_div_const + - Filter.tendsto_atBot_diagonal + - Filter.tendsto_const_mul_atTop_iff + - Filter.tendsto_atTop_embedding + - Filter.Tendsto.atBot_mul_atTop + - Filter.Tendsto.eventually_forall_ge_atTop + - Filter.tendsto_add_atTop_iff_nat + - Filter.tendsto_const_mul_atBot_iff + - Filter.unbounded_of_tendsto_atTop + - Filter.extraction_forall_of_frequently + - Filter.eventually_iff_seq_eventually + - Filter.eventually_atBot_prod_self' + - Filter.tendsto_const_mul_atTop_iff_pos + - Filter.tendsto_mul_const_atBot_iff_neg + - Filter.tendsto_atTop_diagonal + - Filter.Tendsto.atBot_mul_atBot + - OrderIso.tendsto_atTop_iff + - Monotone.piecewise_eventually_eq_iUnion + - Filter.tendsto_Iic_atBot + - Filter.OrderTop.atTop_eq + - Filter.prod_atTop_atTop_eq + - Filter.tendsto_of_subseq_tendsto + - Filter.tendsto_mul_const_atBot_iff_pos + - Filter.tendsto_const_mul_atBot_iff_neg + - Filter.extraction_of_frequently_atTop + - Filter.exists_seq_forall_of_frequently + - Filter.frequently_atTop + - Filter.tendsto_div_const_atTop_of_pos + - Filter.tendsto_mul_const_atTop_iff_neg + - Filter.extraction_forall_of_eventually' + - Filter.exists_seq_tendsto + - Filter.tendsto_atTop_atTop_of_monotone' + - Filter.Tendsto.prod_atBot + - Filter.tendsto_mul_const_atTop_of_pos + - Filter.tendsto_atTop_mono' + - Filter.tendsto_finset_image_atTop_atTop + - Filter.tendsto_comp_val_Iic_atBot + - Filter.tendsto_const_mul_atTop_of_neg + - Filter.extraction_of_frequently_atTop' + - Filter.eventually_atTop_curry + - Filter.eventually_forall_ge_atTop + - Antitone.piecewise_eventually_eq_iInter + - Filter.tendsto_const_mul_pow_atTop_iff + - Filter.comap_abs_atTop + - Filter.atTop_eq_generate_of_forall_exists_le + - OrderIso.comap_atTop + - Filter.tendsto_pow_atTop_iff + - Filter.map_atTop_finset_prod_le_of_prod_eq + - Filter.subseq_forall_of_frequently + - Filter.tendsto_mul_const_atTop_iff_pos + - Filter.tendsto_comp_val_Ioi_atTop + - Filter.tendsto_Iio_atBot + - Filter.tendsto_atTop' + - Filter.map_sub_atTop_eq_nat + - Filter.Tendsto.prod_map_prod_atTop + - Filter.tendsto_zpow_atTop_atTop + - Filter.tendsto_const_mul_atTop_iff_neg + - Filter.map_atTop_eq_of_gc + - Filter.atTop_eq_generate_Ici + - Filter.tendsto_iff_seq_tendsto + - Filter.Subsingleton.atTop_eq + - Filter.tendsto_Ici_atTop + - Filter.exists_eventually_atBot + - Filter.map_val_atTop_of_Ici_subset + - Filter.tendsto_const_mul_atBot_iff_pos + - Filter.tendsto_mul_const_atBot_of_pos + - Filter.Eventually.atTop_of_arithmetic + - Filter.high_scores + - Filter.atTop_finset_eq_iInf + - Filter.frequently_iff_seq_frequently + - Function.Injective.map_atTop_finset_prod_eq + - Filter.frequently_high_scores + - Filter.tendsto_mul_const_atTop_of_neg + - Filter.eventually_atTop_prod_self' + - Filter.atTop_Ici_eq + - Filter.tendsto_Ioi_atTop + - path: Mathlib/Topology/Constructions.lean + theorems: + - map_snd_nhdsWithin + - tendsto_subtype_rng + - isOpenMap_sum_elim + - exists_finset_piecewise_mem_of_mem_nhds + - prod_induced_induced + - isClosedMap_swap + - Sigma.nhds_eq + - isOpen_prod_iff' + - isClosed_sum_iff + - closedEmbedding_subtype_val + - isClosed_sigma_iff + - Filter.Tendsto.update + - CofiniteTopology.nhds_eq + - Filter.HasBasis.prod_nhds + - IsOpen.trans + - Dense.prod + - CofiniteTopology.isClosed_iff + - map_nhds_subtype_coe_eq_nhds + - prod_mem_nhds_iff + - "ContinuousAt.comp\u2082_of_eq" + - inducing_const_prod + - closure_prod_eq + - Inducing.prod_map + - "continuous_inf_dom_right\u2082" + - isOpen_prod_iff + - inducing_sigma_map + - isOpen_sigma_iff + - continuous_sigma_map + - map_nhds_subtype_val + - mem_nhds_prod_iff' + - isClosed_range_inl + - CofiniteTopology.isOpen_iff' + - Filter.Tendsto.fin_insertNth + - IsClosed.trans + - ULift.isOpen_iff + - pi_generateFrom_eq + - induced_to_pi + - continuous_bool_rng + - prod_generateFrom_generateFrom_eq + - inducing_prod_const + - nhds_ne_subtype_eq_bot_iff + - isClosed_range_inr + - Filter.Tendsto.prod_mk_nhds + - IsClosed.setOf_mapsTo + - nhdsWithin_subtype_eq_bot_iff + - mem_nhdsWithin_prod_iff + - isOpenMap_sum + - Filter.eventually_nhdsSet_prod_iff + - frontier_prod_univ_eq + - ULift.closedEmbedding_down + - isOpenMap_sigma + - pi_eq_generateFrom + - isClosed_set_pi + - IsOpenMap.prod + - nhds_prod_eq + - isOpen_set_pi + - isOpenMap_inl + - Pi.induced_precomp' + - DiscreteTopology.preimage_of_continuous_injective + - isOpen_pi_iff' + - isOpen_pi_iff + - Subtype.dense_iff + - tendsto_pi_nhds + - "continuous_inf_dom_left\u2082" + - discreteTopology_subtype_iff + - continuous_pi_iff + - nhdsWithin_prod_eq + - set_pi_mem_nhds + - Embedding.prod_map + - prod_eq_generateFrom + - IsClosed.prod + - mem_nhds_prod_iff + - "continuous_sInf_dom\u2082" + - nhds_ne_subtype_neBot_iff + - openEmbedding_sigma_map + - DenseRange.prod_map + - embedding_sigma_map + - frontier_prod_eq + - "map_mem_closure\u2082" + - IsClosedMap.restrictPreimage + - interior_pi_set + - pi_generateFrom_eq_finite + - Pi.induced_restrict + - isOpenMap_sigmaMk + - inducing_sigma + - Prod.tendsto_iff + - set_pi_mem_nhds_iff + - map_fst_nhdsWithin + - ULift.isClosed_iff + - isOpen_setOf_disjoint_nhds_nhds + - mem_nhds_of_pi_mem_nhds + - Filter.Eventually.curry_nhds + - CofiniteTopology.mem_nhds_iff + - interior_prod_eq + - frontier_univ_prod_eq + - nhds_pi + - nhds_swap + - isOpenMap_inr + - exists_nhds_square + - path: .lake/packages/std/Std/Data/HashMap/WF.lean + theorems: + - Std.HashMap.Imp.WF.filterMap + - Std.HashMap.Imp.WF.mapVal + - Std.HashMap.Imp.Buckets.WF.update + - Std.HashMap.Imp.pairwise_replaceF + - Std.HashMap.Imp.Buckets.WF.mk' + - Std.HashMap.Imp.expand_WF + - Std.HashMap.Imp.Buckets.mk_size + - Std.HashMap.Imp.reinsertAux_size + - Std.HashMap.Imp.modify_size + - Std.HashMap.Imp.erase_WF + - Std.HashMap.Imp.Buckets.update_update + - Std.HashMap.Imp.erase_size + - Std.HashMap.Imp.modify_WF + - Std.HashMap.Imp.expand_size + - Std.HashMap.Imp.insert_WF + - Std.HashMap.Imp.expand_WF.foldl + - Std.HashMap.Imp.mem_replaceF + - Std.HashMap.Imp.insert_size + - Std.HashMap.Imp.WF.out + - path: Mathlib/GroupTheory/Exponent.lean + theorems: + - Monoid.exponent_min' + - Commute.orderOf_mul_pow_eq_lcm + - orderOf_eq_two_iff + - Monoid.exponent_ne_zero_iff_range_orderOf_finite + - Monoid.exponent_eq_max'_orderOf + - Monoid.exp_eq_one_iff + - Monoid.exponent_eq_zero_iff_range_orderOf_infinite + - Monoid.exponent_ne_zero + - Monoid.exponent_dvd_of_monoidHom + - Monoid.exponent_prod + - card_dvd_exponent_pow_rank + - Monoid.ExponentExists.of_finite + - MonoidHom.exponent_dvd + - Monoid.exponent_pi_eq_zero + - Monoid.exponent_eq_zero_iff_forall + - Commute.exists_orderOf_eq_lcm + - Commute.of_orderOf_dvd_two + - Monoid.lcm_orderOf_dvd_exponent + - Monoid.exists_orderOf_eq_exponent + - MulOpposite.exponent + - Monoid.exponent_dvd_iff_forall_pow_eq_one + - Monoid.exponent_eq_prime_iff + - Monoid.one_lt_exponent + - Submonoid.pow_exponent_eq_one + - Monoid.pow_exponent_eq_one + - Monoid.exponent_eq_iSup_orderOf + - Monoid.exponent_min + - Monoid.exponent_dvd + - mul_not_mem_of_orderOf_eq_two + - Monoid.pow_eq_mod_exponent + - Nat.Prime.exists_orderOf_eq_pow_factorization_exponent + - Monoid.ExponentExists.isOfFinOrder + - Monoid.exponent_pi + - path: Mathlib/CategoryTheory/Sites/Closed.lean + theorems: + - CategoryTheory.topologyOfClosureOperator_self + - CategoryTheory.topology_eq_iff_same_sheaves + - CategoryTheory.classifier_isSheaf + - CategoryTheory.le_topology_of_closedSieves_isSheaf + - CategoryTheory.GrothendieckTopology.isClosed_pullback + - CategoryTheory.GrothendieckTopology.pullback_close + - CategoryTheory.GrothendieckTopology.close_eq_top_iff_mem + - CategoryTheory.topologyOfClosureOperator_close + - path: Mathlib/Order/UpperLower/Basic.lean + theorems: + - "UpperSet.mem_iSup\u2082_iff" + - LowerSet.compl_sInf + - LowerSet.mem_iInf_iff + - IsUpperSet.erase + - isUpperSet_iff_Ioi_subset + - lowerClosure_prod + - "UpperSet.mem_iInf\u2082_iff" + - LowerSet.Iic_sInf + - LowerSet.coe_eq_univ + - LowerSet.prod_le_prod_iff + - LowerSet.coe_iInf + - LowerSet.coe_eq_empty + - IsUpperSet.total + - OrderEmbedding.image_Ici + - lowerClosure_sUnion + - UpperSet.iInf_Ici + - LowerSet.map_Iio + - upperClosure_singleton + - IsUpperSet.isLowerSet_preimage_coe + - UpperSet.compl_iSup + - "LowerSet.coe_iInf\u2082" + - UpperSet.coe_eq_univ + - LowerSet.disjoint_coe + - LowerSet.map_Iic + - UpperSet.mem_sInf_iff + - UpperSet.upperClosure_inf_sdiff + - UpperSet.Ici_sSup + - UpperSet.prod_le_prod_iff + - UpperSet.compl_sInf + - LowerSet.Iic_iInf + - UpperSet.map_map + - IsLowerSet.image + - UpperSet.map_refl + - IsLowerSet.erase + - "LowerSet.coe_iSup\u2082" + - LowerSet.compl_sSup + - LowerSet.sdiff_eq_left + - LowerSet.erase_sup_Iic + - LowerSet.mem_map + - UpperSet.sdiff_eq_left + - IsUpperSet.image + - coe_upperClosure + - "UpperSet.compl_iInf\u2082" + - OrderEmbedding.image_Ioi + - lowerClosure_singleton + - bddAbove_lowerClosure + - LowerSet.mem_iSup_iff + - UpperSet.compl_iInf + - IsLowerSet.sdiff_of_isUpperSet + - isUpperSet_iff_forall_lt + - UpperSet.mem_map + - UpperSet.codisjoint_prod + - isUpperSet_iff_Ici_subset + - coe_lowerClosure + - LowerSet.iSup_Iic + - UpperSet.coe_iInf + - LowerSet.disjoint_prod + - LowerSet.symm_map + - UpperSet.compl_sSup + - UpperSet.Ici_inf_erase + - isLowerSet_iff_Iio_subset + - UpperSet.prod_eq_top + - IsLowerSet.isUpperSet_preimage_coe + - UpperSet.coe_eq_empty + - UpperSet.mem_iSup_iff + - isLowerSet_compl + - "UpperSet.coe_iInf\u2082" + - lowerClosure_image + - upperClosure_prod + - IsLowerSet.disjoint_upperClosure_right + - LowerSet.prod_eq_bot + - "LowerSet.mem_iInf\u2082_iff" + - LowerSet.compl_iInf + - LowerSet.erase_eq + - isUpperSet_compl + - IsUpperSet.not_bddAbove + - "UpperSet.Ici_iSup\u2082" + - UpperSet.map_Ioi + - ordConnected_iff_upperClosure_inter_lowerClosure + - LowerSet.compl_iSup + - "LowerSet.Iic_iInf\u2082" + - LowerSet.sdiff_singleton + - LowerSet.map_refl + - UpperSet.coe_iSup + - LowerSet.coe_iSup + - "LowerSet.compl_iSup\u2082" + - UpperSet.sdiff_inf_upperClosure + - UpperSet.Ici_iSup + - "LowerSet.compl_iInf\u2082" + - upperClosure_sUnion + - "UpperSet.compl_iSup\u2082" + - UpperSet.erase_eq + - LowerSet.mem_sSup_iff + - LowerSet.sdiff_sup_lowerClosure + - UpperSet.symm_map + - upperClosure_image + - "UpperSet.coe_iSup\u2082" + - bddBelow_upperClosure + - UpperSet.map_Ici + - isLowerSet_iff_forall_lt + - UpperSet.erase_inf_Ici + - IsUpperSet.sdiff_of_isLowerSet + - LowerSet.lowerClosure_sup_sdiff + - IsLowerSet.disjoint_upperClosure_left + - UpperSet.sdiff_singleton + - UpperSet.mem_iInf_iff + - LowerSet.Iic_sup_erase + - LowerSet.map_map + - IsLowerSet.not_bddBelow + - "LowerSet.mem_iSup\u2082_iff" + - path: Mathlib/Order/Disjoint.lean + theorems: + - Codisjoint.eq_iff + - eq_top_of_isCompl_bot + - codisjoint_inf_left + - Prod.isCompl_iff + - disjoint_left_comm + - Complementeds.isCompl_coe + - IsCompl.inf_left_eq_bot_iff + - Codisjoint.ne + - Disjoint.le_of_codisjoint + - disjoint_assoc + - disjoint_sup_left + - Codisjoint.ne_bot_of_ne_top + - Disjoint.eq_iff + - IsCompl.sup_inf + - codisjoint_inf_right + - Complementeds.coe_le_coe + - disjoint_right_comm + - disjoint_sup_right + - Codisjoint.left_le_of_le_inf_left + - Complementeds.disjoint_coe + - Codisjoint.ne_bot_of_ne_top' + - Prod.disjoint_iff + - IsCompl.disjoint_left_iff + - Complementeds.codisjoint_coe + - Disjoint.ne + - IsCompl.inf_left_le_of_le_sup_right + - path: Mathlib/Probability/Kernel/MeasureCompProd.lean + theorems: + - MeasureTheory.Measure.compProd_apply + - MeasureTheory.Measure.integrable_compProd_iff + - MeasureTheory.Measure.compProd_apply_prod + - MeasureTheory.Measure.compProd_add_right + - MeasureTheory.Measure.set_lintegral_compProd + - MeasureTheory.Measure.compProd_congr + - MeasureTheory.Measure.dirac_unit_compProd + - MeasureTheory.Measure.compProd_zero_left + - MeasureTheory.Measure.integral_compProd + - MeasureTheory.Measure.set_integral_compProd + - MeasureTheory.Measure.lintegral_compProd + - MeasureTheory.Measure.dirac_unit_compProd_const + - MeasureTheory.Measure.compProd_zero_right + - MeasureTheory.Measure.compProd_add_left + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean + theorems: + - EllipticCurve.two_or_three_ne_zero + - "EllipticCurve.coe_inv_variableChange_\u0394'" + - "WeierstrassCurve.variableChange_\u0394" + - EllipticCurve.ofJ0_j + - WeierstrassCurve.VariableChange.id_comp + - WeierstrassCurve.VariableChange.comp_id + - WeierstrassCurve.VariableChange.map_injective + - "WeierstrassCurve.variableChange_b\u2088" + - "WeierstrassCurve.variableChange_b\u2084" + - WeierstrassCurve.VariableChange.comp_assoc + - WeierstrassCurve.twoTorsionPolynomial_disc_isUnit + - WeierstrassCurve.VariableChange.comp_map + - "WeierstrassCurve.ofJ1728_c\u2084" + - WeierstrassCurve.map_injective + - "WeierstrassCurve.variableChange_b\u2082" + - EllipticCurve.ofJ_1728_of_two_eq_zero + - EllipticCurve.variableChange_id + - EllipticCurve.ofJ1728_j + - "WeierstrassCurve.map_c\u2086" + - EllipticCurve.ofJ_1728_of_two_ne_zero + - EllipticCurve.ofJ'_j + - EllipticCurve.ofJ_1728_of_three_eq_zero + - EllipticCurve.ofJ_ne_0_ne_1728 + - WeierstrassCurve.b_relation + - "WeierstrassCurve.variableChange_b\u2086" + - EllipticCurve.map_injective + - "WeierstrassCurve.map_c\u2084" + - EllipticCurve.variableChange_comp + - WeierstrassCurve.map_variableChange + - "WeierstrassCurve.ofJ0_\u0394" + - "WeierstrassCurve.ofJ_\u0394" + - EllipticCurve.map_j + - WeierstrassCurve.twoTorsionPolynomial_disc + - "WeierstrassCurve.map_\u0394" + - WeierstrassCurve.VariableChange.id_map + - "WeierstrassCurve.ofJ1728_\u0394" + - EllipticCurve.ofJ_j + - "WeierstrassCurve.variableChange_c\u2086" + - WeierstrassCurve.c_relation + - EllipticCurve.ofJ_0_of_three_ne_zero + - EllipticCurve.variableChange_j + - "WeierstrassCurve.variableChange_c\u2084" + - EllipticCurve.ofJ_0_of_three_eq_zero + - "WeierstrassCurve.map_b\u2084" + - "WeierstrassCurve.map_b\u2086" + - WeierstrassCurve.VariableChange.comp_left_inv + - "WeierstrassCurve.ofJ0_c\u2084" + - WeierstrassCurve.variableChange_comp + - "WeierstrassCurve.ofJ_c\u2084" + - WeierstrassCurve.variableChange_id + - "WeierstrassCurve.map_b\u2082" + - "WeierstrassCurve.map_b\u2088" + - path: Mathlib/Data/Set/Pointwise/Basic.lean + theorems: + - Set.inv_eq_empty + - Set.one_mem_div_iff + - Set.image_mul_right + - Set.inv_range + - Set.preimage_div + - Group.card_pow_eq_card_pow_card_univ_aux + - Set.div_zero_subset + - Set.image_mul_left + - Set.Nonempty.mul_zero + - Set.univ_div_univ + - Set.isUnit_iff + - Set.zero_div_subset + - Set.image_mul_left' + - Set.preimage_mul_right_one' + - Set.image_op_inv + - Set.mul_zero_subset + - Set.preimage_mul_left_one + - Set.mul_eq_one_iff + - Set.inv_insert + - Set.preimage_mul_right_one + - Set.Nonempty.zero_div + - Set.inv_subset + - Set.preimage_mul_left_singleton + - Set.preimage_mul_right_singleton + - Set.image_mul_right' + - Set.preimage_mul_preimage_subset + - Set.mul_subset_range + - Set.isUnit_iff_singleton + - Set.pow_subset_pow_of_one_mem + - Set.empty_pow + - Set.Nonempty.div_zero + - Set.preimage_div_preimage_subset + - Set.zero_mul_subset + - Set.preimage_mul_left_one' + - Set.preimage_mul + - Set.univ_pow + - Set.Nonempty.zero_mul + - Set.nsmul_univ + - Set.inv_singleton + - Set.inv_mem_inv + - Set.pow_subset_pow + - path: Mathlib/Analysis/InnerProductSpace/Adjoint.lean + theorems: + - ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right + - LinearMap.eq_adjoint_iff_basis_left + - ContinuousLinearMap.isAdjointPair_inner + - Submodule.adjoint_orthogonalProjection + - LinearMap.isSymmetric_adjoint_mul_self + - ContinuousLinearMap.adjoint_id + - LinearMap.eq_adjoint_iff_basis + - ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left + - LinearMap.toMatrix_adjoint + - ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_right + - ContinuousLinearMap.adjointAux_inner_left + - Matrix.toLin_conjTranspose + - ContinuousLinearMap.inner_map_map_iff_adjoint_comp_self + - LinearMap.adjoint_inner_left + - LinearMap.adjoint_inner_right + - IsSelfAdjoint.adjoint_conj + - IsSelfAdjoint.conj_orthogonalProjection + - ContinuousLinearMap.adjoint_comp + - IsSelfAdjoint.conj_adjoint + - LinearMap.adjoint_comp + - IsSelfAdjoint.isSymmetric + - LinearMap.im_inner_adjoint_mul_self_eq_zero + - ContinuousLinearMap.adjointAux_norm + - Submodule.adjoint_subtypeL + - LinearMap.re_inner_adjoint_mul_self_nonneg + - ContinuousLinearMap.eq_adjoint_iff + - ContinuousLinearMap.norm_map_iff_adjoint_comp_self + - LinearMap.eq_adjoint_iff + - ContinuousLinearMap.adjointAux_adjointAux + - LinearMap.IsSymmetric.isSelfAdjoint + - LinearMap.eq_adjoint_iff_basis_right + - LinearMap.isSymmetric_iff_isSelfAdjoint + - LinearMap.adjoint_adjoint + - ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_left + - LinearMap.isAdjointPair_inner + - ContinuousLinearMap.adjointAux_inner_right + - ContinuousLinearMap.norm_adjoint_comp_self + - path: Mathlib/Topology/Basic.lean + theorems: + - accPt_iff_frequently + - isOpen_iff_ultrafilter + - closure_eq_compl_interior_compl + - Dense.interior_compl + - Set.Finite.closure_sUnion + - continuous_iff_isClosed + - frontier_union_subset + - eventuallyEq_zero_nhds + - subset_interior_iff_isOpen + - frontier_inter_subset + - frontier_eq_inter_compl_interior + - DenseRange.comp + - TopologicalSpace.ext_iff + - isClosed_frontier + - interior_iInter_of_finite + - interior_union_isClosed_of_interior_empty + - isOpen_singleton_iff_punctured_nhds + - diff_subset_closure_iff + - TopologicalSpace.ext_iff_isClosed + - isClosed_imp + - mapClusterPt_iff + - not_mem_closure_iff_nhdsWithin_eq_bot + - ClusterPt.of_nhds_le + - mem_closure_iff_comap_neBot + - isOpen_iff_nhds + - closure_iUnion_of_finite + - self_diff_frontier + - Filter.HasBasis.lift'_interior_eq_self + - clusterPt_iff_not_disjoint + - nhds_def' + - mem_closure_iff_nhds_basis' + - dense_closure + - interior_subset_iff + - mapClusterPt_iff_ultrafilter + - mem_closure_iff_frequently + - mem_closure_iff_nhds_basis + - isClosed_iff_frequently + - isClosed_compl_iff + - eventually_nhds_iff + - acc_iff_cluster + - IsClosed.inter + - Set.Finite.closure_biUnion + - Set.Finite.interior_biInter + - IsOpen.inter_closure + - IsOpen.frontier_eq + - le_nhds_iff + - compl_frontier_eq_union_interior + - tendsto_inf_principal_nhds_iff_of_forall_eq + - dense_compl_singleton + - clusterPt_lift'_closure_iff + - closure_union + - Function.Surjective.denseRange + - closure_diff + - exists_open_set_nhds' + - Dense.open_subset_closure_inter + - clusterPt_iff_ultrafilter + - nhds_basis_closeds + - IsClosed.interior_union_right + - isOpen_setOf_eventually_nhds + - subset_interior_iff_nhds + - nhds_le_of_le + - dense_iff_inter_open + - all_mem_nhds + - Continuous.range_subset_closure_image_dense + - Continuous.closure_preimage_subset + - Filter.lift'_closure_eq_bot + - isClosed_iff_ultrafilter + - TopologicalSpace.ext_iff_nhds + - interior_compl + - interior_eq_nhds + - IsOpen.inter_frontier_eq + - dense_compl_singleton_iff_not_open + - IsOpen.union + - IsClosed.union + - clusterPt_iff_lift'_closure + - DenseRange.subset_closure_image_preimage_of_isOpen + - mem_closure_iff_nhds' + - frontier_univ + - isClosed_of_closure_subset + - interior_frontier + - continuousAt_congr + - isOpen_empty + - mem_closure_of_mem_closure_union + - acc_principal_iff_cluster + - IsOpen.closure_inter + - closure_compl + - interior_sInter_subset + - clusterPt_principal_iff_frequently + - frontier_compl + - mem_interior + - isOpen_iff_forall_mem_open + - accPt_iff_nhds + - nhds_basis_opens + - isClosed_iff_nhds + - mem_closure_iff_ultrafilter + - isClosed_sInter + - frequently_frequently_nhds + - isOpen_iff_of_cover + - disjoint_interior_frontier + - isClosed_iUnion_of_finite + - isOpen_singleton_iff_nhds_eq_pure + - frontier_eq_closure_inter_closure + - ContinuousAt.comp_of_eq + - continuous_iff_ultrafilter + - Dense.inter_of_isOpen_left + - frontier_empty + - isClosed_setOf_clusterPt + - isOpen_const + - clusterPt_iff_forall_mem_closure + - tendsto_atTop_nhds + - closure_nonempty_iff + - ClusterPt.of_le_nhds + - IsClosed.frontier_eq + - Dense.denseRange_val + - Set.Finite.isClosed_biUnion + - mapClusterPt_of_comp + - Set.MapsTo.closure + - isClosed_iff_forall_filter + - Filter.Tendsto.lift'_closure + - isClosed_iff_clusterPt + - interior_eq_nhds' + - Set.Finite.isOpen_sInter + - clusterPt_iff_lift'_closure' + - Set.Finite.interior_sInter + - path: Mathlib/Algebra/ModEq.lean + theorems: + - AddCommGroup.sub_modEq_zero + - AddCommGroup.ModEq.nsmul + - AddCommGroup.not_modEq_iff_ne_add_zsmul + - AddCommGroup.modEq_sub_iff_add_modEq + - AddCommGroup.ModEq.trans + - AddCommGroup.modEq_neg + - AddCommGroup.modEq_iff_eq_mod_zmultiples + - AddCommGroup.modEq_comm + - AddCommGroup.self_modEq_zero + - AddCommGroup.ModEq.of_nsmul + - AddCommGroup.add_nsmul_modEq + - AddCommGroup.add_modEq_right + - AddCommGroup.modEq_iff_eq_add_zsmul + - AddCommGroup.ModEq.sub_iff_right + - AddCommGroup.zsmul_add_modEq + - AddCommGroup.modEq_sub_iff_add_modEq' + - AddCommGroup.ModEq.add_iff_left + - AddCommGroup.div_modEq_div + - AddCommGroup.zsmul_modEq_zero + - AddCommGroup.natCast_modEq_natCast + - AddCommGroup.nsmul_add_modEq + - AddCommGroup.add_zsmul_modEq + - AddCommGroup.modEq_zero + - AddCommGroup.nsmul_modEq_nsmul + - AddCommGroup.intCast_modEq_intCast + - AddCommGroup.ModEq.zsmul + - AddCommGroup.add_modEq_left + - AddCommGroup.ModEq.of_zsmul + - AddCommGroup.neg_modEq_neg + - AddCommGroup.modEq_iff_int_modEq + - AddCommGroup.zsmul_modEq_zsmul + - AddCommGroup.ModEq.add_iff_right + - AddCommGroup.ModEq.sub_iff_left + - path: Mathlib/SetTheory/Surreal/Basic.lean + theorems: + - SetTheory.PGame.Numeric.left_lt_right + - SetTheory.PGame.numeric_toPGame + - SetTheory.PGame.numeric_def + - Surreal.bddBelow_of_small + - SetTheory.PGame.lt_def + - SetTheory.PGame.Numeric.add + - SetTheory.PGame.lf_asymm + - SetTheory.PGame.Numeric.moveRight + - SetTheory.PGame.lt_iff_exists_le + - SetTheory.PGame.Numeric.moveLeft + - Surreal.bddBelow_range_of_small + - SetTheory.PGame.Relabelling.numeric_imp + - SetTheory.PGame.le_iff_forall_lt + - Surreal.bddAbove_range_of_small + - Surreal.bddAbove_of_small + - path: Mathlib/ModelTheory/Basic.lean + theorems: + - "FirstOrder.Sequence\u2082.sum_card" + - FirstOrder.Language.card_sum + - FirstOrder.Language.Embedding.ofInjective_toHom + - FirstOrder.Language.Equiv.comp_right_injective + - FirstOrder.Language.card_relations_sum + - FirstOrder.Language.card_eq_card_functions_add_card_relations + - FirstOrder.Language.Equiv.symm_comp_self + - FirstOrder.Language.Equiv.injective_toEmbedding + - FirstOrder.Language.Equiv.symm_comp_self_toHom + - FirstOrder.Language.Equiv.self_comp_symm + - FirstOrder.Language.card_functions_sum + - FirstOrder.Language.Equiv.symm_comp_self_toEmbedding + - "FirstOrder.Sequence\u2082.lift_mk" + - FirstOrder.Language.Equiv.self_comp_symm_toHom + - FirstOrder.Language.Embedding.toHom_comp_injective + - Equiv.toFun_inducedStructureEquiv_Symm + - FirstOrder.Language.Equiv.injective_comp + - FirstOrder.Language.Embedding.comp_injective + - FirstOrder.Language.empty_card + - FirstOrder.Language.Equiv.refl_apply + - "FirstOrder.Language.card_mk\u2082" + - FirstOrder.Language.Equiv.self_comp_symm_toEmbedding + - path: Mathlib/Analysis/Convex/Jensen.lean + theorems: + - StrictConcaveOn.eq_of_map_sum_eq + - ConvexOn.map_sum_le + - ConvexOn.exists_ge_of_mem_convexHull + - ConvexOn.exists_ge_of_centerMass + - StrictConcaveOn.map_sum_eq_iff + - ConvexOn.map_centerMass_le + - StrictConvexOn.eq_of_le_map_sum + - le_sup_of_mem_convexHull + - ConvexOn.map_add_sum_le + - ConvexOn.le_max_of_mem_Icc + - StrictConvexOn.map_sum_eq_iff' + - StrictConvexOn.map_sum_eq_iff + - ConvexOn.le_max_of_mem_segment + - StrictConvexOn.map_sum_lt + - ConcaveOn.min_le_of_mem_segment + - path: Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean + theorems: + - "CategoryTheory.Pretriangulated.contractible_distinguished\u2082" + - "CategoryTheory.Pretriangulated.Triangle.coyoneda_exact\u2082" + - CategoryTheory.Pretriangulated.exists_iso_binaryBiproduct_of_distTriang + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2081_iff" + - "CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism\u2082" + - "CategoryTheory.Pretriangulated.contractible_distinguished\u2081" + - CategoryTheory.Pretriangulated.productTriangle_distinguished + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2083_iff_isIso\u2081" + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2082_of_isZero\u2081\u2083" + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2082_iff" + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2083_iff" + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2081_of_isZero\u2082\u2083" + - "CategoryTheory.Pretriangulated.Triangle.mor\u2081_eq_zero_iff_epi\u2083" + - CategoryTheory.Pretriangulated.Triangle.shift_distinguished + - "CategoryTheory.Pretriangulated.comp_distTriang_mor_zero\u2081\u2082" + - "CategoryTheory.Pretriangulated.Triangle.coyoneda_exact\u2081" + - "CategoryTheory.Pretriangulated.Triangle.yoneda_exact\u2082" + - "CategoryTheory.Pretriangulated.Triangle.mor\u2082_eq_zero_iff_epi\u2081" + - "CategoryTheory.Pretriangulated.isIso\u2083_of_isIso\u2081\u2082" + - CategoryTheory.Pretriangulated.binaryBiproductTriangle_distinguished + - CategoryTheory.Pretriangulated.exists_iso_of_arrow_iso + - "CategoryTheory.Pretriangulated.distinguished_cocone_triangle\u2082" + - "CategoryTheory.Pretriangulated.Triangle.mor\u2081_eq_zero_iff_mono\u2082" + - "CategoryTheory.Pretriangulated.isIso\u2081_of_isIso\u2082\u2083" + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2081_iff_isIso\u2082" + - "CategoryTheory.Pretriangulated.distinguished_cocone_triangle\u2081" + - "CategoryTheory.Pretriangulated.Triangle.isZero\u2083_of_isZero\u2081\u2082" + - "CategoryTheory.Pretriangulated.isIso\u2082_of_isIso\u2081\u2083" + - "CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism\u2081" + - "CategoryTheory.Pretriangulated.Triangle.mor\u2083_eq_zero_iff_epi\u2082" + - "CategoryTheory.Pretriangulated.comp_distTriang_mor_zero\u2083\u2081" + - path: Mathlib/Logic/Small/Basic.lean + theorems: + - small_of_injective_of_exists + - path: Mathlib/CategoryTheory/Limits/Shapes/Images.lean + theorems: + - "CategoryTheory.Limits.image.isoStrongEpiMono_hom_comp_\u03B9" + - CategoryTheory.Limits.epi_image_of_epi + - "CategoryTheory.Limits.imageMonoIsoSource_inv_\u03B9" + - CategoryTheory.Limits.image.map_id + - CategoryTheory.Limits.imageMonoIsoSource_hom_self + - "CategoryTheory.Limits.image.preComp_\u03B9" + - CategoryTheory.Limits.strongEpi_of_strongEpiMonoFactorisation + - CategoryTheory.Limits.image.eq_fac + - CategoryTheory.Limits.image.ext + - CategoryTheory.Limits.ImageMap.map_uniq_aux + - CategoryTheory.Functor.hasStrongEpiMonoFactorisations_imp_of_isEquivalence + - CategoryTheory.Limits.ImageMap.mk.injEq' + - CategoryTheory.Limits.image.map_comp + - CategoryTheory.Limits.ImageMap.map_uniq + - CategoryTheory.Limits.image.preComp_comp + - CategoryTheory.Limits.epi_of_epi_image + - CategoryTheory.Limits.image.factor_map + - CategoryTheory.Limits.IsImage.isoExt_inv_m + - CategoryTheory.Limits.HasImage.uniq + - "CategoryTheory.Limits.image.compIso_hom_comp_image_\u03B9" + - CategoryTheory.Limits.MonoFactorisation.ext + - CategoryTheory.Limits.IsImage.e_isoExt_hom + - CategoryTheory.Limits.IsImage.e_isoExt_inv + - "CategoryTheory.Limits.image.compIso_inv_comp_image_\u03B9" + - CategoryTheory.Limits.ImageMap.factor_map + - CategoryTheory.Limits.IsImage.fac_lift + - CategoryTheory.Limits.IsImage.isoExt_hom_m + - path: Mathlib/Combinatorics/SimpleGraph/Clique.lean + theorems: + - SimpleGraph.isNClique_singleton + - SimpleGraph.IsNClique.map + - SimpleGraph.isNClique_zero + - SimpleGraph.isClique_iff_induce_eq + - SimpleGraph.isNClique_one + - SimpleGraph.CliqueFree.replaceVertex + - SimpleGraph.card_cliqueFinset_le + - SimpleGraph.cliqueSet_eq_empty_iff + - SimpleGraph.cliqueFreeOn_singleton + - SimpleGraph.isNClique_bot_iff + - SimpleGraph.cliqueSet_map_of_equiv + - SimpleGraph.isClique_singleton + - SimpleGraph.cliqueFinset_eq_empty_iff + - SimpleGraph.cliqueFree_iff + - SimpleGraph.cliqueFree_completeMultipartiteGraph + - SimpleGraph.cliqueFreeOn_univ + - SimpleGraph.isClique_empty + - SimpleGraph.cliqueSet_one + - SimpleGraph.IsNClique.mono + - SimpleGraph.cliqueFinset_map + - SimpleGraph.CliqueFree.comap + - SimpleGraph.cliqueSet_map + - SimpleGraph.IsNClique.insert + - SimpleGraph.cliqueFreeOn_two + - SimpleGraph.not_cliqueFree_of_top_embedding + - SimpleGraph.cliqueFree_two + - SimpleGraph.CliqueFreeOn.mono + - SimpleGraph.cliqueFinset_map_of_equiv + - SimpleGraph.CliqueFree.mono + - SimpleGraph.cliqueFreeOn_empty + - SimpleGraph.is3Clique_iff + - SimpleGraph.is3Clique_triple_iff + - SimpleGraph.isNClique_empty + - SimpleGraph.not_cliqueFree_card_of_top_embedding + - SimpleGraph.CliqueFreeOn.of_succ + - SimpleGraph.IsClique.map + - SimpleGraph.CliqueFree.sup_edge + - SimpleGraph.cliqueSet_zero + - SimpleGraph.cliqueFree_bot + - SimpleGraph.cliqueFree_of_card_lt + - path: Mathlib/Data/Set/Pointwise/Interval.lean + theorems: + - Set.image_const_add_uIcc + - Set.preimage_mul_const_Ioc_of_neg + - Set.preimage_add_const_Ico + - Set.preimage_sub_const_uIcc + - Set.preimage_const_add_Icc + - Set.image_const_sub_Ioi + - Set.preimage_sub_const_Iic + - Set.image_sub_const_Icc + - Set.preimage_const_add_Ico + - Set.image_sub_const_Ico + - Set.preimage_mul_const_Ico_of_neg + - Set.preimage_mul_const_uIcc + - Set.image_const_sub_Iic + - Set.image_const_sub_Iio + - Set.Ioc_mul_Ico_subset' + - Set.image_affine_Icc' + - Set.Ioi_mul_Ici_subset' + - Set.image_neg_Ico + - Set.image_neg_Ici + - Set.preimage_const_sub_Ioc + - Set.Icc_mul_Ico_subset' + - Set.image_sub_const_uIcc + - Set.preimage_const_add_Ioo + - Set.image_neg_Iio + - Set.preimage_mul_const_Ioo_of_neg + - Set.image_const_add_Iio + - Set.preimage_const_mul_Ioo_of_neg + - Set.preimage_div_const_uIcc + - Set.preimage_add_const_uIcc + - Set.image_add_const_uIcc + - Set.preimage_const_mul_Iio_of_neg + - Set.image_neg_Icc + - Set.image_const_mul_Ioi_zero + - Set.Iic_mul_Iio_subset' + - Set.preimage_const_mul_Ici_of_neg + - Set.Icc_mul_Icc_subset' + - Set.image_neg_Ioi + - Set.Ico_mul_Ioc_subset' + - Set.Ici_mul_Ioi_subset' + - Set.Iic_mul_Iic_subset' + - Set.image_neg_uIcc + - Set.image_sub_const_Ioc + - Set.image_const_add_Iic + - Set.preimage_sub_const_Ioo + - Set.image_const_sub_Ioc + - Set.preimage_neg_Icc + - Set.preimage_const_mul_Ioi_of_neg + - Set.preimage_sub_const_Ico + - Set.preimage_const_mul_Iic_of_neg + - Set.image_mul_right_Ioo + - Set.inv_Ioi + - Set.preimage_mul_const_Ioo + - Set.preimage_add_const_Ioc + - Set.preimage_mul_const_Ioc + - Set.abs_sub_le_of_uIcc_subset_uIcc + - Set.preimage_neg_Ico + - Set.preimage_const_mul_Icc + - Set.Iio_mul_Iic_subset' + - Set.preimage_const_sub_Ioo + - Set.image_const_sub_uIcc + - Set.preimage_sub_const_Ioc + - Set.image_mul_left_Ioo + - Set.preimage_const_mul_Ico + - Set.preimage_const_sub_uIcc + - Set.preimage_const_mul_Ico_of_neg + - Set.image_sub_const_Iic + - Set.image_const_sub_Icc + - Set.image_div_const_uIcc + - Set.preimage_const_sub_Icc + - Set.preimage_const_mul_Ioc_of_neg + - Set.preimage_mul_const_Icc + - Set.Ico_mul_Icc_subset' + - Set.preimage_const_add_Ioc + - Set.image_sub_const_Ioo + - Set.preimage_const_mul_uIcc + - Set.image_neg_Ioc + - Set.inv_Ioo_0_left + - Set.preimage_const_mul_Icc_of_neg + - Set.preimage_mul_const_Ico + - Set.preimage_sub_const_Icc + - Set.preimage_sub_const_Ioi + - Set.image_add_const_Iic + - Set.image_const_sub_Ici + - Set.image_mul_right_Icc' + - Set.preimage_neg_uIcc + - Set.preimage_add_const_Ioo + - Set.image_mul_left_Icc + - Set.image_const_mul_uIcc + - Set.image_neg_Iic + - Set.preimage_mul_const_Icc_of_neg + - Set.Ici_mul_Ici_subset' + - Set.preimage_const_sub_Ico + - Set.preimage_sub_const_Iio + - Set.preimage_sub_const_Ici + - Set.preimage_neg_Ioo + - Set.preimage_const_mul_Ioo + - Set.image_const_sub_Ioo + - Set.image_sub_const_Iio + - Set.image_mul_left_Icc' + - Set.preimage_const_add_uIcc + - Set.preimage_const_mul_Ioc + - Set.image_sub_const_Ioi + - Set.image_add_const_Iio + - Set.preimage_add_const_Icc + - Set.image_const_sub_Ico + - Set.image_mul_const_uIcc + - path: Mathlib/Data/Polynomial/Module/Basic.lean + theorems: + - PolynomialModule.smul_single_apply + - PolynomialModule.smul_apply + - Module.AEval.of_symm_X_smul + - Module.AEval.comapSubmodule_mapSubmodule + - PolynomialModule.eval_smul + - Module.AEval.mapSubmodule_comapSubmodule + - PolynomialModule.eval_map + - Module.AEval.annihilator_top_eq_ker_aeval + - PolynomialModule.monomial_smul_apply + - PolynomialModule.map_smul + - PolynomialModule.monomial_smul_single + - Module.AEval.C_smul + - PolynomialModule.comp_single + - PolynomialModule.comp_eval + - Module.AEval.comapSubmodule_le_comap + - PolynomialModule.smul_def + - Module.AEval.X_smul_of + - Module.AEval.annihilator_eq_ker_aeval + - Module.AEval.injective_comapSubmodule + - path: Mathlib/Logic/Basic.lean + theorems: + - eq_iff_eq_cancel_left + - eqRec_heq' + - rec_heq_of_heq + - ite_and + - exists_or_forall_not + - ULift.down_inj + - xor_self + - ite_prop_iff_and + - "apply_dite\u2082" + - Prop.forall_iff + - dite_ne_left_iff + - forall_or_exists_not + - "exists\u2082_comm" + - dite_apply + - and_symm_right + - xor_iff_iff_not + - Decidable.and_forall_ne + - xor_iff_not_iff' + - forall_apply_eq_imp_iff' + - xor_true + - if_true_right + - forall_eq_apply_imp_iff' + - beq_ext + - "forall\u2083_true_iff" + - and_symm_left + - if_false_left + - lawful_beq_subsingleton + - ULift.down_injective + - heq_rec_iff_heq + - bex_eq_left + - xor_not_not + - if_false_right + - ite_prop_iff_or + - dite_eq_iff + - dite_dite_comm + - "ExistsUnique.intro\u2082" + - beq_eq_decide + - exists_unique_eq' + - xor_not_left + - ite_ne_left_iff + - exists_unique_prop + - exists_unique_eq + - dite_ne_right_iff + - congr_heq + - bex_imp + - not_xor + - forall_imp_iff_exists_imp + - exists_unique_const + - rec_heq_iff_heq + - ite_eq_iff + - dite_prop_iff_or + - xor_not_right + - ite_ne_right_iff + - "ExistsUnique.unique\u2082" + - Prop.exists_iff + - xor_comm + - Decidable.forall_or_right + - PLift.down_injective + - if_true_left + - dite_prop_iff_and + - eq_iff_eq_cancel_right + - PLift.down_inj + - "forall\u2082_true_iff" + - bex_or_left + - Eq.rec_eq_cast + - eq_true_eq_id + - exists_eq_right' + - "ExistsUnique.elim\u2082" + - imp_and_neg_imp_iff + - path: Mathlib/Data/Finset/Sups.lean + theorems: + - Finset.exists_compls_iff + - Finset.map_sups + - Finset.Nonempty.of_disjSups_left + - Finset.disjSups_union_left + - Finset.filter_infs_le + - Finset.mem_disjSups + - Finset.diffs_compls_eq_infs + - Finset.powerset_inter + - Finset.disjSups_left_comm + - Finset.sups_eq_self + - Finset.disjSups_inter_subset_left + - Finset.compls_infs + - Finset.univ_infs_univ + - Finset.disjSups_right_comm + - Finset.disjSups_inter_subset_right + - Set.Sized.compls + - Finset.disjSups_empty_left + - Finset.disjSups_assoc + - Finset.compls_subset_iff + - Finset.compls_infs_eq_diffs + - Finset.filter_sups_le + - Finset.mem_compls + - Finset.compls_compls + - Finset.compls_sups + - Finset.Nonempty.of_disjSups_right + - Finset.powerset_sups_powerset_self + - Finset.disjSups_singleton + - Finset.disjSups_empty_right + - Finset.infs_self + - Finset.disjSups_subset_sups + - Finset.univ_sups_univ + - Finset.disjSups_comm + - Finset.disjSups_union_right + - Finset.powerset_union + - Finset.compls_univ + - Finset.disjSups_disjSups_disjSups_comm + - Finset.infs_compls_eq_diffs + - Finset.map_infs + - Finset.image_compl + - path: Mathlib/Algebra/Function/Support.lean + theorems: + - Pi.mulSupport_mulSingle + - Pi.mulSupport_mulSingle_one + - Function.mulSupport_const + - Function.mulSupport_extend_one_subset + - Pi.mulSupport_mulSingle_of_ne + - Set.image_inter_mulSupport_eq + - Function.mulSupport_prod_mk + - Function.mulSupport_along_fiber_subset + - Function.mulSupport_eq_iff + - Function.mulSupport_pow + - Function.support_mul_subset_right + - Function.mulSupport_eq_empty_iff + - Function.disjoint_mulSupport_iff + - Function.range_eq_image_or_of_mulSupport_subset + - Function.support_pow + - Function.mulSupport_mul_inv + - Pi.mulSupport_mulSingle_disjoint + - Function.mulSupport_disjoint_iff + - Function.support_div + - Function.mulSupport_comp_eq_of_range_subset + - Function.support_mul_subset_left + - Function.mulSupport_one_sub' + - Function.support_mul + - Function.mulSupport_prod_mk' + - Function.mulSupport_nonempty_iff + - path: Mathlib/Algebra/Module/LocalizedModule.lean + theorems: + - LocalizedModule.subsingleton + - LocalizedModule.induction_on + - IsLocalizedModule.fromLocalizedModule'_smul + - IsLocalizedModule.mk'_surjective + - IsLocalizedModule.ringHom_ext + - LocalizedModule.mk_smul_mk + - LocalizedModule.lift'_smul + - IsLocalizedModule.fromLocalizedModule.inj + - IsLocalizedModule.mk'_smul + - IsLocalizedModule.of_linearEquiv + - LocalizedModule.add_zero' + - IsLocalizedModule.mk'_smul_mk' + - IsLocalizedModule.mk'_eq_iff + - LocalizedModule.lift_unique + - IsLocalizedModule.mk'_sub_mk' + - IsLocalizedModule.mk'_cancel + - IsLocalizedModule.lift_comp + - IsLocalizedModule.mk'_cancel_right + - IsLocalizedModule.mk'_one + - LocalizedModule.lift_comp + - LocalizedModule.algebraMap_mk' + - isLocalizedModule_id + - IsLocalizedModule.mkOfAlgebra + - isLocalizedModule_iff_isLocalization + - IsLocalizedModule.mk'_add_mk' + - LocalizedModule.smul'_mk + - LocalizedModule.one_smul_aux + - LocalizedModule.mul_smul' + - LocalizedModule.smul_zero_aux + - LocalizedModule.divBy_mul_by + - IsLocalizedModule.mk_eq_mk' + - IsLocalizedModule.mk'_mul_mk'_of_map_mul + - LocalizedModule.mul_smul_aux + - LocalizedModule.mk_cancel_common_left + - IsLocalizedModule.eq_iff_exists + - IsLocalizedModule.mk'_cancel_left + - LocalizedModule.algebraMap_mk + - LocalizedModule.lift'_add + - "LocalizedModule.liftOn\u2082_mk" + - LocalizedModule.add_comm' + - IsLocalizedModule.mk'_zero + - LocalizedModule.r.isEquiv + - IsLocalizedModule.iso_symm_apply' + - LocalizedModule.liftOn_mk + - LocalizedModule.mul_by_divBy + - IsLocalizedModule.iso_symm_comp + - LocalizedModule.add_smul_aux + - LocalizedModule.mk_cancel + - LocalizedModule.zero_mk + - IsLocalizedModule.mk'_eq_mk'_iff + - LocalizedModule.zero_smul_aux + - IsLocalizedModule.lift_unique + - IsLocalizedModule.fromLocalizedModule.surj + - IsLocalizedModule.mk'_cancel' + - LocalizedModule.smul'_mul + - IsLocalizedModule.fromLocalizedModule'_add + - LocalizedModule.smul_add_aux + - LocalizedModule.add_assoc' + - IsLocalizedModule.mk'_add + - IsLocalizedModule.mk'_eq_zero + - IsLocalizedModule.mk'_eq_zero' + - LocalizedModule.mk_cancel_common_right + - "LocalizedModule.induction_on\u2082" + - LocalizedModule.mk'_smul_mk + - isLocalizedModule_iff_isLocalization' + - LocalizedModule.zero_add' + - path: Mathlib/GroupTheory/OrderOfElement.lean + theorems: + - IsOfFinOrder.finite_powers + - Subgroup.orderOf_dvd_natCard + - injective_zpow_iff_not_isOfFinOrder + - pow_eq_one_iff_modEq + - Subgroup.pow_index_mem + - not_isOfFinOrder_of_injective_pow + - orderOf_abs_ne_one + - orderOf_eq_orderOf_iff + - orderOf_map_dvd + - Submonoid.isOfFinOrder_coe + - IsOfFinOrder.orderOf_pow + - IsOfFinOrder.mem_powers_iff_mem_zpowers + - Commute.orderOf_mul_eq_mul_orderOf_of_coprime + - orderOf_pow_orderOf_div + - IsOfFinOrder.eq_one + - injective_pow_iff_not_isOfFinOrder + - pow_card_eq_one + - infinite_not_isOfFinOrder + - exists_zpow_eq_one + - isPeriodicPt_mul_iff_pow_eq_one + - orderOf_one + - IsOfFinOrder.mono + - finEquivZPowers_symm_apply + - pow_mod_natCard + - orderOf_inv + - orderOf_eq_prime_pow + - orderOf_pow_of_dvd + - orderOf_eq_of_pow_and_pow_div_prime + - zpow_eq_one_iff_modEq + - orderOf_le_of_pow_eq_one + - orderOf_dvd_card + - finite_powers + - inf_eq_bot_of_coprime + - sum_card_orderOf_eq_card_pow_eq_one + - zpow_eq_zpow_iff_modEq + - orderOf_eq_iff + - orderOf_pos_iff + - zpow_mod_orderOf + - pow_eq_pow_iff_modEq + - MonoidHom.isOfFinOrder + - Commute.orderOf_mul_eq_right_of_forall_prime_mul_dvd + - exists_pow_eq_self_of_coprime + - zpow_mod_card + - LinearOrderedRing.orderOf_le_two + - zpow_pow_orderOf + - orderOf_dvd_iff_pow_eq_one + - isOfFinOrder_iff_pow_eq_one + - exists_orderOf_eq_prime_pow_iff + - pow_inj_iff_of_orderOf_eq_zero + - pow_mod_card + - IsOfFinOrder.apply + - powersEquivPowers_apply + - Commute.orderOf_dvd_lcm_mul + - zpow_mod_natCard + - zpowersEquivZPowers_apply + - Submonoid.orderOf_le_card + - isOfFinOrder_of_finite + - Nat.card_submonoidPowers + - orderOf_eq_zero + - orderOf_dvd_natCard + - image_range_orderOf + - pow_mod_orderOf + - pow_gcd_card_eq_one_iff + - orderOf_pow_dvd + - smul_eq_self_of_mem_zpowers + - IsOfFinOrder.eq_neg_one + - IsOfFinOrder.pow + - IsOfFinOrder.natCard_powers_le_orderOf + - card_zpowers_le + - orderOf_pow' + - finEquivPowers_symm_apply + - orderOf_dvd_iff_zpow_eq_one + - Nat.Coprime.orderOf_pow + - orderOf_eq_zero_iff' + - Function.Injective.isOfFinOrder_iff + - pow_orderOf_eq_one + - orderOf_dvd_of_mem_zpowers + - IsOfFinOrder.of_mem_zpowers + - isOfFinOrder_inv_iff + - IsOfFinOrder.prod_mk + - orderOf_injective + - orderOf_eq_one_iff + - path: Mathlib/NumberTheory/Padics/PadicVal.lean + theorems: + - padicValRat.of_int + - padicValNat_factorial_mul_add + - padicValNat.padicValNat_eq_maxPowDiv + - range_pow_padicValNat_subset_divisors' + - padicValNat.pow + - padicValNat_le_nat_log + - padicValNat_dvd_iff + - padicValRat.div + - padicValNat_def' + - padicValInt_dvd_iff + - padicValNat_mul_div_factorial + - padicValInt.self + - padicValRat.lt_sum_of_lt + - Nat.log_ne_padicValNat_succ + - Nat.max_log_padicValNat_succ_eq_log_succ + - padicValNat.self + - padicValRat.defn + - padicValRat.neg + - dvd_of_one_le_padicValNat + - padicValRat.add_eq_of_lt + - padicValNat.div + - padicValNat.one + - padicValRat.inv + - padicValNat.zero + - padicValRat.self_pow_inv + - padicValRat.padicValRat_le_padicValRat_iff + - padicValRat.of_int_multiplicity + - padicValRat.finite_int_prime_iff + - padicValRat.le_padicValRat_add_of_le + - range_pow_padicValNat_subset_divisors + - padicValNat_factorial_mul + - nat_log_eq_padicValNat_iff + - padicValInt.of_ne_one_ne_zero + - padicValRat.mul + - padicValNat.div_pow + - padicValRat.zero + - padicValRat.add_eq_min + - padicValNat.maxPowDiv_eq_multiplicity_get + - padicValRat_of_nat + - padicValRat.min_le_padicValRat_add + - padicValNat.div' + - sub_one_mul_padicValNat_choose_eq_sub_sum_digits' + - padicValRat.pow + - padicValNat.maxPowDiv_eq_multiplicity + - pow_succ_padicValNat_not_dvd + - padicValNat_self + - padicValNat.prime_pow + - padicValRat.sum_pos_of_pos + - padicValInt.mul + - pow_padicValNat_dvd + - padicValRat.multiplicity_sub_multiplicity + - padicValRat.of_nat + - zero_le_padicValRat_of_nat + - sub_one_mul_padicValNat_factorial + - padicValNat.eq_zero_iff + - padicValRat.one + - padicValInt.one + - padicValRat.self + - sub_one_mul_padicValNat_choose_eq_sub_sum_digits + - padicValNat_dvd_iff_le + - padicValInt_mul_eq_succ + - padicValInt.of_nat + - padicValInt.eq_zero_of_not_dvd + - padicValInt.zero + - padicValInt_dvd + - padicValNat.div_of_dvd + - path: Mathlib/Analysis/BoxIntegral/Partition/Filter.lean + theorems: + - BoxIntegral.IntegrationParams.henstock_le_riemann + - BoxIntegral.IntegrationParams.rCond_of_bRiemann_eq_false + - BoxIntegral.IntegrationParams.tendsto_embedBox_toFilteriUnion_top + - BoxIntegral.IntegrationParams.henstock_le_mcShane + - BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion_top + - BoxIntegral.IntegrationParams.toFilteriUnion_congr + - BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion + - BoxIntegral.IntegrationParams.exists_memBaseSet_le_iUnion_eq + - BoxIntegral.IntegrationParams.MemBaseSet.unionComplToSubordinate + - BoxIntegral.IntegrationParams.MemBaseSet.exists_common_compl + - BoxIntegral.IntegrationParams.biUnionTagged_memBaseSet + - BoxIntegral.IntegrationParams.MemBaseSet.filter + - BoxIntegral.IntegrationParams.hasBasis_toFilter + - BoxIntegral.IntegrationParams.exists_memBaseSet_isPartition + - path: Mathlib/Algebra/Regular/Basic.lean + theorems: + - IsRightRegular.left_of_commute + - IsUnit.isRegular + - isLeftRegular_of_mul_eq_one + - IsRightRegular.mul_right_eq_zero_iff + - IsLeftRegular.of_mul + - IsLeftRegular.ne_zero + - IsRightRegular.of_mul + - IsLeftRegular.mul_left_eq_zero_iff + - isRightRegular_of_mul_eq_one + - not_isRightRegular_zero_iff + - isRegular_mul_and_mul_iff + - IsRightRegular.ne_zero + - isRegular_mul_iff + - not_isLeftRegular_zero_iff + - path: Mathlib/Analysis/InnerProductSpace/Positive.lean + theorems: + - ContinuousLinearMap.IsPositive.conj_orthogonalProjection + - ContinuousLinearMap.isPositive_iff_complex + - ContinuousLinearMap.IsPositive.orthogonalProjection_comp + - ContinuousLinearMap.IsPositive.add + - ContinuousLinearMap.IsPositive.adjoint_conj + - ContinuousLinearMap.IsPositive.inner_nonneg_right + - ContinuousLinearMap.IsPositive.conj_adjoint + - path: Mathlib/Analysis/NormedSpace/ProdLp.lean + theorems: + - WithLp.prod_aux_uniformity_eq + - WithLp.nnnorm_equiv_symm_fst + - WithLp.prod_antilipschitzWith_equiv_aux + - WithLp.prod_lipschitzWith_equiv_aux + - WithLp.prod_norm_sq_eq_of_L2 + - WithLp.edist_equiv_symm_fst + - WithLp.prod_nndist_eq_add + - WithLp.prod_edist_eq_of_L2 + - WithLp.prod_edist_eq_sup + - WithLp.prod_edist_self + - WithLp.prod_nndist_eq_of_L2 + - WithLp.prod_nndist_eq_sup + - WithLp.prod_edist_eq_card + - WithLp.prod_nnnorm_equiv + - WithLp.nndist_equiv_symm_fst + - WithLp.prod_nnnorm_eq_sup + - WithLp.prod_edist_comm + - WithLp.prod_norm_eq_of_nat + - WithLp.prod_nnnorm_eq_add + - WithLp.prod_dist_eq_card + - WithLp.prod_norm_eq_of_L2 + - WithLp.nnnorm_equiv_symm_snd + - WithLp.prod_dist_eq_of_L2 + - WithLp.prod_sup_edist_ne_top_aux + - WithLp.prod_norm_eq_card + - WithLp.prod_dist_eq_sup + - WithLp.prod_nnnorm_eq_of_L2 + - WithLp.nndist_equiv_symm_snd + - WithLp.prod_norm_eq_sup + - WithLp.edist_equiv_symm_snd + - WithLp.prod_infty_equiv_isometry + - path: Mathlib/Order/Filter/NAry.lean + theorems: + - "Filter.map_map\u2082_distrib" + - "Filter.map\u2082_left_identity" + - "Filter.map\u2082_sup_right" + - "Filter.map\u2082_right" + - "Filter.map\u2082_distrib_le_left" + - "Filter.map\u2082_pure" + - "Filter.map\u2082_left_comm" + - "Filter.map\u2082_left" + - "Filter.map_prod_eq_map\u2082" + - "Filter.map_map\u2082_antidistrib" + - "Filter.map\u2082_comm" + - "Filter.map\u2082_pure_left" + - "Filter.map\u2082_right_identity" + - "Filter.map\u2082_swap" + - "Filter.map\u2082_mk_eq_prod" + - "Filter.map\u2082_map_left" + - "Filter.map\u2082_neBot_iff" + - "Filter.map\u2082_sup_left" + - "Filter.map\u2082_map_right" + - "Filter.map\u2082_assoc" + - "Filter.map\u2082_right_comm" + - "Filter.map\u2082_pure_right" + - "Filter.map\u2082_distrib_le_right" + - "Filter.map\u2082_eq_bot_iff" + - path: Mathlib/CategoryTheory/Sites/Grothendieck.lean + theorems: + - CategoryTheory.GrothendieckTopology.arrow_stable + - CategoryTheory.GrothendieckTopology.arrow_trans + - CategoryTheory.GrothendieckTopology.covering_iff_covers_id + - CategoryTheory.GrothendieckTopology.arrow_max + - CategoryTheory.GrothendieckTopology.bot_covers + - CategoryTheory.GrothendieckTopology.top_covers + - CategoryTheory.GrothendieckTopology.arrow_intersect + - CategoryTheory.GrothendieckTopology.ext + - CategoryTheory.GrothendieckTopology.superset_covering + - CategoryTheory.GrothendieckTopology.isGLB_sInf + - CategoryTheory.GrothendieckTopology.intersection_covering + - path: Mathlib/RingTheory/Coprime/Basic.lean + theorems: + - IsCoprime.of_add_mul_right_left + - IsCoprime.intCast + - IsRelPrime.add_mul_left_left + - IsCoprime.of_mul_add_left_left + - IsCoprime.of_add_mul_right_right + - IsRelPrime.of_add_mul_left_right + - isCoprime_one_left + - IsCoprime.of_add_mul_left_right + - IsCoprime.mul_add_left_right + - IsCoprime.of_mul_add_right_right + - IsCoprime.of_mul_add_right_left + - isCoprime_one_right + - IsCoprime.of_mul_left_left + - IsCoprime.mul_right + - IsCoprime.add_mul_left_left + - IsCoprime.of_mul_left_right + - IsCoprime.add_mul_right_right + - IsCoprime.map + - isCoprime_zero_left + - IsCoprime.of_mul_right_left + - IsCoprime.ne_zero + - IsCoprime.mul_dvd + - IsCoprime.dvd_of_dvd_mul_right + - IsCoprime.add_mul_left_right + - IsCoprime.of_add_mul_left_left + - IsCoprime.of_mul_add_left_right + - IsCoprime.sq_add_sq_ne_zero + - IsCoprime.dvd_of_dvd_mul_left + - IsCoprime.symm + - IsCoprime.neg_left + - IsCoprime.add_mul_right_left + - IsCoprime.mul_add_right_left + - IsCoprime.mul_left + - IsCoprime.mul_add_right_right + - isCoprime_group_smul_left + - IsCoprime.mul_add_left_left + - isCoprime_self + - IsCoprime.mul_right_iff + - IsCoprime.of_isCoprime_of_dvd_left + - IsCoprime.ne_zero_or_ne_zero + - path: Mathlib/Data/Ordmap/Ordset.lean + theorems: + - Ordnode.Valid'.balance'_lemma + - Ordnode.Sized.induction + - Ordnode.delta_lt_false + - Ordnode.Raised.right + - Ordnode.Valid'.dual + - Ordset.pos_size_of_mem + - Ordnode.Bounded.weak_left + - "Ordnode.Valid'.node4L_lemma\u2081" + - Ordnode.Sized.balance' + - Ordnode.dual_node4R + - Ordnode.all_balance' + - Ordnode.Valid'.erase_aux + - Ordnode.Valid'.node4L + - Ordnode.balancedSz_up + - Ordnode.dual_node3R + - Ordnode.balancedSz_zero + - Ordnode.Valid'.balanceR + - Ordnode.Sized.dual_iff + - Ordnode.raised_iff + - Ordnode.Balanced.dual + - Ordnode.insert'.valid + - Ordnode.all_balanceL + - Ordnode.balanceR_eq_balance' + - Ordnode.Valid'.balance + - Ordnode.Valid'.merge_aux + - Ordnode.Valid'.rotateL + - Ordnode.Bounded.dual + - Ordnode.all_node4L + - Ordnode.dual_rotateR + - Ordnode.size_eq_realSize + - Ordnode.all_node4R + - Ordnode.Valid'.balanceL_aux + - Ordnode.eraseMin.valid + - Ordnode.Bounded.dual_iff + - Ordnode.dual_rotateL + - Ordnode.dual_insert + - Ordnode.balance_sz_dual + - "Ordnode.Valid'.rotateL_lemma\u2083" + - Ordnode.any_iff_exists + - Ordnode.equiv_iff + - Ordnode.dual_node4L + - Ordnode.Raised.add_right + - Ordnode.dual_node' + - Ordnode.foldr_cons_eq_toList + - Ordnode.emem_iff_mem_toList + - Ordnode.all_rotateL + - Ordnode.toList_node + - Ordnode.Sized.rotateR + - Ordnode.Valid'.glue_aux + - Ordnode.BalancedSz.symm + - "Ordnode.Valid'.rotateL_lemma\u2084" + - Ordnode.Sized.rotateL_size + - Ordnode.Sized.pos + - Ordnode.any_singleton + - "Ordnode.Valid'.node4L_lemma\u2083" + - Ordnode.Valid'.dual_iff + - Ordnode.length_toList' + - Ordnode.Valid'.balanceR_aux + - Ordnode.dual_eraseMax + - Ordnode.all_balanceR + - Ordnode.Bounded.to_sep + - Ordnode.size_balanceR + - Ordnode.all_node3L + - Ordnode.Sized.node4L + - Ordnode.Valid'.map_aux + - Ordset.empty_iff + - Ordnode.Sized.dual + - Ordnode.Valid'.eraseMax_aux + - "Ordnode.Valid'.node4L_lemma\u2084" + - Ordnode.splitMin_eq + - Ordnode.size_balanceL + - Ordnode.Valid'.rotateR + - Ordnode.insert'_eq_insertWith + - Ordnode.Sized.rotateL + - "Ordnode.Valid'.node4L_lemma\u2085" + - Ordnode.Raised.dist_le + - Ordnode.insert.valid + - Ordnode.Sized.eq_node' + - Ordnode.node3R_size + - Ordnode.Valid'.eraseMin_aux + - Ordnode.length_toList + - Ordnode.Sized.rotateR_size + - Ordnode.Valid'.balance'_aux + - Ordnode.insertWith.valid_aux + - Ordnode.eraseMax.valid + - Ordnode.node3L_size + - Ordnode.Valid'.merge_lemma + - Ordnode.dual_dual + - Ordnode.size_erase_of_mem + - Ordnode.dual_balanceR + - Ordnode.Bounded.weak_right + - Ordnode.all_rotateR + - "Ordnode.Valid'.rotateL_lemma\u2082" + - Ordnode.findMax_dual + - Ordnode.dual_balance' + - "Ordnode.Valid'.merge_aux\u2081" + - "Ordnode.Valid'.node4L_lemma\u2082" + - "Ordnode.Valid'.rotateL_lemma\u2081" + - Ordnode.insert_eq_insertWith + - Ordnode.dual_balanceL + - Ordnode.findMax'_dual + - Ordnode.Raised.dist_le' + - Ordnode.Sized.size_eq_zero + - Ordnode.dual_eraseMin + - Ordnode.dual_node3L + - Ordnode.Valid'.balanceL + - Ordnode.splitMax_eq + - Ordnode.balanceL_eq_balance' + - Ordnode.balance_eq_balance' + - Ordnode.merge_nil_left + - Ordnode.size_balance' + - Ordnode.all_iff_forall + - path: Mathlib/Analysis/NormedSpace/Unitization.lean + theorems: + - Unitization.splitMul_apply + - Unitization.splitMul_injective_of_clm_mul_injective + - Unitization.antilipschitzWith_addEquiv + - Unitization.norm_inr + - Unitization.uniformity_eq_aux + - Unitization.lipschitzWith_addEquiv + - Unitization.norm_eq_sup + - path: Mathlib/LinearAlgebra/Trace.lean + theorems: + - LinearMap.trace_comp_comm' + - LinearMap.trace_mul_cycle' + - LinearMap.trace_id + - LinearMap.trace_comp_cycle + - LinearMap.trace_lie + - LinearMap.isNilpotent_trace_of_isNilpotent + - LinearMap.trace_eq_contract_of_basis + - LinearMap.trace_tensorProduct' + - LinearMap.trace_mul_cycle + - LinearMap.trace_mul_comm + - LinearMap.IsProj.trace + - LinearMap.trace_eq_contract_apply + - LinearMap.trace_tensorProduct + - LinearMap.trace_baseChange + - LinearMap.trace_eq_matrix_trace_of_finset + - LinearMap.trace_transpose + - LinearMap.trace_comp_comm + - LinearMap.trace_eq_matrix_trace + - LinearMap.traceAux_eq + - LinearMap.trace_comp_eq_mul_of_commute_of_isNilpotent + - LinearMap.trace_one + - LinearMap.trace_prodMap + - LinearMap.trace_prodMap' + - LinearMap.trace_eq_contract_of_basis' + - LinearMap.trace_conj' + - LinearMap.trace_comp_cycle' + - LinearMap.trace_transpose' + - LinearMap.trace_conj + - path: Mathlib/MeasureTheory/Integral/PeakFunction.lean + theorems: + - tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos + - tendsto_integral_peak_smul_of_integrable_of_tendsto + - tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn + - integrableOn_peak_smul_of_integrableOn_of_tendsto + - tendsto_set_integral_peak_smul_of_integrableOn_of_tendsto + - tendsto_integral_comp_smul_smul_of_integrable + - tendsto_set_integral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn + - tendsto_set_integral_peak_smul_of_integrableOn_of_tendsto_aux + - tendsto_integral_comp_smul_smul_of_integrable' + - path: Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean + theorems: + - Pi.summable + - tsum_setProd_singleton_left + - Pi.hasSum + - hasSum_pi_single + - tsum_op + - HasSum.sigma_of_hasSum + - HasSum.prod_mk + - Summable.ofStar + - tsum_pi_single + - tsum_star + - tsum_comm' + - HasSum.star + - tsum_setProd_singleton_right + - path: .lake/packages/std/Std/Data/List/Lemmas.lean + theorems: + - List.infix_cons_iff + - List.map_sub_range' + - List.of_mem_zip + - List.map_drop + - List.IsInfix.filter + - List.replaceF_cons_of_some + - List.contains_cons + - List.get_take' + - List.length_take_of_le + - List.foldr_hom + - List.append_eq_append_iff + - List.diff_erase + - List.contains_eq_any_beq + - List.length_range + - List.reverse_suffix + - List.range_subset + - List.init_eq_of_concat_eq + - List.nil_diff + - List.prefix_of_prefix_length_le + - List.leftpad_suffix + - List.mem_diff_of_mem + - List.replaceF_of_forall_none + - List.filterMap_filter + - List.findIdx?_of_eq_none + - List.filter_sublist + - List.filterMap_cons_some + - List.length_erase_of_mem + - List.map_add_range' + - List.maximum?_mem + - List.cons_merge_cons_neg + - List.take_append + - List.mem_join + - List.get_cons_length + - List.dropLast_cons_of_ne_nil + - List.infix_concat + - List.partition_eq_filter_filter + - List.singleton_disjoint + - List.sublist.erase + - List.get?_modifyNth + - List.insert_of_mem + - List.Sublist.filter + - List.disjoint_append_right + - List.ne_nil_of_mem + - List.mem_range' + - List.length_eq_one + - List.indexOf_cons + - List.Sublist.diff_right + - List.ext + - List.mem_of_find?_eq_some + - List.get_range' + - List.take_all_of_le + - List.modifyNth_eq_set_get? + - List.length_iota + - List.removeNth_eq_nth_tail + - List.get_of_append + - List.findIdxs_cons + - List.append_of_mem + - List.dropLast_concat + - List.range_succ + - List.disjoint_take_drop + - List.Sublist.eraseP + - List.pairwise_append + - List.suffix_cons_iff + - List.filterMap_map + - List.map_snd_zip + - List.zipWith_map + - List.tail_eq_tailD + - List.concat_append + - List.mem_insert_iff + - List.mem_replicate + - List.takeWhile_append_dropWhile + - List.length_take + - List.mem_union_iff + - List.forIn_eq_bindList + - List.get?_set_eq_of_lt + - List.find?_some + - List.Sublist.reverse + - List.set_set + - List.erase_comm + - List.modifyNth_eq_set_get + - List.take_succ + - List.drop_length_cons + - List.range'_concat + - List.eraseP_append_right + - List.forM_append + - List.findIdx?_replicate + - List.range'_append + - List.concat_ne_nil + - List.map_eq_append_split + - List.filterMap_join + - List.Sublist.erase + - List.length_zip + - List.foldrIdx_start + - List.filter_append + - List.zipWith_eq_nil_iff + - List.last_eq_of_concat_eq + - List.mem_reverseAux + - List.filter_congr' + - List.bind_map + - List.set_eq_modifyNth + - List.range_succ_eq_map + - List.mem_eraseP_of_neg + - List.length_insert_of_mem + - List.reverse_prefix + - List.reverse_infix + - List.Chain.imp' + - List.map_eq_nil + - List.IsInfix.trans + - List.findIdx_of_get?_eq_some + - List.range'_eq_map_range + - List.getLast_cons' + - List.zipWith_append + - List.append_ne_nil_of_ne_nil_right + - List.get_of_append_proof + - List.prefix_append_right_inj + - List.range'_subset_right + - List.dropLast_eq_take + - List.drop_append_of_le_length + - List.take_replicate + - List.modifyNth_eq_take_cons_drop + - List.range'_subset_right_1 + - List.cons_diff_of_mem + - List.mem_merge_left + - List.eraseP_append_left + - List.get?_range' + - List.get?_zero + - List.set_eq_take_cons_drop + - List.cons_union + - List.findIdx?_append + - List.range'_append_1 + - List.forall_mem_map_iff + - List.get?_drop + - List.pairwise_reverse + - List.self_mem_range_succ + - List.length_removeNth + - List.length_take_le' + - List.mem_iff_append + - List.enum_map_fst + - List.foldl_hom + - List.disjoint_left + - List.getLast!_cons + - List.length_filterMap_le + - List.disjoint_cons_right + - List.map_filterMap + - List.exists_mem_cons + - List.tailD_eq_tail? + - List.mem_iota + - List.map_take + - List.range_add + - List.getLast?_cons + - List.Sublist.filterMap + - List.length_replaceF + - List.filterMap_some + - List.insert_of_not_mem + - List.erase_append_right + - List.length_merge + - List.extractP_eq_find?_eraseP + - List.get_drop + - List.dropWhile_append + - List.foldl_map + - List.indexesOf_cons + - List.getLastD_concat + - List.get?_modifyNth_eq + - List.not_mem_range_self + - List.getLast_eq_getLastD + - List.set_comm + - List.filterMap_append + - List.or_all_distrib_right + - List.forall_mem_singleton + - List.map_zipWith + - List.findIdx_lt_length_of_exists + - List.cons_eq_append + - List.zipWith_distrib_drop + - List.cons_subset + - List.exists_of_set + - List.mem_bind + - List.zipWith_distrib_take + - List.mem_iff_get? + - List.and_any_distrib_left + - List.dropLast_take + - List.or_all_distrib_left + - List.all_eq_not_any_not + - List.Sublist.trans + - List.zipWith_map_left + - List.mem_merge_right + - List.foldr_map + - List.length_tail + - List.merge_loop + - List.getLastD_eq_getLast? + - List.maximum?_eq_none_iff + - List.pair_mem_product + - List.get?_set_of_lt + - List.get_append_right' + - List.mem_range'_1 + - List.dropLast_append_cons + - List.range_sublist + - List.maximum?_eq_some_iff' + - List.IsSuffix.isInfix + - List.maximum?_le_iff + - List.filterMap_filterMap + - List.disjoint_append_left + - List.any_eq_not_all_not + - List.infix_append' + - List.erase_cons + - List.filterMap_eq_map + - List.drop_append_eq_append_drop + - List.modifyNthTail_add + - List.merge_loop_nil_right + - List.erase_eq_eraseP + - List.append_ne_nil_of_ne_nil_left + - List.zip_map_right + - List.get?_take + - List.zipWith_map_right + - List.leftpad_prefix + - List.ext_get + - List.cons_merge_cons_pos + - List.findIdx_cons + - List.drop_drop + - List.append_sublist_append_right + - List.get?_set_ne + - List.zip_map + - List.length_take_le + - List.get_of_eq + - List.take_append_eq_append_take + - List.length_eraseP_of_mem + - List.map_filter + - List.disjoint_cons_left + - List.eraseP_of_forall_not + - List.singleton_sublist + - List.zip_map_left + - List.mem_filterMap + - List.indexOf_mem_indexesOf + - List.pairwise_map + - List.drop_left' + - List.filterMap_cons_none + - List.get?_set + - List.range_eq_nil + - List.find?_cons_of_pos + - List.eraseP_cons_of_pos + - List.range_loop_range' + - List.filter_filter + - List.chain_cons + - List.get_range + - List.modify_get?_length + - List.cons_diff_of_not_mem + - List.range_eq_range' + - List.isSublist_iff_sublist + - List.exists_erase_eq + - List.get?_range + - List.get_drop' + - List.nil_union + - List.get_set + - List.take_left' + - List.map_fst_zip + - List.map_filterMap_some_eq_filter_map_is_some + - List.take_eq_nil_iff + - List.get?_inj + - List.append_subset + - List.nil_eq_append + - List.disjoint_nil_right + - List.eraseP_cons_of_neg + - List.mem_nil_iff + - List.cons_subset_cons + - List.lt_length_drop + - List.erase_append_left + - List.cons_merge_cons + - List.length_zipWith + - List.reverse_range' + - List.getLastD_cons + - List.erase_cons_tail + - List.get?_set_eq + - List.maximum?_eq_some_iff + - List.zipWith_foldl_eq_zip_foldl + - List.length_dropLast + - List.set_eq_nil + - List.append_eq_cons + - List.mem_toArray + - List.tail_eq_tail? + - List.dropLast_append_of_ne_nil + - List.filter_eq_self + - List.take_append_of_le_length + - List.get_map + - List.merge_loop_nil_left + - List.getElem?_eq_get? + - List.map_filterMap_of_inv + - List.zip_map' + - List.sublist_or_mem_of_sublist + - List.reverse_take + - List.erase_cons_head + - List.zipWith_distrib_tail + - List.iota_eq_reverse_range' + - List.eraseP_sublist + - List.filter_filterMap + - List.zip_append + - List.range'_1_concat + - List.append_concat + - List.Sublist.erase_diff_erase_sublist + - List.mem_inter_iff + - List.mem_range + - List.getLast_append + - List.eq_replicate_of_mem + - List.not_all_eq_any_not + - List.not_any_eq_all_not + - List.findIdx?_of_eq_some + - List.find?_eq_none + - List.IsSuffix.filter + - List.get?_set_of_lt' + - List.eraseP_map + - List.mem_append + - List.findIdxs_cons_aux + - List.Pairwise.chain + - List.find?_cons_of_neg + - List.exists_of_findSome?_eq_some + - List.leftpad_length + - List.get_append_right_aux + - List.mem_reverse + - List.exists_of_eraseP + - List.diff_append + - List.zipWith_foldr_eq_zip_foldr + - List.get?_modifyNth_ne + - List.drop_append + - List.and_any_distrib_right + - List.IsPrefix.filter + - List.replaceF_cons_of_none + - List.range'_sublist_right + - List.replicate_sublist_replicate + - List.findIdx?_succ + - List.cons_diff + - List.tail_drop + - List.filterMap_eq_filter + - List.disjoint_singleton + - List.take_add + - path: Mathlib/Data/Polynomial/Degree/TrailingDegree.lean + theorems: + - Polynomial.nextCoeffUp_of_pos_natTrailingDegree + - Polynomial.natTrailingDegree_eq_support_min' + - Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq_of_pos + - Polynomial.natTrailingDegree_X_pow + - Polynomial.natTrailingDegree_neg + - Polynomial.natTrailingDegree_mul_X_pow + - Polynomial.Monic.eq_X_pow_of_natTrailingDegree_eq_natDegree + - Polynomial.natTrailingDegree_le_of_ne_zero + - Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree + - Polynomial.le_natTrailingDegree_mul + - Polynomial.natTrailingDegree_nat_cast + - Polynomial.trailingDegree_eq_iff_natTrailingDegree_eq + - Polynomial.natTrailingDegree_le_natDegree + - Polynomial.trailingDegree_le_trailingDegree + - Polynomial.le_trailingDegree_monomial + - Polynomial.le_trailingDegree_mul + - Polynomial.natTrailingDegree_mul' + - Polynomial.trailingDegree_mul' + - Polynomial.trailingDegree_X_pow + - Polynomial.trailingDegree_neg + - Polynomial.trailingDegree_ne_of_natTrailingDegree_ne + - Polynomial.le_trailingDegree_X_pow + - Polynomial.natTrailingDegree_eq_of_trailingDegree_eq_some + - Polynomial.natTrailingDegree_int_cast + - Polynomial.trailingDegree_one_le + - Polynomial.trailingDegree_C_mul_X_pow + - Polynomial.natTrailingDegree_monomial_le + - Polynomial.ne_zero_of_trailingDegree_lt + - Polynomial.natTrailingDegree_le_of_trailingDegree_le + - Polynomial.trailingDegree_monomial + - Polynomial.natTrailingDegree_le_trailingDegree + - Polynomial.le_natTrailingDegree + - Polynomial.trailingDegree_eq_natTrailingDegree + - Polynomial.natTrailingDegree_eq_of_trailingDegree_eq + - Polynomial.le_trailingDegree_C_mul_X_pow + - Polynomial.natTrailingDegree_le_natTrailingDegree + - Polynomial.natTrailingDegree_monomial + - Polynomial.coeff_eq_zero_of_lt_natTrailingDegree + - Polynomial.trailingDegree_eq_top + - path: .lake/packages/std/Std/Data/Int/DivMod.lean + theorems: + - Int.mul_div_assoc' + - Int.eq_mul_of_ediv_eq_left + - Int.mul_div_cancel_of_mod_eq_zero + - Int.le_of_mul_le_mul_left + - Int.div_eq_ediv_of_dvd + - Int.div_sign + - Int.fmod_eq_of_lt + - Int.le_mul_of_ediv_le + - Int.sign_eq_div_abs + - Int.div_add_mod' + - Int.ofNat_dvd_right + - Int.eq_zero_of_ediv_eq_zero + - Int.mul_fmod_right + - Int.fmod_add_fdiv + - Int.mul_fmod_left + - Int.le_of_mul_le_mul_right + - Int.natAbs_dvd + - Int.zero_fdiv + - Int.ofNat_fdiv + - Int.lt_ediv_add_one_mul_self + - Int.mul_ediv_mul_of_pos_left + - Int.mul_mod_left + - Int.ediv_eq_of_eq_mul_left + - Int.eq_zero_of_div_eq_zero + - Int.add_emod_eq_add_emod_left + - Int.dvd_antisymm + - Int.ediv_left_inj + - Int.dvd_iff_dvd_of_dvd_add + - Int.div_self + - Int.ediv_eq_of_eq_mul_right + - Int.mul_div_cancel + - Int.emod_add_cancel_left + - Int.div_left_inj + - Int.dvd_add_left + - Int.fdiv_eq_ediv + - Int.div_eq_of_eq_mul_left + - Int.div_zero + - Int.fdiv_eq_ediv_of_dvd + - Int.div_eq_of_eq_mul_right + - Int.div_dvd_div + - Int.div_one + - Int.ofNat_fmod + - Int.bmod_bmod + - Int.bmod_zero + - Int.fmod_one + - Int.lt_of_mul_lt_mul_left + - Int.mod_add_div' + - Int.sub_ediv_of_dvd_sub + - Int.ediv_emod_unique + - Int.zero_mod + - Int.fmod_self + - Int.bmod_lt + - Int.bmod_le + - Int.ediv_eq_iff_eq_mul_left + - Int.ediv_mul_le + - Int.lt_of_mul_lt_mul_right + - Int.div_mul_cancel_of_mod_eq_zero + - Int.le_bmod + - Int.eq_mul_of_div_eq_right + - Int.zero_div + - Int.mod_neg + - Int.div_eq_iff_eq_mul_left + - Int.neg_div_neg + - Int.mul_mod_right + - Int.fmod_def + - Int.dvd_add_right + - Int.fmod_eq_emod + - Int.mul_div_cancel' + - Int.ediv_pos_of_pos_of_dvd + - Int.dvd_sub_of_emod_eq + - Int.dvd_bmod_sub_self + - Int.emod_bmod + - Int.eq_one_of_mul_eq_one_left + - Int.natAbs_dvd_self + - Int.emod_two_eq + - Int.mod_one + - Int.mul_div_assoc + - Int.dvd_natAbs + - Int.div_eq_ediv + - Int.ediv_le_self + - Int.mod_eq_of_lt + - Int.div_neg + - Int.ediv_sign + - Int.eq_mul_of_div_eq_left + - Int.bmod_natAbs_plus_one + - Int.neg_div + - Int.natAbs_div + - Int.lt_div_add_one_mul_self + - Int.ediv_eq_ediv_of_mul_eq_mul + - Int.emod_neg + - Int.mod_add_div + - Int.mul_ediv_mul_of_pos + - Int.dvd_natAbs_self + - Int.mod_def + - Int.eq_mul_of_ediv_eq_right + - Int.mod_self + - Int.emod_eq_emod_iff_emod_sub_eq_zero + - Int.mul_emod_mul_of_pos + - Int.natAbs_div_le_natAbs + - Int.ediv_add_emod' + - Int.mul_fdiv_cancel + - Int.mod_eq_emod + - Int.zero_fmod + - path: Mathlib/GroupTheory/Perm/Cycle/Factors.lean + theorems: + - Equiv.Perm.cycleOf_zpow_apply_self + - Equiv.Perm.cycleFactorsFinset_eq_finset + - Equiv.Perm.cycleOf_apply_apply_zpow_self + - Equiv.Perm.mem_list_cycles_iff + - Equiv.Perm.SameCycle.exists_pow_eq_of_mem_support + - Equiv.Perm.cycleFactorsFinset_one + - Equiv.Perm.cycleOf_eq_one_iff + - Equiv.Perm.IsCycle.cycleOf_eq + - Equiv.Perm.pow_mod_orderOf_cycleOf_apply + - Equiv.Perm.support_cycleOf_eq_nil_iff + - Equiv.Perm.Disjoint.disjoint_cycleFactorsFinset + - Equiv.Perm.mem_cycleFactorsFinset_support_le + - Equiv.Perm.cycleOf_mul_of_apply_right_eq_self + - Equiv.Perm.Disjoint.cycleFactorsFinset_mul_eq_union + - Equiv.Perm.cycleFactorsFinset_eq_empty_iff + - Equiv.Perm.support_cycleOf_le + - Equiv.Perm.cycle_induction_on + - Equiv.Perm.isCycleOn_support_cycleOf + - Equiv.Perm.mem_support_cycleOf_iff' + - Equiv.Perm.cycleFactorsFinset_mul_inv_mem_eq_sdiff + - Equiv.Perm.SameCycle.cycleOf_eq + - Equiv.Perm.list_cycles_perm_list_cycles + - Equiv.Perm.cycleFactorsFinset_eq_singleton_self_iff + - Equiv.Perm.cycleOf_apply + - Equiv.Perm.isCycle_cycleOf_iff + - Equiv.Perm.cycleOf_pow_apply_self + - Equiv.Perm.cycleOf_apply_apply_self + - Equiv.Perm.disjoint_mul_inv_of_mem_cycleFactorsFinset + - Equiv.Perm.cycleFactorsFinset_eq_singleton_iff + - Equiv.Perm.cycle_is_cycleOf + - Equiv.Perm.Disjoint.cycleOf_mul_distrib + - Equiv.Perm.mem_support_cycleOf_iff + - Equiv.Perm.card_support_cycleOf_pos_iff + - Equiv.Perm.cycleFactorsFinset_eq_list_toFinset + - Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff + - Equiv.Perm.mem_cycleFactorsFinset_iff + - Equiv.Perm.cycleOf_inv + - Equiv.Perm.isCycle_cycleOf + - Equiv.Perm.two_le_card_support_cycleOf_iff + - Equiv.Perm.SameCycle.exists_pow_eq + - Equiv.Perm.cycleOf_apply_apply_pow_self + - Equiv.Perm.pow_mod_card_support_cycleOf_self_apply + - path: Mathlib/RingTheory/DedekindDomain/Different.lean + theorems: + - FractionalIdeal.dual_inv_le + - Submodule.traceDual_top + - Submodule.one_le_traceDual_one + - FractionalIdeal.dual_le_dual + - coeIdeal_differentIdeal + - FractionalIdeal.coe_dual + - FractionalIdeal.one_le_dual_one + - differentialIdeal_le_iff + - FractionalIdeal.dual_inv + - Submodule.traceDual_bot + - traceForm_dualSubmodule_adjoin + - FractionalIdeal.dual_div_dual + - FractionalIdeal.mem_dual + - coeSubmodule_differentIdeal_fractionRing + - conductor_mul_differentIdeal + - differentialIdeal_le_fractionalIdeal_iff + - FractionalIdeal.dual_ne_zero + - FractionalIdeal.le_dual_iff + - FractionalIdeal.dual_eq_mul_inv + - FractionalIdeal.dual_dual + - FractionalIdeal.inv_le_dual + - Submodule.le_traceDual_mul_iff + - FractionalIdeal.dual_zero + - Submodule.le_traceDual_iff_map_le_one + - FractionalIdeal.coe_dual_one + - isIntegral_discr_mul_of_mem_traceDual + - FractionalIdeal.self_mul_dual + - coeSubmodule_differentIdeal + - Submodule.le_traceDual_comm + - FractionalIdeal.le_dual_inv_aux + - FractionalIdeal.dual_mul_self + - map_equiv_traceDual + - path: .lake/packages/std/Std/Data/List/Count.lean + theorems: + - List.countP_cons_of_neg + - List.countP_pos + - List.countP_eq_length + - List.countP_cons_of_pos + - List.length_eq_countP_add_countP + - List.countP_cons + - List.count_erase + - List.countP_false + - List.count_eq_length + - List.count_replicate + - List.Sublist.countP_le + - List.count_singleton + - List.countP_true + - List.countP_le_length + - List.countP_filter + - List.countP_mono_left + - List.count_singleton' + - List.count_filter + - List.le_count_iff_replicate_sublist + - List.count_pos_iff_mem + - List.countP_go_eq_add + - List.count_cons_self + - List.count_tail + - List.countP_eq_zero + - List.count_erase_self + - List.countP_map + - List.countP_append + - List.filter_beq + - List.filter_beq' + - List.count_cons + - List.count_cons_of_ne + - List.countP_eq_length_filter + - List.filter_eq' + - List.count_erase_of_ne + - List.count_concat + - path: Mathlib/Data/Nat/ModEq.lean + theorems: + - Nat.ModEq.cancel_right_div_gcd + - Nat.odd_mul_odd_div_two + - Nat.ModEq.gcd_eq + - Nat.add_div_le_add_div + - Nat.chineseRemainder_modEq_unique + - Nat.ModEq.pow + - Nat.add_div_of_dvd_right + - Nat.add_div + - Nat.add_mod_add_ite + - Nat.ModEq.mul_left_cancel' + - Nat.modEq_iff_dvd + - Nat.add_div_eq_of_add_mod_lt + - Nat.modEq_zero_iff + - Nat.ModEq.cancel_left_of_coprime + - Nat.add_modEq_right + - Nat.ModEq.add + - Nat.add_mod_of_add_mod_lt + - Nat.odd_of_mod_four_eq_one + - Nat.modEq_and_modEq_iff_modEq_mul + - Nat.ModEq.dvd_iff + - Nat.le_mod_add_mod_of_dvd_add_of_not_dvd + - Nat.odd_of_mod_four_eq_three + - Nat.div_mod_eq_mod_mul_div + - Nat.add_div_eq_of_le_mod_add_mod + - Nat.ModEq.cancel_left_div_gcd + - Nat.ModEq.cancel_right_of_coprime + - Nat.ModEq.add_right_cancel + - Nat.ModEq.mul_right_cancel' + - Nat.ModEq.mul_left' + - Nat.ModEq.eq_of_abs_lt + - Nat.ModEq.of_div + - Nat.ModEq.mul_right + - Nat.add_modEq_left + - Nat.coprime_of_mul_modEq_one + - Nat.chineseRemainder'_lt_lcm + - Nat.modEq_iff_dvd' + - Nat.add_div_of_dvd_left + - Nat.modEq_sub + - Nat.ModEq.of_mul_left + - Nat.ModEq.add_left_cancel + - Nat.ModEq.mul_right' + - Nat.modEq_zero_iff_dvd + - path: Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean + theorems: + - SimpleGraph.dart_card_eq_twice_card_edges + - SimpleGraph.dart_fst_fiber + - SimpleGraph.dart_card_eq_sum_degrees + - SimpleGraph.dart_edge_fiber_card + - SimpleGraph.odd_card_odd_degree_vertices_ne + - SimpleGraph.exists_ne_odd_degree_of_exists_odd_degree + - SimpleGraph.Dart.edge_fiber + - SimpleGraph.dart_fst_fiber_card_eq_degree + - SimpleGraph.even_card_odd_degree_vertices + - path: Mathlib/Data/Polynomial/RingDivision.lean + theorems: + - Polynomial.mem_roots_sub_C' + - Polynomial.bUnion_roots_finite + - Polynomial.aroots_pow + - Polynomial.aroots_monomial + - Polynomial.isUnit_of_isUnit_leadingCoeff_of_isUnit_map + - Polynomial.rootMultiplicity_eq_natTrailingDegree' + - Polynomial.count_map_roots_of_injective + - Polynomial.roots.le_of_dvd + - Polynomial.mem_nthRootsFinset + - Polynomial.roots_multiset_prod_X_sub_C + - Polynomial.rootMultiplicity_mul + - Polynomial.roots_C_mul + - Polynomial.mem_roots' + - Polynomial.eq_of_infinite_eval_eq + - Polynomial.ne_zero_of_mem_nthRootsFinset + - Polynomial.rootMultiplicity_le_iff + - Polynomial.nthRoots_two_eq_zero_iff + - Polynomial.rootMultiplicity_X_sub_C + - Polynomial.trailingDegree_mul + - Polynomial.prod_multiset_X_sub_C_dvd + - Polynomial.prime_X_sub_C + - Polynomial.prime_X + - Polynomial.roots_X_sub_C + - Polynomial.aroots_zero + - Polynomial.one_mem_nthRootsFinset + - Polynomial.count_map_roots + - Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq + - Polynomial.card_roots + - Polynomial.Monic.isUnit_leadingCoeff_of_dvd + - Polynomial.Monic.comp + - Polynomial.degree_le_of_dvd + - Polynomial.degree_pos_of_not_isUnit_of_dvd_monic + - Polynomial.eq_zero_of_dvd_of_degree_lt + - Polynomial.prod_multiset_root_eq_finset_root + - Polynomial.Monic.irreducible_iff_degree_lt + - Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le + - Polynomial.roots_C_mul_X_pow + - Polynomial.le_rootMultiplicity_iff + - Polynomial.eq_of_dvd_of_natDegree_le_of_leadingCoeff + - Polynomial.modByMonic_eq_of_dvd_sub + - Polynomial.rootMultiplicity_eq_rootMultiplicity + - Polynomial.natDegree_pow + - Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero + - Polynomial.exists_multiset_roots + - Polynomial.rootSet_maps_to' + - Polynomial.mem_rootSet' + - Polynomial.natDegree_mul + - Polynomial.mem_nonZeroDivisors_of_leadingCoeff + - Polynomial.natDegree_sub_eq_of_prod_eq + - Polynomial.roots_monomial + - Polynomial.irreducible_of_monic + - Polynomial.degree_eq_degree_of_associated + - Polynomial.coeff_coe_units_zero_ne_zero + - Polynomial.aroots_neg + - Polynomial.natDegree_multiset_prod_X_sub_C_eq_card + - Polynomial.finite_setOf_isRoot + - Polynomial.exists_prod_multiset_X_sub_C_mul + - Polynomial.nthRoots_zero_right + - Polynomial.aroots_mul + - Polynomial.X_sub_C_pow_dvd_iff + - Polynomial.not_dvd_of_degree_lt + - Polynomial.rootSet_C + - Polynomial.roots_prod_X_sub_C + - Polynomial.aroots_smul_nonzero + - Polynomial.degree_eq_one_of_irreducible_of_root + - Polynomial.card_roots_sub_C + - Polynomial.rootSet_mapsTo + - Polynomial.comp_eq_zero_iff + - Polynomial.smul_modByMonic + - Polynomial.roots_neg + - Polynomial.associated_of_dvd_of_natDegree_le + - Polynomial.aeval_modByMonic_eq_self_of_root + - Polynomial.card_roots_sub_C' + - Polynomial.map_roots_le_of_injective + - Polynomial.roots_map_of_injective_of_card_eq_natDegree + - Polynomial.rootSet_one + - Polynomial.natDegree_eq_zero_of_isUnit + - Polynomial.aroots_X + - Polynomial.card_roots' + - Polynomial.Monic.irreducible_iff_natDegree + - Polynomial.roots_mul + - Polynomial.mul_mem_nthRootsFinset + - Polynomial.roots_X_pow + - Polynomial.nthRootsFinset_zero + - Polynomial.associated_of_dvd_of_natDegree_le_of_leadingCoeff + - Polynomial.eval_divByMonic_eq_trailingCoeff_comp + - Polynomial.mem_roots_iff_aeval_eq_zero + - Polynomial.count_roots + - Polynomial.map_roots_le + - Polynomial.rootSet_def + - Polynomial.mem_rootSet + - Polynomial.mem_nthRoots + - Polynomial.leadingCoeff_divByMonic_of_monic + - Polynomial.nthRoots_zero + - Polynomial.ne_zero_of_mem_rootSet + - Polynomial.associated_of_dvd_of_degree_eq + - Polynomial.roots_list_prod + - Polynomial.Monic.comp_X_sub_C + - Polynomial.eq_rootMultiplicity_map + - Polynomial.not_isUnit_of_degree_pos + - Polynomial.Monic.irreducible_of_irreducible_map + - Polynomial.card_nthRoots + - Polynomial.roots_prod + - Polynomial.rootMultiplicity_add + - Polynomial.eq_zero_of_forall_eval_zero_of_natDegree_lt_card + - Polynomial.rootSet_zero + - Polynomial.aroots_C_mul + - Polynomial.nthRootsFinset_def + - Polynomial.rootMultiplicity_X_sub_C_pow + - Polynomial.Monic.irreducible_iff_lt_natDegree_lt + - Polynomial.add_modByMonic + - Polynomial.degree_le_mul_left + - Polynomial.card_roots_le_map + - Polynomial.eq_zero_of_dvd_of_natDegree_lt + - Polynomial.card_roots_le_map_of_injective + - Polynomial.roots_multiset_prod + - Polynomial.roots_smul_nonzero + - Polynomial.aroots_C + - Polynomial.rootMultiplicity_mul_X_sub_C_pow + - Polynomial.exists_eval_ne_zero_of_natDegree_lt_card + - Polynomial.natDegree_le_of_dvd + - Polynomial.funext + - Polynomial.mem_aroots + - Polynomial.eq_of_monic_of_dvd_of_natDegree_le + - Polynomial.Monic.not_irreducible_iff_exists_add_mul_eq_coeff + - Polynomial.units_coeff_zero_smul + - Polynomial.rootMultiplicity_mul' + - Polynomial.roots_def + - Polynomial.pow_rootMultiplicity_not_dvd + - Polynomial.aroots_X_pow + - Polynomial.rootSet_neg + - Polynomial.mem_aroots' + - Polynomial.Monic.irreducible_iff_natDegree' + - Polynomial.le_rootMultiplicity_mul + - Polynomial.roots_X + - Polynomial.leadingCoeff_divByMonic_X_sub_C + - Polynomial.Monic.prime_of_degree_eq_one + - Polynomial.Monic.comp_X_add_C + - Multiset.prod_X_sub_C_dvd_iff_le_roots + - Polynomial.le_rootMultiplicity_map + - Polynomial.aroots_C_mul_X_pow + - Polynomial.isCoprime_X_sub_C_of_isUnit_sub + - Polynomial.aroots_X_sub_C + - path: Mathlib/Data/Sym/Sym2.lean + theorems: + - Sym2.eq_iff + - Sym2.other_mem + - Sym2.congr_right + - Sym2.mk_prod_swap_eq + - Sym2.IsDiag.mem_range_diag + - Sym2.out_snd_mem + - Sym2.map_congr + - Sym2.perm_card_two_iff + - Sym2.Rel.trans + - Sym2.other_invol + - Sym2.other_ne + - Sym2.mem_iff' + - Sym2.congr_left + - Sym2.map_id + - Sym2.other_eq_other' + - Sym2.map_map + - Sym2.other_mem' + - Sym2.out_fst_mem + - Sym2.ball + - Sym2.other_invol' + - Sym2.filter_image_mk_isDiag + - Sym2.toRel_symmetric + - Sym2.map.injective + - Sym2.fromRel_bot + - Sym2.fromRel_irreflexive + - Sym2.mem_and_mem_iff + - Sym2.rel_iff + - Sym2.fromRel_ne + - Sym2.filter_image_mk_not_isDiag + - Sym2.other_spec' + - Sym2.map_comp + - Sym2.fromRel_top + - Sym2.mk_eq_mk_iff + - Sym2.Rel.symm + - Sym2.rel_iff' + - "Sym2.inductionOn\u2082" + - path: Mathlib/Data/PFunctor/Univariate/M.lean + theorems: + - PFunctor.M.ext' + - PFunctor.M.mk_dest + - PFunctor.M.agree_iff_agree' + - PFunctor.M.isubtree_cons + - PFunctor.M.corec_unique + - PFunctor.M.iselect_cons + - PFunctor.M.dest_corec + - PFunctor.M.eq_of_bisim + - PFunctor.Approx.agree_children + - PFunctor.M.corec_def + - PFunctor.M.children_mk + - PFunctor.M.agree'_refl + - PFunctor.Approx.head_succ' + - PFunctor.Approx.truncate_eq_of_agree + - PFunctor.M.isPath_cons + - PFunctor.M.nth_of_bisim + - PFunctor.M.ext + - PFunctor.M.bisim + - PFunctor.M.head_mk + - PFunctor.M.ichildren_mk + - PFunctor.Approx.agree_trival + - PFunctor.Approx.P_corec + - PFunctor.M.iselect_eq_default + - PFunctor.M.mk_inj + - PFunctor.M.isPath_cons' + - PFunctor.M.Approx.P_mk + - PFunctor.Approx.approx_eta + - path: Mathlib/Topology/Instances/AddCircle.lean + theorems: + - AddCircle.continuousAt_equivIco + - AddCircle.card_addOrderOf_eq_totient + - AddCircle.coe_image_Ioc_eq + - AddCircle.liftIoc_coe_apply + - AddCircle.addOrderOf_period_div + - AddCircle.liftIco_zero_continuous + - AddCircle.continuousAt_equivIoc + - AddCircle.coe_image_Ico_eq + - AddCircle.equivIccQuot_comp_mk_eq_toIocMod + - AddCircle.addOrderOf_div_of_gcd_eq_one + - AddCircle.gcd_mul_addOrderOf_div_eq + - AddCircle.liftIco_continuous + - AddCircle.finite_setOf_add_order_eq + - AddCircle.liftIco_zero_coe_apply + - continuous_left_toIocMod + - AddCircle.coe_add_period + - AddCircle.liftIco_coe_apply + - AddCircle.coe_eq_zero_iff + - AddCircle.addOrderOf_coe_rat + - continuous_right_toIcoMod + - AddCircle.coe_eq_coe_iff_of_mem_Ico + - AddCircle.liftIco_eq_lift_Icc + - toIcoMod_eventuallyEq_toIocMod + - AddCircle.addOrderOf_eq_pos_iff + - AddCircle.coe_eq_zero_of_pos_iff + - AddCircle.addOrderOf_div_of_gcd_eq_one' + - path: Mathlib/Tactic/NormNum/LegendreSymbol.lean + theorems: + - Mathlib.Meta.NormNum.jacobiSymNat.one_right + - Mathlib.Meta.NormNum.jacobiSymNat.odd_even + - Mathlib.Meta.NormNum.jacobiSymNat.even_even + - Mathlib.Meta.NormNum.jacobiSymNat.zero_left + - "Mathlib.Meta.NormNum.jacobiSymNat.even_odd\u2087" + - Mathlib.Meta.NormNum.LegendreSym.to_jacobiSym + - Mathlib.Meta.NormNum.jacobiSymNat.double_even + - "Mathlib.Meta.NormNum.jacobiSymNat.even_odd\u2085" + - "Mathlib.Meta.NormNum.jacobiSymNat.even_odd\u2083" + - Mathlib.Meta.NormNum.jacobiSymNat.zero_right + - "Mathlib.Meta.NormNum.jacobiSymNat.qr\u2081" + - "Mathlib.Meta.NormNum.jacobiSymNat.qr\u2081'" + - "Mathlib.Meta.NormNum.jacobiSymNat.even_odd\u2081" + - Mathlib.Meta.NormNum.jacobiSymNat.mod_left + - Mathlib.Meta.NormNum.JacobiSym.mod_left + - Mathlib.Meta.NormNum.jacobiSymNat.one_left + - path: Mathlib/MeasureTheory/Order/Lattice.lean + theorems: + - Finset.measurable_range_sup'' + - Finset.measurable_range_sup' + - path: Mathlib/Geometry/Manifold/IntegralCurve.lean + theorems: + - IsIntegralCurveAt.isIntegralCurveOn + - isIntegralCurveAt_iff' + - isIntegralCurveAt_comp_add + - isIntegralCurve_eq_of_contMDiff + - isIntegralCurve_comp_add + - isIntegralCurve_const + - isIntegralCurveOn_comp_add + - isIntegralCurveAt_iff + - IsIntegralCurve.periodic_of_eq + - IsIntegralCurve.comp_mul + - IsIntegralCurveAt.comp_add + - isIntegralCurveAt_comp_mul_ne_zero + - IsIntegralCurveOn.comp_add + - exists_isIntegralCurveAt_of_contMDiffAt + - IsIntegralCurveOn.hasDerivAt + - IsIntegralCurveAt.comp_mul_ne_zero + - IsIntegralCurveOn.comp_mul + - isIntegralCurveOn_comp_mul_ne_zero + - IsIntegralCurveAt.eventually_hasDerivAt + - IsIntegralCurve.comp_add + - IsIntegralCurve.periodic_xor_injective + - isIntegralCurveOn_Ioo_eqOn_of_contMDiff + - isIntegralCurveAt_eventuallyEq_of_contMDiffAt + - isIntegralCurve_comp_mul_ne_zero + - path: Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean + theorems: + - MeasureTheory.Measure.addHaar_preimage_linearMap + - MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates + - MeasureTheory.Measure.addHaar_singleton_add_smul_div_singleton_add_smul + - MeasureTheory.Measure.addHaar_smul + - MeasureTheory.Measure.addHaar_image_linearMap + - MeasureTheory.Measure.addHaar_closedBall_center + - MeasureTheory.Measure.addHaar_ball + - MeasureTheory.Measure.addHaar_preimage_smul + - MeasureTheory.Measure.addHaar_ball_center + - MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one_aux + - MeasureTheory.Measure.addHaar_smul_of_nonneg + - MeasureTheory.addHaarMeasure_eq_volume + - MeasureTheory.Measure.addHaar_closedBall_eq_addHaar_ball + - MeasureTheory.addHaarMeasure_eq_volume_pi + - MeasureTheory.Measure.addHaar_closedBall_mul + - MeasureTheory.Measure.addHaar_submodule + - MeasureTheory.Measure.addHaar_closedBall + - AlternatingMap.measure_parallelepiped + - Basis.parallelepiped_basisFun + - MeasureTheory.Measure.addHaar_sphere_of_ne_zero + - MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar + - Basis.parallelepiped_eq_map + - MeasureTheory.Measure.eventually_nonempty_inter_smul_of_density_one + - MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 + - MeasureTheory.Measure.addHaar_closedBall' + - MeasureTheory.Measure.LinearMap.quasiMeasurePreserving + - MeasureTheory.Measure.addHaar_ball_of_pos + - MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 + - MeasureTheory.Measure.addHaar_image_homothety + - MeasureTheory.Measure.addHaar_ball_mul_of_pos + - Basis.map_addHaar + - MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar + - MeasureTheory.Measure.map_addHaar_smul + - MeasureTheory.Measure.addHaar_preimage_linearEquiv + - MeasureTheory.Measure.quasiMeasurePreserving_smul + - MeasureTheory.Measure.addHaar_parallelepiped + - MeasureTheory.Measure.addHaar_affineSubspace + - MeasureTheory.Measure.addHaar_sphere + - MeasureTheory.Measure.NullMeasurableSet.const_smul + - MeasureTheory.Measure.tendsto_addHaar_inter_smul_one_of_density_one + - MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero + - MeasureTheory.Measure.addHaar_closed_unit_ball_eq_addHaar_unit_ball + - MeasureTheory.Measure.addHaar_ball_mul + - MeasureTheory.Measure.addHaar_eq_zero_of_disjoint_translates_aux + - MeasureTheory.Measure.addHaar_closedBall_mul_of_pos + - path: Mathlib/Algebra/BigOperators/Intervals.lean + theorems: + - Finset.mul_prod_Ioi_eq_prod_Ici + - Finset.prod_range_add_one_eq_factorial + - Finset.prod_range_reflect + - Finset.sum_Ico_Ico_comm' + - Finset.mul_prod_Ico_eq_prod_Icc + - Finset.prod_Ico_id_eq_factorial + - Finset.prod_Ico_succ_top + - Finset.prod_Ico_eq_prod_range + - Finset.prod_range_diag_flip + - Finset.prod_Ico_add + - Finset.mul_prod_Ioc_eq_prod_Icc + - Finset.prod_Ico_eq_div + - Finset.prod_Ico_eq_mul_inv + - Finset.sum_Ico_Ico_comm + - Finset.mul_prod_Iio_eq_prod_Iic + - Finset.prod_eq_prod_Ico_succ_bot + - Finset.prod_range_div_prod_range + - Finset.sum_range_id_mul_two + - Finset.prod_Ioi_mul_eq_prod_Ici + - Finset.sum_range_id + - Finset.prod_Iio_mul_eq_prod_Iic + - Finset.prod_Ioc_consecutive + - Finset.prod_Ioc_succ_top + - path: Mathlib/Data/List/Indexes.lean + theorems: + - List.mapIdxMGo_eq_mapIdxMAuxSpec + - List.mapIdx_append + - List.foldlIdx_eq_foldl_enum + - List.oldMapIdx_append + - List.new_def_eq_old_def + - List.foldrIdx_eq_foldrIdxSpec + - List.mapIdx_eq_ofFn + - List.oldMapIdxCore_eq + - List.nthLe_mapIdx + - List.findIdxs_eq_map_indexesValues + - List.mapIdxGo_append + - List.length_mapIdx + - List.foldrIdx_eq_foldr_enum + - List.mapIdx_eq_nil + - List.foldlIdxM_eq_foldlM_enum + - List.mapIdxGo_length + - List.mapIdx_cons + - List.indexesValues_eq_filter_enum + - List.mapIdx_append_one + - List.mapIdxMAux'_eq_mapIdxMGo + - List.foldrIdxM_eq_foldrM_enum + - List.mapIdx_eq_enum_map + - List.list_reverse_induction + - List.mapIdxM_eq_mmap_enum + - List.oldMapIdxCore_append + - List.map_enumFrom_eq_zipWith + - path: Mathlib/Combinatorics/SimpleGraph/Coloring.lean + theorems: + - SimpleGraph.chromaticNumber_le_one_of_subsingleton + - SimpleGraph.Colorable.mono + - SimpleGraph.chromaticNumber_eq_zero_of_isempty + - SimpleGraph.card_le_chromaticNumber_iff_forall_surjective + - SimpleGraph.IsClique.card_le_of_colorable + - SimpleGraph.chromaticNumber_pos + - SimpleGraph.chromaticNumber_eq_card_iff_forall_surjective + - SimpleGraph.isEmpty_of_colorable_zero + - SimpleGraph.colorable_iff_exists_bdd_nat_coloring + - SimpleGraph.chromaticNumber_ne_top_iff_exists + - SimpleGraph.chromaticNumber_bot + - SimpleGraph.Coloring.colorable + - SimpleGraph.colorable_chromaticNumber + - SimpleGraph.IsClique.card_le_chromaticNumber + - SimpleGraph.cliqueFree_of_chromaticNumber_lt + - SimpleGraph.chromaticNumber_top + - SimpleGraph.Colorable.cliqueFree + - SimpleGraph.colorable_chromaticNumber_of_fintype + - SimpleGraph.Colorable.of_embedding + - SimpleGraph.chromaticNumber_top_eq_top_of_infinite + - SimpleGraph.chromaticNumber_le_of_forall_imp + - SimpleGraph.chromaticNumber_eq_iff_forall_surjective + - SimpleGraph.le_chromaticNumber_iff_forall_surjective + - SimpleGraph.chromaticNumber_le_iff_colorable + - SimpleGraph.CompleteBipartiteGraph.chromaticNumber + - SimpleGraph.Colorable.chromaticNumber_eq_sInf + - SimpleGraph.chromaticNumber_eq_iInf + - SimpleGraph.colorable_of_chromaticNumber_ne_top + - SimpleGraph.IsClique.card_le_of_coloring + - SimpleGraph.Colorable.chromaticNumber_le + - SimpleGraph.isEmpty_of_chromaticNumber_eq_zero + - SimpleGraph.Coloring.card_colorClasses_le + - path: Mathlib/Algebra/Order/Monoid/Lemmas.lean + theorems: + - mulLECancellable_one + - mul_lt_iff_lt_one_right' + - mul_eq_one_iff' + - one_le_of_le_mul_right + - mul_le_iff_le_one_left' + - one_le_of_le_mul_left + - lt_one_of_mul_lt_left + - MulLECancellable.le_mul_iff_one_le_right + - lt_mul_iff_one_lt_left' + - MulLECancellable.mul_le_iff_le_one_right + - exists_square_le + - le_mul_iff_one_le_left' + - MulLECancellable.mul_le_iff_le_one_left + - MulLECancellable.le_mul_iff_one_le_left + - Left.min_le_max_of_mul_le_mul + - MulLECancellable.injective_left + - mul_le_iff_le_one_right' + - one_lt_of_lt_mul_right + - le_one_of_mul_le_right + - le_mul_iff_one_le_right' + - MulLECancellable.mul_le_mul_iff_right + - mul_lt_iff_lt_one_left' + - lt_mul_iff_one_lt_right' + - Right.min_le_max_of_mul_le_mul + - mul_eq_mul_iff_eq_and_eq + - one_lt_of_lt_mul_left + - min_lt_max_of_mul_lt_mul + - mul_le_mul_iff_of_ge + - le_one_of_mul_le_left + - path: Mathlib/Data/Finsupp/BigOperators.lean + theorems: + - Multiset.support_sum_eq + - Multiset.support_sum_subset + - Finset.support_sum_subset + - Multiset.mem_sup_map_support_iff + - Finset.support_sum_eq + - List.support_sum_subset + - List.mem_foldr_sup_support_iff + - List.support_sum_eq + - path: Mathlib/Data/Finmap.lean + theorems: + - Multiset.nodup_keys + - Finmap.disjoint_union_right + - Finmap.insert_insert_of_ne + - Finmap.lookup_insert_of_ne + - Finmap.disjoint_union_left + - Finmap.replace_toFinmap + - Finmap.union_cancel + - Finmap.insert_entries_of_neg + - Finmap.lookup_union_left_of_not_in + - Finmap.keys_erase + - Finmap.mem_lookup_iff + - Finmap.keysLookupEquiv_symm_apply_lookup + - Finmap.keys_ext + - Finmap.insert_union + - Finmap.mem_erase + - Finmap.insert_insert + - Finmap.not_mem_erase_self + - Finmap.liftOn_toFinmap + - Finmap.lookup_singleton_eq + - Finmap.keys_replace + - Finmap.lookup_insert + - Finmap.union_comm_of_disjoint + - Finmap.sigma_keys_lookup + - Finmap.union_assoc + - AList.toFinmap_eq + - Finmap.erase_erase + - Finmap.union_empty + - Finmap.insert_toFinmap + - Finmap.insert_singleton_eq + - Finmap.ext_lookup + - Finmap.induction_on + - Finmap.empty_union + - Finmap.mem_replace + - Finmap.erase_toFinmap + - Finmap.ext + - Finmap.erase_union_singleton + - Finmap.dlookup_list_toFinmap + - Finmap.extract_eq_lookup_erase + - Finmap.mem_list_toFinmap + - "Finmap.liftOn\u2082_toFinmap" + - Finmap.keys_erase_toFinset + - path: Mathlib/Data/Finite/Card.lean + theorems: + - Finite.one_lt_card_iff_nontrivial + - Finite.card_eq_zero_of_surjective + - Set.card_union_le + - PartENat.card_eq_coe_nat_card + - Set.Finite.equiv_image_eq_iff_subset + - Finite.card_pos_iff + - Finite.card_subtype_le + - Finite.card_option + - Nat.card_eq + - Finite.card_le_of_surjective + - Finite.card_eq + - Finite.card_eq_zero_iff + - Finite.card_le_one_iff_subsingleton + - Set.Finite.card_lt_card + - Finite.card_sum + - Finite.card_subtype_lt + - Finite.card_le_of_injective + - path: Mathlib/RingTheory/Coprime/Lemmas.lean + theorems: + - exists_sum_eq_one_iff_pairwise_coprime' + - pairwise_coprime_iff_coprime_prod + - Int.isCoprime_iff_gcd_eq_one + - IsCoprime.pow_right + - Nat.Coprime.cast + - ne_zero_or_ne_zero_of_nat_coprime + - IsRelPrime.prod_left_iff + - exists_sum_eq_one_iff_pairwise_coprime + - pairwise_isRelPrime_iff_isRelPrime_prod + - IsCoprime.prod_right_iff + - IsRelPrime.pow_left + - IsCoprime.prod_left + - IsCoprime.pow_left_iff + - Nat.isCoprime_iff_coprime + - IsRelPrime.prod_left + - IsRelPrime.pow_right + - IsRelPrime.prod_right + - IsCoprime.pow_left + - IsRelPrime.pow_left_iff + - IsRelPrime.prod_right_iff + - IsCoprime.prod_right + - Finset.prod_dvd_of_coprime + - Finset.prod_dvd_of_isRelPrime + - IsCoprime.prod_left_iff + - path: Mathlib/MeasureTheory/Group/Arithmetic.lean + theorems: + - Multiset.measurable_prod' + - List.measurable_prod' + - ae_eq_trim_of_measurable + - Multiset.aemeasurable_prod' + - measurable_const_smul_iff + - nullMeasurableSet_eq_fun + - measurableSet_eq_fun + - aemeasurable_inv_iff + - Finset.measurable_prod + - measurableSet_eq_fun' + - aemeasurable_const_smul_iff + - "measurable_inv_iff\u2080" + - "aemeasurable_inv_iff\u2080" + - measurable_div_const' + - List.measurable_prod + - measurableSet_eq_fun_of_countable + - Multiset.measurable_prod + - Finset.aemeasurable_prod + - List.aemeasurable_prod' + - measurable_inv_iff + - Multiset.aemeasurable_prod + - path: Mathlib/Data/Finset/Pointwise.lean + theorems: + - Finset.image_mul_left + - Finset.smul_finset_neg + - Finset.preimage_mul_right_singleton + - Finset.smul_neg + - Finset.image_mul_right + - Finset.mem_inv' + - Finset.subset_smul_finset_iff + - "Finset.smul_finset_univ\u2080" + - Finset.zero_smul_subset + - Finset.inv_univ + - Finset.preimage_mul_left_one' + - Finset.smul_univ + - Set.card_inv + - Finset.univ_pow + - Finset.empty_pow + - Finset.mem_smul_finset + - Finset.zero_mul_subset + - Finset.inv_zero + - Finset.Nonempty.zero_mul + - Set.card_div_le + - Finset.zero_div_subset + - Finset.one_mem_div_iff + - Finset.mul_eq_one_iff + - "Finset.inv_op_smul_finset_distrib\u2080" + - Finset.preimage_mul_left_one + - Finset.Nonempty.smul_zero + - Finset.mem_pow + - Set.card_mul_le + - Finset.zero_mem_smul_iff + - Finset.mem_prod_list_ofFn + - Finset.Nonempty.zero_smul + - Finset.mem_inv_smul_finset_iff + - Finset.isUnit_iff + - Finset.preimage_inv + - Finset.pow_mem_pow + - Finset.pow_subset_pow + - Finset.neg_smul + - Finset.image_mul_right' + - Finset.coe_zpow + - Finset.coe_pow + - Finset.inv_op_smul_finset_distrib + - Finset.preimage_mul_right_one' + - Finset.mul_zero_subset + - Finset.zero_smul_finset + - Finset.Nonempty.div_zero + - Finset.smul_zero_subset + - Finset.pairwiseDisjoint_smul_iff + - Finset.neg_smul_finset + - Finset.smul_finset_subset_iff + - Finset.div_zero_subset + - Finset.univ_div_univ + - Finset.pow_subset_pow_of_one_mem + - "Finset.inv_smul_finset_distrib\u2080" + - Finset.Nonempty.zero_div + - Finset.zero_mem_smul_finset_iff + - Finset.image_mul_left' + - Finset.isUnit_iff_singleton_aux + - "Finset.smul_univ\u2080'" + - Finset.Nonempty.mul_zero + - Finset.inv_smul_finset_distrib + - Finset.isUnit_coe + - Finset.isUnit_iff_singleton + - path: Mathlib/Order/MinMax.lean + theorems: + - min_lt_min_left_iff + - Monotone.map_max + - min_lt_min_right_iff + - min_eq_iff + - min_choice + - Antitone.map_max + - min_cases + - MonotoneOn.map_max + - path: Mathlib/NumberTheory/Multiplicity.lean + theorems: + - Int.sq_mod_four_eq_one_of_odd + - "dvd_geom_sum\u2082_iff_of_dvd_sub" + - pow_two_pow_sub_pow_two_pow + - Int.two_pow_two_pow_add_two_pow_two_pow + - multiplicity.Nat.pow_add_pow + - padicValNat.pow_two_sub_pow + - Int.two_pow_two_pow_sub_pow_two_pow + - Int.two_pow_sub_pow + - multiplicity.Nat.pow_sub_pow + - multiplicity.pow_sub_pow_of_prime + - multiplicity.pow_prime_pow_sub_pow_prime_pow + - Int.two_pow_sub_pow' + - multiplicity.Int.pow_sub_pow + - "dvd_geom_sum\u2082_iff_of_dvd_sub'" + - "odd_sq_dvd_geom_sum\u2082_sub" + - sq_dvd_add_pow_sub_sub + - "multiplicity.geom_sum\u2082_eq_one" + - Nat.two_pow_sub_pow + - multiplicity.Int.pow_add_pow + - multiplicity.pow_prime_sub_pow_prime + - padicValNat.pow_sub_pow + - path: Mathlib/Combinatorics/SimpleGraph/Density.lean + theorems: + - Rel.edgeDensity_empty_left + - Rel.abs_edgeDensity_sub_edgeDensity_le_two_mul + - Rel.interedges_biUnion_left + - Rel.edgeDensity_empty_right + - Rel.interedges_disjoint_left + - Rel.edgeDensity_nonneg + - Rel.mem_interedges_iff + - Rel.edgeDensity_add_edgeDensity_compl + - Rel.swap_mem_interedges_iff + - SimpleGraph.edgeDensity_add_edgeDensity_compl + - SimpleGraph.card_interedges_add_card_interedges_compl + - Rel.mul_edgeDensity_le_edgeDensity + - Rel.abs_edgeDensity_sub_edgeDensity_le_two_mul_sub_sq + - Rel.edgeDensity_sub_edgeDensity_le_one_sub_mul + - Rel.abs_edgeDensity_sub_edgeDensity_le_one_sub_mul + - Rel.card_interedges_finpartition + - Rel.edgeDensity_le_one + - Rel.edgeDensity_comm + - Rel.card_interedges_finpartition_right + - Rel.interedges_disjoint_right + - Rel.card_interedges_add_card_interedges_compl + - Rel.interedges_empty_left + - Rel.interedges_biUnion + - Rel.interedges_biUnion_right + - Rel.interedges_mono + - Rel.card_interedges_finpartition_left + - path: Mathlib/FieldTheory/SplittingField/IsSplittingField.lean + theorems: + - Polynomial.IsSplittingField.finiteDimensional + - IntermediateField.splits_of_splits + - Polynomial.IsSplittingField.splits_iff + - IsIntegral.mem_intermediateField_of_minpoly_splits + - Polynomial.IsSplittingField.of_algEquiv + - Polynomial.IsSplittingField.mul + - path: Mathlib/Data/Nat/Interval.lean + theorems: + - Nat.card_Iio + - Nat.Icc_pred_right + - Nat.cauchy_induction_mul + - Nat.Ico_pred_singleton + - Nat.image_sub_const_Ico + - Nat.Ico_succ_right + - Finset.range_image_pred_top_sub + - Nat.Icc_succ_left + - Nat.image_Ico_mod + - Nat.card_fintypeIio + - Nat.Iio_eq_range + - Nat.Ico_succ_left + - Nat.Ico_zero_eq_range + - Nat.card_uIcc + - Nat.mod_injOn_Ico + - Finset.range_add_eq_union + - Nat.card_Iic + - Nat.Ico_succ_singleton + - Nat.card_fintypeIoc + - Nat.Ico_succ_succ + - Nat.card_fintypeIic + - Nat.card_fintypeIco + - Nat.multiset_Ico_map_mod + - Nat.Ioc_succ_singleton + - Nat.card_fintypeIcc + - Nat.Ico_image_const_sub_eq_Ico + - Nat.pow_imp_self_of_one_lt + - Nat.Ico_succ_right_eq_insert_Ico + - Nat.card_fintypeIoo + - Nat.cauchy_induction' + - Nat.Ico_succ_left_eq_erase_Ico + - path: Mathlib/GroupTheory/Nilpotent.lean + theorems: + - nilpotent_iff_finite_ascending_central_series + - lowerCentralSeries_map_subtype_le + - upperCentralSeries.map + - IsPGroup.isNilpotent + - nilpotencyClass_pi + - lowerCentralSeries_length_eq_nilpotencyClass + - CommGroup.nilpotencyClass_le_one + - lowerCentralSeries_antitone + - lowerCentralSeries_isDescendingCentralSeries + - is_ascending_rev_series_of_is_descending + - is_decending_rev_series_of_is_ascending + - lowerCentralSeries_succ_eq_bot + - nilpotent_iff_finite_descending_central_series + - ascending_central_series_le_upper + - nilpotent_center_quotient_ind + - lowerCentralSeries_prod + - upperCentralSeries_mono + - isNilpotent_of_product_of_sylow_group + - lowerCentralSeries.map + - nilpotencyClass_le_of_ker_le_center + - lowerCentralSeries_pi_of_finite + - nilpotencyClass_le_of_surjective + - isNilpotent_of_ker_le_center + - nilpotencyClass_eq_quotient_center_plus_one + - isNilpotent_of_finite_tFAE + - comap_center_subst + - nilpotencyClass_zero_iff_subsingleton + - isNilpotent_pi_of_bounded_class + - Subgroup.nilpotencyClass_le + - nilpotencyClass_quotient_center + - comap_upperCentralSeries_quotient_center + - upperCentralSeries_one + - lowerCentralSeries_pi_le + - nilpotencyClass_prod + - least_descending_central_series_length_eq_nilpotencyClass + - nilpotent_iff_lowerCentralSeries + - least_ascending_central_series_length_eq_nilpotencyClass + - lowerCentralSeries_eq_bot_iff_nilpotencyClass_le + - upperCentralSeries_eq_top_iff_nilpotencyClass_le + - of_quotient_center_nilpotent + - derived_le_lower_central + - nilpotent_of_surjective + - normalizerCondition_of_isNilpotent + - upperCentralSeriesStep_eq_comap_center + - lowerCentralSeries_nilpotencyClass + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean + theorems: + - WeierstrassCurve.Affine.CoordinateRing.degree_norm_ne_one + - WeierstrassCurve.Affine.CoordinateRing.basis_zero + - WeierstrassCurve.Affine.CoordinateRing.coe_basis + - WeierstrassCurve.Affine.CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal'_of_Yeq + - WeierstrassCurve.Affine.CoordinateRing.XYIdeal_mul_XYIdeal + - WeierstrassCurve.Affine.Point.add_eq_zero + - WeierstrassCurve.Affine.CoordinateRing.XYIdeal'_mul_inv + - WeierstrassCurve.Affine.CoordinateRing.basis_one + - WeierstrassCurve.Affine.Point.add_comm + - WeierstrassCurve.Affine.CoordinateRing.smul_basis_mul_C + - WeierstrassCurve.Affine.Point.neg_add_eq_zero + - WeierstrassCurve.Affine.Point.add_assoc + - WeierstrassCurve.Affine.CoordinateRing.XYIdeal_add_eq + - WeierstrassCurve.Affine.CoordinateRing.coe_norm_smul_basis + - WeierstrassCurve.Affine.CoordinateRing.XClass_ne_zero + - WeierstrassCurve.Affine.CoordinateRing.smul_basis_eq_zero + - WeierstrassCurve.Affine.CoordinateRing.degree_norm_smul_basis + - WeierstrassCurve.Affine.CoordinateRing.basis_apply + - "WeierstrassCurve.Affine.CoordinateRing.XYIdeal_eq\u2082" + - WeierstrassCurve.Affine.CoordinateRing.XYIdeal_neg_mul + - WeierstrassCurve.Affine.Point.toClass_eq_zero + - "WeierstrassCurve.Affine.CoordinateRing.XYIdeal_eq\u2081" + - WeierstrassCurve.Affine.CoordinateRing.C_addPolynomial + - WeierstrassCurve.Affine.CoordinateRing.YClass_ne_zero + - WeierstrassCurve.Affine.Point.add_left_neg + - WeierstrassCurve.Affine.CoordinateRing.C_addPolynomial_slope + - WeierstrassCurve.Affine.CoordinateRing.norm_smul_basis + - WeierstrassCurve.Affine.Point.toClass_injective + - WeierstrassCurve.Affine.CoordinateRing.smul_basis_mul_Y + - path: Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean + theorems: + - CochainComplex.mappingCone.lift_desc_f + - CochainComplex.mappingCone.ofHom_desc + - CochainComplex.mappingCone.lift_snd + - CochainComplex.mappingCone.ofHom_lift + - "CochainComplex.mappingCone.\u03B4_liftCochain" + - CochainComplex.mappingCone.lift_f_fst_v + - "CochainComplex.mappingCone.\u03B4_descCochain" + - CochainComplex.mappingCone.d_snd_v + - CochainComplex.mappingCone.inl_v_d + - CochainComplex.mappingCone.liftCochain_v_fst_v + - CochainComplex.mappingCone.liftCochain_fst + - CochainComplex.mappingCone.inr_fst_assoc + - CochainComplex.mappingCone.d_snd_v' + - CochainComplex.mappingCone.ext_from_iff + - CochainComplex.mappingCone.ext_cochain_to_iff + - CochainComplex.mappingCone.inl_fst + - CochainComplex.mappingCone.d_fst_v + - CochainComplex.mappingCone.id_X + - CochainComplex.mappingCone.inr_f_desc_f + - CochainComplex.mappingCone.inl_v_desc_f + - CochainComplex.mappingCone.inr_descCochain + - CochainComplex.mappingCone.inr_f_snd_v + - CochainComplex.mappingCone.desc_f + - CochainComplex.mappingCone.liftCochain_snd + - CochainComplex.mappingCone.inr_desc + - CochainComplex.mappingCone.inr_f_d + - CochainComplex.mappingCone.inl_snd + - CochainComplex.mappingCone.liftCochain_v_descCochain_v + - CochainComplex.mappingCone.inl_snd_assoc + - CochainComplex.mappingCone.lift_f_snd_v + - CochainComplex.mappingCone.ext_to_iff + - CochainComplex.mappingCone.inr_f_descCochain_v + - CochainComplex.mappingCone.inl_v_snd_v + - CochainComplex.mappingCone.inl_fst_assoc + - CochainComplex.mappingCone.lift_f + - CochainComplex.mappingCone.liftCochain_v_snd_v + - CochainComplex.mappingCone.ext_from + - "CochainComplex.mappingCone.\u03B4_snd" + - CochainComplex.mappingCone.d_fst_v' + - CochainComplex.mappingCone.inr_fst + - CochainComplex.mappingCone.ext_cochain_from_iff + - "CochainComplex.mappingCone.\u03B4_inl" + - CochainComplex.mappingCone.inl_descCochain + - CochainComplex.mappingCone.inl_v_fst_v + - CochainComplex.mappingCone.inr_snd_assoc + - CochainComplex.mappingCone.lift_fst + - CochainComplex.mappingCone.inl_desc + - CochainComplex.mappingCone.id + - CochainComplex.mappingCone.inr_f_fst_v + - path: Mathlib/AlgebraicTopology/SimplexCategory.lean + theorems: + - SimplexCategory.epi_iff_surjective + - SimplexCategory.mono_iff_injective + - "SimplexCategory.\u03B4_comp_\u03C3_self" + - "SimplexCategory.eq_\u03B4_of_mono" + - "SimplexCategory.\u03B4_comp_\u03B4" + - "SimplexCategory.eq_\u03C3_of_epi" + - SimplexCategory.hom_zero_zero + - SimplexCategory.skeletal + - "SimplexCategory.factor_\u03B4_spec" + - "SimplexCategory.\u03B4_comp_\u03C3_succ" + - "SimplexCategory.eq_comp_\u03B4_of_not_surjective'" + - "SimplexCategory.image_\u03B9_eq" + - SimplexCategory.const_eq_id + - "SimplexCategory.\u03B4_comp_\u03B4'" + - "SimplexCategory.\u03B4_comp_\u03C3_self'" + - "SimplexCategory.eq_\u03C3_comp_of_not_injective" + - "SimplexCategory.eq_comp_\u03B4_of_not_surjective" + - "SimplexCategory.\u03B4_comp_\u03C3_succ'" + - "SimplexCategory.eq_\u03C3_comp_of_not_injective'" + - SimplexCategory.image_eq + - "SimplexCategory.\u03B4_comp_\u03C3_of_le" + - "SimplexCategory.\u03B4_comp_\u03B4''" + - "SimplexCategory.\u03B4_comp_\u03C3_of_gt" + - SimplexCategory.eq_id_of_epi + - "SimplexCategory.\u03B4_comp_\u03C3_of_gt'" + - SimplexCategory.len_le_of_epi + - "SimplexCategory.\u03C3_comp_\u03C3" + - SimplexCategory.eq_id_of_mono + - "SimplexCategory.\u03B4_comp_\u03B4_self'" + - SimplexCategory.len_lt_of_mono + - SimplexCategory.len_le_of_mono + - path: Mathlib/Algebra/Module/Submodule/RestrictScalars.lean + theorems: + - Submodule.restrictScalars_eq_top_iff + - Submodule.restrictScalars_eq_bot_iff + - path: Mathlib/LinearAlgebra/Dimension/Finite.lean + theorems: + - CompleteLattice.Independent.subtype_ne_bot_le_finrank_aux + - finrank_eq_zero_of_basis_imp_false + - finrank_le_one + - Submodule.rank_eq_zero + - rank_zero_iff_forall_zero + - FiniteDimensional.nontrivial_of_finrank_eq_succ + - LinearIndependent.lt_aleph0_of_finite + - FiniteDimensional.finrank_zero_iff + - rank_eq_zero_iff + - Basis.nonempty_fintype_index_of_rank_lt_aleph0 + - finrank_eq_zero_of_basis_imp_not_finite + - FiniteDimensional.finrank_pos_iff + - rank_eq_zero_iff_isTorsion + - Module.exists_nontrivial_relation_of_finrank_lt_card + - exists_mem_ne_zero_of_rank_pos + - FiniteDimensional.finrank_zero_of_subsingleton + - FiniteDimensional.finrank_pos_iff_exists_ne_zero + - FiniteDimensional.finrank_eq_zero_iff + - Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card + - FiniteDimensional.finrank_eq_zero_iff_isTorsion + - LinearIndependent.finset_card_le_finrank + - LinearIndependent.cardinal_mk_le_finrank + - rank_eq_one + - Module.finite_of_rank_eq_zero + - Module.finite_of_rank_eq_nat + - rank_le + - Submodule.bot_eq_top_of_rank_eq_zero + - Submodule.finrank_eq_zero + - LinearIndependent.fintype_card_le_finrank + - FiniteDimensional.finrank_eq_zero_of_rank_eq_zero + - CompleteLattice.Independent.subtype_ne_bot_le_finrank + - rank_pos_iff_exists_ne_zero + - path: Mathlib/RingTheory/Polynomial/Pochhammer.lean + theorems: + - monic_ascPochhammer + - ascPochhammer_pos + - ascPochhammer_succ_right + - descPochhammer_succ_right + - descPochhammer_succ_eval + - ascPochhammer_one + - descPochhammer_map + - descPochhammer_mul + - ascPochhammer_nat_eq_ascFactorial + - ascPochhammer_zero_eval_zero + - ascPochhammer_eval_zero + - descPochhammer_ne_zero_eval_zero + - descPochhammer_eval_cast + - factorial_mul_ascPochhammer + - ascPochhammer_nat_eq_descFactorial + - descPochhammer_eval_eq_ascPochhammer + - ascPochhammer_nat_eval_succ + - descPochhammer_eval_eq_descFactorial + - descPochhammer_eval_zero + - ascPochhammer_eval_neg_eq_descPochhammer + - ascPochhammer_ne_zero_eval_zero + - ascPochhammer_natDegree + - descPochhammer_one + - ascPochhammer_eval_comp + - ascPochhammer_succ_eval + - descPochhammer_succ_left + - ascPochhammer_eval_one + - descPochhammer_succ_comp_X_sub_one + - ascPochhammer_mul + - "ascPochhammer_eval\u2082" + - descPochhammer_int_eq_ascFactorial + - monic_descPochhammer + - descPochhammer_natDegree + - ascPochhammer_succ_left + - ascPochhammer_eval_cast + - ascPochhammer_succ_comp_X_add_one + - ascPochhammer_map + - path: Mathlib/Probability/Kernel/Composition.lean + theorems: + - ProbabilityTheory.kernel.compProdFun_iUnion + - ProbabilityTheory.kernel.compProd_zero_right + - ProbabilityTheory.kernel.compProd_restrict_left + - ProbabilityTheory.kernel.lintegral_compProd' + - ProbabilityTheory.kernel.fst_apply' + - ProbabilityTheory.kernel.fst_compProd + - ProbabilityTheory.kernel.compProd_eq_sum_compProd_right + - ProbabilityTheory.kernel.compProd_eq_tsum_compProd + - ProbabilityTheory.kernel.ae_null_of_compProd_null + - ProbabilityTheory.kernel.compProd_of_not_isSFiniteKernel_right + - ProbabilityTheory.kernel.ae_compProd_of_ae_ae + - ProbabilityTheory.kernel.compProd_apply_univ_le + - ProbabilityTheory.kernel.prodMkLeft_add + - ProbabilityTheory.kernel.comap_id + - ProbabilityTheory.kernel.prod_const + - ProbabilityTheory.kernel.lintegral_compProd + - ProbabilityTheory.kernel.set_lintegral_compProd + - ProbabilityTheory.kernel.set_lintegral_compProd_univ_right + - ProbabilityTheory.kernel.fst_map_prod + - ProbabilityTheory.kernel.compProd_eq_sum_compProd_left + - ProbabilityTheory.kernel.lintegral_map + - ProbabilityTheory.kernel.snd_map_prod + - ProbabilityTheory.kernel.sum_comap_seq + - ProbabilityTheory.kernel.compProd_add_right + - ProbabilityTheory.kernel.compProdFun_empty + - ProbabilityTheory.kernel.compProd_eq_sum_compProd + - ProbabilityTheory.kernel.lintegral_swapRight + - ProbabilityTheory.kernel.comp_assoc + - ProbabilityTheory.kernel.map_zero + - ProbabilityTheory.kernel.compProdFun_eq_tsum + - ProbabilityTheory.kernel.swapRight_apply' + - ProbabilityTheory.kernel.fst_prod + - ProbabilityTheory.kernel.comp_deterministic_eq_comap + - ProbabilityTheory.kernel.prodMkRight_add + - ProbabilityTheory.kernel.sum_map_seq + - ProbabilityTheory.kernel.measurable_compProdFun + - ProbabilityTheory.kernel.snd_apply' + - ProbabilityTheory.kernel.compProdFun_tsum_left + - ProbabilityTheory.kernel.comap_zero + - ProbabilityTheory.kernel.set_lintegral_compProd_univ_left + - ProbabilityTheory.kernel.snd_prod + - ProbabilityTheory.kernel.prodMkRight_zero + - ProbabilityTheory.kernel.lintegral_swapLeft + - ProbabilityTheory.kernel.measurable_compProdFun_of_finite + - ProbabilityTheory.kernel.fst_swapRight + - ProbabilityTheory.kernel.compProd_restrict_right + - ProbabilityTheory.kernel.fst_zero + - ProbabilityTheory.kernel.lintegral_prod + - ProbabilityTheory.kernel.lintegral_comp + - ProbabilityTheory.kernel.lintegral_fst + - ProbabilityTheory.kernel.compProd_add_left + - ProbabilityTheory.kernel.compProdFun_tsum_right + - ProbabilityTheory.kernel.comp_apply' + - ProbabilityTheory.kernel.prod_apply' + - ProbabilityTheory.kernel.deterministic_comp_eq_map + - ProbabilityTheory.kernel.map_id + - ProbabilityTheory.kernel.prodMkLeft_zero + - ProbabilityTheory.kernel.prod_apply + - ProbabilityTheory.kernel.comap_map_comm + - ProbabilityTheory.kernel.compProd_null + - ProbabilityTheory.kernel.compProd_zero_left + - ProbabilityTheory.kernel.map_apply' + - ProbabilityTheory.kernel.compProd_restrict + - ProbabilityTheory.kernel.compProd_of_not_isSFiniteKernel_left + - ProbabilityTheory.kernel.snd_zero + - ProbabilityTheory.kernel.compProd_apply_eq_compProdFun + - ProbabilityTheory.kernel.ae_kernel_lt_top + - "ProbabilityTheory.kernel.lintegral_compProd\u2080" + - ProbabilityTheory.kernel.comp_eq_snd_compProd + - ProbabilityTheory.kernel.lintegral_snd + - ProbabilityTheory.kernel.map_const + - path: Mathlib/Analysis/Calculus/TangentCone.lean + theorems: + - uniqueDiffOn_Ico + - uniqueDiffWithinAt_convex + - uniqueDiffOn_Ici + - subset_tangentCone_prod_left + - uniqueDiffWithinAt_univ + - tangentCone_mono_nhds + - uniqueDiffOn_Iic + - uniqueDiffWithinAt_Iio + - subset_tangentCone_prod_right + - uniqueDiffOn_Icc + - UniqueDiffWithinAt.mono_nhds + - UniqueDiffWithinAt.univ_pi + - uniqueDiffWithinAt_Ioi + - mem_tangentCone_of_openSegment_subset + - UniqueDiffWithinAt.pi + - mapsTo_tangentCone_pi + - tangentConeAt.lim_zero + - mem_tangentConeAt_of_pow_smul + - uniqueDiffOn_Ioc + - tangentCone_mono + - path: Mathlib/RingTheory/NonUnitalSubring/Basic.lean + theorems: + - NonUnitalRingHom.map_range + - NonUnitalSubring.prod_top + - NonUnitalRingHom.range_top_iff_surjective + - NonUnitalSubring.mem_sSup_of_directedOn + - NonUnitalSubring.coe_iInf + - NonUnitalSubring.top_prod + - NonUnitalRingHom.range_eq_map + - NonUnitalSubring.mem_iSup_of_directed + - NonUnitalSubring.coe_sSup_of_directedOn + - NonUnitalSubring.coe_iSup_of_directed + - NonUnitalSubring.mem_bot + - NonUnitalSubring.closure_induction' + - NonUnitalSubring.mem_iInf + - NonUnitalSubring.coe_eq_zero_iff + - NonUnitalSubring.mem_closure_iff + - path: Mathlib/Data/List/Rotate.lean + theorems: + - List.mem_cyclicPermutations_self + - List.IsRotated.symm + - List.Nodup.cyclicPermutations + - List.rotate_injective + - List.length_cyclicPermutations_cons + - List.isRotated_iff_mem_map_range + - List.zipWith_rotate_distrib + - List.rotate_replicate + - List.length_mem_cyclicPermutations + - List.rotate_length_mul + - List.head?_rotate + - List.cyclicPermutations_eq_singleton_iff + - List.rotate_eq_drop_append_take + - List.rotate_eq_iff + - List.reverse_rotate + - List.isRotated_nil_iff + - List.IsRotated.cyclicPermutations + - List.rotate'_length + - List.isRotated_append + - List.map_rotate + - List.isRotated_singleton_iff + - List.get_eq_get_rotate + - List.nthLe_cyclicPermutations + - List.isRotated_iff_mod + - List.Nodup.rotate_eq_self_iff + - List.zipWith_rotate_one + - List.IsRotated.reverse + - List.nil_eq_rotate_iff + - List.isRotated_nil_iff' + - List.isRotated_cyclicPermutations_iff + - List.IsRotated.trans + - List.rotate'_mod + - List.rotate_cons_succ + - List.prod_rotate_eq_one_of_prod_eq_one + - List.singleton_eq_rotate_iff + - List.rotate_eq_singleton_iff + - List.rotate_nil + - List.isRotated_reverse_comm_iff + - List.length_rotate + - List.mem_rotate + - List.rotate'_eq_drop_append_take + - List.length_cyclicPermutations_of_ne_nil + - List.Nodup.rotate_congr + - List.IsRotated.map + - List.IsRotated.refl + - List.rotate_mod + - List.rotate_append_length_eq + - List.rotate_one_eq_self_iff_eq_replicate + - List.isRotated_concat + - List.isRotated_reverse_iff + - List.rotate_eq_drop_append_take_mod + - List.rotate_reverse + - List.length_rotate' + - List.cyclicPermutations_eq_nil_iff + - List.cyclicPermutations_rotate + - List.mem_cyclicPermutations_iff + - List.rotate_length + - List.rotate'_zero + - List.rotate'_rotate' + - List.rotate_eq_self_iff_eq_replicate + - List.rotate_perm + - List.get?_rotate + - List.isRotated_singleton_iff' + - List.cyclicPermutations_of_ne_nil + - List.rotate_zero + - List.rotate_eq_rotate' + - List.rotate'_nil + - path: Mathlib/RingTheory/HahnSeries/Summable.lean + theorems: + - HahnSeries.addVal_le_of_coeff_ne_zero + - HahnSeries.SummableFamily.one_sub_self_mul_hsum_powers + - HahnSeries.SummableFamily.hsum_smul + - HahnSeries.isUnit_iff + - HahnSeries.SummableFamily.hsum_ofFinsupp + - HahnSeries.SummableFamily.embDomain_succ_smul_powers + - HahnSeries.isPWO_iUnion_support_powers + - HahnSeries.SummableFamily.embDomain_notin_range + - HahnSeries.SummableFamily.hsum_embDomain + - HahnSeries.unit_aux + - HahnSeries.SummableFamily.embDomain_image + - HahnSeries.SummableFamily.support_hsum_subset + - HahnSeries.SummableFamily.hsum_sub + - HahnSeries.SummableFamily.hsum_add + - path: Mathlib/Data/Matroid/Basic.lean + theorems: + - Matroid.Base.ncard_diff_comm + - Matroid.Basis.basis_subset + - Matroid.Basis.basis_union_of_subset + - Matroid.indep_or_dep + - Matroid.basis'_iff_basis_inter_ground + - Matroid.indep_iff_not_dep + - Matroid.exists_basis_union_inter_basis + - Matroid.eq_of_indep_iff_indep_forall + - Matroid.not_indep_iff + - Matroid.Dep.nonempty + - Matroid.not_dep_iff + - Matroid.Basis.dep_of_ssubset + - Matroid.Indep.exists_insert_of_not_base + - Matroid.Base.ncard_eq_ncard_of_base + - Matroid.Basis.basis_inter_ground + - Matroid.exists_basis_subset_basis + - Matroid.Base.card_eq_card_of_base + - Matroid.Basis.insert_basis_insert + - Matroid.basis_self_iff_indep + - Matroid.Base.eq_of_subset_indep + - Matroid.basis_empty_iff + - Matroid.basis_iff + - Matroid.basis_ground_iff + - Matroid.Base.exchange_mem + - Matroid.Basis.exists_base + - Matroid.Indep.basis_of_maximal_subset + - Matroid.Indep.subset_basis'_of_subset + - Matroid.setOf_base_eq_maximals_setOf_indep + - Matroid.Base.exchange_base_of_indep + - Matroid.Base.insert_dep + - Matroid.Indep.basis_iff_forall_insert_dep + - Matroid.Base.eq_exchange_of_diff_eq_singleton + - Matroid.ExchangeProperty.encard_diff_le_aux + - Matroid.Basis.basis_union + - Matroid.Indep.exists_insert_of_not_mem_maximals + - Matroid.Basis.exists_basis_inter_eq_of_superset + - Matroid.Base.rkPos_of_nonempty + - Matroid.base_iff_maximal_indep + - Matroid.setOf_basis_eq + - Matroid.Basis.basis_sUnion + - Matroid.Indep.subset_basis_of_subset + - Matroid.Basis.basis_iUnion + - Matroid.Indep.subset_ground + - Matroid.Indep.basis_setOf_insert_basis + - Matroid.Base.base_of_basis_superset + - Matroid.Indep.basis_insert_iff + - Matroid.Basis.union_basis_union + - Matroid.Basis.iUnion_basis_iUnion + - Matroid.eq_iff_indep_iff_indep_forall + - Matroid.Indep.exists_base_subset_union_base + - Matroid.basis'_iff_basis + - Matroid.not_finiteRk + - Matroid.not_infiniteRk + - Matroid.ExchangeProperty.encard_base_eq + - Matroid.Base.nonempty + - Matroid.eq_of_base_iff_base_forall + - Matroid.Indep.rkPos_of_nonempty + - Matroid.Base.exchange_base_of_indep' + - Matroid.basis_iff' + - Matroid.basis_iff_mem_maximals + - Matroid.Base.basis_of_subset + - path: Mathlib/Topology/Algebra/UniformGroup.lean + theorems: + - Filter.HasBasis.uniformity_of_nhds_one_inv_mul + - uniformity_eq_comap_nhds_one + - DenseInducing.extend_Z_bilin_key + - UniformGroup.mk' + - uniformContinuous_of_tendsto_one + - UniformCauchySeqOn.div + - UniformGroup.toUniformSpace_eq + - uniformGroup_iInf + - UniformCauchySeqOn.mul + - uniformContinuous_of_continuousAt_one + - comm_topologicalGroup_is_uniform + - uniformity_eq_comap_inv_mul_nhds_one_swapped + - uniformity_eq_comap_inv_mul_nhds_one + - totallyBounded_iff_subset_finite_iUnion_nhds_one + - Filter.HasBasis.uniformity_of_nhds_one_inv_mul_swapped + - UniformContinuous.mul + - topologicalGroup_is_uniform_of_compactSpace + - Filter.HasBasis.uniformity_of_nhds_one + - tendsto_div_comap_self + - MonoidHom.tendsto_coe_cofinite_of_discrete + - uniformGroup_inf + - DenseInducing.extend_Z_bilin_aux + - UniformContinuous.zpow_const + - DenseInducing.extend_Z_bilin + - UniformGroup.uniformity_countably_generated + - UniformInducing.uniformGroup + - uniformContinuous_monoidHom_of_continuous + - UniformGroup.ext + - UniformContinuous.inv + - uniformEmbedding_translate_mul + - uniformity_eq_comap_nhds_one_swapped + - Filter.HasBasis.uniformity_of_nhds_one_swapped + - path: Mathlib/Topology/Instances/NNReal.lean + theorems: + - HasSum.toNNReal + - NNReal.summable_coe + - NNReal.tsum_mul_right + - NNReal.tendsto_cofinite_zero_of_summable + - NNReal.nhds_zero + - NNReal.coe_tsum + - NNReal.tsum_mul_left + - tendsto_real_toNNReal_atTop + - NNReal.hasSum_nat_add_iff + - NNReal.summable_comp_injective + - NNReal.hasSum_coe + - path: Mathlib/Data/Nat/Size.lean + theorems: + - Nat.lt_size_self + - Nat.shiftLeft'_ne_zero_left + - Nat.size_eq_bits_len + - Nat.shiftLeft'_tt_eq_mul_pow + - Nat.size_bit + - Nat.size_zero + - Nat.size_shiftLeft' + - Nat.lt_size + - Nat.size_pos + - Nat.size_eq_zero + - Nat.size_le + - Nat.size_one + - Nat.size_shiftLeft + - path: Mathlib/ModelTheory/LanguageMap.lean + theorems: + - FirstOrder.Language.LHom.Injective.isExpansionOn_default + - FirstOrder.Language.LHom.map_constants_comp_sumInl + - FirstOrder.Language.LHom.id_comp + - FirstOrder.Language.card_withConstants + - FirstOrder.Language.LHom.funext + - FirstOrder.Language.constantsOnMap_isExpansionOn + - FirstOrder.Language.card_constantsOn + - FirstOrder.Language.LHom.comp_id + - FirstOrder.Language.withConstants_funMap_sum_inr + - path: Mathlib/Data/Set/Card.lean + theorems: + - Set.ncard_exchange' + - Function.Embedding.enccard_le + - Set.ncard_le_ncard_diff_add_ncard + - Set.Infinite.ncard + - Set.ncard_empty + - Set.one_lt_encard_iff + - Set.Finite.encard_lt_top + - Set.encard_image_le + - Set.encard_univ + - Set.ncard_le_ncard + - Set.encard_tsub_one_le_encard_diff_singleton + - Set.InjOn.encard_image + - Set.ncard_lt_ncard_iff_ncard_diff_lt_ncard_diff + - Set.encard_le_encard_of_injOn + - Set.le_ncard_diff + - Set.finite_iff_finite_of_encard_eq_encard + - Set.ncard_le_one_iff_subset_singleton + - Set.encard_preimage_of_injective_subset_range + - Set.encard_ne_top_iff + - Set.inj_on_of_surj_on_of_ncard_le + - Set.encard_univ_coe + - Set.Infinite.exists_superset_ncard_eq + - Set.ncard_diff_singleton_lt_of_mem + - Set.Finite.injOn_of_encard_image_eq + - Set.exists_subset_or_subset_of_two_mul_lt_ncard + - Set.encard_eq_encard_iff_encard_diff_eq_encard_diff + - Set.encard_diff_add_encard_of_subset + - Set.ncard_le_one_of_subsingleton + - Set.encard_pos + - Set.ncard_insert_eq_ite + - Set.exists_superset_subset_encard_eq + - Set.ncard_lt_ncard + - Set.encard_diff_add_encard + - Set.ncard_diff_add_ncard + - Set.ncard_eq_succ + - Set.encard_eq_one + - Set.ncard_inter_add_ncard_union + - Set.pred_ncard_le_ncard_diff_singleton + - Set.exists_intermediate_Set + - Set.encard_pair + - Set.diff_nonempty_of_ncard_lt_ncard + - Set.ncard_union_eq + - Set.Nat.card_coe_set_eq + - Set.two_lt_ncard + - Set.ncard_univ + - Set.ncard_eq_three + - Set.ncard_eq_toFinset_card + - Set.ncard_singleton + - Set.eq_insert_of_ncard_eq_succ + - Set.sep_of_ncard_eq + - Set.ncard_diff_singleton_add_one + - Set.encard_diff_add_encard_inter + - Set.ncard_add_ncard_compl + - Set.finite_of_encard_le_coe + - Set.one_lt_ncard + - Set.encard_eq_two + - Set.Finite.exists_injOn_of_encard_le + - Set.ncard_eq_ncard_iff_ncard_diff_eq_ncard_diff + - Set.ncard_inter_add_ncard_diff_eq_ncard + - Set.surj_on_of_inj_on_of_ncard_le + - Set.ncard_insert_of_mem + - Set.encard_le_one_iff + - Set.encard_ne_zero + - Set.encard_union_eq + - Set.ncard_coe_Finset + - Set.nonempty_of_ncard_ne_zero + - Set.encard_insert_le + - Set.encard_singleton + - Set.encard_exchange + - Set.ncard_eq_zero + - Set.encard_eq_top_iff + - Set.exists_subset_encard_eq + - Set.exists_ne_map_eq_of_ncard_lt_of_maps_to + - Set.encard_lt_encard_iff_encard_diff_lt_encard_diff + - Set.encard_diff_singleton_of_mem + - Set.ncard_congr + - Set.encard_add_encard_compl + - Set.ncard_subtype + - Set.encard_union_add_encard_inter + - Set.ncard_le_one + - Set.encard_empty + - Set.ncard_preimage_of_injective_subset_range + - Set.injOn_of_ncard_image_eq + - Set.encard_congr + - Set.ncard_union_le + - Set.ncard_insert_of_not_mem + - Set.encard_le_coe_iff + - Set.ncard_eq_one + - Set.encard_le_one_iff_eq + - Set.ncard_eq_two + - Set.nonempty_of_encard_ne_zero + - Set.encard_le_card + - Set.exists_eq_insert_iff_ncard + - Set.ncard_diff_add_ncard_of_subset + - Set.ncard_union_add_ncard_inter + - Set.Finite.encard_eq_coe_toFinset_card + - Set.ncard_diff + - Set.ncard_le_ncard_iff_ncard_diff_le_ncard_diff + - Set.exists_ne_of_one_lt_ncard + - Set.Infinite.encard_eq + - Set.ncard_pos + - Set.encard_eq_add_one_iff + - Set.ncard_le_ncard_insert + - Set.ncard_le_one_iff_eq + - Set.two_lt_ncard_iff + - Set.Finite.cast_ncard_eq + - Set.eq_empty_or_encard_eq_top_or_encard_diff_singleton_lt + - Set.exists_ne_of_one_lt_encard + - Set.encard_insert_of_not_mem + - Set.ncard_diff_singleton_le + - Set.ncard_le_one_iff + - Set.Finite.eq_of_subset_of_encard_le + - Set.tsub_encard_le_encard_diff + - Set.encard_diff_singleton_add_one + - Set.exists_intermediate_set' + - Set.ncard_eq_of_bijective + - Set.Infinite.exists_subset_ncard_eq + - Set.Finite.eq_insert_of_subset_of_encard_eq_succ + - Set.encard_singleton_inter + - Set.ncard_image_le + - Set.Finite.exists_bijOn_of_encard_eq + - Set.exists_smaller_set + - Set.infinite_iff_infinite_of_encard_eq_encard + - Set.encard_eq_coe_toFinset_card + - Set.encard_coe_eq_coe_finsetCard + - Set.ncard_le_ncard_of_injOn + - Set.encard_exchange' + - Set.ncard_eq_toFinset_card' + - Set.encard_le_coe_iff_finite_ncard_le + - Set.le_ncard_of_inj_on_range + - Set.encard_eq_zero + - Set.encard_le_encard_iff_encard_diff_le_encard_diff + - path: Mathlib/Data/MvPolynomial/Basic.lean + theorems: + - "MvPolynomial.eval\u2082_mul_C" + - MvPolynomial.coeff_X' + - MvPolynomial.coeff_monomial_mul' + - MvPolynomial.aeval_unique + - MvPolynomial.coeff_mul_X' + - "MvPolynomial.eval\u2082_eq_eval_map" + - MvPolynomial.constantCoeff_comp_C + - MvPolynomial.C_mul_X_eq_monomial + - MvPolynomial.mem_support_iff + - MvPolynomial.monomial_add_single + - MvPolynomial.coeff_mul_monomial' + - MvPolynomial.support_symmDiff_support_subset_support_add + - MvPolynomial.C_eq_smul_one + - MvPolynomial.monomial_eq + - MvPolynomial.support_X + - MvPolynomial.coeff_X + - MvPolynomial.induction_on' + - "MvPolynomial.eval\u2082Hom_zero" + - "MvPolynomial.map_eval\u2082" + - MvPolynomial.map_map + - MvPolynomial.map_leftInverse + - "MvPolynomial.eval\u2082_map" + - MvPolynomial.map_surjective + - MvPolynomial.constantCoeff_C + - MvPolynomial.support_sdiff_support_subset_support_add + - MvPolynomial.monomial_single_add + - "MvPolynomial.eval\u2082_C" + - "MvPolynomial.comp_eval\u2082Hom" + - "MvPolynomial.eval\u2082_monomial" + - "MvPolynomial.eval\u2082Hom_congr" + - MvPolynomial.coeff_map + - MvPolynomial.monomial_sum_index + - "MvPolynomial.eval\u2082_eq'" + - "MvPolynomial.eval\u2082_mul_monomial" + - MvPolynomial.comp_aeval + - MvPolynomial.aevalTower_id + - "MvPolynomial.eval\u2082_comp_right" + - MvPolynomial.support_mul_X + - MvPolynomial.C_dvd_iff_map_hom_eq_zero + - MvPolynomial.ne_zero_iff + - MvPolynomial.coeff_X_mul' + - MvPolynomial.constantCoeff_monomial + - MvPolynomial.isRegular_X + - "MvPolynomial.eval\u2082_eta" + - "MvPolynomial.eval\u2082_pow" + - MvPolynomial.coeff_X_pow + - MvPolynomial.map_mapRange_eq_iff + - MvPolynomial.sum_def + - MvPolynomial.adjoin_range_X + - "MvPolynomial.eval\u2082Hom_eq_zero" + - MvPolynomial.not_mem_support_iff + - MvPolynomial.X_pow_eq_monomial + - "MvPolynomial.eval\u2082_add" + - MvPolynomial.aevalTower_ofId + - "MvPolynomial.eval\u2082_assoc" + - MvPolynomial.coeff_zero_X + - "MvPolynomial.eval\u2082Hom_monomial" + - MvPolynomial.eval_assoc + - Algebra.adjoin_eq_range + - MvPolynomial.C_eq_coe_nat + - MvPolynomial.support_monomial + - MvPolynomial.coeff_C_mul + - MvPolynomial.support_map_subset + - MvPolynomial.support_X_pow + - "MvPolynomial.eval\u2082_mul" + - "MvPolynomial.map_eval\u2082Hom" + - "MvPolynomial.eval\u2082_comp_left" + - MvPolynomial.X_ne_zero + - MvPolynomial.ringHom_ext + - MvPolynomial.eq_C_of_isEmpty + - MvPolynomial.constantCoeff_comp_map + - MvPolynomial.ext_iff + - MvPolynomial.eq_zero_iff + - MvPolynomial.map_aeval + - MvPolynomial.monic_monomial_eq + - "MvPolynomial.eval\u2082_congr" + - MvPolynomial.support_map_of_injective + - "MvPolynomial.eval\u2082_X" + - MvPolynomial.map_injective + - MvPolynomial.C_dvd_iff_dvd_coeff + - MvPolynomial.C_mul_X_pow_eq_monomial + - MvPolynomial.C_surjective + - MvPolynomial.smul_eval + - MvPolynomial.support_X_mul + - MvPolynomial.constantCoeff_X + - MvPolynomial.C_mul_monomial + - MvPolynomial.eval_map + - "MvPolynomial.eval\u2082_mem" + - path: Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean + theorems: + - MeasureTheory.snormEssSup_piecewise + - "Continuous.mem\u2112p_top_of_hasCompactSupport" + - MeasureTheory.snorm'_const + - MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top + - MeasureTheory.snorm_le_of_ae_bound + - MeasureTheory.snorm_eq_lintegral_rpow_nnnorm + - MeasureTheory.snorm_const + - MeasureTheory.snorm_eq_snorm' + - MeasureTheory.snorm_le_of_ae_nnnorm_bound + - "MeasureTheory.mem\u2112p_zero_iff_aestronglyMeasurable" + - MeasurableEmbedding.snorm_map_measure + - MeasureTheory.snorm_eq_zero_and_zero_of_ae_le_mul_neg + - MeasureTheory.snorm_mono_nnnorm_ae + - MeasureTheory.snorm_one_eq_lintegral_nnnorm + - MeasureTheory.snorm'_measure_zero_of_exponent_zero + - "MeasureTheory.mem\u2112p_top_of_bound" + - MeasureTheory.snormEssSup_eq_zero_iff + - MeasureTheory.snorm'_mono_nnnorm_ae + - MeasureTheory.snorm'_const' + - MeasureTheory.snorm'_eq_zero_of_ae_zero + - MeasureTheory.snorm_norm_rpow + - MeasureTheory.snorm_one_add_measure + - MeasureTheory.snormEssSup_smul_measure + - "MeasureTheory.Mem\u2112p.re" + - MeasureTheory.snorm'_mono_measure + - "MeasureTheory.mem\u2112p_congr_ae" + - MeasureTheory.snormEssSup_const_smul + - MeasureTheory.ae_bdd_liminf_atTop_rpow_of_snorm_bdd + - MeasureTheory.snorm_smul_measure_of_ne_zero_of_ne_top + - MeasureTheory.ae_eq_zero_of_snorm'_eq_zero + - "MeasureTheory.Mem\u2112p.im" + - MeasureTheory.snormEssSup_le_nnreal_smul_snormEssSup_of_ae_le_mul + - MeasureTheory.le_mul_iff_eq_zero_of_nonneg_of_neg_of_nonneg + - MeasureTheory.lintegral_rpow_nnnorm_eq_rpow_snorm' + - MeasureTheory.snorm_map_measure + - MeasureTheory.snorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top + - MeasureTheory.snorm_one_smul_measure + - MeasureTheory.AEEqFun.snorm_compMeasurePreserving + - MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top + - MeasureTheory.snorm_smul_measure_of_ne_top + - MeasureTheory.snorm_indicator_ge_of_bdd_below + - MeasureTheory.snorm_const_smul + - "MeasureTheory.zero_mem\u2112p" + - MeasureTheory.snorm'_le_nnreal_smul_snorm'_of_ae_le_mul + - "MeasureTheory.Mem\u2112p.of_measure_le_smul" + - "MeasurableEmbedding.mem\u2112p_map_measure_iff" + - MeasureTheory.snormEssSup_measure_zero + - MeasureTheory.snorm'_norm + - MeasureTheory.snorm'_zero' + - MeasureTheory.snormEssSup_const + - MeasureTheory.snorm_eq_zero_iff + - MeasureTheory.ae_bdd_liminf_atTop_of_snorm_bdd + - MeasureTheory.snorm'_zero + - MeasureTheory.snorm'_const_smul + - MeasureTheory.snorm'_smul_measure + - MeasureTheory.snorm_const_lt_top_iff + - MeasureTheory.snorm'_exponent_zero + - MeasureTheory.snorm_exponent_zero + - "MeasureTheory.mem\u2112p_const_iff" + - MeasureTheory.snorm_const' + - MeasureTheory.snormEssSup_zero + - MeasureTheory.snorm_le_nnreal_smul_snorm_of_ae_le_mul + - MeasureTheory.snorm'_const_of_isProbabilityMeasure + - MeasureTheory.snorm'_eq_zero_of_ae_zero' + - MeasureTheory.snorm_zero + - "MeasureTheory.mem\u2112p_norm_iff" + - MeasureTheory.snorm_smul_measure_of_ne_zero + - MeasureTheory.snorm'_norm_rpow + - "MeasureTheory.mem\u2112p_top_const" + - "MeasureTheory.Mem\u2112p.neg" + - MeasureTheory.snorm'_neg + - "MeasureTheory.mem\u2112p_map_measure_iff" + - MeasureTheory.snorm_neg + - MeasureTheory.snorm_exponent_top + - MeasureTheory.snorm'_congr_nnnorm_ae + - MeasureTheory.snorm_measure_zero + - MeasureTheory.snorm'_measure_zero_of_pos + - MeasureTheory.snorm_indicator_sub_indicator + - MeasureTheory.snorm_mono_measure + - "MeasureTheory.Mem\u2112p.norm" + - path: Mathlib/Algebra/Divisibility/Units.lean + theorems: + - Units.dvd_mul_left + - isRelPrime_mul_unit_left + - IsRelPrime.mul_dvd_of_right_isPrimal + - IsRelPrime.mul_right + - IsRelPrime.mul_dvd_of_left_isPrimal + - isRelPrime_mul_unit_right_left + - isUnit_iff_dvd_one + - IsUnit.dvd_mul_left + - Units.dvd_mul_right + - IsUnit.dvd_mul_right + - IsUnit.mul_left_dvd + - Units.mul_right_dvd + - IsRelPrime.of_mul_right_left + - IsRelPrime.mul_left + - IsRelPrime.of_dvd_left + - Units.coe_dvd + - IsUnit.mul_right_dvd + - IsRelPrime.dvd_of_dvd_mul_right_of_isPrimal + - IsUnit.dvd + - Units.mul_left_dvd + - isRelPrime_mul_unit_right + - path: Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean + theorems: + - Real.smoothTransition.one_of_one_le + - Real.smoothTransition.projIcc + - expNegInvGlue.contDiff + - expNegInvGlue.nonneg + - expNegInvGlue.zero_of_nonpos + - Real.smoothTransition.zero_iff_nonpos + - expNegInvGlue.tendsto_polynomial_inv_mul_zero + - expNegInvGlue.contDiff_polynomial_eval_inv_mul + - path: Mathlib/Data/Real/NNReal.lean + theorems: + - Real.nat_cast_le_toNNReal + - NNReal.coe_iSup + - NNReal.iSup_mul + - NNReal.exists_pow_lt_of_lt_one + - Real.le_toNNReal_iff_coe_le' + - NNReal.lt_iff_exists_rat_btwn + - Real.toNNReal_lt_nat_cast + - Real.toNNReal_mul + - Real.cast_natAbs_eq_nnabs_cast + - Real.one_le_toNNReal + - NNReal.le_inv_iff_mul_le + - NNReal.le_iInf_add_iInf + - NNReal.coe_iInf + - Real.toNNReal_lt_one + - Real.one_lt_toNNReal + - Real.toNNReal_div + - Real.nnabs_coe + - Real.toNNReal_lt_nat_cast' + - Real.toNNReal_add + - Real.le_toNNReal_iff_coe_le + - NNReal.iInf_mul + - Real.toNNReal_le_toNNReal_iff + - NNReal.coe_sSup + - Real.toNNReal_inv + - Real.nat_cast_le_toNNReal' + - Real.toNNReal_le_nat_cast + - NNReal.sInf_empty + - NNReal.mul_iInf + - NNReal.le_mul_iInf + - Real.toNNReal_eq_zero + - Real.toNNReal_eq_toNNReal_iff + - Real.toNNReal_div' + - NNReal.iInf_const_zero + - NNReal.le_iInf_mul + - NNReal.mul_le_iff_le_inv + - NNReal.inv_le_of_le_mul + - NNReal.div_le_of_le_mul + - NNReal.iInf_empty + - NNReal.toNNReal_coe_nat + - NNReal.iSup_mul_le + - NNReal.one_le_coe + - NNReal.div_lt_one_of_lt + - Set.OrdConnected.image_real_toNNReal + - NNReal.inv_le + - NNReal.coe_le_one + - Real.toNNReal_abs + - Real.nat_cast_lt_toNNReal + - Real.toNNReal_prod_of_nonneg + - NNReal.mul_iSup + - NNReal.mul_lt_of_lt_div + - NNReal.coe_sInf + - Real.toNNReal_of_nonneg + - Real.toNNReal_pow + - Real.toNNReal_le_toNNReal_iff_of_pos + - NNReal.lt_inv_iff_mul_lt + - NNReal.coe_lt_one + - Real.toNNReal_le_toNNReal_iff' + - NNReal.le_of_forall_lt_one_mul_le + - NNReal.coe_sub + - Real.nnabs_of_nonneg + - NNReal.finset_sup_div + - NNReal.coe_eq_zero + - NNReal.one_lt_coe + - NNReal.coe_eq_one + - NNReal.mul_iSup_le + - NNReal.inv_lt_one_iff + - Real.toNNReal_le_one + - Real.toNNReal_lt_iff_lt_coe + - path: Mathlib/MeasureTheory/Measure/Portmanteau.lean + theorems: + - MeasureTheory.tendsto_measure_of_le_liminf_measure_of_limsup_measure_le + - MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto + - MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto + - MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure + - MeasureTheory.exists_null_frontier_thickening + - MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto + - MeasureTheory.limsup_measure_closed_le_iff_liminf_measure_open_ge + - MeasureTheory.integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure + - MeasureTheory.le_measure_compl_liminf_of_limsup_measure_le + - MeasureTheory.limsup_measure_compl_le_of_le_liminf_measure + - MeasureTheory.lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure + - MeasureTheory.exists_null_frontiers_thickening + - MeasureTheory.tendsto_of_forall_isOpen_le_liminf + - path: Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean + theorems: + - Complex.hasSum_taylorSeries_neg_log + - Complex.norm_log_one_add_sub_self_le + - Complex.norm_one_add_mul_inv_le + - Complex.norm_log_one_sub_inv_add_logTaylor_neg_le + - Complex.hasDerivAt_logTaylor + - Complex.continuousOn_one_add_mul_inv + - Complex.norm_log_one_sub_inv_sub_self_le + - Complex.integrable_pow_mul_norm_one_add_mul_inv + - Complex.hasSum_taylorSeries_log + - Complex.log_eq_integral + - Complex.logTaylor_at_zero + - Complex.logTaylor_succ + - Complex.logTaylor_zero + - Complex.hasDerivAt_log_sub_logTaylor + - Complex.log_inv_eq_integral + - Complex.norm_log_sub_logTaylor_le + - path: Mathlib/Combinatorics/SetFamily/Shatter.lean + theorems: + - Finset.Shatters.mono_right + - Finset.Shatters.of_compression + - Finset.Shatters.subset_iff + - Finset.mem_shatterer + - Finset.shatterer_mono + - Finset.card_shatterer_le_sum_vcDim + - Finset.card_le_card_shatterer + - Finset.shatterer_idem + - Finset.shatterer_eq + - Finset.shatters_iff + - Finset.shatters_shatterer + - Finset.isLowerSet_shatterer + - Finset.shatters_univ + - Finset.shatterer_compress_subset_shatterer + - Finset.aux + - path: Mathlib/RingTheory/Polynomial/Content.lean + theorems: + - Polynomial.aeval_primPart_eq_zero + - Polynomial.isUnit_primPart_C + - Polynomial.isPrimitive_iff_content_eq_one + - Polynomial.content_X_pow + - Polynomial.IsPrimitive.primPart_eq + - Polynomial.natDegree_primPart + - Polynomial.content_C + - Polynomial.isPrimitive_primPart + - Polynomial.content_C_mul + - Polynomial.eq_C_content_mul_primPart + - Polynomial.content_X + - Polynomial.degree_gcd_le_left + - Polynomial.Monic.isPrimitive + - Polynomial.degree_gcd_le_right + - Polynomial.content_monomial + - Polynomial.IsPrimitive.mul + - Polynomial.content_mul + - Polynomial.gcd_content_eq_of_dvd_sub + - Polynomial.dvd_iff_content_dvd_content_and_primPart_dvd_primPart + - Polynomial.dvd_content_iff_C_dvd + - Polynomial.exists_primitive_lcm_of_isPrimitive + - Polynomial.content_eq_zero_iff + - Polynomial.IsPrimitive.ne_zero + - Polynomial.content_eq_gcd_range_of_lt + - "Polynomial.eval\u2082_primPart_eq_zero" + - Polynomial.content_zero + - Polynomial.content_eq_gcd_leadingCoeff_content_eraseLead + - Polynomial.content_dvd_coeff + - Polynomial.IsPrimitive.dvd_primPart_iff_dvd + - Polynomial.content_X_mul + - Polynomial.content_mul_aux + - Polynomial.content_one + - path: Mathlib/RingTheory/WittVector/Compare.lean + theorems: + - TruncatedWittVector.zmodEquivTrunc_apply + - TruncatedWittVector.commutes_symm' + - WittVector.toZModPow_compat + - TruncatedWittVector.commutes' + - WittVector.toPadicInt_comp_fromPadicInt_ext + - WittVector.zmodEquivTrunc_compat + - TruncatedWittVector.commutes_symm + - TruncatedWittVector.card_zmod + - WittVector.toPadicInt_comp_fromPadicInt + - WittVector.fromPadicInt_comp_toPadicInt + - WittVector.fromPadicInt_comp_toPadicInt_ext + - TruncatedWittVector.eq_of_le_of_cast_pow_eq_zero + - path: Mathlib/MeasureTheory/Constructions/Cylinders.lean + theorems: + - MeasureTheory.isPiSystem_squareCylinders + - MeasureTheory.cylinder_univ + - MeasureTheory.union_mem_measurableCylinders + - MeasureTheory.union_cylinder + - MeasureTheory.diff_cylinder_same + - MeasureTheory.inter_cylinder + - MeasureTheory.cylinder_empty + - MeasureTheory.empty_mem_measurableCylinders + - MeasureTheory.compl_cylinder + - MeasureTheory.inter_mem_measurableCylinders + - MeasureTheory.generateFrom_measurableCylinders + - MeasureTheory.cylinder_mem_measurableCylinders + - MeasureTheory.compl_mem_measurableCylinders + - MeasureTheory.diff_mem_measurableCylinders + - MeasureTheory.inter_cylinder_same + - MeasureTheory.generateFrom_squareCylinders + - MeasureTheory.cylinder_eq_cylinder_union + - MeasureTheory.squareCylinders_eq_iUnion_image + - MeasureTheory.union_cylinder_same + - MeasureTheory.comap_eval_le_generateFrom_squareCylinders_singleton + - MeasureTheory.mem_measurableCylinders + - MeasureTheory.univ_mem_measurableCylinders + - MeasureTheory.cylinder_eq_empty_iff + - MeasureTheory.eq_of_cylinder_eq_of_subset + - path: Mathlib/LinearAlgebra/BilinearForm/Hom.lean + theorems: + - BilinForm.sum_repr_mul_repr_mul + - BilinForm.sum_apply + - BilinForm.compRight_id + - BilinForm.comp_id_id + - BilinForm.comp_inj + - BilinForm.comp_id_left + - BilinForm.compLeft_id + - BilinForm.congr_symm + - BilinForm.comp_id_right + - path: Mathlib/Probability/Martingale/Basic.lean + theorems: + - MeasureTheory.martingale_of_condexp_sub_eq_zero_nat + - MeasureTheory.martingale_zero + - MeasureTheory.Supermartingale.smul_nonpos + - MeasureTheory.Martingale.add + - MeasureTheory.Submartingale.add + - MeasureTheory.Martingale.sub + - MeasureTheory.Submartingale.zero_le_of_predictable + - MeasureTheory.Submartingale.condexp_sub_nonneg + - MeasureTheory.Submartingale.neg + - MeasureTheory.Submartingale.sum_mul_sub + - MeasureTheory.supermartingale_nat + - MeasureTheory.Supermartingale.le_zero_of_predictable + - MeasureTheory.Supermartingale.set_integral_le + - MeasureTheory.Submartingale.set_integral_le + - MeasureTheory.supermartingale_of_set_integral_succ_le + - MeasureTheory.Martingale.eq_zero_of_predictable + - MeasureTheory.Martingale.smul + - MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat + - MeasureTheory.Martingale.set_integral_eq + - MeasureTheory.Submartingale.sub_supermartingale + - MeasureTheory.martingale_const + - MeasureTheory.Submartingale.smul_nonneg + - MeasureTheory.submartingale_of_set_integral_le_succ + - MeasureTheory.submartingale_nat + - MeasureTheory.Submartingale.sup + - MeasureTheory.martingale_const_fun + - MeasureTheory.submartingale_of_condexp_sub_nonneg_nat + - MeasureTheory.submartingale_of_condexp_sub_nonneg + - MeasureTheory.Supermartingale.add + - MeasureTheory.Supermartingale.sub_submartingale + - MeasureTheory.Supermartingale.neg + - MeasureTheory.Submartingale.smul_nonpos + - MeasureTheory.submartingale_of_set_integral_le + - MeasureTheory.Supermartingale.smul_nonneg + - path: Mathlib/CategoryTheory/Shift/Basic.lean + theorems: + - CategoryTheory.shiftFunctorAdd_add_zero_hom_app + - CategoryTheory.shift_shiftFunctorCompIsoId_hom_app + - CategoryTheory.ShiftMkCore.add_zero_inv_app + - CategoryTheory.shiftFunctorZero_inv_app_shift + - CategoryTheory.shiftFunctorAdd'_add_zero + - CategoryTheory.shiftFunctorComm_hom_app_comp_shift_shiftFunctorAdd_hom_app + - CategoryTheory.shift_shiftFunctorCompIsoId_neg_add_self_inv_app + - CategoryTheory.shiftFunctorAdd'_assoc + - CategoryTheory.ShiftMkCore.shiftFunctorZero_eq + - CategoryTheory.map_hasShiftOfFullyFaithful_zero_inv_app + - CategoryTheory.shiftFunctorComm_eq + - CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd + - CategoryTheory.shiftComm' + - CategoryTheory.shiftComm_symm + - CategoryTheory.shiftFunctorAdd_assoc_inv_app + - CategoryTheory.shiftFunctorAdd'_assoc_inv_app + - CategoryTheory.shiftFunctorAdd_add_zero_inv_app + - CategoryTheory.shiftFunctorAdd'_zero_add + - CategoryTheory.shiftFunctorAdd_zero_add_inv_app + - CategoryTheory.shiftFunctorComm_zero_hom_app + - CategoryTheory.shift_shiftFunctorCompIsoId_add_neg_self_hom_app + - CategoryTheory.shift_shiftFunctorCompIsoId_add_neg_self_inv_app + - CategoryTheory.shift_shiftFunctorCompIsoId_neg_add_self_hom_app + - CategoryTheory.shiftFunctorAdd_zero_add_hom_app + - CategoryTheory.shiftFunctorAdd'_assoc_hom_app + - CategoryTheory.shiftFunctorAdd'_add_zero_inv_app + - CategoryTheory.ShiftMkCore.assoc_inv_app + - CategoryTheory.shiftFunctorAdd'_zero_add_hom_app + - CategoryTheory.map_hasShiftOfFullyFaithful_add_hom_app + - CategoryTheory.shift_shift' + - CategoryTheory.ShiftMkCore.zero_add_inv_app + - CategoryTheory.shiftFunctorAdd_assoc_hom_app + - CategoryTheory.ShiftMkCore.shiftFunctor_eq + - CategoryTheory.shiftFunctorZero_hom_app_shift + - CategoryTheory.shiftFunctorComm_eq_refl + - CategoryTheory.shiftComm_hom_comp + - CategoryTheory.shiftFunctorAdd'_zero_add_inv_app + - CategoryTheory.ShiftMkCore.shiftFunctorAdd_eq + - CategoryTheory.shiftZero' + - CategoryTheory.shiftFunctorAdd'_add_zero_hom_app + - CategoryTheory.shiftFunctorComm_symm + - CategoryTheory.shiftFunctorAdd_assoc + - CategoryTheory.map_hasShiftOfFullyFaithful_zero_hom_app + - CategoryTheory.map_hasShiftOfFullyFaithful_add_inv_app + - CategoryTheory.shift_shiftFunctorCompIsoId_inv_app + - path: Mathlib/RingTheory/Perfection.lean + theorems: + - Perfection.coeff_frobenius + - ModP.v_p_lt_preVal + - ModP.mul_ne_zero_of_pow_p_ne_zero + - Perfection.coeff_add_ne_zero + - PreTilt.map_eq_zero + - PerfectionMap.mk' + - Perfection.coeff_iterate_frobenius + - Perfection.frobenius_pthRoot + - ModP.preVal_eq_zero + - PreTilt.valAux_mul + - PerfectionMap.map_eq_map + - PreTilt.valAux_one + - ModP.preVal_add + - PreTilt.valAux_eq + - ModP.preVal_mk + - Perfection.coeff_pow_p + - Perfection.pthRoot_frobenius + - ModP.preVal_mul + - ModP.v_p_lt_val + - PerfectionMap.id + - PreTilt.valAux_add + - path: Mathlib/Data/Nat/Digits.lean + theorems: + - Nat.getLast_digit_ne_zero + - Nat.ofDigits_lt_base_pow_length + - Nat.eleven_dvd_of_palindrome + - Nat.modEq_digits_sum + - Nat.ofDigits_div_pow_eq_ofDigits_drop + - Nat.ofDigits_append + - Nat.digits_zero + - Nat.coe_ofDigits + - Nat.ofDigits_digits_append_digits + - Nat.digits_add + - Nat.digits_getLast + - Nat.ofDigits_modEq' + - Nat.ofDigits_monotone + - Nat.modEq_three_digits_sum + - Nat.modEq_nine_digits_sum + - Nat.digits_of_two_le_of_pos + - Nat.digits_two_eq_bits + - Nat.digits_append_digits + - Nat.dvd_iff_dvd_digits_sum + - Nat.digits_len + - Nat.NormDigits.digits_one + - Nat.three_dvd_iff + - Nat.digits_def' + - Nat.digit_sum_le + - Nat.ofDigits_neg_one + - Nat.repr_length + - Nat.nat_repr_len_aux + - Nat.ofDigits_singleton + - Nat.ofDigits_digits + - Nat.ofDigits_lt_base_pow_length' + - Nat.ofDigits_eq_foldr + - Nat.pow_length_le_mul_ofDigits + - Nat.digits_lt_base' + - Nat.digits_eq_nil_iff_eq_zero + - Nat.NormDigits.digits_succ + - Nat.digits_ofDigits + - Nat.modEq_eleven_digits_sum + - Nat.base_pow_length_digits_le' + - Nat.digits_add_two_add_one + - Nat.dvd_iff_dvd_ofDigits + - Nat.ofDigits_one + - Nat.eleven_dvd_iff + - Nat.toDigitsCore_lens_eq_aux + - Nat.sub_one_mul_sum_log_div_pow_eq_sub_sum_digits + - Nat.ofDigits_zmodeq' + - Nat.ofDigits_eq_sum_mapIdx + - Nat.lt_base_pow_length_digits' + - Nat.digitsAux_def + - Nat.digits_eq_cons_digits_div + - Nat.sub_one_mul_sum_div_pow_eq_sub_sum_digits + - Nat.digitsAux_zero + - Nat.digits_len_le_digits_len_succ + - Nat.nine_dvd_iff + - Nat.digits_lt_base + - Nat.toDigitsCore_length + - Nat.coe_int_ofDigits + - Nat.lt_base_pow_length_digits + - Nat.self_div_pow_eq_ofDigits_drop + - Nat.ofDigits_eq_sum_map_with_index_aux + - Nat.dvd_ofDigits_sub_ofDigits + - Nat.zmodeq_ofDigits_digits + - Nat.base_pow_length_digits_le + - Nat.ofDigits_div_eq_ofDigits_tail + - Nat.digits_of_lt + - path: Mathlib/MeasureTheory/Integral/Lebesgue.lean + theorems: + - MeasureTheory.lintegral_add_measure + - MeasureTheory.lintegral_mul_const + - MeasureTheory.lintegral_countable + - MeasureTheory.lintegral_iSup' + - MeasureTheory.set_lintegral_dirac' + - MeasureTheory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt + - MeasureTheory.lintegral_strict_mono + - MeasureTheory.lintegral_count' + - MeasureTheory.exists_pos_set_lintegral_lt_of_measure_lt + - "MeasureTheory.lintegral_rw\u2082" + - MeasureTheory.set_lintegral_max + - MeasureTheory.lintegral_indicator + - MeasureTheory.lintegral_finset + - MeasureTheory.MeasurePreserving.lintegral_comp_emb + - MeasureTheory.lintegral_iInf_ae + - MeasureTheory.MeasurePreserving.lintegral_comp + - MeasureTheory.lintegral_mono_ae + - MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone_aux + - MeasureTheory.lintegral_trim + - MeasureTheory.lintegral_zero + - MeasureTheory.set_lintegral_eq_const + - MeasureTheory.lintegral_congr + - MeasureTheory.lintegral_sub_le' + - MeasureTheory.lintegral_indicator_const_comp + - MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero + - MeasureTheory.lintegral_nnnorm_eq_of_ae_nonneg + - MeasureTheory.set_lintegral_lt_top_of_bddAbove + - MeasureTheory.lintegral_insert + - MeasureTheory.ae_lt_top + - MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage_emb + - MeasureTheory.tendsto_set_lintegral_zero + - MeasureTheory.set_lintegral_const + - MeasureTheory.set_lintegral_congr + - MeasureTheory.lintegral_iSup_directed_of_measurable + - MeasureTheory.lintegral_max + - MeasureTheory.lintegral_eq_iSup_eapprox_lintegral + - "MeasureTheory.mul_meas_ge_le_lintegral\u2080" + - MeasureTheory.lintegral_zero_measure + - MeasureTheory.lintegral_sub' + - MeasureTheory.exists_pos_lintegral_lt_of_sigmaFinite + - MeasureTheory.lintegral_singleton' + - MeasureTheory.lintegral_union + - MeasureTheory.lintegral_add_compl + - MeasureTheory.tendsto_lintegral_of_dominated_convergence' + - MeasureTheory.lintegral_mul_const'' + - MeasureTheory.set_lintegral_empty + - MeasureTheory.set_lintegral_univ + - MeasureTheory.set_lintegral_const_lt_top + - MeasureTheory.le_iInf_lintegral + - "MeasureTheory.lintegral_iUnion\u2080" + - MeasureTheory.lintegral_le_of_forall_fin_meas_le' + - MeasureTheory.MeasurePreserving.set_lintegral_comp_emb + - MeasureTheory.lintegral_fintype + - MeasureTheory.lintegral_singleton + - MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero + - MeasureTheory.set_lintegral_dirac + - "MeasureTheory.le_iInf\u2082_lintegral" + - MeasureTheory.lintegral_one + - MeasurableEmbedding.lintegral_map + - "MeasureTheory.iSup\u2082_lintegral_le" + - MeasureTheory.lintegral_const_mul_le + - MeasureTheory.lintegral_add_left + - MeasureTheory.set_lintegral_one + - MeasureTheory.lintegral_const_lt_top + - MeasureTheory.lintegral_mono' + - MeasureTheory.lintegral_sum_measure + - "MeasureTheory.lintegral_indicator\u2080" + - MeasureTheory.le_lintegral_add + - MeasureTheory.limsup_lintegral_le + - MeasureTheory.lintegral_union_le + - MeasureTheory.lintegral_iUnion_le + - MeasureTheory.lintegral_count + - MeasureTheory.lintegral_map + - MeasureTheory.SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral + - MeasureTheory.lintegral_tsum + - ENNReal.tsum_const_eq + - MeasureTheory.lintegral_const_mul'' + - MeasureTheory.ae_eq_of_ae_le_of_lintegral_le + - MeasureTheory.lintegral_const_mul + - MeasureTheory.lintegral_dirac + - MeasureTheory.lintegral_tendsto_of_tendsto_of_monotone + - MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone + - IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal + - MeasureTheory.lintegral_smul_measure + - MeasureTheory.exists_measurable_le_lintegral_eq + - MeasureTheory.lintegral_le_of_forall_fin_meas_le_of_measurable + - MeasureTheory.iSup_lintegral_measurable_le_eq_lintegral + - MeasureTheory.set_lintegral_subtype + - MeasureTheory.lintegral_const + - MeasureTheory.meas_ge_le_lintegral_div + - MeasureTheory.lintegral_eq_zero_iff' + - MeasureTheory.lintegral_of_isEmpty + - MeasureTheory.MeasurePreserving.lintegral_map_equiv + - MeasureTheory.lintegral_finset_sum_measure + - "MeasureTheory.meas_le_lintegral\u2080" + - MeasureTheory.univ_le_of_forall_fin_meas_le + - MeasureTheory.lintegral_eq_nnreal + - MeasureTheory.lintegral_map_le + - MeasureTheory.set_lintegral_smul_measure + - MeasureTheory.lintegral_add_mul_meas_add_le_le_lintegral + - MeasureTheory.lintegral_mul_const_le + - "MeasureTheory.lintegral_biUnion\u2080" + - MeasureTheory.set_lintegral_strict_mono + - MeasureTheory.ae_lt_top' + - MeasureTheory.lintegral_mul_const' + - MeasureTheory.tendsto_of_lintegral_tendsto_of_antitone + - MeasureTheory.lintegral_ofReal_le_lintegral_nnnorm + - MeasureTheory.lintegral_le_meas + - MeasureTheory.lintegral_iSup_directed + - MeasureTheory.lintegral_trim_ae + - MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos + - "MeasureTheory.lintegral_biUnion_finset\u2080" + - MeasureTheory.lintegral_countable' + - MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence + - MeasureTheory.lintegral_finset_sum' + - MeasureTheory.lintegral_le_of_forall_fin_meas_le + - MeasureTheory.set_lintegral_measure_zero + - MeasureTheory.lintegral_dirac' + - MeasureTheory.lintegral_subtype_comap + - MeasureTheory.tendsto_measure_of_ae_tendsto_indicator + - MeasureTheory.lintegral_inter_add_diff + - MeasureTheory.set_lintegral_congr_fun + - "MeasureTheory.lintegral_indicator_const\u2080" + - MeasureTheory.lintegral_liminf_le' + - MeasureTheory.lintegral_add_left' + - MeasureTheory.lintegral_map' + - MeasureTheory.lintegral_pos_iff_support + - MeasureTheory.SimpleFunc.lintegral_eq_lintegral + - "MeasureTheory.lintegral_rw\u2081" + - MeasureTheory.MeasurePreserving.set_lintegral_comp_preimage + - MeasureTheory.lintegral_tendsto_of_tendsto_of_antitone + - MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral + - MeasureTheory.lintegral_iSup_ae + - MeasureTheory.set_lintegral_map + - MeasureTheory.lintegral_add_aux + - MeasureTheory.lintegral_add_right' + - MeasureTheory.lintegral_iInf' + - MeasureTheory.lintegral_const_mul' + - MeasureTheory.iSup_lintegral_le + - MeasureTheory.lintegral_iInf_directed_of_measurable + - path: Mathlib/Algebra/Algebra/Operations.lean + theorems: + - Submodule.prod_span + - Submodule.one_mul + - Submodule.one_eq_span + - Submodule.pow_eq_span_pow_set + - Submodule.mem_span_singleton_mul + - Submodule.mul_toAddSubmonoid + - Submodule.mem_smul_iff_inv_mul_mem + - Submodule.pow_toAddSubmonoid + - Submodule.mem_span_mul_finite_of_mem_mul + - Submodule.mul_induction_on + - Submodule.mul_induction_on' + - Submodule.map_one + - SubMulAction.mem_one' + - Submodule.mul_mem_smul_iff + - Submodule.restrictScalars_mul + - Submodule.map_op_one + - Submodule.le_pow_toAddSubmonoid + - Submodule.comap_op_one + - Submodule.map_mul + - Submodule.mem_div_iff_smul_subset + - Submodule.singleton_smul + - Submodule.map_unop_pow + - Submodule.comap_unop_one + - Submodule.le_div_iff_mul_le + - Submodule.map_unop_mul + - Submodule.map_div + - Submodule.one_le_one_div + - Submodule.pow_induction_on_left' + - Submodule.map_unop_one + - Submodule.mul_one + - Submodule.map_op_mul + - Submodule.prod_span_singleton + - Submodule.le_self_mul_one_div + - Submodule.comap_unop_mul + - Submodule.mul_one_div_le_one + - Submodule.map_op_pow + - Submodule.mul_smul_mul_eq_smul_mul_smul + - Submodule.mem_mul_span_singleton + - Submodule.mem_span_mul_finite_of_mem_span_mul + - Submodule.comap_op_mul + - Submodule.one_le + - Submodule.span_pow + - Submodule.pow_induction_on_right' + - path: Mathlib/Topology/MetricSpace/PseudoMetric.lean + theorems: + - Metric.uniformity_edist + - Metric.denseRange_iff + - Metric.sphere_union_ball + - Metric.ne_of_mem_sphere + - dist_le_pi_dist + - Metric.emetric_closedBall + - nndist_le_pi_nndist + - Metric.continuousWithinAt_iff + - Metric.exists_lt_mem_ball_of_mem_ball + - Metric.ball_zero + - Metric.inseparable_iff + - Real.dist_right_le_of_mem_uIcc + - dist_triangle_left + - Metric.tendstoUniformly_iff + - Inducing.isSeparable_preimage + - Metric.isOpen_singleton_iff + - Metric.tendsto_atTop + - Metric.isBounded_iff + - ball_pi' + - Metric.mem_ball' + - Metric.tendstoUniformlyOnFilter_iff + - Metric.eventually_nhds_prod_iff + - NNReal.le_add_nndist + - Metric.finite_approx_of_totallyBounded + - Metric.dense_iff + - Metric.ball_eq_empty + - Real.closedBall_eq_Icc + - Metric.closedBall_diff_sphere + - closedBall_prod_same + - dist_edist + - closedBall_pi + - dist_le_Ico_sum_dist + - ball_prod_same + - dist_pi_le_iff + - Metric.continuousOn_iff' + - edist_lt_ofReal + - nndist_pi_lt_iff + - Metric.ball_disjoint_closedBall + - closedBall_pi' + - Real.Icc_eq_closedBall + - Metric.continuousAt_iff' + - Metric.secondCountable_of_almost_dense_set + - tendsto_iff_dist_tendsto_zero + - Metric.totallyBounded_of_finite_discretization + - nndist_pi_eq_iff + - Metric.iUnion_inter_closedBall_nat + - Metric.tendsto_nhdsWithin_nhdsWithin + - Metric.tendstoLocallyUniformlyOn_iff + - Metric.tendstoUniformlyOn_iff + - Metric.mem_sphere_comm + - edist_le_coe + - tendsto_uniformity_iff_dist_tendsto_zero + - Fin.dist_insertNth_insertNth + - Metric.emetric_closedBall_nnreal + - eventually_closedBall_subset + - Real.dist_0_eq_abs + - Metric.emetric_ball_nnreal + - lebesgue_number_lemma_of_metric + - Metric.forall_of_forall_mem_closedBall + - PseudoMetricSpace.replaceBornology_eq + - dist_pi_eq_iff + - Metric.closedBall_eq_empty + - Metric.closedBall_eq_bInter_ball + - Metric.forall_of_forall_mem_ball + - cauchySeq_iff_tendsto_dist_atTop_0 + - Real.dist_le_of_mem_pi_Icc + - dist_triangle4_left + - dist_pi_lt_iff + - dist_triangle4_right + - Metric.uniformity_edist_aux + - Metric.dist_lt_add_of_nonempty_ball_inter_closedBall + - Metric.mem_closure_range_iff_nat + - Metric.mk_uniformity_basis_le + - Metric.tendsto_atTop' + - Metric.mk_uniformity_basis + - Dense.exists_dist_lt + - Real.dist_le_of_mem_Icc_01 + - NNReal.nndist_eq + - Metric.mem_closedBall' + - Metric.uniformity_eq_comap_nhds_zero + - Metric.mem_closedBall_comm + - PseudoMetricSpace.ext + - Metric.mem_of_closed' + - Metric.mem_closure_range_iff + - dist_prod_same_right + - Metric.mem_sphere' + - Real.dist_left_le_of_mem_uIcc + - dist_pi_const + - lebesgue_number_lemma_of_metric_sUnion + - PseudoMetricSpace.replaceUniformity_eq + - nndist_edist + - nndist_pi_le_iff + - Metric.closedBall_diff_ball + - Metric.exists_isCompact_closedBall + - Metric.tendstoLocallyUniformly_iff + - NNReal.nndist_zero_eq_val + - Real.dist_le_of_mem_uIcc + - dist_prod_same_left + - PseudoMetricSpace.replaceTopology_eq + - dist_pi_le_iff' + - Metric.isOpen_iff + - edist_lt_coe + - Metric.emetric_ball + - Metric.uniformity_basis_dist + - Metric.mem_ball_self + - Metric.mem_ball_comm + - dist_triangle_right + - Metric.uniformEmbedding_iff + - sphere_prod + - Real.ball_eq_Ioo + - Metric.ball_eq_ball' + - Metric.uniformInducing_iff + - Metric.ball_half_subset + - edist_nndist + - NNReal.nndist_zero_eq_val' + - Metric.eventually_prod_nhds_iff + - Metric.exists_ball_subset_ball + - Fin.nndist_insertNth_insertNth + - Metric.continuousAt_iff + - Metric.isBounded_iff_nndist + - ContinuousOn.isSeparable_image + - Metric.mem_closedBall_self + - dist_nonneg' + - Metric.mem_closure_iff + - ball_pi + - sphere_pi + - Metric.continuousOn_iff + - Metric.ball_subset + - Real.dist_le_of_mem_Icc + - Metric.tendsto_nhdsWithin_nhds + - Metric.continuousWithinAt_iff' + - path: Mathlib/NumberTheory/PythagoreanTriples.lean + theorems: + - pythagoreanTriple_comm + - PythagoreanTriple.mul_iff + - PythagoreanTriple.zero + - sq_ne_two_fin_zmod_four + - PythagoreanTriple.isPrimitiveClassified_of_coprime_of_odd_of_pos + - PythagoreanTriple.isPrimitiveClassified_of_coprime + - PythagoreanTriple.mul_isClassified + - PythagoreanTriple.symm + - PythagoreanTriple.coprime_classification' + - PythagoreanTriple.classification + - PythagoreanTriple.isPrimitiveClassified_of_coprime_of_zero_left + - PythagoreanTriple.mul + - PythagoreanTriple.normalize + - coprime_sq_sub_mul + - PythagoreanTriple.coprime_of_coprime + - PythagoreanTriple.even_odd_of_coprime + - coprime_sq_sub_sq_add_of_even_odd + - coprime_sq_sub_mul_of_odd_even + - PythagoreanTriple.isPrimitiveClassified_of_coprime_of_pos + - PythagoreanTriple.ne_zero_of_coprime + - coprime_sq_sub_sq_sum_of_odd_odd + - coprime_sq_sub_mul_of_even_odd + - PythagoreanTriple.coprime_classification + - PythagoreanTriple.isClassified_of_isPrimitiveClassified + - PythagoreanTriple.classified + - Int.sq_ne_two_mod_four + - coprime_sq_sub_sq_add_of_odd_even + - PythagoreanTriple.isClassified_of_normalize_isPrimitiveClassified + - PythagoreanTriple.isPrimitiveClassified_aux + - PythagoreanTriple.gcd_dvd + - path: Mathlib/MeasureTheory/Group/FundamentalDomain.lean + theorems: + - MeasureTheory.fundamentalInterior_smul + - MeasureTheory.IsFundamentalDomain.set_lintegral_eq_tsum + - MeasureTheory.IsFundamentalDomain.sum_restrict_of_ac + - MeasureTheory.IsFundamentalDomain.set_integral_eq + - MeasureTheory.IsFundamentalDomain.set_integral_eq_tsum + - MeasureTheory.IsFundamentalDomain.integral_eq_tsum_of_ac + - MeasureTheory.IsFundamentalDomain.measure_set_eq + - MeasureTheory.IsFundamentalDomain.measure_eq + - MeasureTheory.IsFundamentalDomain.exists_ne_one_smul_eq + - MeasureTheory.IsFundamentalDomain.mk_of_measure_univ_le + - MeasureTheory.IsFundamentalDomain.measure_ne_zero + - MeasureTheory.IsFundamentalDomain.measure_eq_tsum + - MeasureTheory.IsFundamentalDomain.image_of_equiv + - MeasureTheory.IsFundamentalDomain.smul + - MeasureTheory.IsFundamentalDomain.essSup_measure_restrict + - MeasureTheory.IsFundamentalDomain.set_lintegral_eq_tsum' + - MeasureTheory.IsFundamentalDomain.integral_eq_tsum + - MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant + - MeasureTheory.mem_fundamentalInterior + - MeasureTheory.IsFundamentalDomain.fundamentalInterior + - MeasureTheory.IsFundamentalDomain.aEStronglyMeasurable_on_iff + - MeasureTheory.IsFundamentalDomain.measure_eq_tsum_of_ac + - MeasureTheory.IsFundamentalDomain.preimage_of_equiv + - MeasureTheory.mem_fundamentalFrontier + - MeasureTheory.IsFundamentalDomain.measure_fundamentalFrontier + - MeasureTheory.fundamentalFrontier_smul + - MeasureTheory.IsFundamentalDomain.hasFiniteIntegral_on_iff + - MeasureTheory.IsFundamentalDomain.mk' + - MeasureTheory.IsFundamentalDomain.set_lintegral_eq + - MeasureTheory.IsFundamentalDomain.set_integral_eq_tsum' + - MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum_of_ac + - MeasureTheory.IsFundamentalDomain.measure_eq_card_smul_of_smul_ae_eq_self + - MeasureTheory.pairwise_disjoint_fundamentalInterior + - path: Mathlib/Data/List/DropRight.lean + theorems: + - List.rtakeWhile_eq_self_iff + - List.rdropWhile_prefix + - List.rdropWhile_eq_nil_iff + - List.rtake_zero + - List.rtakeWhile_concat_pos + - List.rdrop_concat_succ + - List.rtake_eq_reverse_take_reverse + - List.rdropWhile_concat_neg + - List.dropWhile_eq_self_iff + - List.rdropWhile_last_not + - List.rtakeWhile_concat_neg + - List.rtakeWhile_suffix + - List.dropWhile_idempotent + - List.rtake_nil + - List.rdropWhile_eq_self_iff + - List.mem_rtakeWhile_imp + - List.rdropWhile_singleton + - List.rdropWhile_nil + - List.rdropWhile_concat_pos + - List.rdrop_nil + - List.rdropWhile_concat + - List.rtakeWhile_concat + - List.rdrop_zero + - List.rdrop_eq_reverse_drop_reverse + - List.rtake_concat_succ + - List.rtakeWhile_nil + - List.rtakeWhile_eq_nil_iff + - path: Mathlib/CategoryTheory/Limits/HasLimits.lean + theorems: + - CategoryTheory.Limits.colimit.desc_extend + - CategoryTheory.Limits.colimit.pre_pre + - CategoryTheory.Limits.limit.pre_eq + - "CategoryTheory.Limits.limit.isoLimitCone_inv_\u03C0" + - CategoryTheory.Limits.colimit.map_desc + - CategoryTheory.Limits.colimit.pre_map + - CategoryTheory.Limits.limit.map_post + - "CategoryTheory.Limits.colimit.\u03B9_pre" + - "CategoryTheory.Limits.colimit.isoColimitCocone_\u03B9_hom" + - "CategoryTheory.Limits.limit.isoLimitCone_hom_\u03C0" + - CategoryTheory.Limits.hasLimitOfEquivalenceComp + - CategoryTheory.Limits.limit.lift_post + - "CategoryTheory.Limits.limit.post_\u03C0" + - CategoryTheory.Limits.colimit.pre_desc + - CategoryTheory.Limits.hasColimitsOfShape_of_equivalence + - CategoryTheory.Limits.limit.lift_extend + - CategoryTheory.Limits.colimit.pre_id + - CategoryTheory.Limits.colimit.map_post + - "CategoryTheory.Limits.HasColimit.isoOfEquivalence_inv_\u03C0" + - CategoryTheory.Limits.limit.map_pre + - CategoryTheory.Limits.limit.post_post + - CategoryTheory.Limits.limit.id_pre + - CategoryTheory.Limits.limit.pre_pre + - "CategoryTheory.Limits.HasLimit.isoOfEquivalence_hom_\u03C0" + - CategoryTheory.Limits.limit.lift_pre + - "CategoryTheory.Limits.HasLimit.isoOfEquivalence_inv_\u03C0" + - CategoryTheory.Limits.limit.pre_post + - CategoryTheory.Limits.colimit.pre_eq + - "CategoryTheory.Limits.colimit.\u03B9_post" + - CategoryTheory.Limits.hasLimitsOfShape_of_equivalence + - "CategoryTheory.Limits.limit.pre_\u03C0" + - "CategoryTheory.Limits.limit.coneMorphism_\u03C0" + - CategoryTheory.Limits.limit.map_pre' + - CategoryTheory.Limits.colimit.post_desc + - CategoryTheory.Limits.limit.lift_map + - "CategoryTheory.Limits.colimit.\u03B9_coconeMorphism" + - CategoryTheory.Limits.colimit.pre_map' + - "CategoryTheory.Limits.colimit.isoColimitCocone_\u03B9_inv" + - CategoryTheory.Limits.hasColimit_of_equivalence_comp + - "CategoryTheory.Limits.HasColimit.isoOfEquivalence_hom_\u03C0" + - "CategoryTheory.Limits.colimit.\u03B9_map" + - path: Mathlib/Analysis/Calculus/Deriv/Add.lean + theorems: + - HasDerivAtFilter.neg + - HasStrictDerivAt.sum + - differentiableAt_comp_neg_iff + - HasStrictDerivAt.neg + - HasStrictDerivAt.const_sub + - derivWithin.neg + - derivWithin_add_const + - HasStrictDerivAt.sub + - HasDerivAtFilter.sub + - deriv.neg + - deriv_const_add + - derivWithin_const_sub + - HasStrictDerivAt.add + - deriv_add_const + - deriv_const_sub + - HasDerivAtFilter.add + - HasDerivAtFilter.sum + - derivWithin_const_add + - HasDerivAtFilter.const_sub + - HasDerivAtFilter.sub_const + - not_differentiableAt_abs_zero + - derivWithin_sub_const + - path: Mathlib/Topology/MetricSpace/HausdorffDimension.lean + theorems: + - IsometryEquiv.dimH_univ + - LipschitzOnWith.dimH_image_le + - dimH_bUnion + - dimH_union + - dimH_mono + - ContinuousLinearEquiv.dimH_univ + - dimH_range_le_of_locally_holder_on + - dimH_image_le_of_locally_lipschitzOn + - le_dimH_of_hausdorffMeasure_eq_top + - Real.dimH_ball_pi_fin + - dimH_def + - dimH_sUnion + - Real.dimH_univ + - dimH_image_le_of_locally_holder_on + - HolderOnWith.dimH_image_le + - ContinuousLinearEquiv.dimH_image + - dimH_iUnion + - ContinuousLinearEquiv.dimH_preimage + - IsometryEquiv.dimH_preimage + - Real.dimH_of_mem_nhds + - exists_mem_nhdsWithin_lt_dimH_of_lt_dimH + - bsupr_limsup_dimH + - dense_compl_of_dimH_lt_finrank + - dimH_subsingleton + - hausdorffMeasure_of_lt_dimH + - dimH_countable + - ContDiff.dimH_range_le + - AntilipschitzWith.dimH_preimage_le + - Real.dimH_univ_pi + - dimH_range_le_of_locally_lipschitzOn + - iSup_limsup_dimH + - hausdorffMeasure_of_dimH_lt + - Real.dimH_ball_pi + - Real.dimH_univ_pi_fin + - path: Mathlib/Order/CompleteLattice.lean + theorems: + - isGLB_biInf + - iSup_sum + - iSup_eq_dif + - iSup_range' + - iSup_eq_top + - iInf_pair + - isGLB_iff_sInf_eq + - Prod.swap_iSup + - iSup_emptyset + - iSup_image + - isLUB_iff_sSup_eq + - biSup_prod + - iSup_const + - iInf_inf + - iSup_congr_Prop + - iSup_option + - biSup_congr' + - iInf_univ + - le_iSup_iff + - iSup_singleton + - iSup_prod + - iSup_union + - iSup_apply + - OrderIso.map_sSup + - eq_singleton_bot_of_sSup_eq_bot_of_nonempty + - Function.Surjective.iSup_comp + - sSup_image + - iSup_sigma + - biInf_congr' + - iSup_pair + - iSup_ne_bot_subtype + - inf_iInf + - OrderIso.map_sSup_eq_sSup_symm_preimage + - isLUB_biSup + - sup_biSup + - iInf_false + - "iSup\u2082_le_iff" + - sup_iSup_nat_succ + - sSup_image2 + - sup_iSup + - iSup_univ + - iInf_singleton + - iSup_bool_eq + - iInf_le_iff + - iInf_emptyset + - iSup_ge_eq_iSup_nat_add + - iSup_iSup_eq_right + - Function.Surjective.iSup_congr + - iInf_eq_bot + - iSup_iSup_eq_left + - "le_iInf\u2082_iff" + - iSup_dite + - sInf_image2 + - iSup_sup + - "iSup\u2082_comm" + - iSup_split + - biSup_const + - Monotone.le_map_sSup + - binary_relation_sInf_iff + - iSup_extend_bot + - biSup_sup + - "iInf\u2082_eq_top" + - binary_relation_sSup_iff + - biInf_le + - sup_eq_iSup + - iSup_false + - Equiv.biSup_comp + - OrderIso.map_sInf_eq_sInf_symm_preimage + - iSup_iInf_ge_nat_add + - iSup_range + - unary_relation_sInf_iff + - OrderIso.map_iSup + - Antitone.map_sSup_le + - sSup_eq_iSup' + - unary_relation_sSup_iff + - sSup_image' + - iInf_nat_gt_zero_eq + - "iSup\u2082_eq_bot" + - Prod.swap_iInf + - path: Mathlib/Algebra/Order/WithZero.lean + theorems: + - "mul_lt_right\u2080" + - "mul_inv_lt_of_lt_mul\u2080" + - "lt_of_mul_lt_mul_of_le\u2080" + - pow_pos_iff + - mul_inv_le_of_le_mul + - le_of_le_mul_right + - "OrderIso.mulRight\u2080'_symm" + - "mul_le_mul_left\u2080" + - "inv_mul_lt_of_lt_mul\u2080" + - "div_le_div\u2080" + - le_mul_inv_of_mul_le + - "div_le_div_left\u2080" + - "mul_lt_mul_of_lt_of_le\u2080" + - "le_div_iff\u2080" + - "div_le_iff\u2080" + - pow_lt_pow_succ + - "OrderIso.mulLeft\u2080'_symm" + - zero_le' + - "pow_lt_pow_right\u2080" + - "div_le_div_right\u2080" + - path: Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean + theorems: + - AffineSubspace.coe_affineSpan_singleton + - AffineSubspace.Parallel.direction_eq + - vectorSpan_range_eq_span_range_vsub_right + - AffineSubspace.affineSpan_coe + - AffineSubspace.subsingleton_of_subsingleton_span_eq_top + - AffineSubspace.comap_span + - direction_affineSpan + - AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot + - vadd_left_mem_affineSpan_pair + - vectorSpan_range_eq_span_range_vsub_left_ne + - AffineSubspace.direction_eq_top_iff_of_nonempty + - AffineSubspace.comap_top + - AffineSubspace.affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty + - affineSpan_pair_le_of_mem_of_mem + - AffineSubspace.eq_bot_or_nonempty + - AffineSubspace.ext_of_direction_eq + - AffineSubspace.mem_direction_iff_eq_vsub_right + - AffineSubspace.directionOfNonempty_eq_direction + - AffineSubspace.map_direction + - AffineSubspace.direction_affineSpan_insert + - vectorSpan_image_eq_span_vsub_set_right_ne + - vectorSpan_range_eq_span_range_vsub_left + - AffineSubspace.vadd_mem_of_mem_direction + - AffineSubspace.vadd_mem_iff_mem_of_mem_direction + - AffineSubspace.Parallel.vectorSpan_eq + - AffineSubspace.vsub_right_mem_direction_iff_mem + - AffineSubspace.nonempty_iff_ne_bot + - AffineSubspace.direction_lt_of_nonempty + - AffineSubspace.mem_mk'_iff_vsub_mem + - AffineMap.vectorSpan_image_eq_submodule_map + - AffineSubspace.parallel_bot_iff_eq_bot + - AffineSubspace.map_eq_bot_iff + - AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top + - AffineSubspace.direction_bot + - vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints + - AffineSubspace.vadd_mem_iff_mem_direction + - vectorSpan_eq_span_vsub_set_right_ne + - AffineSubspace.mem_map + - AffineMap.span_eq_top_of_surjective + - AffineEquiv.span_eq_top_iff + - AffineSubspace.card_pos_of_affineSpan_eq_top + - AffineSubspace.direction_sup + - mem_vectorSpan_pair_rev + - AffineSubspace.direction_mk' + - mem_vectorSpan_pair + - AffineSubspace.coe_direction_eq_vsub_set_left + - vectorSpan_pair_rev + - AffineMap.lineMap_mem + - AffineSubspace.coe_direction_eq_vsub_set_right + - affineSpan_insert_affineSpan + - Submodule.toAffineSubspace_direction + - AffineSubspace.vectorSpan_eq_top_of_affineSpan_eq_top + - vectorSpan_singleton + - AffineSubspace.bot_parallel_iff_eq_bot + - AffineSubspace.inter_eq_singleton_of_nonempty_of_isCompl + - vectorSpan_eq_span_vsub_set_right + - AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty + - AffineSubspace.eq_univ_of_subsingleton_span_eq_top + - AffineSubspace.nonempty_of_affineSpan_eq_top + - AffineSubspace.mem_affineSpan_singleton + - AffineSubspace.Parallel.refl + - AffineSubspace.lt_iff_le_and_exists + - vadd_right_mem_affineSpan_pair + - AffineSubspace.direction_top + - AffineSubspace.direction_inf + - AffineSubspace.sup_direction_le + - AffineSubspace.Parallel.trans + - AffineMap.map_top_of_surjective + - AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial + - vectorSpan_pair + - AffineSubspace.vsub_left_mem_direction_iff_mem + - vectorSpan_eq_span_vsub_finset_right_ne + - vectorSpan_image_eq_span_vsub_set_left_ne + - AffineSubspace.bot_ne_top + - vectorSpan_empty + - AffineSubspace.map_span + - vectorSpan_eq_span_vsub_set_left_ne + - bot_lt_affineSpan + - AffineSubspace.mem_direction_iff_eq_vsub + - AffineSubspace.mem_affineSpan_insert_iff + - spanPoints_nonempty + - AffineSubspace.direction_le + - affineSpan_eq_bot + - AffineSubspace.affineSpan_pair_parallel_iff_vectorSpan_eq + - vectorSpan_range_eq_span_range_vsub_right_ne + - vectorSpan_insert_eq_vectorSpan + - affineSpan_singleton_union_vadd_eq_top_of_span_eq_top + - affineSpan_induction' + - AffineSubspace.mem_direction_iff_eq_vsub_left + - AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty + - AffineSubspace.direction_inf_of_mem + - AffineSubspace.Parallel.symm + - affineSpan_coe_preimage_eq_top + - affineSpan_insert_eq_affineSpan + - vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan + - path: Mathlib/Algebra/GroupPower/Ring.lean + theorems: + - pow_dvd_pow_iff + - mul_neg_one_pow_eq_zero_iff + - neg_one_pow_eq_pow_mod_two + - sub_sq' + - neg_one_pow_mul_eq_zero_iff + - neg_one_pow_eq_or + - min_pow_dvd_add + - neg_sq + - Commute.sq_sub_sq + - Commute.sq_eq_sq_iff_eq_or_eq_neg + - sub_sq + - neg_pow_bit0 + - zpow_bit0_neg + - Units.sq_eq_sq_iff_eq_or_eq_neg + - neg_one_sq + - add_sq' + - Ring.inverse_pow + - sq_eq_one_iff + - neg_pow_bit1 + - path: Mathlib/Data/Multiset/Basic.lean + theorems: + - Multiset.filter_comm + - Multiset.card_sub + - Multiset.sub_eq_fold_erase + - Multiset.rel_cons_right + - Multiset.card_eq_card_of_rel + - Multiset.inter_add_distrib + - Multiset.mem_nsmul + - Multiset.exists_mem_of_ne_zero + - Multiset.sizeOf_lt_sizeOf_of_mem + - Multiset.filter_cons + - Multiset.map_set_pairwise + - Multiset.coe_inter + - Multiset.card_eq_zero + - Multiset.nsmul_cons + - Multiset.filterMap_eq_filter + - Multiset.le_count_iff_replicate_le + - Multiset.add_union_distrib + - Multiset.disjoint_union_right + - Multiset.map_count_True_eq_filter_card + - Multiset.count_eq_card + - Multiset.attach_cons + - Multiset.strongInductionOn_eq + - Multiset.filter_singleton + - Multiset.count_eq_card_filter_eq + - Multiset.map_filter + - Multiset.le_cons_of_not_mem + - Multiset.filter_cons_of_neg + - Multiset.countP_attach + - Multiset.le_singleton + - Multiset.countP_le_of_le + - Multiset.singleton_subset + - Multiset.length_toList + - Multiset.eq_of_mem_map_const + - Multiset.disjoint_singleton + - Multiset.card_pair + - Multiset.singleton_eq_cons_iff + - Multiset.map_comp_cons + - Multiset.le_add_right + - Multiset.countP_sub + - Multiset.zero_union + - Multiset.filterMap_filter + - Multiset.map_eq_zero + - Multiset.count_inter + - Multiset.exists_mem_of_rel_of_mem + - Multiset.rel_flip_eq + - Multiset.foldr_induction' + - Multiset.mem_toList + - Multiset.map_mk_eq_map_mk_of_rel + - Multiset.singleton_inj + - Multiset.card_erase_eq_ite + - Multiset.inter_le_right + - Multiset.count_erase_of_ne + - Multiset.rel_map_right + - Multiset.mem_inter + - Multiset.count_replicate_self + - Multiset.map_surjective_of_surjective + - Multiset.countP_eq_card + - Multiset.erase_lt + - Multiset.filter_nsmul + - Multiset.erase_attach_map_val + - Multiset.inter_replicate + - Multiset.map_lt_map + - Multiset.map_eq_map + - Multiset.nsmul_singleton + - Multiset.filter_cons_of_pos + - Multiset.card_nsmul + - Multiset.replicate_inter + - Multiset.count_attach + - Multiset.filter_congr + - Multiset.count_ne_zero + - Multiset.monotone_filter_right + - Multiset.count_map_eq_count' + - Multiset.singleton_disjoint + - Multiset.toList_eq_singleton_iff + - Multiset.erase_cons_tail_of_mem + - Multiset.add_eq_union_iff_disjoint + - Multiset.zero_ssubset + - Multiset.count_filter + - Multiset.countP_congr + - Multiset.count_map_eq_count + - Multiset.disjoint_add_left + - Multiset.countP_cons + - Multiset.union_add_inter + - Multiset.disjoint_cons_left + - Multiset.countP_map + - Multiset.countP_eq_countP_filter_add + - Multiset.card_eq_countP_add_countP + - Multiset.replicate_le_replicate + - Multiset.filter_add_not + - Multiset.rel_eq + - Multiset.rel_map_left + - Multiset.lt_singleton + - Multiset.sub_inter + - Multiset.sub_le_iff_le_add + - Multiset.count_sub + - Multiset.countP_filter + - Multiset.cons_add + - Multiset.count_replicate + - Multiset.le_cons_erase + - Multiset.mem_singleton_self + - Multiset.map_erase_of_mem + - Multiset.Rel.trans + - Multiset.rel_refl_of_refl_on + - Multiset.strongDownwardInductionOn_eq + - Multiset.add_singleton_eq_iff + - Multiset.ext + - Multiset.countP_eq_zero + - Multiset.foldl_induction' + - Multiset.erase_attach_map + - Multiset.addHom_ext + - Multiset.cons_union_distrib + - Multiset.Rel.countP_eq + - Multiset.add_inter_distrib + - Multiset.map_union + - Multiset.map_eq_cons + - Multiset.mem_singleton + - Multiset.map_eq_singleton + - Multiset.inter_eq_zero_iff_disjoint + - Multiset.cons_inj_right + - Multiset.disjoint_cons_right + - Multiset.coe_count + - Multiset.filter_filter + - Multiset.count_map + - Multiset.countP_add + - Multiset.induction + - Multiset.ssubset_singleton_iff + - Multiset.count_filter_of_pos + - Multiset.rel_add_right + - Multiset.disjoint_union_left + - Multiset.rel_replicate_left + - Multiset.map_erase + - Multiset.count_pos + - Multiset.card_singleton + - Multiset.le_iff_count + - Multiset.cons_eq_cons + - Multiset.union_add_distrib + - Multiset.count_nsmul + - Multiset.count_singleton + - Multiset.disjoint_map_map + - Multiset.Rel.mono + - Multiset.le_inter + - Multiset.eq_union_right + - Multiset.erase_le_iff_le_cons + - Multiset.count_erase_self + - Multiset.union_zero + - Multiset.map_congr + - Multiset.rel_zero_left + - Multiset.rel_of_forall + - Multiset.disjoint_add_right + - Multiset.map_hcongr + - Multiset.lt_cons_self + - Multiset.filter_eq' + - Multiset.rel_zero_right + - Multiset.add_cons + - Multiset.rel_add_left + - Multiset.filter_add_filter + - Multiset.sub_add_inter + - Multiset.rel_cons_left + - Multiset.card_filter_le_iff + - Multiset.filter_eq + - Multiset.map_replicate + - Multiset.eq_zero_of_forall_not_mem + - Multiset.le_add_left + - Multiset.strongDownwardInduction_eq + - Multiset.map_le_map_iff + - Multiset.mem_of_mem_nsmul + - Multiset.coe_eq_singleton + - Multiset.disjoint_iff_ne + - Multiset.cons_inter_distrib + - Multiset.filter_sub + - Multiset.countP_nsmul + - Multiset.erase_add_left_neg + - Multiset.cons_sub_of_le + - Multiset.toList_eq_nil + - path: Mathlib/Order/RelIso/Basic.lean + theorems: + - injective_of_increasing + - RelIso.cast_trans + - Function.Surjective.wellFounded_iff + - RelHomClass.acc + - "wellFounded_lift\u2082_iff" + - RelIso.toEquiv_injective + - RelEmbedding.acc + - "acc_lift\u2082_iff" + - RelIso.rel_symm_apply + - RelIso.symm_apply_rel + - RelEmbedding.eq_preimage + - path: Mathlib/SetTheory/ZFC/Basic.lean + theorems: + - ZFSet.toSet_singleton + - Class.ofSet.inj + - Class.toSet_of_ZFSet + - Class.mem_of_mem_sInter + - PSet.equiv_of_isEmpty + - ZFSet.mem_pairSep + - ZFSet.toSet_union + - ZFSet.toSet_equiv_aux + - Class.mem_wf + - ZFSet.sInter_singleton + - PSet.not_nonempty_empty + - ZFSet.toSet_pair + - ZFSet.toSet_insert + - ZFSet.toSet_sUnion + - ZFSet.mem_powerset + - ZFSet.hereditarily_iff + - ZFSet.toSet_range + - ZFSet.sUnion_lem + - ZFSet.singleton_injective + - ZFSet.map_unique + - Class.sInter_apply + - ZFSet.toSet_sdiff + - ZFSet.mem_prod + - ZFSet.sUnion_singleton + - Class.mem_sUnion + - ZFSet.mem_pair + - ZFSet.toSet_image + - PSet.mem_wf_aux + - ZFSet.map_fval + - ZFSet.eq_empty + - ZFSet.mem_of_mem_sInter + - Class.congToClass_empty + - Class.iota_val + - ZFSet.Hereditarily.empty + - ZFSet.choice_mem + - Class.sUnion_empty + - ZFSet.sUnion_empty + - Class.sInter_empty + - ZFSet.mem_range + - ZFSet.mem_union + - Class.eq_univ_of_powerset_subset + - ZFSet.choice_mem_aux + - Class.sUnion_apply + - ZFSet.mem_funs + - Class.mem_sInter + - ZFSet.omega_succ + - ZFSet.pair_mem_prod + - ZFSet.toSet_empty + - ZFSet.not_nonempty_empty + - ZFSet.toSet_inter + - ZFSet.mem_sInter + - PSet.toSet_empty + - ZFSet.pair_injective + - PSet.func_mem + - PSet.toSet_sUnion + - ZFSet.ext_iff + - Class.classToCong_empty + - ZFSet.nonempty_mk_iff + - ZFSet.mem_sep + - PSet.mem_sUnion + - ZFSet.sInter_empty + - ZFSet.toSet_sInter + - ZFSet.eq_empty_or_nonempty + - path: Mathlib/Data/MvPolynomial/Comap.lean + theorems: + - MvPolynomial.comap_eq_id_of_eq_id + - MvPolynomial.comap_id_apply + - MvPolynomial.comap_comp + - MvPolynomial.comap_comp_apply + - MvPolynomial.comap_rename + - MvPolynomial.comap_id + - path: Mathlib/Data/Nat/Dist.lean + theorems: + - Nat.dist_tri_right' + - Nat.dist_eq_max_sub_min + - Nat.dist_mul_left + - Nat.dist.triangle_inequality + - Nat.dist_comm + - Nat.dist_tri_left' + - Nat.dist_eq_sub_of_le_right + - Nat.dist_eq_sub_of_le + - Nat.dist_add_add_left + - Nat.dist_mul_right + - Nat.dist_eq_zero + - Nat.dist_self + - Nat.dist_eq_intro + - Nat.dist_succ_succ + - Nat.dist_pos_of_ne + - Nat.dist_tri_right + - Nat.dist_add_add_right + - path: Mathlib/Analysis/Complex/Angle.lean + theorems: + - Complex.angle_exp_one + - Complex.angle_div_left_eq_angle_mul_right + - Complex.angle_one_right + - Complex.angle_eq_abs_arg + - Complex.angle_one_left + - Complex.angle_mul_right + - Complex.angle_div_right_eq_angle_mul_left + - Complex.angle_exp_exp + - Complex.angle_mul_left + - path: Mathlib/NumberTheory/Padics/PadicNumbers.lean + theorems: + - padicNormE.add_eq_max_of_ne + - PadicSeq.norm_values_discrete + - Padic.valuation_p + - padicNormE.eq_of_norm_add_lt_left + - PadicSeq.norm_eq_of_equiv_aux + - PadicSeq.norm_eq_of_add_equiv_zero + - Padic.exi_rat_seq_conv_cauchy + - padicNormE.defn + - PadicSeq.not_equiv_zero_const_of_nonzero + - padicNormE.is_rat + - Padic.rat_dense' + - padicNormE.norm_int_le_one + - padicNormE.norm_p + - Padic.AddValuation.map_mul + - padicNormE.norm_p_lt_one + - padicNormE.image + - PadicSeq.norm_eq_norm_app_of_nonzero + - padicNormE.norm_rat_le_one + - PadicSeq.norm_one + - Padic.AddValuation.map_one + - Padic.exi_rat_seq_conv + - Padic.addValuation.apply + - PadicSeq.norm_zero_iff + - PadicSeq.stationary + - Padic.coe_inj + - Padic.valuation_map_add + - Padic.const_equiv + - Padic.norm_eq_pow_val + - PadicSeq.norm_mul + - Padic.AddValuation.map_add + - Padic.norm_le_pow_iff_norm_lt_pow_add_one + - PadicSeq.norm_eq_of_equiv + - Padic.norm_lt_pow_iff_norm_le_pow_sub_one + - PadicSeq.lift_index_right + - padicNormE.norm_int_le_pow_iff_dvd + - PadicSeq.equiv_zero_of_val_eq_of_equiv_zero + - PadicSeq.norm_nonneg + - padicNormE.eq_padicNorm + - PadicSeq.norm_neg + - PadicSeq.norm_const + - Padic.complete' + - padicNormE.norm_p_zpow + - Padic.norm_le_one_iff_val_nonneg + - PadicSeq.lift_index_left_left + - Padic.complete'' + - PadicSeq.val_eq_iff_norm_eq + - padicNormE.norm_int_lt_one_iff_dvd + - padicNormE.mul + - padicNormE.nonarchimedean + - Padic.rat_dense + - PadicSeq.add_eq_max_of_ne + - PadicSeq.norm_nonarchimedean + - PadicSeq.lift_index_left + - PadicSeq.norm_eq_pow_val + - Padic.valuation_map_mul + - PadicSeq.norm_eq + - Padic.AddValuation.map_zero + - PadicSeq.norm_equiv + - padicNormE.norm_p_pow + - path: Mathlib/RingTheory/WittVector/Defs.lean + theorems: + - WittVector.zero_coeff + - WittVector.constantCoeff_wittSub + - WittVector.v2_coeff + - WittVector.constantCoeff_wittNSMul + - WittVector.wittOne_pos_eq_zero + - WittVector.constantCoeff_wittMul + - WittVector.constantCoeff_wittAdd + - WittVector.add_coeff_zero + - WittVector.wittOne_zero_eq_one + - WittVector.wittMul_zero + - WittVector.mul_coeff_zero + - WittVector.neg_coeff + - WittVector.ext_iff + - WittVector.wittSub_zero + - WittVector.constantCoeff_wittZSMul + - WittVector.constantCoeff_wittNeg + - WittVector.wittAdd_zero + - WittVector.ext + - WittVector.one_coeff_eq_of_pos + - WittVector.wittNeg_zero + - WittVector.wittZero_eq_zero + - WittVector.one_coeff_zero + - path: Mathlib/Order/Lattice.lean + theorems: + - lt_sup_iff + - sup_sup_distrib_right + - sup_assoc + - sup_eq_left + - inf_lt_sup + - sup_le_inf + - sup_comm + - inf_eq_left + - Monotone.map_sup + - AntitoneOn.map_sup + - sup_left_idem + - sup_eq_iff_inf_eq + - sup_idem + - sup_inf_right + - Function.update_inf + - sup_sup_sup_comm + - inf_eq_minDefault + - sup_eq_right + - inf_sup_left + - SemilatticeSup.ext + - SemilatticeInf.ext_inf + - sup_sup_distrib_left + - inf_idem + - inf_sup_self + - inf_eq_sup + - semilatticeSup_mk'_partialOrder_eq_semilatticeInf_mk'_partialOrder + - sup_left_right_swap + - SemilatticeSup.ext_sup + - sup_right_idem + - sup_left_comm + - Function.update_sup + - Lattice.ext + - MonotoneOn.map_sup + - Monotone.of_map_inf + - sup_eq_maxDefault + - le_iff_exists_sup + - inf_right_idem + - sup_ind + - AntitoneOn.of_map_inf + - inf_left_idem + - SemilatticeInf.ext + - sup_inf_self + - le_of_inf_le_sup_le + - MonotoneOn.of_map_inf + - le_sup_iff + - inf_sup_right + - inf_eq_and_sup_eq_iff + - path: Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean + theorems: + - IsPrimitiveRoot.sub_one_norm_two + - IsPrimitiveRoot.sub_one_norm_isPrimePow + - IsPrimitiveRoot.pow_sub_one_norm_prime_pow_ne_two + - IsPrimitiveRoot.norm_of_cyclotomic_irreducible + - IsPrimitiveRoot.sub_one_norm_prime_ne_two + - IsPrimitiveRoot.pow_sub_one_norm_prime_pow_of_ne_zero + - IsPrimitiveRoot.pow_sub_one_norm_two + - IsPrimitiveRoot.sub_one_norm_eq_eval_cyclotomic + - IsCyclotomicExtension.zeta_isRoot + - IsPrimitiveRoot.norm_eq_neg_one_pow + - IsPrimitiveRoot.minpoly_sub_one_eq_cyclotomic_comp + - IsCyclotomicExtension.aeval_zeta + - IsPrimitiveRoot.sub_one_norm_prime + - IsPrimitiveRoot.norm_eq_one + - IsCyclotomicExtension.finrank + - IsPrimitiveRoot.powerBasis_gen_mem_adjoin_zeta_sub_one + - IsPrimitiveRoot.pow_sub_one_norm_prime_ne_two + - IsPrimitiveRoot.norm_eq_one_of_linearly_ordered + - path: Mathlib/Data/Nat/Prime.lean + theorems: + - Nat.prime_of_coprime + - Nat.Prime.dvd_iff_eq + - Nat.Prime.even_iff + - Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul + - Nat.minFac_le_of_dvd + - Nat.factors_lemma + - Nat.prime_two + - Nat.coprime_or_dvd_of_prime + - Nat.prime_iff_prime_int + - Nat.minFac_lemma + - Nat.Prime.even_sub_one + - Nat.Prime.five_le_of_ne_two_of_ne_three + - Nat.dvd_prime_pow + - Nat.Prime.pow_eq_iff + - Nat.minFac_pos + - Nat.eq_prime_pow_of_dvd_least_prime_pow + - Nat.eq_one_iff_not_exists_prime_dvd + - Nat.minFac_has_prop + - Nat.minFac_eq_one_iff + - Nat.Prime.pow_inj + - Nat.Prime.not_dvd_mul + - Nat.Prime.eq_one_or_self_of_dvd + - Nat.Prime.mul_eq_prime_sq_iff + - Nat.Prime.dvd_factorial + - Nat.minFacAux_has_prop + - Nat.prime_five + - Nat.prime_def_le_sqrt + - Nat.Prime.coprime_iff_not_dvd + - Nat.prime_three + - Nat.prime_def_lt' + - Nat.ne_one_iff_exists_prime_dvd + - Nat.coprime_of_dvd + - Nat.Prime.not_coprime_iff_dvd + - Nat.prime_def_lt'' + - Nat.le_minFac' + - Nat.coprime_two_left + - Nat.not_bddAbove_setOf_prime + - Nat.minFac_eq_two_iff + - Nat.Prime.eq_two_or_odd + - Nat.not_prime_mul + - Nat.dvd_of_forall_prime_mul_dvd + - Nat.minFac_dvd + - Nat.pow_minFac + - Nat.Prime.mod_two_eq_one_iff_ne_two + - Nat.le_minFac + - Nat.Prime.pow_minFac + - path: Mathlib/Data/Set/Pointwise/SMul.lean + theorems: + - Set.smul_set_compl + - Set.smul_inter_ne_empty_iff' + - Set.smul_zero_subset + - Set.add_smul_subset + - Set.op_smul_set_smul_eq_smul_smul_set + - Set.inv_op_smul_set_distrib + - Set.inv_zero + - Set.op_smul_inter_ne_empty_iff + - Set.inv_smul_set_distrib + - Set.zero_smul_subset + - "Set.inv_smul_set_distrib\u2080" + - Set.zero_smul_set + - Set.zero_mem_smul_set_iff + - Set.iUnion_smul_eq_setOf_exists + - Set.neg_smul_set + - Set.zero_mem_smul_iff + - Set.image_op_smul + - Set.Nonempty.zero_smul + - Set.smul_set_neg + - Set.smul_neg + - Set.smul_inter_ne_empty_iff + - Set.mem_inv_smul_set_iff + - "Set.inv_op_smul_set_distrib\u2080" + - Set.smul_set_disjoint_iff + - Set.Nonempty.smul_zero + - Set.neg_smul + - path: Mathlib/LinearAlgebra/Basis.lean + theorems: + - Basis.singleton_repr + - Basis.equivFun_symm_apply + - Basis.eq_ofRepr_eq_repr + - Basis.repr_support_subset_of_mem_span + - Basis.singleton_apply + - Basis.equivFun_self + - Basis.repr_reindex_apply + - Basis.mem_submodule_iff + - Basis.map_equiv + - Basis.reindex_refl + - Basis.mem_span_repr_support + - Basis.ext + - Basis.forall_coord_eq_zero_iff + - Basis.reindexRange_apply + - Basis.sum_equivFun + - Basis.equiv_trans + - Basis.sumCoords_reindex + - Basis.mk_coord_apply_ne + - Basis.self_mem_span_image + - Basis.equivFun_ofEquivFun + - Module.card_fintype + - Basis.prod_apply + - basis_finite_of_finite_spans + - Basis.equiv_refl + - Basis.reindexFinsetRange_repr_self + - Basis.basis_singleton_iff + - Basis.equiv_symm + - Basis.coe_sumCoords_eq_finsum + - Basis.sumCoords_self_apply + - Basis.index_nonempty + - Basis.sum_repr_mul_repr + - Basis.constr_basis + - Basis.restrictScalars_repr_apply + - Basis.mem_submodule_iff' + - Basis.eq_bot_of_rank_eq_zero + - Basis.noZeroSMulDivisors + - Basis.constr_range + - Basis.reindex_apply + - Basis.coe_ofEquivFun + - Basis.reindexFinsetRange_apply + - Basis.maximal + - Basis.reindexFinsetRange_repr + - Basis.constr_comp + - Basis.repr_apply_eq + - Basis.coe_mkFinCons + - Basis.repr_symm_single + - Basis.repr_symm_apply + - Basis.restrictScalars_apply + - Basis.coe_sumCoords_of_fintype + - Basis.equiv_apply + - Basis.prod_apply_inl_snd + - Basis.coord_repr_symm + - Basis.prod_apply_inr_snd + - Basis.mapCoeffs_apply + - Basis.range_reindex + - Basis.finTwoProd_zero + - Basis.repr_sum_self + - Basis.dvd_coord_smul + - Basis.reindexRange_repr' + - Basis.prod_apply_inr_fst + - Basis.linearIndependent + - Basis.mk_coord_apply + - Basis.ext_elem_iff + - Basis.coord_unitsSMul + - Basis.groupSMul_span_eq_top + - Basis.finTwoProd_one + - Basis.repr_range + - Basis.mk_coord_apply_eq + - Basis.total_repr + - Basis.mem_center_iff + - Basis.mk_apply + - infinite_basis_le_maximal_linearIndependent' + - Basis.reindexRange_self + - Basis.repr_injective + - Basis.constr_apply_fintype + - Basis.ofEquivFun_equivFun + - union_support_maximal_linearIndependent_eq_range_basis + - Basis.reindexFinsetRange_self + - Basis.prod_apply_inl_fst + - Basis.repr_reindex + - Basis.mem_span_iff_repr_mem + - Basis.repr_total + - path: Mathlib/CategoryTheory/Monoidal/Category.lean + theorems: + - CategoryTheory.MonoidalCategory.inv_whiskerRight + - CategoryTheory.MonoidalCategory.associator_inv_naturality_right + - CategoryTheory.MonoidalCategory.prodMonoidal_leftUnitor_inv_snd + - CategoryTheory.MonoidalCategory.tensor_hom_inv_id + - CategoryTheory.MonoidalCategory.whiskerLeft_inv_hom + - CategoryTheory.MonoidalCategory.tensor_right_iff + - CategoryTheory.MonoidalCategory.associator_inv_naturality_middle + - CategoryTheory.MonoidalCategory.triangle_assoc_comp_right + - CategoryTheory.MonoidalCategory.tensor_hom_inv_id' + - CategoryTheory.MonoidalCategory.whiskerLeft_hom_inv' + - CategoryTheory.MonoidalCategory.leftUnitor_tensor + - CategoryTheory.MonoidalCategory.tensor_inv_hom_id' + - CategoryTheory.MonoidalCategory.whiskerRight_id_symm + - CategoryTheory.MonoidalCategory.pentagon_inv_hom_hom_hom_inv + - CategoryTheory.MonoidalCategory.inv_hom_whiskerRight' + - CategoryTheory.MonoidalCategory.hom_inv_id_tensor' + - CategoryTheory.MonoidalCategory.pentagon_hom_hom_inv_inv_hom + - CategoryTheory.MonoidalCategory.id_tensorHom + - CategoryTheory.MonoidalCategory.id_tensor_comp + - CategoryTheory.MonoidalCategory.tensor_dite + - CategoryTheory.MonoidalCategory.whisker_assoc + - CategoryTheory.MonoidalCategory.whiskerLeft_comp + - CategoryTheory.MonoidalCategory.tensor_left_iff + - CategoryTheory.MonoidalCategory.pentagon_inv_inv_hom_inv_inv + - CategoryTheory.MonoidalCategory.pentagon_inv_hom_hom_hom_hom + - CategoryTheory.MonoidalCategory.leftUnitor_inv_whiskerRight + - CategoryTheory.MonoidalCategory.eqToHom_whiskerRight + - CategoryTheory.MonoidalCategory.comp_whiskerRight + - CategoryTheory.MonoidalCategory.id_whiskerLeft_symm + - CategoryTheory.MonoidalCategory.pentagon_hom_inv_inv_inv_inv + - CategoryTheory.MonoidalCategory.whiskerLeft_dite + - CategoryTheory.MonoidalCategory.hom_inv_id_tensor + - CategoryTheory.MonoidalCategory.associator_conjugation + - CategoryTheory.MonoidalCategory.whiskerLeft_iff + - CategoryTheory.MonoidalCategory.pentagon_hom_hom_inv_hom_hom + - CategoryTheory.MonoidalCategory.tensorRightTensor_inv_app + - CategoryTheory.MonoidalCategory.prodMonoidal_rightUnitor_hom_snd + - CategoryTheory.MonoidalCategory.dite_tensor + - CategoryTheory.MonoidalCategory.whisker_assoc_symm + - CategoryTheory.MonoidalCategory.whisker_exchange + - CategoryTheory.MonoidalCategory.rightUnitor_tensor + - CategoryTheory.MonoidalCategory.associator_inv_naturality_left + - CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv + - CategoryTheory.MonoidalCategory.hom_inv_whiskerRight' + - CategoryTheory.MonoidalCategory.whiskerRight_id + - CategoryTheory.MonoidalCategory.pentagon_inv_inv_hom_hom_inv + - CategoryTheory.MonoidalCategory.prodMonoidal_rightUnitor_inv_snd + - CategoryTheory.MonoidalCategory.hom_inv_whiskerRight + - CategoryTheory.MonoidalCategory.rightUnitor_inv_naturality + - CategoryTheory.MonoidalCategory.dite_whiskerRight + - CategoryTheory.MonoidalCategory.prodMonoidal_leftUnitor_hom_snd + - CategoryTheory.MonoidalCategory.pentagon_hom_inv_inv_inv_hom + - CategoryTheory.MonoidalCategory.leftUnitor_whiskerRight + - CategoryTheory.MonoidalCategory.rightUnitor_tensor_inv + - CategoryTheory.MonoidalCategory.tensorHom_id + - CategoryTheory.MonoidalCategory.prodMonoidal_leftUnitor_inv_fst + - CategoryTheory.MonoidalCategory.inv_tensor + - CategoryTheory.MonoidalCategory.inv_hom_id_tensor + - CategoryTheory.MonoidalCategory.leftUnitor_inv_naturality + - CategoryTheory.MonoidalCategory.inv_hom_id_tensor' + - CategoryTheory.MonoidalCategory.tensor_whiskerLeft_symm + - CategoryTheory.MonoidalCategory.whiskerRight_tensor + - CategoryTheory.MonoidalCategory.id_tensor_comp_tensor_id + - CategoryTheory.MonoidalCategory.associator_inv_naturality + - CategoryTheory.MonoidalCategory.prodMonoidal_rightUnitor_hom_fst + - CategoryTheory.MonoidalCategory.whiskerRight_iff + - CategoryTheory.MonoidalCategory.prodMonoidal_rightUnitor_inv_fst + - CategoryTheory.MonoidalCategory.associator_naturality_middle + - CategoryTheory.MonoidalCategory.triangle_assoc_comp_right_inv + - CategoryTheory.MonoidalCategory.whiskerLeft_hom_inv + - CategoryTheory.MonoidalCategory.whiskerLeft_rightUnitor_inv + - CategoryTheory.MonoidalCategory.id_tensor_associator_naturality + - CategoryTheory.MonoidalCategory.whiskerLeft_inv_hom' + - CategoryTheory.MonoidalCategory.id_tensor_associator_inv_naturality + - CategoryTheory.MonoidalCategory.tensorLeftTensor_inv_app + - CategoryTheory.MonoidalCategory.inv_hom_whiskerRight + - CategoryTheory.MonoidalCategory.tensor_id_comp_id_tensor + - CategoryTheory.MonoidalCategory.tensor_whiskerLeft + - CategoryTheory.MonoidalCategory.inv_whiskerLeft + - CategoryTheory.MonoidalCategory.associator_inv_conjugation + - CategoryTheory.MonoidalCategory.whiskerLeft_rightUnitor + - CategoryTheory.MonoidalCategory.tensor_inv_hom_id + - CategoryTheory.MonoidalCategory.prodMonoidal_leftUnitor_hom_fst + - CategoryTheory.MonoidalCategory.pentagon_inv + - path: Mathlib/LinearAlgebra/Dual.lean + theorems: + - LinearMap.dualMap_surjective_iff + - Module.DualBases.mem_of_mem_span + - Basis.dualBasis_apply_self + - Basis.linearEquiv_dual_iff_finiteDimensional + - "LinearMap.flip_injective_iff\u2081" + - Basis.toDual_eq_equivFun + - Submodule.exists_dual_map_eq_bot_of_nmem + - Module.DualBases.coe_basis + - Module.comap_eval_surjective + - LinearMap.ker_dualMap_eq_dualAnnihilator_range + - Subspace.dualPairing_nondegenerate + - LinearMap.dualCoannihilator_range_eq_ker_flip + - LinearMap.dualMap_injective_of_surjective + - Submodule.range_dualMap_mkQ_eq + - LinearMap.range_dualMap_dual_eq_span_singleton + - Submodule.mem_dualCoannihilator + - Basis.dual_rank_eq + - Basis.toDual_injective + - Subspace.comap_dualAnnihilator_dualAnnihilator + - Basis.sum_dual_apply_smul_coord + - Subspace.dualAnnihilator_iInf_eq + - Module.map_eval_injective + - Module.subsingleton_dual_iff + - Subspace.finrank_dualCoannihilator_eq + - Module.eval_apply_eq_zero_iff + - Subspace.dualCopairing_nondegenerate + - Submodule.dualAnnihilator_gc + - LinearEquiv.dualMap_refl + - Module.apply_evalEquiv_symm_apply + - Basis.toDual_total_right + - Subspace.isCompl_dualAnnihilator + - Subspace.dualPairing_eq + - Basis.total_coord + - Basis.eval_ker + - Basis.toDual_apply_right + - "LinearMap.flip_bijective_iff\u2081" + - Module.equiv + - Module.DualBases.lc_coeffs + - Basis.eval_range + - Module.eval_ker + - Basis.toDual_total_left + - Submodule.sup_dualAnnihilator_le_inf + - Module.DualBases.coeffs_lc + - Subspace.dual_finrank_eq + - LinearMap.dualMap_bijective_iff + - TensorProduct.dualDistrib_dualDistribInvOfBasis_left_inverse + - Module.symm_dualMap_evalEquiv + - Module.finite_dual_iff + - Submodule.quotDualCoannihilatorToDual_nondegenerate + - Subspace.forall_mem_dualAnnihilator_apply_eq_zero_iff + - Basis.coe_toDual_self + - LinearMap.dualMap_id + - Basis.coe_dualBasis + - LinearMap.finrank_range_dualMap_eq_finrank_range + - Basis.toDual_toDual + - LinearMap.range_dualMap_eq_dualAnnihilator_ker_of_subtype_range_surjective + - Basis.dualBasis_equivFun + - Module.DualBases.dual_lc + - Basis.toDual_range + - Basis.toDual_apply + - LinearMap.dualAnnihilator_ker_eq_range_flip + - Module.DualBases.coe_dualBasis + - Subspace.dualAnnihilator_inf_eq + - Subspace.dualRestrict_leftInverse + - TensorProduct.dualDistrib_dualDistribInvOfBasis_right_inverse + - LinearMap.ker_dualMap_eq_dualCoannihilator_range + - Module.nontrivial_dual_iff + - Basis.total_dualBasis + - Subspace.dualAnnihilator_dualAnnihilator_eq_map + - Dual.apply_one_mul_eq + - LinearMap.dualMap_injective_iff + - Subspace.dualAnnihilator_dualCoannihilator_eq + - Basis.dualBasis_repr + - Basis.toDual_inj + - Subspace.dualAnnihilator_dualAnnihilator_eq + - "LinearMap.flip_injective_iff\u2082" + - Subspace.dualCoannihilator_top + - Submodule.mem_dualAnnihilator + - Basis.toDual_apply_left + - Subspace.quotAnnihilatorEquiv_apply + - Submodule.dualAnnihilator_top + - path: Mathlib/Analysis/Calculus/Deriv/Basic.lean + theorems: + - norm_derivWithin_eq_norm_fderivWithin + - Filter.EventuallyEq.hasDerivAtFilter_iff + - norm_deriv_eq_norm_fderiv + - norm_deriv_le_of_lipschitz + - deriv_eqOn + - hasStrictFDerivAt_iff_hasStrictDerivAt + - HasDerivAt.le_of_lipschitz + - derivWithin_zero_of_isolated + - HasDerivAtFilter.isBigO_sub_rev + - hasDerivWithinAt_Iio_iff_Iic + - Filter.EventuallyEq.derivWithin_eq + - deriv_fderiv + - derivWithin_inter + - deriv_zero_of_not_differentiableAt + - derivWithin_zero_of_not_differentiableWithinAt + - norm_deriv_le_of_lipschitzOn + - derivWithin_of_mem_nhds + - HasDerivAt.le_of_lipschitzOn + - HasDerivAtFilter.congr_of_eventuallyEq + - derivWithin_zero_of_nmem_closure + - hasDerivWithinAt_Ioi_iff_Ici + - hasFDerivAtFilter_iff_hasDerivAtFilter + - norm_deriv_le_of_lip' + - deriv_mem_iff + - derivWithin_univ + - derivWithin_congr + - derivWithin_mem_iff + - derivWithin_fderivWithin + - derivWithin_congr_set' + - derivWithin_Ioi_eq_Ici + - Filter.EventuallyEq.deriv_eq + - HasDerivAt.le_of_lip' + - derivWithin_congr_set + - path: Mathlib/Data/Polynomial/HasseDeriv.lean + theorems: + - Polynomial.hasseDeriv_apply_one + - Polynomial.hasseDeriv_eq_zero_of_lt_natDegree + - Polynomial.hasseDeriv_zero' + - Polynomial.hasseDeriv_one' + - Polynomial.hasseDeriv_C + - Polynomial.hasseDeriv_X + - Polynomial.hasseDeriv_apply + - Polynomial.hasseDeriv_comp + - Polynomial.hasseDeriv_mul + - Polynomial.natDegree_hasseDeriv + - Polynomial.hasseDeriv_coeff + - Polynomial.hasseDeriv_monomial + - Polynomial.natDegree_hasseDeriv_le + - Polynomial.factorial_smul_hasseDeriv + - path: Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean + theorems: + - tendsto_rpow_atTop_of_base_gt_one + - Complex.isTheta_cpow_const_rpow + - tendsto_rpow_atTop_of_base_lt_one + - isLittleO_log_rpow_nhds_zero + - NNReal.tendsto_rpow_atTop + - tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero + - tendsto_log_mul_self_nhds_zero_left + - isLittleO_log_rpow_rpow_atTop + - isLittleO_zpow_exp_pos_mul_atTop + - Complex.isBigO_cpow_rpow + - tendsto_rpow_div + - Asymptotics.IsLittleO.rpow + - isLittleO_abs_log_rpow_rpow_nhds_zero + - tendsto_exp_mul_div_rpow_atTop + - isLittleO_pow_exp_pos_mul_atTop + - ENNReal.tendsto_rpow_at_top + - isLittleO_exp_neg_mul_rpow_atTop + - tendsto_rpow_atTop + - tendsto_rpow_neg_div + - tendsto_exp_div_rpow_atTop + - Complex.isTheta_cpow_rpow + - isLittleO_rpow_exp_atTop + - isLittleO_rpow_exp_pos_mul_atTop + - tendsto_rpow_atBot_of_base_lt_one + - tendsto_rpow_atBot_of_base_gt_one + - tendsto_rpow_div_mul_add + - tendsto_log_mul_rpow_nhds_zero + - Asymptotics.IsBigOWith.rpow + - path: Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean + theorems: + - Finset.truncatedSup_infs + - Finset.map_truncatedSup + - Finset.truncatedInf_union_left + - Finset.truncatedSup_infs_of_not_mem + - Finset.truncatedSup_of_isAntichain + - binomial_sum_eq + - Fintype.sum_div_mul_card_choose_card + - Finset.truncatedInf_sups + - AhlswedeZhang.supSum_union_add_supSum_infs + - Finset.truncatedInf_le + - Finset.truncatedSup_singleton + - Finset.infs_aux + - Finset.truncatedSup_empty + - Finset.truncatedInf_sups_of_not_mem + - Finset.truncatedInf_union_right + - Finset.truncatedInf_of_isAntichain + - Finset.truncatedInf_union_of_not_mem + - Finset.map_truncatedInf + - AhlswedeZhang.supSum_of_not_univ_mem + - AhlswedeZhang.infSum_union_add_infSum_sups + - Finset.card_truncatedSup_union_add_card_truncatedSup_infs + - AhlswedeZhang.IsAntichain.le_infSum + - Finset.upper_aux + - Finset.truncatedSup_union_right + - Finset.truncatedSup_union + - Finset.truncatedInf_empty + - AhlswedeZhang.infSum_compls_add_supSum + - Finset.truncatedInf_singleton + - Finset.card_truncatedInf_union_add_card_truncatedInf_sups + - Finset.lower_aux + - AhlswedeZhang.infSum_eq_one + - Finset.le_truncatedSup + - AhlswedeZhang.supSum_singleton + - Finset.truncatedSup_union_left + - path: Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean + theorems: + - Polynomial.monic_generator_eq_minpoly + - Polynomial.span_minpoly_eq_annihilator + - Polynomial.mem_iff_annIdealGenerator_dvd + - Polynomial.span_singleton_annIdealGenerator + - Polynomial.mem_iff_eq_smul_annIdealGenerator + - Polynomial.annIdealGenerator_eq_minpoly + - Polynomial.annIdealGenerator_eq_zero_iff + - path: Mathlib/Data/Set/Intervals/Monoid.lean + theorems: + - Set.Ico_add_bij + - Set.image_const_add_Ioo + - Set.Ioi_add_bij + - Set.Ici_add_bij + - Set.Ioc_add_bij + - Set.Icc_add_bij + - Set.image_const_add_Icc + - Set.image_const_add_Ico + - Set.image_const_add_Ioc + - Set.Ioo_add_bij + - Set.image_const_add_Ici + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean + theorems: + - CliffordAlgebra.forall_mul_self_eq_iff + - "TensorAlgebra.toClifford_\u03B9" + - CliffordAlgebra.map_surjective + - CliffordAlgebra.leftInverse_map_of_leftInverse + - "CliffordAlgebra.mul_\u03B9_mul_\u03B9_of_isOrtho" + - "CliffordAlgebra.comp_\u03B9_sq_scalar" + - CliffordAlgebra.mul_add_swap_eq_polar_of_forall_mul_self_eq + - CliffordAlgebra.equivOfIsometry_refl + - CliffordAlgebra.map_id + - CliffordAlgebra.map_comp_map + - "CliffordAlgebra.\u03B9_mul_\u03B9_add_swap_of_isOrtho" + - "CliffordAlgebra.\u03B9_mul_\u03B9_mul_\u03B9" + - "CliffordAlgebra.\u03B9_mul_\u03B9_mul_of_isOrtho" + - CliffordAlgebra.lift_unique + - "CliffordAlgebra.\u03B9_range_map_lift" + - "CliffordAlgebra.\u03B9_sq_scalar" + - CliffordAlgebra.hom_ext + - CliffordAlgebra.equivOfIsometry_trans + - path: .lake/packages/std/Std/Data/AssocList.lean + theorems: + - Std.AssocList.toList_mapKey + - Std.AssocList.toList_modify + - Std.AssocList.find?_eq_findEntry? + - Std.AssocList.toList_eraseP + - Std.AssocList.foldl_eq + - Std.AssocList.foldlM_eq + - List.toList_toAssocList + - Std.AssocList.toList_eq_toListTR + - Std.AssocList.all_eq + - List.length_toAssocList + - Std.AssocList.length_modify + - Std.AssocList.length_replace + - Std.AssocList.toList_mapVal + - Std.AssocList.findEntryP?_eq + - Std.AssocList.forIn_eq + - Std.AssocList.toList_replace + - Std.AssocList.toList_toAssocList + - Std.AssocList.find?_eq + - Std.AssocList.forM_eq + - Std.AssocList.isEmpty_eq + - Std.AssocList.length_mapVal + - Std.AssocList.any_eq + - Std.AssocList.length_mapKey + - path: Mathlib/FieldTheory/Adjoin.lean + theorems: + - IntermediateField.isSplittingField_iff + - PowerBasis.equivAdjoinSimple_symm_gen + - IntermediateField.isAlgebraic_adjoin + - IntermediateField.algHomAdjoinIntegralEquiv_symm_apply_gen + - IntermediateField.mem_adjoin_simple_iff + - IntermediateField.adjoin.range_algebraMap_subset + - IntermediateField.biSup_adjoin_simple + - IntermediateField.extendScalars_adjoin + - IntermediateField.induction_on_adjoin_finset + - IntermediateField.adjoin_toSubalgebra_of_isAlgebraic + - IntermediateField.lift_adjoin_simple + - IntermediateField.eq_adjoin_of_eq_algebra_adjoin + - IntermediateField.adjoin.finrank + - IntermediateField.adjoin_minpoly_coeff_of_exists_primitive_element + - IntermediateField.bot_eq_top_of_rank_adjoin_eq_one + - IntermediateField.finrank_bot + - IntermediateField.exists_finset_of_mem_iSup + - PowerBasis.equivAdjoinSimple_symm_aeval + - IntermediateField.isSplittingField_iSup + - IntermediateField.bot_eq_top_of_finrank_adjoin_le_one + - IntermediateField.iInf_toSubfield + - IntermediateField.coe_iInf + - IntermediateField.exists_lt_finrank_of_infinite_dimensional + - IntermediateField.AdjoinSimple.isIntegral_gen + - IntermediateField.restrictScalars_adjoin + - IntermediateField.adjoin_int + - isSplittingField_iff_intermediateField + - IntermediateField.adjoin_le_subfield + - IntermediateField.rank_adjoin_simple_eq_one_iff + - IntermediateField.iInf_toSubalgebra + - IntermediateField.fG_of_fG_toSubalgebra + - IntermediateField.lift_top + - IntermediateField.adjoin_simple_le_iff + - IntermediateField.induction_on_adjoin_fg + - IntermediateField.mem_adjoin_iff + - IntermediateField.isAlgebraic_iSup + - IntermediateField.fg_bot + - IntermediateField.fieldRange_comp_val + - IntermediateField.rank_bot' + - IntermediateField.toSubalgebra_iSup_of_directed + - IntermediateField.sInf_toSubalgebra + - IntermediateField.sInf_toSubfield + - IntermediateField.adjoin_simple_eq_bot_iff + - IntermediateField.sup_toSubalgebra_of_isAlgebraic_right + - IntermediateField.adjoin_map + - IntermediateField.restrictScalars_adjoin_of_algEquiv + - IntermediateField.rank_eq_one_iff + - IntermediateField.adjoin_adjoin_left + - IntermediateField.aeval_gen_minpoly + - IntermediateField.adjoin_eq_bot_iff + - IntermediateField.adjoin_simple_isCompactElement + - Polynomial.irreducible_comp + - IntermediateField.equivOfEq_rfl + - IntermediateField.exists_finset_of_mem_supr' + - IntermediateField.finrank_adjoin_simple_eq_one_iff + - IntermediateField.adjoin_finset_isCompactElement + - IntermediateField.botEquiv_def + - IntermediateField.sup_toSubalgebra_of_isAlgebraic_left + - IntermediateField.card_algHom_adjoin_integral + - IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one + - IntermediateField.adjoin_algebraic_toSubalgebra + - IntermediateField.rank_bot + - IntermediateField.minpoly_gen + - IntermediateField.finrank_eq_one_iff + - IntermediateField.exists_finset_of_mem_adjoin + - IntermediateField.finiteDimensional_adjoin + - IntermediateField.adjoin_adjoin_comm + - IntermediateField.adjoin_rank_le_of_isAlgebraic + - path: Mathlib/RingTheory/Polynomial/Quotient.lean + theorems: + - "MvPolynomial.eval\u2082_C_mk_eq_zero" + - MvPolynomial.quotientEquivQuotientMvPolynomial_leftInverse + - Ideal.polynomialQuotientEquivQuotientPolynomial_symm_mk + - Ideal.eq_zero_of_polynomial_mem_map_range + - Ideal.polynomialQuotientEquivQuotientPolynomial_map_mk + - Ideal.quotient_map_C_eq_zero + - "Ideal.eval\u2082_C_mk_eq_zero" + - MvPolynomial.quotientEquivQuotientMvPolynomial_rightInverse + - MvPolynomial.quotient_map_C_eq_zero + - path: Mathlib/Algebra/Order/Monoid/WithTop.lean + theorems: + - WithTop.add_coe_eq_top_iff + - WithTop.le_of_add_le_add_right + - WithTop.add_lt_top + - WithTop.add_right_cancel_iff + - WithTop.add_lt_add_right + - WithTop.add_lt_add_left + - WithTop.add_top + - WithTop.add_eq_coe + - WithBot.add_bot + - WithTop.add_left_cancel_iff + - WithTop.coe_add_eq_top_iff + - WithTop.le_of_add_le_add_left + - WithTop.add_eq_top + - path: Mathlib/RingTheory/Ideal/Norm.lean + theorems: + - Ideal.spanNorm_mul + - Ideal.span_singleton_absNorm_le + - Ideal.exists_mul_add_mem_pow_succ + - Ideal.absNorm_top + - Ideal.absNorm_dvd_norm_of_mem + - Ideal.absNorm_bot + - Ideal.natAbs_det_basis_change + - Ideal.irreducible_of_irreducible_absNorm + - Ideal.absNorm_eq_one_iff + - cardQuot_pow_of_prime + - Ideal.spanNorm_mul_spanNorm_le + - Ideal.norm_dvd_iff + - Ideal.spanNorm_eq_bot_iff + - Ideal.spanNorm_localization + - Ideal.relNorm_top + - Ideal.finite_setOf_absNorm_eq + - Ideal.mem_prime_of_mul_mem_pow + - Ideal.span_singleton_absNorm + - Ideal.mul_add_mem_pow_succ_unique + - cardQuot_mul_of_coprime + - Ideal.spanNorm_mul_of_bot_or_top + - Ideal.spanNorm_top + - Ideal.absNorm_span_insert + - Ideal.absNorm_mem + - cardQuot_mul + - Ideal.absNorm_span_singleton + - Ideal.map_spanNorm + - Ideal.mul_add_mem_pow_succ_inj + - Submodule.cardQuot_apply + - Ideal.spanNorm_singleton + - Submodule.cardQuot_eq_one_iff + - Ideal.spanNorm_bot + - Ideal.natAbs_det_equiv + - Ideal.relNorm_bot + - path: Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean + theorems: + - ModuleCat.MonoidalCategory.pentagon + - ModuleCat.MonoidalCategory.tensor_comp + - ModuleCat.MonoidalCategory.triangle + - ModuleCat.MonoidalCategory.leftUnitor_naturality + - ModuleCat.MonoidalCategory.pentagon_aux + - ModuleCat.MonoidalCategory.rightUnitor_naturality + - ModuleCat.MonoidalCategory.tensor_id + - ModuleCat.MonoidalCategory.associator_naturality_aux + - path: Mathlib/Algebra/Module/PointwisePi.lean + theorems: + - smul_pi_subset + - smul_univ_pi + - path: Mathlib/Topology/NhdsSet.lean + theorems: + - nhdsSet_iUnion + - nhdsSet_insert + - disjoint_nhdsSet_principal + - nhdsSet_union + - mem_nhdsSet_iff_exists + - hasBasis_nhdsSet + - nhdsSet_empty + - mem_nhdsSet_empty + - subset_interior_iff_mem_nhdsSet + - Continuous.tendsto_nhdsSet_nhds + - nhdsSet_eq_principal_iff + - eventually_nhdsSet_iUnion + - IsClosed.nhdsSet_le_sup + - disjoint_principal_nhdsSet + - nhdsSet_univ + - "eventually_nhdsSet_iUnion\u2082" + - IsOpen.mem_nhdsSet + - mem_nhdsSet_iff_forall + - Filter.Eventually.union_nhdsSet + - nhdsSet_le + - union_mem_nhdsSet + - IsClosed.nhdsSet_le_sup' + - nhdsSet_singleton + - path: Mathlib/Dynamics/OmegaLimit.lean + theorems: + - "mem_omegaLimit_iff_frequently\u2082" + - Flow.isInvariant_omegaLimit + - eventually_mapsTo_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset + - omegaLimit_union + - Flow.omegaLimit_image_subset + - mem_omegaLimit_iff_frequently + - eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset' + - nonempty_omegaLimit_of_isCompact_absorbing + - omegaLimit_iUnion + - Flow.omegaLimit_omegaLimit + - mapsTo_omegaLimit' + - mem_omegaLimit_singleton_iff_map_cluster_point + - omegaLimit_eq_iInter_inter + - eventually_mapsTo_of_isOpen_of_omegaLimit_subset + - omegaLimit_image_eq + - omegaLimit_subset_of_tendsto + - omegaLimit_subset_closure_fw_image + - Flow.omegaLimit_image_eq + - path: Mathlib/Analysis/Convex/Function.lean + theorems: + - ConvexOn.comp_affineMap + - neg_concaveOn_iff + - ConvexOn.comp_linearMap + - StrictConvexOn.eq_of_isMinOn + - LinearOrder.strictConvexOn_of_lt + - ConvexOn.lt_left_of_right_lt + - ConvexOn.translate_left + - ConvexOn.sup + - ConvexOn.convex_epigraph + - StrictConvexOn.lt_on_open_segment' + - OrderIso.convexOn_symm + - convexOn_iff_div + - StrictConvexOn.translate_left + - ConvexOn.lt_right_of_left_lt' + - OrderIso.concaveOn_symm + - ConvexOn.translate_right + - ConvexOn.le_left_of_right_le + - StrictConvexOn.translate_right + - StrictConvexOn.add_convexOn + - StrictConvexOn.convex_lt + - neg_strictConvexOn_iff + - ConvexOn.openSegment_subset_strict_epigraph + - convexOn_iff_forall_pos + - ConvexOn.le_on_segment' + - StrictConvexOn.sup + - concaveOn_id + - convexOn_iff_pairwise_pos + - ConvexOn.add + - neg_strictConcaveOn_iff + - ConvexOn.convex_le + - StrictConvexOn.add + - ConvexOn.lt_right_of_left_lt + - neg_convexOn_iff + - strictConvexOn_iff_div + - LinearOrder.convexOn_of_lt + - OrderIso.strictConvexOn_symm + - ConvexOn.le_right_of_left_le + - convexOn_id + - LinearMap.concaveOn + - ConvexOn.le_right_of_left_le' + - ConvexOn.smul + - LinearMap.convexOn + - OrderIso.strictConcaveOn_symm + - path: Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean + theorems: + - "MeasureTheory.Mem\u2112p.finStronglyMeasurable_of_stronglyMeasurable" + - path: Mathlib/LinearAlgebra/AffineSpace/Matrix.lean + theorems: + - AffineBasis.affineIndependent_of_toMatrix_right_inv + - AffineBasis.toMatrix_self + - AffineBasis.toMatrix_row_sum_one + - AffineBasis.toMatrix_vecMul_coords + - AffineBasis.isUnit_toMatrix_iff + - AffineBasis.toMatrix_mul_toMatrix + - AffineBasis.toMatrix_inv_vecMul_toMatrix + - AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv + - path: Mathlib/Algebra/DirectLimit.lean + theorems: + - Module.DirectLimit.toModule_totalize_of_le + - Ring.DirectLimit.of.zero_exact_aux + - Ring.DirectLimit.of.zero_exact_aux2 + - Ring.DirectLimit.map_id + - Ring.DirectLimit.congr_symm_apply_of + - Ring.DirectLimit.lift_unique + - Module.DirectLimit.of.zero_exact + - Module.DirectLimit.exists_of + - Field.DirectLimit.mul_inv_cancel + - AddCommGroup.DirectLimit.map_comp + - Module.DirectLimit.map_comp + - Ring.DirectLimit.map_comp + - Module.DirectLimit.congr_symm_apply_of + - AddCommGroup.DirectLimit.lift_injective + - Module.DirectLimit.map_id + - Module.DirectLimit.of.zero_exact_aux + - Ring.DirectLimit.exists_of + - Ring.DirectLimit.lift_injective + - Ring.DirectLimit.of.zero_exact + - AddCommGroup.DirectLimit.lift_unique + - Field.DirectLimit.inv_mul_cancel + - Module.DirectLimit.lift_injective + - Field.DirectLimit.exists_inv + - Module.DirectLimit.lift_unique + - Ring.DirectLimit.Polynomial.exists_of + - Ring.DirectLimit.of_injective + - AddCommGroup.DirectLimit.map_id + - path: Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean + theorems: + - MeasureTheory.Measure.rnDeriv_withDensity_right_of_absolutelyContinuous + - MeasureTheory.Measure.rnDeriv_pos + - MeasureTheory.integrable_rnDeriv_smul_iff + - MeasureTheory.Measure.set_integral_toReal_rnDeriv' + - MeasureTheory.Measure.rnDeriv_withDensity_left_of_absolutelyContinuous + - MeasureTheory.Measure.rnDeriv_mul_rnDeriv + - MeasureTheory.integral_rnDeriv_smul + - MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left + - MeasureTheory.Measure.rnDeriv_pos' + - MeasureTheory.Measure.withDensity_rnDeriv_eq + - MeasureTheory.Measure.set_lintegral_rnDeriv + - MeasureTheory.Measure.set_integral_toReal_rnDeriv_le + - MeasureTheory.Measure.integrableOn_toReal_rnDeriv + - MeasureTheory.Measure.rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular + - MeasureTheory.Measure.inv_rnDeriv + - MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular + - MeasureTheory.lintegral_rnDeriv_mul + - MeasureTheory.Measure.set_integral_toReal_rnDeriv_eq_withDensity + - MeasureTheory.Measure.integral_toReal_rnDeriv + - "MeasureTheory.SignedMeasure.withDensity\u1D65_rnDeriv_eq" + - MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_right + - MeasureTheory.Measure.set_lintegral_rnDeriv' + - MeasureTheory.Measure.set_integral_toReal_rnDeriv + - MeasureTheory.Measure.inv_rnDeriv' + - MeasureTheory.Measure.rnDeriv_withDensity_left + - MeasureTheory.set_lintegral_rnDeriv_mul + - MeasureTheory.Measure.rnDeriv_withDensity_rnDeriv + - "MeasureTheory.withDensity\u1D65_rnDeriv_smul" + - MeasureTheory.Measure.rnDeriv_withDensity_right + - MeasureTheory.Measure.rnDeriv_restrict + - MeasureTheory.Measure.lintegral_rnDeriv + - MeasureTheory.Measure.set_integral_toReal_rnDeriv_eq_withDensity' + - MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular' + - path: Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean + theorems: + - le_rpow_one_add_norm_iff_norm_le + - sqrt_one_add_norm_sq_le + - finite_integral_one_add_norm + - integrable_rpow_neg_one_add_norm_sq + - finite_integral_rpow_sub_one_pow_aux + - closedBall_rpow_sub_one_eq_empty_aux + - integrable_one_add_norm + - rpow_neg_one_add_norm_sq_le + - one_add_norm_le_sqrt_two_mul_sqrt + - path: Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean + theorems: + - integral_univ_inv_one_add_sq + - integral_Ioi_rpow_of_lt + - integral_exp_neg_Ioi_zero + - integral_Ioi_cpow_of_lt + - integrableOn_Ioi_cpow_iff + - integrableOn_Ioi_cpow_of_lt + - not_integrableOn_Ioi_cpow + - integral_exp_Iic + - integral_exp_neg_Ioi + - not_integrableOn_Ioi_rpow + - integrable_inv_one_add_sq + - integrableOn_Ioi_rpow_of_lt + - integrableOn_Ioi_rpow_iff + - path: Mathlib/Data/Fintype/Basic.lean + theorems: + - exists_seq_of_forall_finset_exists' + - Fin.univ_succAbove + - Finset.coe_filter_univ + - Set.disjoint_toFinset + - Finset.insert_compl_insert + - Set.toFinset_insert + - Finset.Nonempty.eq_univ + - Set.toFinset_nonempty + - Fin.univ_succ + - Finset.subtype_univ + - Finset.compl_erase + - Finset.inter_univ + - Finset.compl_ne_univ_iff_nonempty + - Finset.compl_filter + - Function.Injective.invOfMemRange_surjective + - Fin.image_succ_univ + - Set.toFinset_inter + - Set.toFinset_congr + - Finset.compl_singleton + - Finset.subset_compl_singleton + - Finset.univ_filter_exists + - Set.mem_toFinset + - mem_image_univ_iff_mem_range + - Set.toFinset_ssubset_univ + - Set.toFinset_inj + - Set.toFinset_eq_empty + - Finset.univ_nonempty_iff + - Finset.insert_inj_on' + - Set.toFinset_setOf + - Fin.univ_castSuccEmb + - Finset.piecewise_erase_univ + - Finset.univ_inter + - Function.Embedding.invOfMemRange_surjective + - Set.toFinset_eq_univ + - Function.Embedding.invFun_restrict + - Fin.image_castSucc + - Finset.eq_univ_iff_forall + - Set.toFinset_subset_toFinset + - Set.toFinset_ofFinset + - Fin.image_succAbove_univ + - Set.toFinset_symmDiff + - Finset.piecewise_univ + - Set.toFinset_compl + - List.toFinset_finRange + - Set.toFinset_diff + - Finset.insert_compl_self + - Finset.not_mem_compl + - Set.toFinset_ssubset_toFinset + - Set.toFinset_singleton + - Set.subset_toFinset + - Finset.coe_eq_univ + - Set.toFinset_range + - Finset.filter_univ_mem + - Finset.compl_insert + - Set.toFinset_subset + - Finset.subtype_eq_univ + - Set.filter_mem_univ_eq_toFinset + - Finset.piecewise_compl + - Set.ssubset_toFinset + - Multiset.map_univ_val_equiv + - Set.toFinset_union + - Set.toFinset_ssubset + - exists_seq_of_forall_finset_exists + - Finset.mem_compl + - Finset.univ_filter_mem_range + - Finset.coe_univ + - Finset.univ_map_equiv_to_embedding + - Fintype.coe_image_univ + - Set.toFinset_univ + - Finset.univ_eq_empty_iff + - Set.toFinset_empty + - Function.Injective.invFun_restrict + - Fintype.finsetEquivSet_symm_apply + - Finset.codisjoint_left + - Fintype.univ_Prop + - Finset.singleton_eq_univ + - path: Mathlib/LinearAlgebra/Multilinear/Basic.lean + theorems: + - MultilinearMap.map_update + - MultilinearMap.comp_linearEquiv_eq_zero_iff + - MultilinearMap.domDomRestrict_aux + - MultilinearMap.map_sum_finset_aux + - MultilinearMap.mkPiRing_zero + - MultilinearMap.curryFinFinset_apply_const + - MultilinearMap.snoc_add + - MultilinearMap.map_add_sub_map_add_sub_linearDeriv + - MultilinearMap.sum_apply + - MultilinearMap.curryFinFinset_symm_apply_piecewise_const_aux + - LinearMap.compMultilinearMap_domDomCongr + - MultilinearMap.curryFinFinset_symm_apply_piecewise_const + - MultilinearMap.map_update_sum + - MultilinearMap.map_piecewise_smul + - MultilinearMap.cons_smul + - MultilinearMap.map_sub_map_piecewise + - MultilinearMap.mkPiAlgebraFin_apply_const + - MultilinearMap.mkPiRing_eq_zero_iff + - MultilinearMap.mkPiRing_eq_iff + - MultilinearMap.curry_uncurryRight + - MultilinearMap.compLinearMap_injective + - MultilinearMap.map_smul_univ + - MultilinearMap.mkPiRing_apply_one_eq_self + - MultilinearMap.map_piecewise_add + - MultilinearMap.uncurry_curryRight + - MultilinearMap.uncurry_curryLeft + - MultilinearMap.map_sub + - LinearMap.curry_uncurryLeft + - MultilinearMap.curryFinFinset_apply_const_aux + - MultilinearMap.map_add_eq_map_add_linearDeriv_add + - MultilinearMap.map_add_univ + - MultilinearMap.cons_add + - MultilinearMap.map_piecewise_sub_map_piecewise + - MultilinearMap.map_zero + - MultilinearMap.map_coord_zero + - MultilinearMap.map_update_smul + - MultilinearMap.snoc_smul + - MultilinearMap.map_neg + - MultilinearMap.linearDeriv_apply + - path: Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean + theorems: + - NonUnitalAlgebra.coe_iInf + - NonUnitalAlgebra.mem_sInf + - NonUnitalAlgebra.sInf_toSubmodule + - NonUnitalSubalgebra.toSubmodule_injective + - NonUnitalSubalgebra.toNonUnitalSubsemiring_injective + - NonUnitalAlgebra.sInf_toNonUnitalSubsemiring + - NonUnitalAlgebra.mem_bot + - NonUnitalAlgebra.mem_iInf + - NonUnitalSubalgebra.iSupLift_mk + - NonUnitalAlgebra.toSubmodule_bot + - NonUnitalAlgebra.iInf_toSubmodule + - NonUnitalAlgebra.mem_sup_right + - NonUnitalSubalgebra.iSupLift_comp_inclusion + - NonUnitalAlgebra.eq_top_iff + - Set.smul_mem_center + - NonUnitalAlgebra.mem_sup_left + - NonUnitalSubalgebra.prod_top + - NonUnitalAlgebra.adjoin_induction' + - NonUnitalSubalgebra.iSupLift_inclusion + - NonUnitalAlgebra.map_bot + - NonUnitalAlgebra.adjoin_empty + - NonUnitalAlgebra.coe_bot + - NonUnitalSubalgebra.iSupLift_of_mem + - NonUnitalSubalgebra.toNonUnitalSubring_injective + - path: Mathlib/Data/Int/Parity.lean + theorems: + - Int.four_dvd_add_or_sub_of_odd + - Int.even_mul_pred_self + - Int.even_sub' + - Int.natAbs_odd + - Int.odd_sub' + - Int.even_add + - Int.odd_add' + - Int.not_even_iff + - Int.even_add_one + - Int.even_or_odd' + - Int.odd_mul + - Int.odd_coe_nat + - Int.even_sub_one + - Int.even_mul_succ_self + - Int.emod_two_ne_zero + - Int.even_sub + - Int.even_xor'_odd + - Int.not_even_one + - Int.odd_add + - Int.not_odd_iff + - Int.even_pow + - Int.odd_sub + - Int.odd_pow + - Int.odd_iff + - Int.odd_iff_not_even + - Int.isCompl_even_odd + - Int.even_mul + - Int.two_mul_ediv_two_add_one_of_odd + - Int.natAbs_even + - Int.even_iff_not_odd + - Int.even_add' + - Int.even_coe_nat + - Int.not_even_bit1 + - Int.even_iff + - Int.emod_two_ne_one + - Int.two_not_dvd_two_mul_add_one + - Int.ediv_two_mul_two_add_one_of_odd + - Int.even_xor'_odd' + - path: Mathlib/Analysis/BoxIntegral/Partition/Basic.lean + theorems: + - BoxIntegral.Prepartition.IsPartition.inf + - BoxIntegral.Prepartition.mem_inf + - BoxIntegral.Prepartition.mem_restrict + - BoxIntegral.Prepartition.restrict_boxes_of_le + - BoxIntegral.Prepartition.sum_fiberwise + - BoxIntegral.Prepartition.iUnion_ofWithBot + - BoxIntegral.Prepartition.isPartitionDisjUnionOfEqDiff + - BoxIntegral.Prepartition.IsPartition.restrict + - BoxIntegral.Prepartition.sum_biUnion_boxes + - BoxIntegral.Prepartition.biUnion_assoc + - BoxIntegral.Prepartition.iUnion_eq_empty + - BoxIntegral.Prepartition.isPartition_single_iff + - BoxIntegral.Prepartition.iUnion_top + - BoxIntegral.Prepartition.mem_biUnionIndex + - BoxIntegral.Prepartition.injective_boxes + - BoxIntegral.Prepartition.biUnionIndex_le + - BoxIntegral.Prepartition.injOn_setOf_mem_Icc_setOf_lower_eq + - BoxIntegral.Prepartition.mem_restrict' + - BoxIntegral.Prepartition.iUnion_restrict + - BoxIntegral.Prepartition.biUnion_congr + - BoxIntegral.Prepartition.iUnion_single + - BoxIntegral.Prepartition.iUnion_inf + - BoxIntegral.Prepartition.mem_iUnion + - BoxIntegral.Prepartition.le_biUnion_iff + - BoxIntegral.Prepartition.iUnion_disjUnion + - BoxIntegral.Prepartition.isPartition_iff_iUnion_eq + - BoxIntegral.Prepartition.restrict_biUnion + - BoxIntegral.Prepartition.card_filter_mem_Icc_le + - BoxIntegral.Prepartition.le_ofWithBot + - BoxIntegral.Prepartition.biUnion_top + - BoxIntegral.Prepartition.biUnion_le_iff + - BoxIntegral.Prepartition.iUnion_filter_not + - BoxIntegral.Prepartition.restrict_mono + - BoxIntegral.Prepartition.filter_of_true + - BoxIntegral.Prepartition.IsPartition.existsUnique + - BoxIntegral.Prepartition.mem_biUnion + - BoxIntegral.Prepartition.le_iff_nonempty_imp_le_and_iUnion_subset + - BoxIntegral.Prepartition.biUnionIndex_mem + - BoxIntegral.Prepartition.iUnion_biUnion + - path: Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean + theorems: + - strictConcaveOn_cos_Icc + - strictConvexOn_zpow + - deriv_sqrt_mul_log + - strictConvexOn_pow + - deriv2_sqrt_mul_log + - int_prod_range_nonneg + - int_prod_range_pos + - Finset.prod_nonneg_of_card_nonpos_even + - Even.strictConvexOn_pow + - strictConcaveOn_sqrt_mul_log_Ioi + - strictConcaveOn_sin_Icc + - path: .lake/packages/std/Std/Data/RBMap/Lemmas.lean + theorems: + - Std.RBNode.foldlM_eq_foldlM_toList + - Std.RBNode.Path.ordered_iff + - Std.RBNode.Path.ins_toList + - Std.RBNode.Stream.toList_cons + - Std.RBNode.lowerBound?_le' + - Std.RBNode.Stream.forIn_eq_forIn_toList + - Std.RBNode.foldr_eq_foldr_toList + - Std.RBNode.Stream.foldl_eq_foldl_toList + - Std.RBNode.Stream.foldr_eq_foldr_toList + - Std.RBNode.Any_def + - Std.RBNode.toStream_toList' + - Std.RBNode.zoom_toList + - Std.RBNode.forIn_eq_forIn_toList + - Std.RBNode.Path.insert_toList + - Std.RBNode.balRight_toList + - Std.RBSet.find?_insert + - Std.RBSet.toStream_toList + - Std.RBNode.Ordered.lowerBound?_lt + - Std.RBNode.insert_toList_zoom + - Std.RBNode.Ordered.find?_some + - Std.RBNode.mem_congr + - Std.RBNode.Ordered.lowerBound?_least + - Std.RBNode.mem_insert_self + - Std.RBSet.mem_toList_insert + - Std.RBNode.exists_insert_toList_zoom_nil + - Std.RBNode.Ordered.lowerBound?_exists + - Std.RBNode.find?_some_mem + - Std.RBNode.mem_insert + - Std.RBNode.isOrdered_iff + - Std.RBNode.balance2_toList + - Std.RBNode.size_lt_depth + - Std.RBNode.Balanced.le_size + - Std.RBNode.balance1_toList + - Std.RBSet.mem_iff_mem_toList + - Std.RBNode.toList_node + - Std.RBNode.foldr_cons + - Std.RBNode.lowerBound?_of_some + - Std.RBNode.ordered_iff + - Std.RBSet.mem_insert + - Std.RBNode.mem_insert_of_mem + - Std.RBNode.Path.rootOrdered_iff + - Std.RBSet.find?_some + - Std.RBNode.Ordered.lowerBound?_least_lb + - Std.RBNode.Ordered.memP_iff_lowerBound? + - Std.RBNode.find?_some_eq_eq + - Std.RBSet.find?_insert_of_ne + - Std.RBNode.size_eq + - Std.RBNode.All_def + - Std.RBNode.setRed_toList + - Std.RBNode.forIn_visit_eq_bindList + - Std.RBNode.Ordered.unique + - Std.RBSet.find?_congr + - Std.RBNode.Ordered.memP_iff_find? + - Std.RBNode.toStream_toList + - Std.RBNode.setBlack_toList + - Std.RBNode.foldl_eq_foldl_toList + - Std.RBNode.Path.fill_toList + - Std.RBNode.Path.insertNew_toList + - Std.RBNode.IsCut.congr + - Std.RBNode.forM_eq_forM_toList + - Std.RBNode.isOrdered_iff' + - Std.RBNode.exists_insert_toList_zoom_node + - Std.RBNode.balLeft_toList + - Std.RBNode.All.lowerBound?_lb + - Std.RBNode.mem_toList + - Std.RBNode.Stream.foldr_cons + - path: Mathlib/Data/Multiset/Pi.lean + theorems: + - Multiset.Pi.cons_injective + - Multiset.Nodup.pi + - Multiset.pi.cons_eta + - Multiset.mem_pi + - Multiset.card_pi + - Multiset.Pi.cons_swap + - path: Mathlib/Logic/IsEmpty.lean + theorems: + - isEmpty_pi + - isEmpty_sum + - isEmpty_pprod + - isEmpty_psum + - isEmpty_fun + - isEmpty_subtype + - isEmpty_plift + - isEmpty_psigma + - isEmpty_sigma + - isEmpty_ulift + - nonempty_fun + - isEmpty_Prop + - isEmpty_prod + - path: Mathlib/CategoryTheory/Monoidal/End.lean + theorems: + - "CategoryTheory.obj_\u03BC_app" + - CategoryTheory.associativity_app + - "CategoryTheory.\u03B5_inv_naturality" + - "CategoryTheory.obj_\u03B5_inv_app" + - CategoryTheory.right_unitality_app + - "CategoryTheory.obj_zero_map_\u03BC_app" + - "CategoryTheory.\u03B5_app_obj" + - "CategoryTheory.obj_\u03B5_app" + - "CategoryTheory.\u03BC_inv_naturality\u1D63" + - "CategoryTheory.\u03BC_naturality\u1D63" + - "CategoryTheory.obj_\u03BC_zero_app" + - "CategoryTheory.\u03B5_inv_app_obj" + - "CategoryTheory.obj_\u03BC_inv_app" + - CategoryTheory.left_unitality_app + - "CategoryTheory.\u03BC_naturality\u2082" + - "CategoryTheory.\u03BC_naturality\u2097" + - path: Mathlib/Analysis/Convex/Basic.lean + theorems: + - convex_iff_pointwise_add_subset + - Convex.vadd + - Directed.convex_iUnion + - DirectedOn.convex_sUnion + - Convex.add_smul + - convex_segment + - convex_openSegment + - stdSimplex_unique + - Set.OrdConnected.convex_of_chain + - single_mem_stdSimplex + - AffineSubspace.convex + - Convex.smul_mem_of_zero_mem + - Submodule.convex + - convex_stdSimplex + - Convex.affine_image + - Convex.add_smul_mem + - Convex.sub + - Convex.mem_smul_of_zero_mem + - Convex.mapsTo_lineMap + - Convex.translate_preimage_left + - convex_hyperplane + - convex_iff_pairwise_pos + - Convex.exists_mem_add_smul_eq + - Convex.affinity + - stdSimplex_of_isEmpty_index + - Convex_subadditive_le + - convex_Iio + - Convex.add + - Convex.translate_preimage_right + - Convex.linear_preimage + - stdSimplex_fin_two + - ite_eq_mem_stdSimplex + - Convex.add_smul_sub_mem + - convex_iff_ordConnected + - stdSimplex_eq_inter + - Convex.linear_image + - convex_vadd + - path: Mathlib/Analysis/Normed/Group/Hom.lean + theorems: + - NormedAddGroupHom.SurjectiveOnWith.mono + - NormedAddGroupHom.coe_injective + - "NormedAddGroupHom.Equalizer.\u03B9_comp_lift" + - NormedAddGroupHom.ext_iff + - NormedAddGroupHom.sum_apply + - NormedAddGroupHom.ker_zero + - NormedAddGroupHom.le_of_opNorm_le + - NormedAddGroupHom.norm_comp_le_of_le + - NormedAddGroupHom.NormNoninc.normNoninc_iff_norm_le_one + - NormedAddGroupHom.coe_inj + - NormedAddGroupHom.toAddMonoidHom_injective + - NormedAddGroupHom.SurjectiveOnWith.exists_pos + - NormedAddGroupHom.comp_range + - NormedAddGroupHom.range_comp_incl_top + - NormedAddGroupHom.le_opNorm + - NormedAddGroupHom.opNorm_neg + - "NormedAddGroupHom.Equalizer.comp_\u03B9_eq" + - NormedAddGroupHom.norm_comp_le_of_le' + - NormedAddGroupHom.lipschitz + - NormedAddGroupHom.opNorm_zero_iff + - NormedAddGroupHom.opNorm_le_of_lipschitz + - exists_pos_bound_of_bound + - NormedAddGroupHom.Equalizer.map_id + - NormedAddGroupHom.antilipschitz_of_norm_ge + - NormedAddGroupHom.NormNoninc.zero + - NormedAddGroupHom.norm_id_le + - NormedAddGroupHom.comp_assoc + - NormedAddGroupHom.norm_id + - NormedAddGroupHom.incl_range + - NormedAddGroupHom.mkNormedAddGroupHom_norm_le' + - NormedAddGroupHom.le_opNorm_of_le + - NormedAddGroupHom.Equalizer.map_comp_map + - NormedAddGroupHom.norm_id_of_nontrivial_seminorm + - NormedAddGroupHom.opNorm_zero + - NormedAddGroupHom.ker.incl_comp_lift + - "NormedAddGroupHom.Equalizer.comm_sq\u2082" + - NormedAddGroupHom.zero_comp + - NormedAddGroupHom.comp_zero + - NormedAddGroupHom.NormNoninc.neg_iff + - path: Mathlib/Analysis/SpecialFunctions/Bernstein.lean + theorems: + - bernsteinApproximation.le_of_mem_S_compl + - bernsteinApproximation.lt_of_mem_S + - bernstein_apply + - bernsteinApproximation.apply + - bernstein_nonneg + - bernstein.variance + - bernsteinApproximation_uniform + - bernstein.probability + - path: Mathlib/Analysis/NormedSpace/QuaternionExponential.lean + theorems: + - Quaternion.re_exp + - Quaternion.hasSum_expSeries_of_imaginary + - Quaternion.norm_exp + - Quaternion.im_exp + - Quaternion.exp_eq + - Quaternion.expSeries_even_of_imaginary + - Quaternion.normSq_exp + - Quaternion.expSeries_odd_of_imaginary + - Quaternion.exp_of_re_eq_zero + - path: Mathlib/Computability/DFA.lean + theorems: + - DFA.evalFrom_of_pow + - DFA.evalFrom_append_singleton + - DFA.evalFrom_split + - DFA.mem_accepts + - path: Mathlib/Data/Int/Bitwise.lean + theorems: + - Int.bit_coe_nat + - Int.shiftLeft_add + - Int.zero_shiftLeft + - Int.shiftRight_eq_div_pow + - Int.bodd_add_div2 + - Int.bit_negSucc + - Int.bit1_ne_zero + - Int.lnot_bit + - Int.testBit_lor + - Int.testBit_ldiff + - Int.testBit_bit_succ + - Int.shiftRight_add + - Int.bit0_ne_bit1 + - Int.bodd_subNatNat + - Int.shiftLeft_coe_nat + - Int.ldiff_bit + - Int.lor_bit + - Int.shiftRight_negSucc + - Int.one_shiftLeft + - Int.bitwise_bit + - Int.bodd_negOfNat + - Int.bodd_mul + - Int.shiftRight_neg + - Int.shiftRight_coe_nat + - Int.bitwise_or + - Int.bitwise_xor + - Int.bitwise_and + - Int.bodd_neg + - Int.testBit_land + - Int.testBit_lnot + - Int.shiftLeft_eq_mul_pow + - Int.testBit_lxor + - Int.bodd_add + - Int.bodd_bit + - Int.testBit_bitwise + - Int.land_bit + - Int.lxor_bit + - Int.bitwise_diff + - path: Mathlib/Data/Set/Intervals/UnorderedInterval.lean + theorems: + - Set.uIcc_of_le + - Set.mem_uIcc + - Set.uIcc_injective_right + - Set.eq_of_mem_uIcc_of_mem_uIcc' + - Set.uIcc_eq_union + - Set.uIcc_comm + - Set.uIoc_eq_union + - Set.uIcc_of_ge + - Set.forall_uIoc_iff + - Set.uIcc_prod_eq + - Set.mem_uIoc + - Set.uIoc_injective_left + - Set.monotoneOn_or_antitoneOn_iff_uIcc + - Set.uIoc_comm + - AntitoneOn.mapsTo_uIcc + - Set.not_mem_uIoc + - Set.uIcc_self + - Set.left_mem_uIoc + - Set.eq_of_mem_uIoc_of_mem_uIoc' + - Set.eq_of_mem_uIoc_of_mem_uIoc + - MonotoneOn.mapsTo_uIcc + - Set.uIcc_injective_left + - Set.uIoc_of_le + - Set.uIoc_of_lt + - Set.monotone_or_antitone_iff_uIcc + - Set.eq_of_not_mem_uIoc_of_not_mem_uIoc + - Set.uIoc_injective_right + - Set.uIcc_subset_uIcc_union_uIcc + - Set.right_mem_uIoc + - path: Mathlib/MeasureTheory/Function/LpSpace.lean + theorems: + - MeasureTheory.Lp.nnnorm_zero + - "MeasureTheory.mem\u2112p_indicator_iff_restrict" + - BoundedContinuousFunction.range_toLpHom + - MeasureTheory.norm_indicatorConstLp' + - "MeasureTheory.Lp.tendsto_Lp_iff_tendsto_\u2112p'" + - MeasureTheory.Lp.norm_measure_zero + - "MeasureTheory.Mem\u2112p.piecewise" + - MeasureTheory.Lp.norm_le_of_ae_bound + - "MeasureTheory.mem\u2112p_re_im_iff" + - MeasureTheory.exists_snorm_indicator_le + - "MeasureTheory.Mem\u2112p.snorm_mk_lt_top" + - MeasureTheory.norm_indicatorConstLp_top + - MeasureTheory.snorm_indicator_le + - MeasureTheory.Lp.snorm_exponent_top_lim_eq_essSup_liminf + - LipschitzWith.compLp_zero + - MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum + - MeasureTheory.snorm_indicator_const_le + - MeasureTheory.Lp.lipschitzWith_pos_part + - MeasureTheory.Lp.norm_le_norm_of_ae_le + - ContinuousMap.range_toLp + - MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm + - MeasureTheory.Lp.continuous_negPart + - MeasureTheory.Lp.norm_toLp + - MeasureTheory.snormEssSup_indicator_eq_snormEssSup_restrict + - MeasureTheory.Lp.eq_zero_iff_ae_eq_zero + - LipschitzWith.lipschitzWith_compLp + - MeasureTheory.Lp.coeFn_negPart_eq_max + - MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top + - LipschitzWith.norm_compLp_sub_le + - "MeasureTheory.snorm_indicator_const\u2080" + - "MeasureTheory.mem\u2112p_norm_rpow_iff" + - ContinuousLinearMap.add_compLp + - "MeasureTheory.Mem\u2112p.toLp_congr" + - MeasureTheory.Lp.nnnorm_le_of_ae_bound + - MeasureTheory.edist_indicatorConstLp_eq_nnnorm + - MeasureTheory.Lp.norm_const + - MeasureTheory.Lp.nnnorm_le_mul_nnnorm_of_ae_le_mul + - MeasureTheory.indicatorConstLp_disjoint_union + - MeasureTheory.snormEssSup_indicator_const_eq + - BoundedContinuousFunction.toLp_inj + - "MeasureTheory.Lp.tendsto_Lp_iff_tendsto_\u2112p''" + - MeasureTheory.Lp.edist_toLp_zero + - MeasureTheory.Lp.coeFn_negPart + - MeasureTheory.snormEssSup_indicator_le + - MeasureTheory.norm_indicatorConstLp + - "MeasureTheory.Mem\u2112p.mem\u2112p_of_exponent_le_of_measure_support_ne_top" + - MeasureTheory.Lp.ext_iff + - MeasureTheory.Lp.const_smul_mem_Lp + - MeasureTheory.Lp.coe_nnnorm_toLp + - MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm' + - MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum + - "LipschitzWith.comp_mem\u2112p" + - MeasureTheory.Lp.nnnorm_eq_zero_iff + - "MeasureTheory.Lp.completeSpace_lp_of_cauchy_complete_\u2112p" + - MeasureTheory.Lp.edist_toLp_toLp + - MeasureTheory.AEEqFun.compMeasurePreserving_mem_Lp + - MeasureTheory.snormEssSup_indicator_const_le + - MeasureTheory.Lp.norm_le_mul_norm_of_ae_le_mul + - MeasureTheory.snorm_indicator_eq_snorm_restrict + - ContinuousLinearMap.smul_compLpL + - BoundedContinuousFunction.mem_Lp + - MeasureTheory.snorm_indicator_const' + - ContinuousMap.hasSum_of_hasSum_Lp + - MeasureTheory.indicatorConstLp_eq_toSpanSingleton_compLp + - MeasureTheory.Lp.norm_constL_le + - "MeasureTheory.Mem\u2112p.toLp_eq_toLp_iff" + - MeasureTheory.Lp.snorm_lim_le_liminf_snorm + - "MeasureTheory.Mem\u2112p.norm_rpow" + - MeasureTheory.Lp.edist_dist + - BoundedContinuousFunction.Lp_nnnorm_le + - MeasureTheory.Lp.ext + - MeasureTheory.Lp.snorm'_lim_eq_lintegral_liminf + - MeasureTheory.Lp.toLp_coeFn + - MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' + - "MeasureTheory.Lp.mem\u2112p_of_cauchy_tendsto" + - "MeasureTheory.Mem\u2112p.norm_rpow_div" + - MeasureTheory.norm_indicatorConstLp_le + - "MeasureTheory.mem\u2112p_add_of_disjoint" + - "MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_\u2112p" + - LipschitzWith.norm_compLp_le + - "MeasureTheory.Lp.tendsto_Lp_iff_tendsto_\u2112p" + - "MeasureTheory.Lp.mem_Lp_iff_mem\u2112p" + - MeasureTheory.Lp.nnnorm_neg + - MeasureTheory.indicatorConstLp_univ + - MeasureTheory.Lp.cauchy_tendsto_of_tendsto + - MeasureTheory.Lp.norm_const' + - MeasureTheory.Lp.snorm_exponent_top_lim_le_liminf_snorm_exponent_top + - ContinuousMap.toLp_norm_le + - MeasureTheory.Lp.norm_exponent_zero + - "MeasureTheory.mem\u2112p_indicator_const" + - MeasureTheory.Lp.norm_const_le + - MeasureTheory.Lp.snorm'_lim_le_liminf_snorm' + - "MeasureTheory.Mem\u2112p.of_comp_antilipschitzWith" + - path: Mathlib/Analysis/InnerProductSpace/Basic.lean + theorems: + - norm_sub_sq + - inner_eq_one_iff_of_norm_one + - re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four + - norm_add_sq_real + - inner_im_symm + - parallelogram_law_with_norm + - norm_inner_eq_norm_tfae + - inner_sum_smul_sum_smul_of_sum_eq_zero + - LinearIsometry.inner_map_map + - OrthogonalFamily.independent + - Orthonormal.inner_right_finsupp + - parallelogram_law + - norm_sub_eq_sqrt_iff_real_inner_eq_zero + - inner_lt_one_iff_real_of_norm_one + - inner_neg_left + - inner_sub_right + - InnerProductSpace.Core.inner_add_add_self + - real_inner_self_eq_norm_sq + - real_inner_div_norm_mul_norm_eq_neg_one_iff + - norm_sub_mul_self_real + - Orthonormal.equiv_refl + - linearIndependent_of_ne_zero_of_inner_eq_zero + - norm_add_sq + - orthonormal_iUnion_of_directed + - inner_smul_real_right + - LinearIsometryEquiv.inner_map_eq_flip + - dist_div_norm_sq_smul + - InnerProductSpace.Core.inner_neg_left + - real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul + - InnerProductSpace.Core.inner_sub_sub_self + - ext_inner_right + - Orthonormal.orthogonalFamily + - inner_eq_sum_norm_sq_div_four + - norm_inner_symm + - real_inner_smul_self_left + - im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four + - InnerProductSpace.Core.inner_smul_right + - Orthonormal.inner_left_sum + - Orthonormal.orthonormal_of_forall_eq_or_eq_neg + - real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul + - InnerProductSpace.Core.inner_self_im + - InnerProductSpace.Core.inner_zero_left + - InnerProductSpace.Core.inner_sub_left + - OrthogonalFamily.orthonormal_sigma_orthonormal + - OrthogonalFamily.norm_sq_diff_sum + - OrthogonalFamily.norm_sum + - InnerProductSpace.Core.inner_sub_right + - InnerProductSpace.Core.inner_zero_right + - inner_map_polarization' + - norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero + - real_inner_self_eq_norm_mul_norm + - Orthonormal.inner_left_right_finset + - ext_inner_left + - Orthonormal.ne_zero + - inner_sub_left + - isBoundedBilinearMap_inner + - norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero + - inner_eq_zero_symm + - inner_neg_neg + - re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two + - exists_maximal_orthonormal + - orthonormal_iff_ite + - inner_map_polarization + - InnerProductSpace.Core.inner_re_symm + - norm_add_eq_sqrt_iff_real_inner_eq_zero + - Finsupp.inner_sum + - InnerProductSpace.Core.inner_self_eq_zero + - inner_zero_right + - InnerProductSpace.Core.inner_im_symm + - inner_map_complex + - orthonormal_sUnion_of_directed + - norm_add_mul_self_real + - inner_self_im + - Orthonormal.inner_sum + - innerSL_apply_norm + - InnerProductSpace.Core.normSq_eq_zero + - Orthonormal.inner_finsupp_eq_sum_left + - Orthonormal.tsum_inner_products_le + - InnerProductSpace.Core.inner_neg_right + - OrthogonalFamily.inner_sum + - inner_self_eq_norm_sq + - inner_re_zero_left + - norm_inner_le_norm + - Orthonormal.comp_linearIsometry + - orthonormal_empty + - Orthonormal.linearIndependent + - OrthogonalFamily.eq_ite + - InnerProductSpace.Core.norm_inner_le_norm + - OrthogonalFamily.inner_right_dfinsupp + - LinearIsometry.orthonormal_comp_iff + - InnerProductSpace.Core.inner_self_eq_norm_mul_norm + - inner_sub_sub_self + - norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul + - re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two + - inner_self_eq_zero + - real_inner_smul_self_right + - real_inner_mul_inner_self_le + - real_inner_I_smul_self + - real_inner_add_sub_eq_zero_iff + - OrthogonalFamily.inner_right_fintype + - real_inner_sub_sub_self + - DirectSum.IsInternal.collectedBasis_orthonormal + - InnerProductSpace.Core.inner_add_right + - inner_add_add_self + - Orthonormal.inner_products_summable + - inner_mul_symm_re_eq_norm + - Orthonormal.inner_left_finsupp + - inner_self_eq_norm_sq_to_K + - norm_add_mul_self + - norm_inner_div_norm_mul_norm_eq_one_iff + - orthonormal_subtype_iff_ite + - inner_map_self_eq_zero + - inner_self_nonpos + - Orthonormal.inner_finsupp_eq_zero + - inner_self_eq_norm_mul_norm + - inner_neg_right + - InnerProductSpace.Core.inner_mul_inner_self_le + - eq_of_norm_le_re_inner_eq_norm_sq + - ContinuousLinearMap.toSesqForm_apply_norm_le + - inner_eq_norm_mul_iff + - inner_smul_real_left + - inner_self_re_eq_norm + - Finsupp.sum_inner + - ContinuousLinearMap.reApplyInnerSelf_smul + - DFinsupp.sum_inner + - "flip_inner\u2097" + - inner_re_symm + - InnerProductSpace.Core.ofReal_normSq_eq_inner_self + - InnerProductSpace.Core.norm_inner_symm + - norm_sub_mul_self + - real_inner_add_add_self + - Orthonormal.inner_right_sum + - inner_re_zero_right + - inner_zero_left + - Orthonormal.equiv_trans + - InnerProductSpace.Core.inner_mul_symm_re_eq_norm + - Orthonormal.equiv_symm + - abs_real_inner_div_norm_mul_norm_le_one + - Orthonormal.comp + - Orthonormal.inner_finsupp_eq_sum_right + - orthonormal_subtype_range + - real_inner_div_norm_mul_norm_eq_one_iff + - norm_sub_eq_norm_add + - InnerProductSpace.Core.cauchy_schwarz_aux + - inner_smul_right + - ext_inner_map + - inner_self_ofReal_norm + - UniformSpace.Completion.continuous_inner + - path: Mathlib/Topology/Algebra/Module/FiniteDimension.lean + theorems: + - unique_topology_of_t2 + - isClosedMap_smul_left + - ContinuousLinearMap.coe_toContinuousLinearEquivOfDetNeZero + - LinearMap.continuous_of_finiteDimensional + - LinearMap.closedEmbedding_of_injective + - LinearMap.continuous_of_nonzero_on_open + - LinearEquiv.toLinearEquiv_toContinuousLinearEquiv + - LinearMap.isOpenMap_of_finiteDimensional + - continuous_equivFun_basis_aux + - FiniteDimensional.complete + - LinearEquiv.toLinearEquiv_toContinuousLinearEquiv_symm + - path: Mathlib/Data/Matrix/ColumnRowPartitioned.lean + theorems: + - Matrix.sum_elim_vecMul_fromRows + - Matrix.fromColumns_toColumns + - Matrix.fromRows_fromColumn_eq_fromBlocks + - Matrix.transpose_fromColumns + - Matrix.transpose_fromRows + - Matrix.fromRows_mulVec + - Matrix.fromColumns_inj + - Matrix.fromColumns_mul_fromRows_eq_one_comm + - Matrix.fromColumns_mul_fromRows + - Matrix.fromColumns_mul_fromBlocks + - Matrix.fromColumns_fromRows_eq_fromBlocks + - Matrix.fromColumns_zero + - Matrix.fromRows_zero + - Matrix.fromRows_toRows + - Matrix.fromRows_inj + - Matrix.fromRows_mul + - Matrix.vecMul_fromColumns + - Matrix.conjTranspose_fromColumns_eq_fromRows_conjTranspose + - Matrix.mul_fromColumns + - Matrix.fromColumns_mulVec_sum_elim + - Matrix.fromRows_mul_fromColumns + - Matrix.conjTranspose_fromRows_eq_fromColumns_conjTranspose + - Matrix.fromBlocks_mul_fromRows + - path: Mathlib/RingTheory/Polynomial/ScaleRoots.lean + theorems: + - Polynomial.isCoprime_scaleRoots + - "Polynomial.scaleRoots_eval\u2082_mul_of_commute" + - Polynomial.monic_scaleRoots_iff + - Polynomial.scaleRoots_dvd + - Polynomial.scaleRoots_one + - Polynomial.scaleRoots_C + - Polynomial.mul_scaleRoots_of_noZeroDivisors + - Polynomial.mul_scaleRoots + - Polynomial.support_scaleRoots_le + - "Polynomial.scaleRoots_eval\u2082_eq_zero_of_eval\u2082_div_eq_zero" + - "Polynomial.scaleRoots_eval\u2082_eq_zero" + - Polynomial.scaleRoots_dvd' + - Polynomial.add_scaleRoots_of_natDegree_eq + - Polynomial.scaleRoots_aeval_eq_zero + - Polynomial.one_scaleRoots + - Polynomial.coeff_scaleRoots_natDegree + - Polynomial.zero_scaleRoots + - Polynomial.scaleRoots_zero + - Polynomial.degree_scaleRoots + - Polynomial.natDegree_scaleRoots + - Polynomial.scaleRoots_ne_zero + - Polynomial.scaleRoots_dvd_iff + - Polynomial.mul_scaleRoots' + - Polynomial.map_scaleRoots + - Polynomial.coeff_scaleRoots + - Polynomial.scaleRoots_mul + - path: Mathlib/CategoryTheory/Limits/Final.lean + theorems: + - CategoryTheory.Functor.Initial.limit_cone_comp_aux + - CategoryTheory.Functor.initial_of_adjunction + - CategoryTheory.Functor.initial_comp + - CategoryTheory.Functor.Final.colimit_pre_is_iso_aux + - CategoryTheory.Functor.cofinal_of_isTerminal_colimit_comp_yoneda + - CategoryTheory.Functor.Final.colimit_cocone_comp_aux + - CategoryTheory.IsFilteredOrEmpty.of_final + - CategoryTheory.Functor.Final.zigzag_of_eqvGen_quot_rel + - CategoryTheory.Functor.final_comp + - CategoryTheory.Functor.Initial.limit_pre_is_iso_aux + - CategoryTheory.Functor.final_of_final_comp + - CategoryTheory.Functor.final_of_adjunction + - CategoryTheory.Functor.initial_of_initial_comp + - CategoryTheory.Functor.cofinal_of_colimit_comp_coyoneda_iso_pUnit + - path: Mathlib/Data/Set/NAry.lean + theorems: + - Set.image2_mk_eq_prod + - Set.image2_right + - Set.image2_left_comm + - Set.image2_right_identity + - Set.image2_distrib_subset_right + - Set.image_image2_antidistrib_left + - Set.image2_subset + - Set.image2_image_left + - Set.image_prod + - Set.image2_subset_iff_left + - Set.image2_comm + - Set.image_image2_distrib_right + - Set.image2_image_right + - Set.image2_union_right + - Set.image2_congr + - Set.image2_union_left + - Set.image2_union_inter_subset_union + - Set.image2_right_comm + - Set.image2_insert_left + - Set.image2_inter_right + - Set.Subsingleton.image2 + - Set.image2_image2_image2_comm + - Set.image2_assoc + - Set.mem_image2_iff + - Set.image2_inter_left + - Set.image2_swap + - Set.image2_empty_right + - Set.image2_singleton_left + - Set.image_image2_antidistrib + - Set.image2_inter_union_subset_union + - Set.image2_singleton + - Set.image2_distrib_subset_left + - Set.image2_empty_left + - Set.image2_curry + - Set.image_image2_distrib_left + - Set.image2_left + - Set.image2_left_identity + - Set.image2_insert_right + - Set.image2_singleton_right + - Set.image_image2_antidistrib_right + - Set.image2_union_inter_subset + - Set.image2_eq_empty_iff + - Set.image_image2_distrib + - Set.image2_inter_union_subset + - Set.image2_subset_iff_right + - Set.image2_range + - Set.image_image2 + - path: Mathlib/Analysis/Normed/Group/Basic.lean + theorems: + - tendsto_norm_div_self + - mem_ball_one_iff + - norm_div_pos_iff + - nnnorm_prod_le + - comap_norm_nhdsWithin_Ioi_zero' + - dist_self_mul_right + - dist_mul_self_left + - nnnorm_eq_zero' + - edist_eq_coe_nnnorm_div + - norm_pow_le_mul_norm + - ball_one_eq + - Int.norm_coe_nat + - NormedCommGroup.cauchySeq_iff + - tendsto_one_iff_norm_tendsto_zero + - tendsto_norm' + - LipschitzWith.norm_le_mul' + - nnnorm_multiset_prod_le + - preimage_mul_closedBall + - norm_nonneg' + - AntilipschitzWith.le_mul_norm_div + - Real.toNNReal_eq_nnnorm_of_nonneg + - norm_zpow_le_mul_norm + - pow_mem_ball + - LipschitzWith.div + - ball_eq' + - norm_multiset_prod_le + - dist_self_div_right + - norm_prod_le + - lipschitzWith_iff_norm_div_le + - HasCompactMulSupport.exists_bound_of_continuous + - Filter.Tendsto.op_one_isBoundedUnder_le' + - MonoidHomClass.isometry_iff_norm + - NormedCommGroup.nhds_basis_norm_lt + - norm_div_sub_norm_div_le_norm_div + - mem_ball_iff_norm'' + - dist_inv + - lipschitzWith_one_norm' + - edist_mul_mul_le + - norm_div_rev + - tendsto_norm_one + - mem_closedBall_iff_norm'' + - edist_eq_coe_nnnorm' + - Real.ennnorm_eq_ofReal + - Int.abs_le_floor_nnreal_iff + - dist_div_eq_dist_mul_right + - dist_one_right + - preimage_mul_ball + - isBounded_iff_forall_norm_le' + - dist_mul_mul_le + - norm_inv' + - mul_mem_closedBall_iff_norm + - mul_mem_closedBall_mul_iff + - lipschitzOnWith_iff_norm_div_le + - smul_ball'' + - norm_lt_of_mem_ball' + - dist_le_norm_add_norm' + - preimage_mul_sphere + - dist_div_eq_dist_mul_left + - norm_div_eq_zero_iff + - nnnorm_pow_le_mul_norm + - cauchySeq_prod_of_eventually_eq + - mem_closure_one_iff_norm + - mem_sphere_iff_norm' + - NormedCommGroup.tendsto_nhds_one + - comap_norm_nhds_one + - LipschitzOnWith.norm_div_le_of_le + - edist_mulIndicator + - pi_norm_le_iff_of_nonneg' + - SeminormedCommGroup.mem_closure_iff + - Real.nnnorm_abs + - ne_one_of_norm_ne_zero + - Continuous.bounded_above_of_compact_support + - NormedCommGroup.tendsto_nhds_nhds + - dist_eq_norm_div' + - inseparable_one_iff_norm + - dist_div_div_le + - continuous_norm' + - norm_le_norm_add_norm_div' + - MonoidHomClass.antilipschitz_of_bound + - Real.ennnorm_eq_ofReal_abs + - abs_dist_sub_le_dist_mul_mul + - norm_le_mul_norm_add + - mem_emetric_ball_one_iff + - tendsto_iff_norm_div_tendsto_zero + - controlled_prod_of_mem_closure_range + - AntilipschitzWith.mul_lipschitzWith + - mem_closedBall_iff_norm''' + - dist_mulIndicator + - mul_mem_ball_iff_norm + - comap_norm_atTop' + - controlled_prod_of_mem_closure + - Real.nnnorm_two + - Int.norm_eq_abs + - pi_norm_const' + - norm_of_subsingleton' + - norm_div_le_norm_div_add_norm_div + - norm_le_of_mem_closedBall' + - AntilipschitzWith.mul_div_lipschitzWith + - eq_of_norm_div_le_zero + - norm_mul_le' + - mem_sphere_one_iff_norm + - nnnorm_zpow_le_mul_norm + - smul_closedBall'' + - Real.ofReal_le_ennnorm + - pow_mem_closedBall + - norm_one' + - SeminormedGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_one + - HasCompactMulSupport.exists_pos_le_norm + - mem_ball_iff_norm''' + - AntilipschitzWith.le_mul_norm' + - norm_le_norm_add_norm_div + - ne_one_of_nnnorm_ne_zero + - NNReal.coe_natAbs + - ne_one_of_mem_sphere + - dist_mul_self_right + - pi_norm_le_iff_of_nonempty' + - Bornology.IsBounded.exists_pos_norm_lt' + - pi_norm_lt_iff' + - dist_self_mul_left + - norm_le_zero_iff''' + - zero_lt_one_add_norm_sq' + - SeminormedGroup.tendstoUniformlyOn_one + - NormedCommGroup.nhds_one_basis_norm_lt + - Int.norm_cast_rat + - mul_mem_ball_mul_iff + - Bornology.IsBounded.exists_pos_norm_le' + - mem_closedBall_one_iff + - LipschitzWith.norm_div_le_of_le + - dist_self_div_left + - SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one + - norm_div_le + - dist_one_left + - norm_pos_iff''' + - abs_norm_sub_norm_le' + - Isometry.norm_map_of_map_one + - tendsto_norm_atTop_iff_cobounded' + - path: Mathlib/Topology/UniformSpace/Basic.lean + theorems: + - mem_uniform_prod + - UniformContinuous.prod_mk + - Filter.HasBasis.biInter_biUnion_ball + - uniformSpace_comap_id + - mem_ball_symmetry + - UniformSpace.ext + - uniformity_hasBasis_closed + - closure_eq_uniformity + - uniformContinuous_iInf_dom + - symmetric_symmetrizeRel + - uniformity_lift_le_comp + - nhds_eq_uniformity_prod + - Filter.HasBasis.uniformContinuousOn_iff + - Dense.biUnion_uniformity_ball + - nhds_basis_uniformity + - symmetrize_mem_uniformity + - subset_iterate_compRel + - closure_eq_inter_uniformity + - UniformSpace.ball_mem_nhds + - "uniformContinuous_inf_dom_right\u2082" + - uniformContinuous_iInf_rng + - Disjoint.exists_uniform_thickening_of_basis + - UniformContinuousOn.continuousOn + - uniformContinuousOn_iff_restrict + - Filter.HasBasis.uniformContinuous_iff + - uniformContinuous_of_const + - tendsto_of_uniformContinuous_subtype + - lebesgue_number_lemma_sUnion + - comp_open_symm_mem_uniformity_sets + - "uniformContinuous_sInf_dom\u2082" + - UniformSpace.toCore_toTopologicalSpace + - UniformSpace.hasBasis_nhds_prod + - comp_comp_symm_mem_uniformity_sets + - UniformSpace.hasBasis_nhds + - UniformSpace.toTopologicalSpace_sInf + - Uniform.tendsto_nhds_left + - idRel_subset + - Filter.HasBasis.mem_uniformity_iff + - mem_uniformity_of_uniformContinuous_invariant + - ball_eq_of_symmetry + - UniformSpace.comap_iInf + - uniformity_hasBasis_open_symmetric + - id_compRel + - SymmetricRel.inter + - swap_idRel + - nhdset_of_mem_uniformity + - uniformContinuous_sInf_dom + - Uniform.exists_is_open_mem_uniformity_of_forall_mem_eq + - mem_comp_comp + - nhds_le_uniformity + - UniformSpace.mem_nhds_iff_symm + - le_iff_uniformContinuous_id + - UniformSpace.mem_closure_iff_ball + - mem_nhds_uniformity_iff_right + - UniformSpace.mem_closure_iff_symm_ball + - uniformity_prod_eq_comap_prod + - interior_mem_uniformity + - "uniformContinuous_inf_dom_left\u2082" + - Uniform.tendsto_nhds_right + - lift_nhds_right + - "uniformContinuous\u2082_curry" + - Uniform.continuousOn_iff'_left + - compRel_assoc + - comp_symm_mem_uniformity_sets + - eventually_uniformity_iterate_comp_subset + - UniformSpace.ball_mem_nhdsWithin + - Filter.Tendsto.uniformity_trans + - IsCompact.nhdsSet_basis_uniformity + - map_uniformity_set_coe + - isOpen_iff_ball_subset + - mem_comp_of_mem_ball + - Uniform.continuousWithinAt_iff'_right + - mem_nhds_uniformity_iff_left + - uniformity_eq_uniformity_interior + - isOpen_uniformity + - uniformity_prod_eq_prod + - Uniform.continuousAt_iff'_left + - UniformSpace.comap_comap + - UniformSpace.Core.nhds_toTopologicalSpace + - Uniform.continuousWithinAt_iff'_left + - Uniform.continuousOn_iff'_right + - UniformSpace.ext_iff + - Uniform.continuousAt_iff'_right + - nhds_nhds_eq_uniformity_uniformity_prod + - lebesgue_number_lemma + - uniformContinuous_sInf_rng + - nhds_basis_uniformity' + - uniformContinuousOn_univ + - isOpen_iff_open_ball_subset + - Filter.HasBasis.lebesgue_number_lemma + - nhdsWithin_eq_comap_uniformity_of_mem + - lift_nhds_left + - UniformSpace.mem_nhds_iff + - path: Mathlib/MeasureTheory/Integral/Average.lean + theorems: + - MeasureTheory.measure_average_le_pos + - MeasureTheory.average_zero + - MeasureTheory.laverage_add_measure + - MeasureTheory.ofReal_setAverage + - MeasureTheory.exists_not_mem_null_le_laverage + - MeasureTheory.average_add_measure + - MeasureTheory.toReal_laverage + - MeasureTheory.exists_not_mem_null_lintegral_le + - MeasureTheory.laverage_union_mem_segment + - MeasureTheory.setAverage_eq + - MeasureTheory.toReal_setLaverage + - MeasureTheory.setAverage_congr + - MeasureTheory.average_mem_openSegment_compl_self + - MeasureTheory.exists_le_lintegral + - MeasureTheory.tendsto_integral_smul_of_tendsto_average_norm_sub + - MeasureTheory.average_eq_integral + - MeasureTheory.setLaverage_congr_fun + - MeasureTheory.measure_smul_setAverage + - MeasureTheory.exists_lintegral_le + - MeasureTheory.average_union_mem_openSegment + - MeasureTheory.exists_le_integral + - MeasureTheory.exists_not_mem_null_laverage_le + - MeasureTheory.measure_le_integral_pos + - MeasureTheory.average_const + - MeasureTheory.exists_not_mem_null_le_lintegral + - MeasureTheory.laverage_lt_top + - MeasureTheory.laverage_mem_openSegment_compl_self + - MeasureTheory.laverage_const + - MeasureTheory.measure_le_laverage_pos + - MeasureTheory.measure_integral_le_pos + - MeasureTheory.laverage_eq_lintegral + - MeasureTheory.measure_setAverage_le_pos + - MeasureTheory.exists_not_mem_null_le_integral + - MeasureTheory.average_zero_measure + - MeasureTheory.measure_mul_setLaverage + - MeasureTheory.ofReal_average + - MeasureTheory.setLaverage_congr + - MeasureTheory.measure_mul_laverage + - MeasureTheory.laverage_zero_measure + - MeasureTheory.laverage_union_mem_openSegment + - MeasureTheory.exists_not_mem_null_average_le + - MeasureTheory.laverage_zero + - MeasureTheory.lintegral_laverage + - MeasureTheory.exists_not_mem_null_le_average + - MeasureTheory.setLaverage_const + - MeasureTheory.exists_integral_le + - MeasureTheory.setLaverage_eq' + - MeasureTheory.average_eq + - MeasureTheory.measure_laverage_le_pos + - MeasureTheory.measure_le_setLaverage_pos + - MeasureTheory.average_congr + - MeasureTheory.measure_le_setAverage_pos + - MeasureTheory.laverage_union + - MeasureTheory.setLaverage_eq + - MeasureTheory.integral_average + - MeasureTheory.exists_not_mem_null_integral_le + - MeasureTheory.average_union_mem_segment + - MeasureTheory.setAverage_eq' + - MeasureTheory.measure_smul_average + - MeasureTheory.average_union + - MeasureTheory.laverage_congr + - MeasureTheory.integral_sub_average + - MeasureTheory.measure_le_lintegral_pos + - MeasureTheory.integral_average_sub + - MeasureTheory.setAverage_congr_fun + - MeasureTheory.measure_le_average_pos + - MeasureTheory.measure_lintegral_le_pos + - path: Mathlib/LinearAlgebra/AffineSpace/Combination.lean + theorems: + - Finset.weightedVSubOfPoint_map + - Finset.sum_smul_const_vsub_eq_vsub_affineCombination + - mem_affineSpan_iff_eq_weightedVSubOfPoint_vadd + - Finset.weightedVSubVSubWeights_self + - Finset.sum_centroidWeights_eq_one_of_cast_card_ne_zero + - Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one + - Finset.sum_affineCombinationLineMapWeights + - Finset.weightedVSubOfPoint_sdiff + - Finset.weightedVSubOfPoint_apply_const + - Finset.affineCombination_affineCombinationLineMapWeights + - Finset.weightedVSub_eq_linear_combination + - Finset.attach_affineCombination_coe + - Finset.eq_weightedVSubOfPoint_subset_iff_eq_weightedVSubOfPoint_subtype + - Finset.centroid_singleton + - Finset.weightedVSub_weightedVSubVSubWeights + - Finset.sum_smul_const_vsub_eq_sub_weightedVSubOfPoint + - Finset.centroid_eq_centroid_image_of_inj_on + - Finset.affineCombinationLineMapWeights_self + - Finset.centroid_univ + - Finset.sum_smul_vsub_eq_affineCombination_vsub + - Finset.weightedVSubVSubWeights_apply_left + - Finset.weightedVSub_apply_const + - Finset.affineCombination_apply_const + - affineSpan_eq_affineSpan_lineMap_units + - mem_vectorSpan_iff_eq_weightedVSub + - Finset.sum_weightedVSubVSubWeights + - Finset.centroid_pair + - Finset.sum_affineCombinationSingleWeights + - Finset.affineCombination_vsub + - Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub + - Finset.affineCombination_congr + - Finset.affineCombinationLineMapWeights_apply_left + - Finset.attach_affineCombination_of_injective + - Finset.sum_smul_vsub_const_eq_weightedVSub + - Finset.weightedVSubOfPoint_eq_of_weights_eq + - Finset.weightedVSubOfPoint_sdiff_sub + - Finset.weightedVSub_apply + - Finset.weightedVSubOfPoint_indicator_subset + - Finset.affineCombinationLineMapWeights_apply_right + - Finset.eq_affineCombination_subset_iff_eq_affineCombination_subtype + - Finset.affineCombination_map + - Finset.affineCombinationSingleWeights_apply_of_ne + - Finset.affineCombination_affineCombinationSingleWeights + - Finset.centroid_pair_fin + - Finset.weightedVSub_vadd_affineCombination + - Finset.weightedVSubOfPoint_eq_of_sum_eq_zero + - Finset.sum_centroidWeights_eq_one_of_card_ne_zero + - Finset.univ_fin2 + - Finset.centroid_eq_of_inj_on_of_image_eq + - Finset.weightedVSubOfPoint_subtype_eq_filter + - Finset.weightedVSubOfPoint_erase + - Finset.affineCombination_subtype_eq_filter + - Finset.map_affineCombination + - eq_affineCombination_of_mem_affineSpan_of_fintype + - eq_affineCombination_of_mem_affineSpan + - Finset.affineCombination_eq_linear_combination + - Finset.affineCombination_of_eq_one_of_eq_zero + - Finset.sum_centroidWeightsIndicator_eq_one_of_nonempty + - Finset.weightedVSubOfPoint_const_smul + - Finset.weightedVSubVSubWeights_apply_of_ne + - Finset.weightedVSubOfPoint_filter_of_ne + - Finset.weightedVSub_empty + - Finset.sum_centroidWeightsIndicator_eq_one_of_card_ne_zero + - Finset.affineCombination_indicator_subset + - Finset.sum_smul_const_vsub_eq_neg_weightedVSub + - Finset.affineCombination_sdiff_sub + - Finset.affineCombination_filter_of_ne + - Finset.affineCombinationLineMapWeights_apply_of_ne + - Finset.sum_centroidWeightsIndicator_eq_one_of_card_eq_add_one + - Finset.weightedVSubOfPoint_congr + - Finset.weightedVSubVSubWeights_apply_right + - weightedVSub_mem_vectorSpan + - Finset.weightedVSubOfPoint_apply + - affineCombination_mem_affineSpan + - Finset.sum_smul_vsub_const_eq_weightedVSubOfPoint_sub + - Finset.centroid_map + - Finset.weightedVSubOfPoint_insert + - mem_affineSpan_iff_eq_affineCombination + - Finset.sum_smul_vsub_const_eq_affineCombination_vsub + - path: Mathlib/CategoryTheory/Limits/IsLimit.lean + theorems: + - CategoryTheory.Limits.IsLimit.conePointsIsoOfNatIso_inv_comp + - CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_fac + - CategoryTheory.Limits.IsLimit.OfNatIso.coneOfHom_homOfCone + - CategoryTheory.Limits.IsColimit.OfNatIso.cocone_fac + - CategoryTheory.Limits.IsColimit.hom_desc + - CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso_inv_desc + - CategoryTheory.Limits.IsLimit.hom_ext + - CategoryTheory.Limits.IsColimit.coconePointsIsoOfNatIso_hom_desc + - CategoryTheory.Limits.IsLimit.OfNatIso.cone_fac + - CategoryTheory.Limits.IsLimit.conePointsIsoOfNatIso_hom_comp + - CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso_hom_desc + - CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_homOfCocone + - CategoryTheory.Limits.IsColimit.comp_coconePointsIsoOfNatIso_hom + - CategoryTheory.Limits.IsLimit.lift_comp_conePointsIsoOfNatIso_inv + - CategoryTheory.Limits.IsLimit.lift_comp_conePointsIsoOfNatIso_hom + - CategoryTheory.Limits.IsColimit.comp_coconePointsIsoOfNatIso_inv + - CategoryTheory.Limits.IsLimit.uniq_cone_morphism + - CategoryTheory.Limits.IsColimit.uniq_cocone_morphism + - CategoryTheory.Limits.IsLimit.lift_comp_conePointUniqueUpToIso_inv + - CategoryTheory.Limits.IsColimit.hom_ext + - CategoryTheory.Limits.IsColimit.coconePointsIsoOfNatIso_inv_desc + - CategoryTheory.Limits.IsLimit.OfNatIso.coneOfHom_fac + - CategoryTheory.Limits.IsLimit.lift_comp_conePointUniqueUpToIso_hom + - path: Mathlib/Data/Set/Prod.lean + theorems: + - Set.range_const_eq_diagonal + - Set.prod_insert + - Set.offDiag_singleton + - Set.mk_preimage_prod_left + - Set.univ_prod + - Set.diagonal_eq_univ_iff + - Set.pi_inter_distrib + - Set.mk_preimage_prod_right_eq_empty + - Set.preimage_coe_coe_diagonal + - Set.prod_singleton + - Set.eval_preimage + - Set.univ_pi_update + - Set.prod_range_univ_eq + - Set.offDiag_empty + - Set.pi_eq_empty_iff + - Set.prod_inter_prod + - Set.pi_inter_compl + - Set.mk_preimage_prod_right_fn_eq_if + - Set.univ_pi_subset_univ_pi_iff + - Set.exists_prod_set + - Set.pi_if + - Set.univ_pi_eq_empty_iff + - Set.pi_update_of_not_mem + - Set.prod_range_range_eq + - Set.range_diag + - Set.prod_sub_preimage_iff + - Set.diag_image + - Set.offDiag_nonempty + - Set.image_prod_mk_subset_prod_left + - Set.univ_pi_ite + - Set.pi_nonempty_iff + - Equiv.sumPiEquivProdPi_symm_preimage_univ_pi + - Set.pi_update_of_mem + - Set.univ_pi_nonempty_iff + - Set.prod_eq_prod_iff_of_nonempty + - Set.pi_eq_empty_iff' + - Set.subset_eval_image_pi + - Set.disjoint_pi + - Set.union_pi + - Set.image_prod_mk_subset_prod_right + - Set.inter_prod + - Set.mk_preimage_prod_left_eq_empty + - Equiv.piCongrLeft_preimage_pi + - Set.preimage_swap_prod + - Set.image_swap_prod + - Set.offDiag_union + - Set.uniqueElim_preimage + - Set.univ_pi_singleton + - range_toPullbackDiag + - Set.union_pi_inter + - Set.prod_univ + - Set.range_dcomp + - Set.prod_univ_range_eq + - Set.singleton_pi + - Function.Injective.preimage_pullbackDiagonal + - Set.prod_eq_empty_iff + - Set.empty_pi + - Set.offDiag_inter + - Set.offDiag_eq_empty + - Set.singleton_prod + - Set.pi_subset_pi_iff + - Set.compl_prod_eq_union + - Set.prod_image_image_eq + - Set.prod_eq_prod_iff + - Set.pi_eq_empty + - Set.prod_empty + - Set.offDiag_insert + - Set.pi_univ_ite + - Set.insert_prod + - Set.insert_pi + - Set.image_prod_mk_subset_prod + - Set.mk_preimage_prod_right + - Set.prod_diff_prod + - Set.prod_eq_iff_eq + - Equiv.piCongrLeft_symm_preimage_pi + - preimage_map_fst_pullbackDiagonal + - Set.prod_eq_univ + - Set.update_preimage_pi + - Set.singleton_prod_singleton + - Set.univ_pi_update_univ + - Equiv.piCongrLeft_preimage_univ_pi + - image_toPullbackDiag + - Set.prod_union + - Set.disjoint_prod + - Set.union_prod + - Set.prod_inter + - Set.mk_preimage_prod_left_eq_if + - Set.mk_preimage_prod_right_eq_if + - Set.eval_preimage' + - Set.disjoint_univ_pi + - Set.mk_preimage_prod_left_fn_eq_if + - Set.prod_subset_prod_iff + - Set.diagonal_subset_iff + - Equiv.piCongrLeft_symm_preimage_univ_pi + - Set.empty_prod + - Set.offDiag_univ + - path: Mathlib/RingTheory/WittVector/Frobenius.lean + theorems: + - "WittVector.bind\u2081_frobeniusPolyRat_wittPolynomial" + - WittVector.frobenius_zmodp + - WittVector.ghostComponent_frobeniusFun + - WittVector.map_frobeniusPoly + - "WittVector.map_frobeniusPoly.key\u2082" + - WittVector.frobenius_eq_map_frobenius + - WittVector.coeff_frobeniusFun + - WittVector.coeff_frobenius_charP + - "WittVector.map_frobeniusPoly.key\u2081" + - "WittVector.bind\u2081_frobeniusPoly_wittPolynomial" + - WittVector.frobeniusPolyAux_eq + - WittVector.frobeniusPoly_zmod + - path: Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean + theorems: + - GaussianInt.toComplex_eq_zero + - GaussianInt.toComplex_def' + - GaussianInt.normSq_div_sub_div_lt_one + - GaussianInt.normSq_le_normSq_of_re_le_of_im_le + - GaussianInt.nat_cast_natAbs_norm + - GaussianInt.toComplex_inj + - GaussianInt.toComplex_star + - GaussianInt.div_def + - GaussianInt.to_real_im + - "GaussianInt.toComplex_def\u2082" + - GaussianInt.toComplex_div_re + - GaussianInt.norm_le_norm_mul_left + - GaussianInt.sq_add_sq_of_nat_prime_of_not_irreducible + - GaussianInt.toComplex_im + - GaussianInt.int_cast_complex_norm + - GaussianInt.norm_pos + - GaussianInt.natAbs_norm_eq + - GaussianInt.norm_nonneg + - GaussianInt.norm_mod_lt + - GaussianInt.to_real_re + - GaussianInt.natAbs_norm_mod_lt + - GaussianInt.int_cast_real_norm + - GaussianInt.norm_eq_zero + - GaussianInt.toComplex_div_im + - path: Mathlib/RepresentationTheory/Rep.lean + theorems: + - Rep.unit_iso_comm + - "Rep.\u03C1_self_inv_apply" + - "Rep.linearization_\u03BC_inv_hom" + - Rep.leftRegularHom_apply + - Rep.linearization_of + - "Rep.\u03C1_inv_self_apply" + - Rep.leftRegularHomEquiv_symm_single + - Rep.to_Module_monoidAlgebra_map_aux + - Rep.linearization_single + - path: Mathlib/Data/Set/Intervals/Basic.lean + theorems: + - Set.Iio_diff_Iic + - Set.Ioo_union_Ioo + - Set.Ioo_eq_empty_iff + - Set.Ioc_diff_right + - Set.Ico_diff_Ioo_same + - Set.Ioo_union_right + - Set.Icc_diff_Ioo_same + - Set.left_mem_Ico + - Set.subsingleton_Icc_iff + - Set.Iio_union_Ioo + - Set.Icc_inter_Icc_eq_singleton + - Set.Icc_union_Icc + - Set.mem_Ici_Ioi_of_subset_of_subset + - Set.Iic_union_Ioc + - Set.Icc_diff_both + - Set.Ioc_union_left + - Set.Ioo_union_left + - Set.Ioo_inter_Ioc_of_right_lt + - Set.Iic_union_Icc' + - Set.left_mem_Ici + - Set.Icc_union_Icc' + - Set.Ioc_union_Ioc + - Set.Ico_bot + - Set.Ioc_subset_Ioc_iff + - Set.Ioc_union_Ioc_symm + - Set.Iic_inter_Iic + - Set.Iic_diff_Iio_same + - Set.Ioc_insert_left + - Set.Icc_union_Ici' + - Set.Ioo_subset_Ioo_iff + - Set.Ioc_union_Ioc_union_Ioc_cycle + - Set.right_mem_Icc + - Set.Ioi_subset_Ioi_iff + - Set.Ico_eq_Ico_iff + - Set.Icc_prod_eq + - Set.Ico_union_Ico' + - Set.Ici_inter_Ici + - Set.Iio_union_Ioo' + - Set.Ioc_union_Ioc_right + - Set.Ico_union_right + - Set.Ici_diff_Ioi + - Set.Icc_diff_left + - Set.right_mem_Ioo + - Set.Ici_diff_left + - Set.Iic_union_Ioc' + - Set.Ico_diff_left + - Set.Icc_union_Ici + - Set.Ioc_diff_Ioo_same + - Set.Ico_inter_Ico + - Set.Iio_diff_Iio + - Set.right_mem_Ioc + - Set.Icc_bot_top + - Set.Ico_insert_right + - Set.Iic_union_Icc + - Set.Ioo_insert_left + - Set.Ioi_diff_Ioi + - Set.Ico_diff_Iio + - Set.Ioo_inter_Ioc_of_left_le + - Set.Ioi_diff_Ici + - Set.Ico_union_Ici' + - Set.Iic_diff_right + - Set.Ioo_inter_Ioo + - Set.Iic_diff_Iic + - Set.Icc_eq_singleton_iff + - Set.Iio_subset_Iic_iff + - Set.Ioo_union_Ioi + - Set.Ico_inter_Ici + - Set.Iio_subset_Iio_iff + - Set.right_mem_Ico + - Set.Icc_diff_Ico_same + - Set.Ioi_subset_Ici_iff + - Set.Ici_diff_Ioi_same + - Set.Ioc_union_Ioi' + - Set.Ioc_union_Ioi + - Set.Ioc_top + - Set.Icc_bot + - Set.Ioo_union_Ioi' + - Set.Ioc_diff_Iic + - Set.Icc_eq_empty_iff + - Set.Ioc_inter_Iic + - Set.Icc_prod_Icc + - Set.Ioc_eq_empty_iff + - Set.Icc_diff_right + - Set.Ioi_union_left + - Set.Ioo_insert_right + - Set.Icc_inter_Icc + - Set.Ioo_union_Ioo' + - Set.Ico_inter_Iio + - Set.Ioc_diff_Ioi + - Set.Ioc_inter_Ioi + - Set.left_mem_Icc + - Set.Ioc_inter_Ioc + - Set.Ici_eq_singleton_iff_isTop + - Set.Ico_union_Ico + - Set.left_mem_Ioo + - Set.left_mem_Ioc + - Set.Icc_self + - Set.Ioo_union_both + - Set.Icc_top + - Set.Iic_diff_Iio + - Set.Ici_diff_Ici + - Set.Icc_diff_Ioc_same + - Set.Ioc_union_Ioc' + - Set.Iio_union_Ico + - Set.Ioc_inter_Ioo_of_left_lt + - Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset + - Set.Ico_union_Ici + - Set.Ico_eq_empty_iff + - Set.Ioc_union_Ioc_left + - Set.Ioc_inter_Ioo_of_right_le + - Set.Iio_union_Ico' + - path: Mathlib/MeasureTheory/Integral/Periodic.lean + theorems: + - Function.Periodic.intervalIntegral_add_eq_of_pos + - Function.Periodic.intervalIntegral_add_eq_add + - Function.Periodic.tendsto_atBot_intervalIntegral_of_pos + - Function.Periodic.sInf_add_zsmul_le_integral_of_pos + - AddCircle.lintegral_preimage + - UnitAddCircle.measure_univ + - AddCircle.intervalIntegral_preimage + - AddCircle.measure_univ + - Function.Periodic.tendsto_atTop_intervalIntegral_of_pos + - Function.Periodic.integral_le_sSup_add_zsmul_of_pos + - isAddFundamentalDomain_Ioc' + - Function.Periodic.intervalIntegral_add_eq + - isAddFundamentalDomain_Ioc + - Function.Periodic.intervalIntegral_add_zsmul_eq + - AddCircle.integral_preimage + - AddCircle.measurePreserving_mk + - AddCircle.volume_closedBall + - path: Mathlib/Data/Nat/Defs.lean + theorems: + - Nat.div_mul_div_comm + - Nat.pred_one_add + - Nat.div_lt_iff_lt_mul' + - Nat.two_mul_ne_two_mul_add_one + - Nat.eq_zero_of_dvd_of_div_eq_zero + - Nat.mul_div_le_mul_div_assoc + - Nat.pred_eq_succ_iff + - Nat.lt_one_add_iff + - Nat.leRecOn_succ_left + - Nat.le_induction + - Nat.div_pos + - Nat.le_find_iff + - Nat.pred_eq_of_eq_succ + - Nat.decreasingInduction_succ_left + - Nat.mul_eq_right + - Nat.find_le_iff + - Nat.one_add_le_iff + - Nat.div_pow + - Nat.forall_lt_succ + - Nat.leRecOn_self + - Nat.div_left_inj + - Nat.decreasingInduction_succ' + - Nat.add_mod_eq_add_mod_left + - Nat.decreasingInduction_self + - Nat.mul_eq_left + - Nat.div_le_iff_le_mul_add_pred + - Nat.lt_iff_add_one_le + - Nat.mul_add_mod' + - Nat.lt_find_iff + - Nat.mul_dvd_of_dvd_div + - Nat.mod_eq_iff_lt + - Nat.div_mod_unique + - Nat.or_exists_succ + - Nat.find_comp_succ + - Nat.find_eq_iff + - Nat.lt_mul_div_succ + - Nat.decreasingInduction_trans + - Nat.one_lt_iff_ne_zero_and_ne_one + - Nat.succ_succ_ne_one + - Nat.findGreatest_eq + - Nat.div_add_mod' + - Nat.add_mod_eq_add_mod_right + - Nat.mul_dvd_mul_iff_left + - Nat.leRecOn_injective + - Nat.le_succ_iff + - Nat.lt_div_mul_add + - Nat.one_le_div_iff + - Nat.leRecOn_succ' + - Nat.strongRecOn'_beta + - Nat.succ_add_sub_one + - Nat.self_add_sub_one + - Nat.mul_add_mod_of_lt + - Nat.mod_add_div' + - Nat.one_lt_mul_iff + - Nat.leRecOn_succ + - Nat.mul_dvd_mul_iff_right + - Nat.eq_mul_of_div_eq_left + - Nat.findGreatest_of_not + - Nat.exists_lt_succ + - Nat.eq_of_dvd_of_div_eq_one + - Nat.find_eq_zero + - Nat.le_one_iff_eq_zero_or_eq_one + - Nat.leRecOn_trans + - Nat.pred_eq_self_iff + - Nat.leRecOn_surjective + - Nat.decreasingInduction_succ + - path: Mathlib/FieldTheory/RatFunc.lean + theorems: + - RatFunc.ofFractionRing_algebraMap + - RatFunc.map_apply_ofFractionRing_mk + - RatFunc.num_dvd + - RatFunc.mul_inv_cancel + - RatFunc.intDegree_add_le + - RatFunc.toFractionRing_smul + - RatFunc.eval_algebraMap + - RatFunc.algebraMap_eq_zero_iff + - RatFunc.denom_div_dvd + - RatFunc.num_denom_neg + - RatFunc.ofFractionRing_one + - RatFunc.num_algebraMap + - RatFunc.eval_mul + - RatFunc.toFractionRing_eq + - RatFunc.eval_zero + - RatFunc.div_smul + - RatFunc.map_apply_div + - RatFunc.denom_one + - RatFunc.mk_smul + - RatFunc.ofFractionRing_mul + - RatFunc.eval_C + - RatFunc.monic_denom + - RatFunc.smul_eq_C_smul + - RatFunc.mk_coe_def + - RatFunc.denom_div + - RatFunc.liftOn_ofFractionRing_mk + - RatFunc.num_zero + - RatFunc.num_denom_add + - RatFunc.ofFractionRing_smul + - RatFunc.mk_zero + - RatFunc.coe_X + - RatFunc.denom_zero + - RatFunc.natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree + - RatFunc.smul_eq_C_mul + - RatFunc.intDegree_neg + - RatFunc.liftMonoidWithZeroHom_apply_div + - RatFunc.liftMonoidWithZeroHom_apply + - RatFunc.num_mul_dvd + - RatFunc.denom_mul_dvd + - RatFunc.map_apply_div_ne_zero + - RatFunc.liftOn'_mk + - RatFunc.map_injective + - RatFunc.map_apply + - RatFunc.mk_eq_localization_mk + - RatFunc.toFractionRingRingEquiv_symm_eq + - RatFunc.mk_one' + - RatFunc.denom_dvd + - RatFunc.ofFractionRing_zero + - RatFunc.num_div_dvd' + - RatFunc.num_div_dvd + - RatFunc.toFractionRing_injective + - RatFunc.liftMonoidWithZeroHom_injective + - RatFunc.denom_algebraMap + - RatFunc.mk_def_of_mem + - RatFunc.eval_X + - RatFunc.liftOn_condition_of_liftOn'_condition + - RatFunc.numDenom_div + - RatFunc.ofFractionRing_eq + - RatFunc.isCoprime_num_denom + - RatFunc.mk_eq_div' + - RatFunc.intDegree_zero + - RatFunc.intDegree_X + - RatFunc.intDegree_one + - RatFunc.num_mul_denom_add_denom_mul_num_ne_zero + - RatFunc.intDegree_C + - RatFunc.num_one + - RatFunc.ofFractionRing_add + - RatFunc.coe_C + - RatFunc.denom_add_dvd + - RatFunc.induction_on' + - RatFunc.liftMonoidWithZeroHom_apply_div' + - RatFunc.intDegree_polynomial + - RatFunc.mk_eq_mk + - RatFunc.eval_add + - RatFunc.ofFractionRing_neg + - RatFunc.intDegree_mul + - RatFunc.liftOn'_div + - RatFunc.num_denom_mul + - "RatFunc.eval_eq_zero_of_eval\u2082_denom_eq_zero" + - RatFunc.coe_smul + - RatFunc.eval_one + - RatFunc.liftOn_div + - RatFunc.ofFractionRing_div + - RatFunc.mk_eq_div + - RatFunc.num_div_denom + - RatFunc.liftOn_mk + - RatFunc.algebraMap_apply + - RatFunc.num_div + - RatFunc.algebraMap_injective + - RatFunc.num_div' + - RatFunc.algebraMap_apply_div + - RatFunc.ofFractionRing_inv + - path: Mathlib/MeasureTheory/Decomposition/Jordan.lean + theorems: + - MeasureTheory.SignedMeasure.toJordanDecomposition_smul + - MeasureTheory.JordanDecomposition.real_smul_posPart_nonneg + - MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive + - MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition + - MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive + - MeasureTheory.SignedMeasure.totalVariation_absolutelyContinuous_iff + - MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart + - MeasureTheory.JordanDecomposition.real_smul_negPart_neg + - MeasureTheory.SignedMeasure.totalVariation_neg + - MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real + - MeasureTheory.SignedMeasure.null_of_totalVariation_zero + - MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_negative + - MeasureTheory.SignedMeasure.subset_negative_null_set + - MeasureTheory.JordanDecomposition.real_smul_posPart_neg + - MeasureTheory.SignedMeasure.toJordanDecomposition_spec + - MeasureTheory.SignedMeasure.subset_positive_null_set + - MeasureTheory.JordanDecomposition.exists_compl_positive_negative + - MeasureTheory.SignedMeasure.toJordanDecomposition_smul_real_nonneg + - MeasureTheory.SignedMeasure.mutuallySingular_iff + - MeasureTheory.JordanDecomposition.real_smul_negPart_nonneg + - MeasureTheory.SignedMeasure.toJordanDecomposition_neg + - MeasureTheory.SignedMeasure.totalVariation_zero + - MeasureTheory.SignedMeasure.mutuallySingular_ennreal_iff + - MeasureTheory.JordanDecomposition.toSignedMeasure_neg + - MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_negative + - MeasureTheory.JordanDecomposition.toSignedMeasure_smul + - MeasureTheory.JordanDecomposition.toJordanDecomposition_toSignedMeasure + - MeasureTheory.JordanDecomposition.toSignedMeasure_injective + - MeasureTheory.JordanDecomposition.coe_smul + - MeasureTheory.SignedMeasure.toJordanDecomposition_eq + - MeasureTheory.SignedMeasure.toJordanDecomposition_zero + - MeasureTheory.JordanDecomposition.toSignedMeasure_zero + - path: Mathlib/Logic/Encodable/Lattice.lean + theorems: + - "Encodable.iUnion_decode\u2082_disjoint_on" + - "Encodable.iSup_decode\u2082" + - "Encodable.iUnion_decode\u2082_cases" + - path: Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean + theorems: + - "AlgebraicTopology.DoldKan.Compatibility.equivalence\u2081UnitIso_eq" + - "AlgebraicTopology.DoldKan.Compatibility.equivalence\u2081CounitIso_eq" + - AlgebraicTopology.DoldKan.Compatibility.equivalenceUnitIso_eq + - "AlgebraicTopology.DoldKan.Compatibility.equivalence\u2082UnitIso_eq" + - "AlgebraicTopology.DoldKan.Compatibility.equivalence\u2082CounitIso_eq" + - AlgebraicTopology.DoldKan.Compatibility.equivalenceCounitIso_eq + - path: Mathlib/NumberTheory/Zsqrtd/Basic.lean + theorems: + - Zsqrtd.hom_ext + - Zsqrtd.coe_int_re + - Zsqrtd.Nonneg.add + - Zsqrtd.norm_eq_of_associated + - Zsqrtd.coe_int_inj + - Zsqrtd.le_antisymm + - Zsqrtd.le_total + - Zsqrtd.nonneg_mul_lem + - Zsqrtd.nonneg_add_lem + - Zsqrtd.decompose + - Zsqrtd.sqLe_smul + - Zsqrtd.dmuld + - Zsqrtd.le_of_add_le_add_left + - Zsqrtd.not_divides_sq + - Zsqrtd.smul_im + - Zsqrtd.eq_zero_or_eq_zero_of_mul_eq_zero + - Zsqrtd.muld_val + - Zsqrtd.norm_zero + - Zsqrtd.nonneg_mul + - Zsqrtd.nonnegg_pos_neg + - Zsqrtd.exists_coprime_of_gcd_pos + - Zsqrtd.norm_eq_one_iff' + - Zsqrtd.nonneg_antisymm + - Zsqrtd.sqLe_cancel + - Zsqrtd.norm_eq_one_iff_mem_unitary + - Zsqrtd.norm_neg + - Zsqrtd.bit1_im + - Zsqrtd.le_arch + - Zsqrtd.eq_of_smul_eq_smul_left + - Zsqrtd.norm_nonneg + - Zsqrtd.norm_eq_zero_iff + - Zsqrtd.smuld_val + - Zsqrtd.gcd_eq_zero_iff + - Zsqrtd.norm_one + - Zsqrtd.norm_conj + - Zsqrtd.smul_val + - Zsqrtd.mul_nonneg + - Zsqrtd.norm_eq_mul_conj + - Zsqrtd.sqLe_mul + - Zsqrtd.nonneg_smul + - Zsqrtd.mul_star + - Zsqrtd.norm_mul + - Zsqrtd.nonnegg_neg_pos + - Zsqrtd.coe_int_dvd_iff + - Zsqrtd.ofInt_eq_coe + - Zsqrtd.norm_int_cast + - Zsqrtd.add_le_add_left + - Zsqrtd.sqLe_add_mixed + - Zsqrtd.lift_injective + - Zsqrtd.nonneg_iff_zero_le + - Zsqrtd.norm_eq_one_iff + - Zsqrtd.coe_int_val + - Zsqrtd.isUnit_iff_norm_isUnit + - Zsqrtd.smul_re + - Zsqrtd.nonneg_muld + - Zsqrtd.norm_eq_zero + - Zsqrtd.divides_sq_eq_zero_z + - Zsqrtd.divides_sq_eq_zero + - Zsqrtd.sqLe_add + - Zsqrtd.coe_int_im + - Zsqrtd.coe_int_dvd_coe_int + - Zsqrtd.coprime_of_dvd_coprime + - path: Mathlib/Data/Nat/PartENat.lean + theorems: + - PartENat.add_eq_top_iff + - PartENat.toWithTop_zero' + - PartENat.pos_iff_one_le + - PartENat.coe_add_get + - PartENat.lt_add_one + - PartENat.withTopEquiv_symm_lt + - PartENat.add_one_le_iff_lt + - PartENat.add_lt_add_iff_left + - PartENat.find_le + - PartENat.withTopEquiv_symm_le + - PartENat.lt_coe_succ_iff_le + - PartENat.lt_add_iff_pos_right + - PartENat.dom_of_le_natCast + - PartENat.withTopEquiv_symm_top + - PartENat.add_top + - PartENat.toWithTop_le + - PartENat.toWithTop_natCast + - PartENat.lt_add_one_iff_lt + - PartENat.ofENat_toWithTop + - PartENat.coe_le_iff + - PartENat.ne_top_iff + - PartENat.add_lt_add_right + - PartENat.add_one_le_of_lt + - PartENat.lt_def + - PartENat.ne_top_iff_dom + - PartENat.eq_natCast_sub_of_add_eq_natCast + - PartENat.withTopEquiv_symm_ofNat + - PartENat.withTopEquiv_top + - PartENat.le_of_lt_add_one + - PartENat.casesOn + - PartENat.withTopEquiv_lt + - PartENat.withTopEquiv_symm_zero + - PartENat.natCast_lt_top + - PartENat.coe_lt_iff + - PartENat.ofENat_le + - PartENat.withTopEquiv_symm_coe + - PartENat.lt_wf + - PartENat.add_left_cancel_iff + - PartENat.coe_succ_le_iff + - PartENat.lt_find_iff + - PartENat.le_coe_iff + - PartENat.toWithTop_natCast' + - PartENat.withTopEquiv_zero + - PartENat.toWithTop_add + - PartENat.toWithTop_top' + - PartENat.toWithTop_one' + - PartENat.withTopEquiv_one + - PartENat.add_right_cancel_iff + - PartENat.get_natCast' + - PartENat.toWithTop_ofENat + - PartENat.natCast_get + - PartENat.withTopEquiv_symm_one + - PartENat.withTopEquiv_le + - PartENat.eq_top_iff_forall_lt + - PartENat.lt_find + - PartENat.get_eq_iff_eq_coe + - PartENat.withTopEquiv_natCast + - PartENat.ofENat_lt + - PartENat.lt_coe_iff + - path: Mathlib/Combinatorics/Configuration.lean + theorems: + - Configuration.ProjectivePlane.one_lt_order + - Configuration.HasLines.lineCount_eq_pointCount + - Configuration.ProjectivePlane.lineCount_eq_pointCount + - Configuration.HasLines.pointCount_le_lineCount + - Configuration.ProjectivePlane.lineCount_eq + - Configuration.HasLines.exists_bijective_of_card_eq + - Configuration.ProjectivePlane.card_points + - Configuration.ProjectivePlane.pointCount_eq_pointCount + - Configuration.ProjectivePlane.two_lt_pointCount + - Configuration.sum_lineCount_eq_sum_pointCount + - Configuration.Nondegenerate.exists_injective_of_card_le + - Configuration.ProjectivePlane.two_lt_lineCount + - Configuration.ProjectivePlane.lineCount_eq_lineCount + - Configuration.HasLines.card_le + - path: Mathlib/Algebra/Polynomial/BigOperators.lean + theorems: + - Polynomial.coeff_multiset_prod_of_natDegree_le + - Polynomial.multiset_prod_X_sub_C_coeff_card_pred + - Polynomial.natDegree_prod_le + - Polynomial.natDegree_prod_of_monic + - Polynomial.leadingCoeff_prod' + - Polynomial.coeff_zero_multiset_prod + - Polynomial.natDegree_sum_le_of_forall_le + - Polynomial.natDegree_prod' + - Polynomial.natDegree_multiset_sum_le + - Polynomial.prod_X_sub_C_coeff_card_pred + - Polynomial.natDegree_multiset_prod_of_monic + - Polynomial.degree_multiset_prod_le + - Polynomial.degree_prod_le + - Polynomial.natDegree_prod + - Polynomial.leadingCoeff_multiset_prod' + - Polynomial.leadingCoeff_prod + - Polynomial.coeff_prod_of_natDegree_le + - Polynomial.natDegree_multiset_prod_le + - Polynomial.degree_list_sum_le + - Polynomial.natDegree_sum_le + - Polynomial.prod_X_sub_C_nextCoeff + - Polynomial.coeff_zero_prod + - Polynomial.leadingCoeff_multiset_prod + - Polynomial.natDegree_multiset_prod + - Polynomial.multiset_prod_X_sub_C_nextCoeff + - Polynomial.natDegree_list_sum_le + - Polynomial.coeff_list_prod_of_natDegree_le + - Polynomial.natDegree_list_prod_le + - Polynomial.degree_list_prod_le + - Polynomial.natDegree_multiset_prod' + - path: Mathlib/MeasureTheory/Integral/SetToL1.lean + theorems: + - MeasureTheory.L1.SimpleFunc.setToL1S_mono + - MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left' + - MeasureTheory.setToFun_finset_sum' + - MeasureTheory.L1.SimpleFunc.norm_eq_sum_mul + - MeasureTheory.setToFun_congr_smul_measure + - MeasureTheory.FinMeasAdditive.of_smul_measure + - MeasureTheory.setToFun_congr_measure_of_add_left + - MeasureTheory.SimpleFunc.setToSimpleFunc_add + - MeasureTheory.setToFun_const + - MeasureTheory.L1.SimpleFunc.norm_setToL1S_le + - MeasureTheory.L1.SimpleFunc.setToL1S_add + - MeasureTheory.SimpleFunc.setToSimpleFunc_congr_left + - MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left + - MeasureTheory.norm_setToFun_le_mul_norm + - MeasureTheory.SimpleFunc.setToSimpleFunc_zero' + - MeasureTheory.SimpleFunc.setToSimpleFunc_neg + - MeasureTheory.SimpleFunc.map_setToSimpleFunc + - MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left + - MeasureTheory.tendsto_setToFun_approxOn_of_measurable_of_range_subset + - MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg + - MeasureTheory.tendsto_setToFun_filter_of_dominated_convergence + - MeasureTheory.SimpleFunc.setToSimpleFunc_add_left + - MeasureTheory.FinMeasAdditive.smul_measure + - MeasureTheory.continuous_setToFun + - MeasureTheory.setToFun_mono_left' + - MeasureTheory.L1.setToL1_indicatorConstLp + - MeasureTheory.setToFun_congr_ae + - MeasureTheory.setToFun_add_left' + - MeasureTheory.setToFun_sub + - MeasureTheory.setToFun_nonneg + - MeasureTheory.setToFun_congr_measure + - MeasureTheory.SimpleFunc.setToSimpleFunc_congr' + - MeasureTheory.L1.SimpleFunc.setToL1S_neg + - MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_filter + - MeasureTheory.L1.SimpleFunc.setToL1S_smul_real + - MeasureTheory.norm_setToFun_le + - MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left' + - MeasureTheory.continuousOn_setToFun_of_dominated + - MeasureTheory.setToFun_smul_left' + - MeasureTheory.continuous_L1_toL1 + - MeasureTheory.FinMeasAdditive.map_empty_eq_zero + - MeasureTheory.L1.setToL1_zero_left + - MeasureTheory.L1.setToL1_nonneg + - MeasureTheory.tendsto_setToFun_of_dominated_convergence + - MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left + - MeasureTheory.setToFun_add + - MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_opNorm + - MeasureTheory.setToFun_mono + - MeasureTheory.SimpleFunc.setToSimpleFunc_const' + - MeasureTheory.SimpleFunc.setToSimpleFunc_indicator + - MeasureTheory.L1.setToL1_zero_left' + - MeasureTheory.SimpleFunc.setToSimpleFunc_zero + - MeasureTheory.SimpleFunc.setToSimpleFunc_mono + - MeasureTheory.norm_setToFun_le_mul_norm' + - MeasureTheory.SimpleFunc.setToSimpleFunc_smul + - MeasureTheory.DominatedFinMeasAdditive.smul + - MeasureTheory.setToFun_finset_sum + - MeasureTheory.L1.setToL1_simpleFunc_indicatorConst + - MeasureTheory.setToFun_congr_left + - MeasureTheory.L1.SimpleFunc.setToL1S_nonneg + - MeasureTheory.L1.SimpleFunc.setToL1S_smul + - MeasureTheory.L1.setToL1_add_left + - MeasureTheory.SimpleFunc.setToSimpleFunc_const + - MeasureTheory.DominatedFinMeasAdditive.zero + - MeasureTheory.setToFun_neg + - MeasureTheory.setToFun_top_smul_measure + - MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg' + - MeasureTheory.setToFun_congr_measure_of_integrable + - MeasureTheory.DominatedFinMeasAdditive.of_measure_le + - MeasureTheory.SimpleFunc.setToSimpleFunc_smul_real + - MeasureTheory.setToFun_congr_measure_of_add_right + - MeasureTheory.FinMeasAdditive.zero + - MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm + - MeasureTheory.setToFun_indicator_const + - MeasureTheory.setToFun_congr_left' + - MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_mul_norm_of_integrable + - MeasureTheory.DominatedFinMeasAdditive.add + - MeasureTheory.SimpleFunc.setToSimpleFunc_sub + - MeasureTheory.SimpleFunc.setToSimpleFunc_congr + - MeasureTheory.DominatedFinMeasAdditive.eq_zero_of_measure_zero + - MeasureTheory.DominatedFinMeasAdditive.of_smul_measure + - MeasureTheory.L1.setToL1_mono + - MeasureTheory.SimpleFunc.setToSimpleFunc_add_left' + - MeasureTheory.setToFun_zero + - MeasureTheory.L1.setToFun_eq_setToL1 + - MeasureTheory.norm_setToFun_le' + - MeasureTheory.L1.setToL1_congr_left' + - MeasureTheory.L1.setToL1_congr_left + - MeasureTheory.setToFun_zero_left + - MeasureTheory.setToFun_zero_left' + - MeasureTheory.L1.setToL1_add_left' + - MeasureTheory.L1.setToL1_smul_left' + - MeasureTheory.L1.setToL1_smul_left + - MeasureTheory.setToFun_measure_zero + - MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM + - MeasureTheory.DominatedFinMeasAdditive.eq_zero + - MeasureTheory.L1.setToL1_smul + - MeasureTheory.setToFun_smul_left + - MeasureTheory.L1.SimpleFunc.setToL1S_congr_measure + - MeasureTheory.SimpleFunc.setToSimpleFunc_zero_apply + - MeasureTheory.L1.setToL1_mono_left' + - MeasureTheory.FinMeasAdditive.map_iUnion_fin_meas_set_eq_sum + - MeasureTheory.L1.SimpleFunc.setToL1S_sub + - MeasureTheory.FinMeasAdditive.smul + - MeasureTheory.setToFun_add_left + - MeasureTheory.L1.SimpleFunc.setToL1S_indicatorConst + - MeasureTheory.setToFun_smul + - MeasureTheory.FinMeasAdditive.add + - path: Mathlib/GroupTheory/Perm/List.lean + theorems: + - List.formPerm_eq_formPerm_iff + - List.formPerm_pow_apply_nthLe + - List.formPerm_apply_of_not_mem + - List.mem_of_formPerm_apply_mem + - List.formPerm_eq_head_iff_eq_getLast + - List.support_formPerm_le' + - List.support_formPerm_le + - List.formPerm_apply_mem_of_mem + - List.zipWith_swap_prod_support + - List.formPerm_apply_lt + - List.formPerm_apply_mem_eq_self_iff + - List.formPerm_cons_concat_apply_last + - List.formPerm_pow_apply_head + - List.formPerm_pow_length_eq_one_of_nodup + - List.support_formPerm_of_nodup' + - List.zipWith_swap_prod_support' + - List.mem_of_formPerm_ne_self + - List.formPerm_apply_getLast + - List.formPerm_eq_one_iff + - List.formPerm_apply_head + - List.formPerm_apply_nthLe + - List.formPerm_reverse + - List.form_perm_zpow_apply_mem_imp_mem + - List.formPerm_eq_of_isRotated + - List.formPerm_apply_mem_ne_self_iff + - List.formPerm_apply_nthLe_zero + - List.formPerm_rotate_one + - List.formPerm_apply_nthLe_length + - List.formPerm_ext_iff + - List.support_formPerm_of_nodup + - List.formPerm_rotate + - path: Mathlib/RingTheory/Filtration.lean + theorems: + - Ideal.Filtration.submodule_eq_span_le_iff_stable_ge + - Ideal.Filtration.pow_smul_le + - Ideal.iInf_pow_eq_bot_of_localRing + - Ideal.Filtration.inf_submodule + - Ideal.Filtration.submodule_closure_single + - Ideal.Filtration.Stable.exists_pow_smul_eq + - Ideal.Filtration.submodule_span_single + - Ideal.Filtration.submodule_fg_iff_stable + - Ideal.iInf_pow_eq_bot_of_isDomain + - Ideal.Filtration.pow_smul_le_pow_smul + - Ideal.stableFiltration_stable + - Ideal.Filtration.stable_iff_exists_pow_smul_eq_of_ge + - Ideal.mem_iInf_smul_pow_eq_bot_iff + - Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge + - Ideal.iInf_pow_smul_eq_bot_of_localRing + - Ideal.Filtration.Stable.exists_forall_le + - Ideal.Filtration.Stable.of_le + - Ideal.Filtration.Stable.bounded_difference + - path: Mathlib/Geometry/RingedSpace/PresheafedSpace.lean + theorems: + - AlgebraicGeometry.PresheafedSpace.ext + - AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf + - AlgebraicGeometry.PresheafedSpace.congr_app + - AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c + - AlgebraicGeometry.PresheafedSpace.id_c_app + - AlgebraicGeometry.PresheafedSpace.hext + - path: Mathlib/Data/List/Card.lean + theorems: + - List.card_map_eq_of_inj_on + - List.remove_eq_of_not_mem + - List.card_le_card_cons + - List.card_remove_of_mem + - List.card_subset_le + - List.card_union_disjoint + - List.card_cons_of_mem + - List.card_insert_of_mem + - List.insert_equiv_cons + - List.union_equiv_append + - List.card_cons_of_not_mem + - List.card_insert_of_not_mem + - List.card_map_le + - List.mem_of_mem_remove + - List.mem_remove_iff + - List.card_append_disjoint + - path: Mathlib/Algebra/Category/ModuleCat/Subobject.lean + theorems: + - "ModuleCat.cokernel_\u03C0_imageSubobject_ext" + - ModuleCat.toKernelSubobject_arrow + - path: Mathlib/Data/DFinsupp/Lex.lean + theorems: + - DFinsupp.lex_lt_of_lt + - DFinsupp.lex_lt_of_lt_of_preorder + - DFinsupp.toLex_monotone + - path: Mathlib/Computability/Primrec.lean + theorems: + - Nat.Primrec.prec1 + - Nat.Primrec.swap' + - Primrec.option_orElse + - Primrec.list_findIdx + - Primrec.of_equiv_symm_iff + - Primrec.option_iget + - "Nat.Primrec'.unpair\u2081" + - Nat.Primrec.mul + - Primrec.nat_bodd + - Primrec.comp + - Primrec.nat_lt + - Nat.Primrec'.prec' + - Primrec.nat_iterate + - "Nat.Primrec'.unpair\u2082" + - "Primrec.fin_curry\u2081" + - Nat.Primrec'.to_prim + - Primrec.list_length + - Primrec.subtype_val + - Primrec.list_ofFn + - Nat.Primrec.pow + - Nat.Primrec'.encode + - PrimrecPred.not + - Primrec.option_get + - Primrec.subtype_val_iff + - Primrec.of_equiv_iff + - Primrec.list_join + - Nat.Primrec.add + - Primrec.vector_tail + - Primrec.encode_iff + - Primrec.dom_bool + - Primrec.cond + - Primrec.fin_curry + - Nat.Primrec'.add + - Primrec.snd + - "Primrec\u2082.uncurry" + - Primrec.encode + - "Primrec\u2082.natPair" + - "Primrec.dom_bool\u2082" + - Nat.Primrec'.head + - Primrec.option_some + - Primrec.option_getD + - Primrec.id + - Primrec.fin_val_iff + - Nat.Primrec.swap + - Primrec.fin_app + - Primrec.of_equiv_symm + - Nat.Primrec'.sqrt + - "Primrec\u2082.unpaired" + - Nat.Primrec'.of_prim + - list_reverse' + - Primrec.beq + - Primrec.vector_cons + - "Nat.Primrec'.comp\u2082" + - "Primrec.list_get?\u2081" + - list_foldl' + - Primrec.nat_findGreatest + - Primrec.vector_get + - Primrec.list_foldr + - "Primrec\u2082.curry" + - Primrec.list_map + - "Primrec\u2082.nat_iff" + - Primrec.pair + - Primrec.fin_succ + - Nat.Primrec'.mul + - Primrec.nat_strong_rec + - list_casesOn' + - Primrec.list_append + - Primrec.ite + - Nat.Primrec'.sub + - Primrec.list_get? + - "Primrec.list_findIdx\u2081" + - Primrec.unpair + - Nat.Primrec.id + - "Nat.Primrec'.prim_iff\u2082" + - Nat.Primrec'.pred + - Primrec.option_isSome + - Primrec.list_range + - "Primrec.decode\u2082" + - Nat.Primrec.casesOn' + - PrimrecPred.or + - Primrec.option_map + - Primrec.nat_rec + - Primrec.fst + - Primrec.vector_ofFn + - Primrec.sum_casesOn + - Nat.Primrec'.tail + - Primrec.nat_div + - Primrec.nat_le + - Primrec.list_rec + - Nat.Primrec.casesOn1 + - "Primrec\u2082.ofNat_iff" + - Primrec.of_graph + - Nat.Primrec.pred + - Primrec.nat_mod + - Nat.Primrec'.cons + - Primrec.list_head? + - Primrec.list_tail + - Nat.Primrec'.vec_iff + - "Primrec\u2082.of_eq" + - Primrec.dom_fintype + - Nat.Primrec'.comp' + - Primrec.option_bind + - Primrec.list_getD + - "Nat.Primrec'.prim_iff\u2081" + - Nat.Primrec'.if_lt + - Primrec.dom_denumerable + - PrimrecPred.and + - Primrec.map_decode_iff + - Primrec.option_casesOn + - path: Mathlib/Data/Multiset/Powerset.lean + theorems: + - Multiset.powersetCardAux_cons + - Multiset.mem_powersetCard + - Multiset.powersetCard_map + - Multiset.powersetCard_one + - Multiset.card_powersetCard + - Multiset.powersetCard_cons + - Multiset.map_single_le_powerset + - Multiset.powersetCard_mono + - Multiset.mem_powersetCardAux + - Multiset.bind_powerset_len + - Multiset.powerset_aux'_perm + - Multiset.revzip_powersetAux' + - Multiset.powersetCardAux_perm + - Multiset.powersetCardAux_zero + - Multiset.revzip_powersetAux_perm_aux' + - Multiset.card_powerset + - Multiset.revzip_powersetAux + - Multiset.revzip_powersetAux_perm + - Multiset.powerset_cons + - Multiset.powersetAux'_cons + - Multiset.revzip_powersetAux_lemma + - Multiset.powersetCard_le_powerset + - Multiset.powersetAux_perm_powersetAux' + - Multiset.mem_powerset + - Multiset.powersetCardAux_eq_map_coe + - Multiset.mem_powersetAux + - Multiset.nodup_powerset + - path: Mathlib/Analysis/LocallyConvex/WithSeminorms.lean + theorems: + - SeminormFamily.basisSets_intersect + - Seminorm.isBounded_sup + - SeminormFamily.basisSets_smul + - Seminorm.cont_withSeminorms_normedSpace + - Seminorm.continuous_of_continuous_comp + - norm_withSeminorms + - SeminormFamily.basisSets_smul_right + - LinearMap.withSeminorms_induced + - WithSeminorms.mem_nhds_iff + - WithSeminorms.topologicalAddGroup + - Seminorm.isBounded_const + - SeminormFamily.basisSets_nonempty + - SeminormFamily.finset_sup_comp + - WithSeminorms.continuous_seminorm + - WithSeminorms.congr + - SeminormFamily.withSeminorms_iff_nhds_eq_iInf + - Seminorm.bound_of_continuous + - WithSeminorms.separating_of_T1 + - WithSeminorms.isOpen_iff_mem_balls + - SeminormFamily.basisSets_smul_left + - WithSeminorms.tendsto_nhds + - WithSeminorms.first_countable + - SeminormFamily.basisSets_iff + - SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf + - SeminormFamily.basisSets_zero + - WithSeminorms.hasBasis_zero_ball + - SeminormFamily.basisSets_singleton_mem + - WithSeminorms.isVonNBounded_iff_seminorm_bounded + - WithSeminorms.tendsto_nhds' + - SeminormFamily.withSeminorms_of_nhds + - WithSeminorms.toLocallyConvexSpace + - WithSeminorms.finset_sups + - SeminormFamily.filter_eq_iInf + - Inducing.withSeminorms + - WithSeminorms.hasBasis_ball + - WithSeminorms.congr_equiv + - Seminorm.continuous_from_bounded + - WithSeminorms.image_isVonNBounded_iff_seminorm_bounded + - withSeminorms_iInf + - SeminormFamily.withSeminorms_iff_uniformSpace_eq_iInf + - Seminorm.bound_of_continuous_normedSpace + - WithSeminorms.T1_of_separating + - WithSeminorms.isVonNBounded_iff_finset_seminorm_bounded + - SeminormFamily.basisSets_neg + - WithSeminorms.separating_iff_T1 + - WithSeminorms.image_isVonNBounded_iff_finset_seminorm_bounded + - WithSeminorms.continuousSMul + - WithSeminorms.tendsto_nhds_atTop + - WithSeminorms.equicontinuous_TFAE + - Seminorm.const_isBounded + - Seminorm.cont_normedSpace_to_withSeminorms + - path: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean + theorems: + - IntervalIntegrable.comp_sub_left + - intervalIntegral.integral_const_of_cdf + - intervalIntegrable_iff_integrableOn_Ioo_of_le + - IntervalIntegrable.mul_continuousOn + - MeasureTheory.Integrable.hasSum_intervalIntegral + - intervalIntegrable_iff_integrableOn_Ico_of_le + - intervalIntegral.inv_smul_integral_comp_add_div + - intervalIntegrable_iff + - intervalIntegral.integral_comp_sub_div + - IntervalIntegrable.refl + - IntervalIntegrable.continuousOn_mul + - intervalIntegral.norm_integral_le_of_norm_le_const_ae + - intervalIntegral.integral_comp_sub_mul + - intervalIntegral.intervalIntegral_pos_of_pos_on + - intervalIntegral.integral_symm + - intervalIntegrable_iff_integrableOn_Ioc_of_le + - intervalIntegral.norm_integral_eq_norm_integral_Ioc + - intervalIntegral.integral_interval_add_interval_comm + - intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt + - intervalIntegral.integral_comp_mul_sub + - intervalIntegral.integral_comp_mul_add + - intervalIntegral.integral_smul_const + - intervalIntegral.sum_integral_adjacent_intervals_Ico + - intervalIntegral.integral_comp_div_sub + - intervalIntegrable_iff' + - intervalIntegral.integral_mono_interval + - intervalIntegral.sum_integral_adjacent_intervals + - intervalIntegral.abs_integral_le_integral_abs + - intervalIntegral.integral_zero + - intervalIntegral.smul_integral_comp_sub_mul + - intervalIntegral.integral_indicator + - intervalIntegral.integral_smul + - intervalIntegral.integral_comp_add_div + - IntervalIntegrable.comp_sub_right + - intervalIntegral.norm_integral_le_of_norm_le + - intervalIntegral.integral_comp_sub_left + - intervalIntegral.norm_integral_le_integral_norm + - intervalIntegral.inv_smul_integral_comp_div + - intervalIntegral.smul_integral_comp_mul_sub + - intervalIntegral.integral_finset_sum + - IntervalIntegrable.comp_mul_right + - intervalIntegral.integral_comp_add_left + - intervalIntegral.integral_comp_add_mul + - MonotoneOn.intervalIntegrable + - intervalIntegral.integral_interval_sub_interval_comm' + - IsROrC.interval_integral_ofReal + - IntervalIntegrable.div_const + - intervalIntegral.integral_smul_measure + - intervalIntegral.integral_non_aestronglyMeasurable + - intervalIntegral.integral_congr_ae' + - intervalIntegral.norm_integral_min_max + - intervalIntegral.integral_comp_add_right + - intervalIntegral.integral_mono_ae_restrict + - IntervalIntegrable.comp_add_left + - intervalIntegral.smul_integral_comp_mul_left + - ContinuousLinearMap.intervalIntegral_comp_comm + - intervalIntegral.integral_comp_mul_left + - intervalIntegral.integral_mul_const + - intervalIntegral.integral_eq_zero_iff_of_nonneg_ae + - intervalIntegral.inv_smul_integral_comp_div_add + - intervalIntegral.integral_mono_ae + - intervalIntegral.integral_comp_neg + - intervalIntegral.integral_const + - intervalIntegral.integral_Iic_sub_Iic + - IntervalIntegrable.iff_comp_neg + - intervalIntegral.integral_Iic_add_Ioi + - intervalIntegral.norm_intervalIntegral_eq + - intervalIntegrable_iff_integrableOn_Icc_of_le + - intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_ae + - intervalIntegral.integral_lt_integral_of_ae_le_of_measure_setOf_lt_ne_zero + - IntervalIntegrable.trans_iterate_Ico + - intervalIntegral.integral_mono_on + - intervalIntegral.integral_comp_div_add + - intervalIntegral.integral_eq_integral_of_support_subset + - ContinuousLinearMap.intervalIntegral_apply + - intervalIntegral.integral_add_adjacent_intervals_cancel + - intervalIntegral.integral_sub + - intervalIntegral.inv_smul_integral_comp_sub_div + - intervalIntegral.integral_const' + - intervalIntegral.inv_smul_integral_comp_div_sub + - intervalIntegral.integral_nonneg_of_ae_restrict + - intervalIntegral.integral_comp_div + - intervalIntegral.integral_add + - IntervalIntegrable.intervalIntegrable_norm_iff + - intervalIntegral.integral_undef + - intervalIntegrable_const_iff + - intervalIntegral.integral_comp_mul_right + - intervalIntegral.norm_integral_le_abs_integral_norm + - intervalIntegral.smul_integral_comp_mul_add + - MeasureTheory.Integrable.hasSum_intervalIntegral_comp_add_int + - IntervalIntegrable.comp_mul_left + - intervalIntegral.integral_div + - intervalIntegral.integral_Iio_add_Ici + - intervalIntegral.integral_neg + - intervalIntegral.integral_of_ge + - intervalIntegral.intervalIntegral_eq_integral_uIoc + - intervalIntegral.integral_pos_iff_support_of_nonneg_ae' + - intervalIntegral.integral_non_aestronglyMeasurable_of_le + - intervalIntegral.smul_integral_comp_mul_right + - intervalIntegral.integral_comp_sub_right + - IntervalIntegrable.comp_mul_left_iff + - intervalIntegral.integral_of_le + - intervalIntegral.integral_add_adjacent_intervals + - path: Mathlib/Tactic/NormNum/Basic.lean + theorems: + - Mathlib.Meta.NormNum.isNat_cast + - Mathlib.Meta.NormNum.isRat_neg + - Mathlib.Meta.NormNum.isInt_cast + - Mathlib.Meta.NormNum.isRat_div + - Mathlib.Meta.NormNum.isNat_natAbs_neg + - Mathlib.Meta.NormNum.isNat_natDiv + - Mathlib.Meta.NormNum.isRat_eq_true + - Mathlib.Meta.NormNum.isNat_natSub + - Mathlib.Meta.NormNum.isNat_int_cast + - Mathlib.Meta.NormNum.isNat_natSucc + - Mathlib.Meta.NormNum.isRat_mul + - Mathlib.Meta.NormNum.isRat_sub + - Mathlib.Meta.NormNum.isNat_natMod + - Mathlib.Meta.NormNum.isRat_add + - Mathlib.Meta.NormNum.ble_eq_false + - path: Mathlib/NumberTheory/Modular.lean + theorems: + - ModularGroup.three_lt_four_mul_im_sq_of_mem_fdo + - ModularGroup.exists_eq_T_zpow_of_c_eq_zero + - ModularGroup.re_T_smul + - ModularGroup.smul_eq_lcRow0_add + - ModularGroup.coe_T_zpow_smul_eq + - ModularGroup.im_lt_im_S_smul + - ModularGroup.abs_two_mul_re_lt_one_of_mem_fdo + - ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo + - ModularGroup.tendsto_normSq_coprime_pair + - ModularGroup.bottom_row_surj + - ModularGroup.exists_smul_mem_fd + - ModularGroup.c_eq_zero + - ModularGroup.bottom_row_coprime + - ModularGroup.tendsto_abs_re_smul + - ModularGroup.tendsto_lcRow0 + - ModularGroup.g_eq_of_c_eq_one + - ModularGroup.one_lt_normSq_T_zpow_smul + - ModularGroup.abs_c_le_one + - ModularGroup.re_T_inv_smul + - ModularGroup.normSq_S_smul_lt_one + - ModularGroup.im_T_inv_smul + - ModularGroup.exists_max_im + - ModularGroup.eq_zero_of_mem_fdo_of_T_zpow_mem_fdo + - ModularGroup.im_T_smul + - ModularGroup.im_T_zpow_smul + - ModularGroup.exists_row_one_eq_and_min_re + - path: Mathlib/Topology/Homeomorph.lean + theorems: + - Homeomorph.symm_trans_self + - Homeomorph.isPreconnected_image + - Homeomorph.image_closure + - Homeomorph.comp_continuousAt_iff' + - Homeomorph.comap_nhds_eq + - Equiv.toHomeomorph_symm + - Homeomorph.self_trans_symm + - Homeomorph.inducing + - Homeomorph.isOpen_image + - Homeomorph.map_punctured_nhds_eq + - Homeomorph.isSigmaCompact_preimage + - Homeomorph.locallyConnectedSpace + - Homeomorph.isClosed_image + - Homeomorph.image_interior + - Homeomorph.isPreconnected_preimage + - Homeomorph.comp_isOpenMap_iff' + - Homeomorph.isCompact_preimage + - Homeomorph.quotientMap + - Homeomorph.isClosed_preimage + - Homeomorph.comp_isOpenMap_iff + - Homeomorph.map_nhds_eq + - Homeomorph.image_frontier + - Homeomorph.map_cocompact + - Homeomorph.isConnected_preimage + - Continuous.continuous_symm_of_equiv_compact_to_t2 + - Homeomorph.image_connectedComponentIn + - path: Mathlib/FieldTheory/Minpoly/Field.lean + theorems: + - minpoly.coeff_zero_ne_zero + - minpoly.dvd_map_of_isScalarTower' + - minpoly.root + - minpoly.zero + - minpoly.prime + - minpoly.one + - minpoly_algEquiv_toLinearMap + - minpoly.isRadical + - minpoly.ker_aeval_eq_span_minpoly + - minpoly_algHom_toLinearMap + - minpoly.dvd_map_of_isScalarTower + - minpoly.dvd + - minpoly.coeff_zero_eq_zero + - minpoly.degree_le_of_ne_zero + - minpoly.dvd_iff + - minpoly.unique + - minpoly.add_algebraMap + - minpoly.aux_inj_roots_of_min_poly + - minpoly.eq_of_irreducible + - minpoly.sub_algebraMap + - path: Mathlib/LinearAlgebra/Contraction.lean + theorems: + - homTensorHomEquiv_apply + - dualTensorHomEquivOfBasis_symm_cancel_right + - rTensorHomEquivHomRTensor_apply + - toMatrix_dualTensorHom + - dualTensorHom_prodMap_zero + - dualTensorHomEquivOfBasis_symm_cancel_left + - lTensorHomEquivHomLTensor_toLinearMap + - zero_prodMap_dualTensorHom + - transpose_dualTensorHom + - map_dualTensorHom + - rTensorHomEquivHomRTensor_toLinearMap + - lTensorHomEquivHomLTensor_apply + - path: Mathlib/Data/List/Prime.lean + theorems: + - mem_list_primes_of_dvd_prod + - Prime.dvd_prod_iff + - perm_of_prod_eq_prod + - path: Mathlib/Topology/UniformSpace/Equicontinuity.lean + theorems: + - equicontinuousAt_iff_range + - UniformEquicontinuous.closure' + - equicontinuousWithinAt_finite + - EquicontinuousOn.closure' + - equicontinuousWithinAt_iInf_rng + - uniformEquicontinuousOn_iInf_rng + - Filter.HasBasis.uniformEquicontinuous_iff_left + - Filter.HasBasis.equicontinuousWithinAt_iff + - equicontinuousAt_iff_continuousAt + - Filter.HasBasis.equicontinuousAt_iff + - uniformEquicontinuousOn_iff_range + - equicontinuousAt_iff_pair + - equicontinuous_restrict_iff + - equicontinuousWithinAt_univ + - equicontinuousOn_iff_continuousOn + - equicontinuousAt_finite + - UniformEquicontinuousOn.closure' + - uniformEquicontinuousOn_iff_uniformContinuousOn + - equicontinuousWithinAt_iff_range + - Filter.HasBasis.equicontinuousWithinAt_iff_right + - Filter.HasBasis.uniformEquicontinuousOn_iff + - uniformEquicontinuous_iff_range + - Filter.HasBasis.equicontinuousAt_iff_right + - uniformEquicontinuousOn_iInf_dom + - equicontinuousOn_finite + - equicontinuousAt_iInf_rng + - UniformInducing.equicontinuousWithinAt_iff + - UniformInducing.uniformEquicontinuous_iff + - uniformEquicontinuous_finite + - EquicontinuousAt.closure' + - equicontinuousWithinAt_iInf_dom + - Filter.Tendsto.continuousWithinAt_of_equicontinuousWithinAt + - Filter.HasBasis.uniformEquicontinuous_iff_right + - Filter.Tendsto.uniformContinuous_of_uniformEquicontinuous + - equicontinuousAt_restrict_iff + - equicontinuous_iInf_rng + - UniformInducing.equicontinuousOn_iff + - Filter.Tendsto.continuousAt_of_equicontinuousAt + - uniformEquicontinuous_iInf_dom + - UniformInducing.equicontinuous_iff + - uniformEquicontinuous_iff_uniformContinuous + - Filter.HasBasis.equicontinuousAt_iff_left + - uniformEquicontinuousOn_univ + - uniformEquicontinuousOn_finite + - equicontinuousAt_iInf_dom + - equicontinuousOn_iInf_rng + - EquicontinuousWithinAt.closure' + - equicontinuousWithinAt_iff_continuousWithinAt + - Filter.HasBasis.uniformEquicontinuous_iff + - equicontinuous_finite + - UniformInducing.uniformEquicontinuousOn_iff + - equicontinuousWithinAt_iff_pair + - uniformEquicontinuous_iInf_rng + - UniformInducing.equicontinuousAt_iff + - EquicontinuousAt.tendsto_of_mem_closure + - uniformEquicontinuous_restrict_iff + - equicontinuousOn_univ + - equicontinuous_iff_continuous + - Filter.HasBasis.equicontinuousWithinAt_iff_left + - Filter.HasBasis.uniformEquicontinuousOn_iff_right + - Filter.Tendsto.uniformContinuousOn_of_uniformEquicontinuousOn + - path: Mathlib/Data/List/BigOperators/Basic.lean + theorems: + - List.headI_add_tail_sum + - List.alternatingProd_cons + - List.prod_append + - List.prod_lt_prod' + - List.alternatingProd_cons' + - List.prod_take_mul_prod_drop + - List.exists_le_of_prod_le' + - List.prod_eq_one + - List.Sublist.prod_le_prod' + - List.one_lt_prod_of_one_lt + - List.prod_le_prod' + - List.prod_join + - List.prod_reverse_noncomm + - List.tail_sum + - List.prod_eq_zero_iff + - List.eq_of_prod_take_eq + - List.prod_range_div' + - List.prod_nat_mod + - List.one_le_prod_of_one_le + - List.prod_int_mod + - List.sum_int_mod + - List.prod_hom_rel + - List.prod_set' + - List.prod_set + - List.prod_concat + - List.prod_erase_of_comm + - List.prod_map_hom + - List.length_pos_of_prod_ne_one + - List.get?_zero_mul_tail_prod + - List.prod_take_succ + - CanonicallyOrderedCommSemiring.list_prod_pos + - List.prod_map_erase + - List.single_le_prod + - "List.Forall\u2082.prod_le_prod'" + - List.prod_inv + - "List.prod_hom\u2082" + - List.prod_induction + - List.prod_cons + - List.prod_range_div + - List.exists_mem_ne_one_of_prod_ne_one + - List.prod_pos + - List.prod_eq_zero + - List.prod_eq_pow_card + - List.alternatingProd_cons_cons + - List.exists_lt_of_prod_lt' + - List.prod_replicate + - List.headI_mul_tail_prod_of_ne_nil + - List.prod_inv_reverse + - List.prod_isUnit + - List.prod_eq_foldr + - List.monotone_prod_take + - List.prod_le_pow_card + - List.prod_hom + - List.sum_le_foldr_max + - List.sum_nat_mod + - path: Mathlib/Analysis/Convex/Segment.lean + theorems: + - Pi.openSegment_subset + - Pi.image_update_openSegment + - segment_eq_Icc' + - segment_subset_uIcc + - openSegment_eq_image' + - Prod.image_mk_openSegment_left + - mem_openSegment_translate + - Prod.openSegment_subset + - openSegment_eq_image_lineMap + - right_mem_openSegment_iff + - mem_segment_iff_div + - Convex.mem_Ico + - Prod.image_mk_openSegment_right + - segment_subset_Icc + - sameRay_of_mem_segment + - segment_eq_image + - image_segment + - segment_inter_eq_endpoint_of_linearIndependent_of_ne + - segment_eq_image_lineMap + - Convex.mem_Ioc + - openSegment_subset_iff_segment_subset + - mem_openSegment_iff_div + - segment_eq_image' + - mem_openSegment_of_ne_left_right + - Convex.mem_Ioo + - segment_same + - Prod.segment_subset + - mem_segment_add_sub + - insert_endpoints_openSegment + - openSegment_same + - midpoint_mem_segment + - openSegment_eq_image + - Prod.image_mk_segment_left + - "openSegment_eq_image\u2082" + - Pi.image_update_segment + - Icc_subset_segment + - "segment_eq_image\u2082" + - image_openSegment + - openSegment_subset_union + - segment_inter_eq_endpoint_of_linearIndependent_sub + - Pi.segment_subset + - openSegment_subset_Ioo + - left_mem_openSegment_iff + - mem_segment_iff_sameRay + - Prod.image_mk_segment_right + - left_mem_segment + - Convex.mem_Icc + - path: .lake/packages/std/Std/Data/Array/Lemmas.lean + theorems: + - Array.get_append_right + - Array.forIn_eq_data_forIn + - Array.get_extract_loop_lt_aux + - Array.get_swap_left + - Array.eq_empty_of_size_eq_zero + - Array.get_extract_aux + - Array.size_ofFn + - Array.extract_empty_of_size_le_start + - Array.getElem_setD + - Array.fin_cast_val + - Array.all_eq_true + - Array.extract_loop_eq_aux + - Array.all_eq_not_any_not + - Array.mem_filter + - Array.getElem_ofFn_go + - Array.foldr_induction + - Array.filter_filter + - Array.get?_size + - Array.any_eq_true + - Array.get?_set + - Array.get_extract_loop_ge + - Array.get?_set_ne + - Array.all_eq_true_iff_forall_mem + - Array.extract_loop_succ + - Array.size_swap! + - Array.get_modify + - Array.getElem_ofFn + - Array.get?_push_eq + - Array.extract_loop_eq + - Array.all_iff_forall + - Array.get_append_left + - Array.filter_data + - Array.mem_filterMap + - Array.set_set + - Array.data_pop + - Array.append_assoc + - Array.swap_swap + - Array.zipWith_eq_zipWith_data + - Array.get_set_eq + - Array.mapM_eq_mapM_data + - Array.map_induction + - Array.mem_append + - Array.get?_swap + - Array.get?_len_le + - Array.get!_eq_get? + - Array.anyM_loop_iff_exists + - Array.size_ofFn_go + - Array.size_append + - Array.pop_push + - Array.all_def + - Array.getElem_map + - Array.get_swap + - Array.filterMap_data + - Array.mapM_map_eq_foldl + - Array.extract_loop_of_ge + - Array.getElem?_mem + - Array.swap_def + - Array.mapIdx_induction + - Array.extract_empty_of_stop_le_start + - Array.get_set + - Array.size_modify + - Array.map_spec + - Array.size_reverse + - Array.get_extract_loop_ge_aux + - Array.size_extract + - Array.get_swap_of_ne + - Array.reverse_data + - Array.get_swap' + - getElem_mkArray + - Array.swapAt!_def + - Array.extract_loop_zero + - Array.foldl_induction + - Array.get_extract + - Array.join_data + - Array.getElem_mem_data + - Array.extract_all + - Array.any_iff_exists + - Array.mem_map + - Array.back_eq_back? + - Array.get?_set_eq + - Array.back_push + - Array.getElem?_eq_data_get? + - Array.nil_append + - Array.size_range + - Array.size_extract_loop + - Array.get_set_ne + - Array.back?_push + - Array.eq_push_pop_back_of_size_ne_zero + - Array.get?_push + - Array.get_swap_right + - Array.mem_join + - Array.size_filter_le + - Array.swap_comm + - Array.any_def + - Array.getElem_mapIdx + - Array.append_nil + - path: Mathlib/Order/Filter/Partial.lean + theorems: + - Filter.tendsto_iff_ptendsto_univ + - Filter.rtendsto'_def + - Filter.rtendsto_iff_le_rcomap + - Filter.tendsto_iff_rtendsto' + - Filter.tendsto_iff_rtendsto + - Filter.ptendsto_of_ptendsto' + - Filter.pmap_res + - Filter.tendsto_iff_ptendsto + - Filter.rcomap_rcomap + - Filter.rmap_rmap + - Filter.rcomap'_rcomap' + - Filter.ptendsto'_of_ptendsto + - path: Mathlib/Tactic/ComputeDegree.lean + theorems: + - Mathlib.Tactic.ComputeDegree.degree_eq_of_le_of_coeff_ne_zero' + - Mathlib.Tactic.ComputeDegree.natDegree_eq_of_le_of_coeff_ne_zero' + - Mathlib.Tactic.ComputeDegree.coeff_mul_add_of_le_natDegree_of_eq_ite + - Mathlib.Tactic.ComputeDegree.coeff_add_of_eq + - Mathlib.Tactic.ComputeDegree.coeff_sub_of_eq + - Mathlib.Tactic.ComputeDegree.coeff_pow_of_natDegree_le_of_eq_ite' + - path: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Restrict.lean + theorems: + - SpectrumRestricts.compactSpace + - SpectrumRestricts.cfc + - path: Mathlib/RepresentationTheory/Character.lean + theorems: + - FdRep.char_iso + - FdRep.char_linHom + - FdRep.char_orthonormal + - FdRep.char_one + - FdRep.char_tensor + - FdRep.average_char_eq_finrank_invariants + - FdRep.char_tensor' + - FdRep.char_conj + - FdRep.char_mul_comm + - path: Mathlib/Algebra/Homology/Homotopy.lean + theorems: + - Homotopy.nullHomotopicMap_f + - homology'_map_eq_of_homotopy + - Homotopy.prevD_chainComplex + - "Homotopy.mkCoinductiveAux\u2083" + - Homotopy.nullHomotopicMap_f_of_not_rel_right + - Homotopy.nullHomotopicMap'_f_of_not_rel_left + - Homotopy.nullHomotopicMap_f_of_not_rel_left + - dNext_nat + - Homotopy.comp_nullHomotopicMap' + - Homotopy.nullHomotopicMap'_f_of_not_rel_right + - Homotopy.nullHomotopicMap'_comp + - Homotopy.nullHomotopicMap_comp + - prevD_eq + - dNext_eq + - Homotopy.nullHomotopicMap_f_eq_zero + - prevD_nat + - Homotopy.nullHomotopicMap'_f_eq_zero + - Homotopy.map_nullHomotopicMap' + - Homotopy.prevD_succ_cochainComplex + - prevD_comp_right + - "Homotopy.mkInductiveAux\u2083" + - Homotopy.map_nullHomotopicMap + - prevD_eq_zero + - Homotopy.prevD_zero_cochainComplex + - dNext_eq_zero + - Homotopy.dNext_zero_chainComplex + - Homotopy.dNext_cochainComplex + - Homotopy.dNext_succ_chainComplex + - path: Mathlib/Data/Complex/Exponential.lean + theorems: + - Complex.tanh_neg + - Complex.cosh_add_sinh + - Real.cos_add_cos + - Real.cos_neg + - Complex.tan_conj + - Real.exp_bound_div_one_sub_of_interval + - Complex.sum_div_factorial_le + - Real.exp_approx_succ + - Complex.sin_mul_I + - Complex.abs_exp_ofReal + - Complex.ofReal_sinh_ofReal_re + - Complex.cosh_three_mul + - Real.exp_bound + - Complex.sin_ofReal_im + - Complex.inv_one_add_tan_sq + - Complex.tan_sq_div_one_add_tan_sq + - Real.exp_sub + - Real.cos_one_le + - Real.sinh_sub + - Real.sin_two_mul + - Real.inv_sqrt_one_add_tan_sq + - Real.cos_two_mul + - Complex.sinh_sub + - Complex.ofReal_sin_ofReal_re + - Complex.cos_conj + - Real.cos_two_mul' + - Real.add_one_lt_exp_of_pos + - Real.cos_abs + - Real.abs_exp_sub_one_le + - Complex.tan_zero + - Complex.exp_nat_mul + - Complex.sinh_add + - Real.pow_div_factorial_le_exp + - Real.exp_add + - Complex.exp_conj + - Real.cos_two_neg + - Complex.cosh_mul_I + - Real.expNear_succ + - Real.add_one_le_exp + - Real.sinh_two_mul + - Complex.ofReal_tan_ofReal_re + - Complex.cos_add_cos + - Complex.exp_add + - Real.one_lt_exp_iff + - Complex.cosh_conj + - Real.cosh_eq + - Complex.cos_add + - Complex.cosh_add_aux + - Complex.sin_three_mul + - Complex.sin_sub + - Real.cosh_sq + - Real.cos_sq_le_one + - Real.abs_sin_le_one + - Complex.isCauSeq_abs_exp + - Real.cosh_add_sinh + - Real.sin_sub + - Real.sin_sq_eq_half_sub + - Complex.exp_neg + - Real.add_one_lt_exp + - Real.sin_sub_sin + - Complex.abs_cos_add_sin_mul_I + - Real.abs_sin_eq_sqrt_one_sub_cos_sq + - Complex.sin_neg + - Complex.cosh_add + - Real.sin_three_mul + - Complex.tanh_mul_I + - Complex.exp_ofReal_mul_I_re + - Complex.sin_sq_add_cos_sq + - Complex.tanh_zero + - Real.cos_add + - Complex.cosh_neg + - Real.exp_pos + - Real.sinh_neg + - Real.sinh_add_cosh + - Real.sinh_eq + - Real.exp_zero + - Real.sinh_zero + - Real.exp_abs_le + - Complex.sin_sq + - Real.sin_sq_le_one + - Complex.abs_exp_sub_one_sub_id_le + - Complex.ofReal_exp_ofReal_re + - Complex.cosh_two_mul + - Real.exp_strictMono + - Complex.tan_neg + - Real.cosh_three_mul + - Complex.cos_add_aux + - Complex.sinh_mul_I + - Real.tanh_zero + - Complex.sinh_zero + - Real.tanh_neg + - Complex.cos_two_mul' + - Real.sinh_three_mul + - Real.tan_sq_div_one_add_tan_sq + - Complex.tanh_ofReal_im + - Complex.tan_mul_cos + - Real.cos_sub + - Complex.exp_im + - Complex.cos_add_sin_mul_I_pow + - Real.exp_ne_zero + - Complex.abs_exp_ofReal_mul_I + - Complex.exp_bound + - Complex.sinh_sub_cosh + - Complex.cos_ofReal_im + - Complex.sin_two_mul + - Real.tan_zero + - Complex.cos_mul_I + - Real.exp_bound_div_one_sub_of_interval' + - Complex.sinh_conj + - Complex.cos_three_mul + - Complex.sin_add + - Real.cosh_add + - Complex.cosh_sub_sinh + - Complex.sinh_ofReal_im + - Real.tan_mul_cos + - Complex.exp_add_mul_I + - Complex.abs_exp_eq_iff_re_eq + - Complex.ofReal_cos_ofReal_re + - Real.sin_zero + - Complex.sin_sub_sin + - Complex.cosh_zero + - Complex.cosh_ofReal_im + - Real.sin_bound + - Real.abs_cos_eq_sqrt_one_sub_sin_sq + - Complex.exp_re + - Real.exp_approx_end + - Real.sinh_sub_cosh + - Real.tan_div_sqrt_one_add_tan_sq + - Real.exp_lt_one_iff + - Complex.abs_exp_sub_one_le + - Complex.sinh_add_aux + - Real.cosh_zero + - Complex.sin_eq + - Real.add_one_le_exp_of_nonneg + - Complex.cos_sub_cos + - Real.tan_eq_sin_div_cos + - Real.expNear_sub + - Complex.cos_sub + - Real.sinh_sq + - Complex.sinh_sq + - Complex.abs_exp + - Real.cos_sub_cos + - Complex.sin_conj + - Real.cos_zero + - Complex.cos_eq + - Real.cos_sq' + - Real.cos_pos_of_le_one + - Real.cosh_sq_sub_sinh_sq + - Real.cosh_two_mul + - Real.exp_neg + - Real.cosh_abs + - Complex.cosh_sub + - Real.sin_sq_add_cos_sq + - Complex.sinh_three_mul + - Real.inv_one_add_tan_sq + - Real.one_sub_div_pow_le_exp_neg + - Real.tan_neg + - Complex.cos_add_mul_I + - Real.sin_neg + - Complex.exp_bound' + - Complex.cosh_sq + - Complex.cos_sq' + - Real.cos_bound + - Real.one_le_exp + - Real.exp_bound' + - Real.sin_pos_of_pos_of_le_two + - Real.abs_cos_le_one + - Real.sin_add + - Complex.cos_neg + - Complex.sinh_neg + - Complex.tan_mul_I + - Complex.tan_ofReal_im + - Real.abs_exp_sub_one_sub_id_le + - Complex.sin_add_mul_I + - Complex.cosh_sq_sub_sinh_sq + - Complex.cos_add_sin_I + - Real.cos_sq + - Complex.tanh_conj + - Real.cos_three_mul + - Real.cos_sq_add_sin_sq + - Complex.exp_ofReal_im + - Real.cosh_neg + - Complex.cos_zero + - Complex.exp_eq_exp_re_mul_sin_add_cos + - Real.sinh_add + - Real.exp_nat_mul + - Complex.sinh_add_cosh + - Complex.exp_zero + - Complex.exp_ofReal_mul_I_im + - Real.sin_pos_of_pos_of_le_one + - Real.sum_le_exp_of_nonneg + - Real.cosh_sub_sinh + - Complex.ofReal_cosh_ofReal_re + - Real.exp_approx_end' + - Real.tanh_eq_sinh_div_cosh + - Complex.exp_sub + - Complex.ofReal_tanh_ofReal_re + - Complex.exp_ne_zero + - Complex.cos_sq_add_sin_sq + - Complex.sinh_two_mul + - Real.expNear_zero + - Complex.cos_sub_sin_I + - Complex.exp_int_mul + - path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean + theorems: + - CochainComplex.HomComplex.Cochain.ofHom_comp + - "CochainComplex.HomComplex.Cochain.ext\u2080" + - CochainComplex.HomComplex.Cochain.add_comp + - "CochainComplex.HomComplex.\u03B4_comp_zero_cocycle" + - "CochainComplex.HomComplex.\u03B4_v" + - CochainComplex.HomComplex.Cochain.comp_add + - CochainComplex.HomComplex.Cochain.comp_units_smul + - CochainComplex.HomComplex.Cocycle.coe_zero + - CochainComplex.HomComplex.Cochain.smul_comp + - CochainComplex.HomComplex.Cocycle.ofHom_homOf_eq_self + - CochainComplex.HomComplex.Cochain.ofHoms_comp + - CochainComplex.HomComplex.Cochain.map_comp + - CochainComplex.HomComplex.Cochain.comp_assoc + - CochainComplex.HomComplex.Cochain.ofHoms_v_comp_d + - "CochainComplex.HomComplex.\u03B4_zero_cocycle_comp" + - CochainComplex.HomComplex.Cochain.ofHom_neg + - CochainComplex.HomComplex.Cochain.comp_assoc_of_third_is_zero_cochain + - CochainComplex.HomComplex.Cochain.d_comp_ofHoms_v + - CochainComplex.HomComplex.Cochain.neg_comp + - CochainComplex.HomComplex.Cochain.ofHoms_zero + - CochainComplex.HomComplex.Cochain.comp_id + - CochainComplex.HomComplex.Cochain.map_add + - "CochainComplex.HomComplex.\u03B4_neg_one_cochain" + - "CochainComplex.HomComplex.\u03B4_ofHom" + - CochainComplex.HomComplex.Cochain.comp_assoc_of_second_is_zero_cochain + - "CochainComplex.HomComplex.\u03B4_zero_cochain_v" + - "CochainComplex.HomComplex.\u03B4_\u03B4" + - CochainComplex.HomComplex.Cocycle.mem_iff + - CochainComplex.HomComplex.Cochain.congr_v + - CochainComplex.HomComplex.Cochain.map_sub + - "CochainComplex.HomComplex.\u03B4_ofHomotopy" + - "CochainComplex.HomComplex.\u03B4_zero_cochain_comp" + - CochainComplex.HomComplex.Cochain.zero_comp + - CochainComplex.HomComplex.Cochain.comp_assoc_of_second_degree_eq_neg_third_degree + - CochainComplex.HomComplex.Cochain.id_comp + - CochainComplex.HomComplex.Cochain.map_ofHom + - CochainComplex.HomComplex.Cochain.ofHom_zero + - CochainComplex.HomComplex.Cocycle.homOf_ofHom_eq_self + - "CochainComplex.HomComplex.\u03B4_ofHom_comp" + - CochainComplex.HomComplex.Cochain.d_comp_ofHom_v + - CochainComplex.HomComplex.Cochain.comp_v + - "CochainComplex.HomComplex.\u03B4_comp_zero_cochain" + - CochainComplex.HomComplex.Cochain.sub_comp + - "CochainComplex.HomComplex.\u03B4_comp" + - CochainComplex.HomComplex.Cochain.comp_smul + - "CochainComplex.HomComplex.\u03B4_map" + - CochainComplex.HomComplex.Cochain.ofHom_v_comp_d + - CochainComplex.HomComplex.Cochain.comp_neg + - CochainComplex.HomComplex.Cochain.map_zero + - CochainComplex.HomComplex.Cochain.comp_zero + - "CochainComplex.HomComplex.\u03B4_shape" + - CochainComplex.HomComplex.Cochain.v_comp_XIsoOfEq_hom + - "CochainComplex.HomComplex.\u03B4_comp_ofHom" + - CochainComplex.HomComplex.Cocycle.cochain_ofHom_homOf_eq_coe + - CochainComplex.HomComplex.Cochain.ofHom_sub + - CochainComplex.HomComplex.Cochain.v_comp_XIsoOfEq_inv + - CochainComplex.HomComplex.Cochain.ofHoms_v + - CochainComplex.HomComplex.Cochain.ofHom_v + - CochainComplex.HomComplex.Cochain.ofHom_add + - CochainComplex.HomComplex.Cochain.map_neg + - CochainComplex.HomComplex.Cochain.units_smul_comp + - CochainComplex.HomComplex.Cochain.ext + - CochainComplex.HomComplex.Cochain.comp_sub + - "CochainComplex.HomComplex.Cocycle.\u03B4_eq_zero" + - path: Mathlib/Data/Multiset/Sum.lean + theorems: + - Multiset.inr_mem_disjSum + - Multiset.card_disjSum + - Multiset.Nodup.disjSum + - Multiset.inl_mem_disjSum + - path: .lake/packages/std/Std/Data/Nat/Gcd.lean + theorems: + - Nat.lcm_ne_zero + - Nat.gcd_mul_right_left + - Nat.gcd_mul_right + - Nat.gcd_eq_zero_iff + - Nat.gcd_eq_right + - Nat.gcd_mul_right_right + - Nat.gcd_div + - Nat.lcm_zero_right + - Nat.coprime_mul_iff_right + - Nat.coprime_self + - Nat.gcd_eq_right_iff_dvd + - Nat.Coprime.eq_one_of_dvd + - Nat.not_coprime_of_dvd_of_dvd + - Nat.lcm_dvd + - Nat.gcd_mul_dvd_mul_gcd + - Nat.lcm_self + - Nat.lcm_zero_left + - Nat.Coprime.gcd_mul_left_cancel + - Nat.Coprime.dvd_of_dvd_mul_right + - Nat.coprime_zero_left + - Nat.gcd_gcd_self_left_right + - Nat.coprime_div_gcd_div_gcd + - Nat.lcm_one_right + - Nat.eq_zero_of_gcd_eq_zero_right + - Nat.gcd_gcd_self_right_right + - Nat.gcd_eq_left_iff_dvd + - Nat.coprime_zero_right + - Nat.Coprime.gcd_mul_right_cancel_right + - Nat.coprime_mul_iff_left + - Nat.lcm_one_left + - Nat.Coprime.gcd_mul_right_cancel + - Nat.gcd_mul_left + - Nat.Coprime.coprime_dvd_left + - Nat.gcd_mul_left_right + - Nat.dvd_lcm_left + - Nat.gcd_eq_iff + - Nat.Coprime.coprime_div_left + - Nat.Coprime.pow_left + - Nat.gcd_mul_gcd_of_coprime_of_mul_eq_mul + - Nat.Coprime.dvd_of_dvd_mul_left + - Nat.gcd_gcd_self_left_left + - Nat.gcd_add_mul_self + - path: Mathlib/Data/Matrix/PEquiv.lean + theorems: + - PEquiv.toPEquiv_mul_matrix + - PEquiv.toMatrix_injective + - PEquiv.single_mul_single_right + - PEquiv.single_mul_single_of_ne + - PEquiv.toMatrix_swap + - PEquiv.single_mul_single + - PEquiv.toMatrix_trans + - PEquiv.toMatrix_symm + - PEquiv.mul_toPEquiv_toMatrix + - PEquiv.toMatrix_refl + - PEquiv.matrix_mul_apply + - PEquiv.mul_matrix_apply + - path: Mathlib/Data/Set/Sigma.lean + theorems: + - Set.sigma_univ_range_eq + - Set.insert_sigma + - Set.mk_preimage_sigma_fn_eq_if + - Set.mk_preimage_sigma_eq_if + - biSup_sigma + - Set.image_sigmaMk_preimage_sigmaMap + - Set.sigma_singleton + - Set.singleton_sigma + - Set.sigma_eq_empty_iff + - Set.sigma_insert + - Set.singleton_sigma_singleton + - Set.preimage_image_sigmaMk_of_ne + - Set.sigma_diff_sigma + - Set.sigma_inter_sigma + - path: Mathlib/Data/Nat/Choose/Cast.lean + theorems: + - Nat.cast_choose_two + - Nat.cast_add_choose + - Nat.cast_choose_eq_ascPochhammer_div + - path: Mathlib/LinearAlgebra/LinearPMap.lean + theorems: + - LinearPMap.eq_of_eq_graph + - LinearPMap.smul_graph + - LinearPMap.le_graph_of_le + - LinearPMap.graph_map_snd_eq_range + - LinearPMap.neg_graph + - LinearPMap.ext_iff + - LinearPMap.sSup_aux + - LinearPMap.inverse_graph + - LinearPMap.le_of_le_graph + - LinearPMap.graph_map_fst_eq_domain + - LinearPMap.mem_graph_iff + - Submodule.toLinearPMap_graph_eq + - Submodule.mem_graph_toLinearPMap + - LinearPMap.inverse_range + - Submodule.toLinearPMap_range + - LinearPMap.left_le_sup + - LinearPMap.mkSpanSingleton'_apply_self + - LinearPMap.sup_h_of_disjoint + - LinearPMap.mem_inverse_graph + - LinearPMap.sSup_apply + - LinearPMap.mem_graph_snd_inj' + - LinearPMap.sup_aux + - Submodule.toLinearPMap_apply_aux + - LinearPMap.mem_domain_iff_of_eq_graph + - LinearPMap.ext + - LinearPMap.mem_domain_iff + - LinearPMap.mem_graph_iff' + - LinearPMap.domRestrict_apply + - Submodule.existsUnique_from_graph + - LinearPMap.inverse_apply_eq + - LinearPMap.domRestrict_le + - LinearPMap.supSpanSingleton_apply_mk + - LinearPMap.right_le_sup + - LinearPMap.mem_range_iff + - LinearPMap.mem_domain_of_mem_graph + - LinearPMap.mem_graph + - LinearPMap.image_iff + - LinearPMap.inverse_domain + - LinearPMap.mkSpanSingleton'_apply + - LinearPMap.mem_inverse_graph_snd_eq_zero + - path: Mathlib/Control/Traversable/Equiv.lean + theorems: + - Equiv.id_traverse + - Equiv.isLawfulTraversable' + - Equiv.lawfulFunctor' + - Equiv.comp_traverse + - Equiv.comp_map + - Equiv.traverse_eq_map_id + - Equiv.id_map + - Equiv.naturality + - path: Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean + theorems: + - Complex.betaIntegral_recurrence + - Complex.Gamma_eq_zero_iff + - Complex.GammaSeq_add_one_left + - Complex.Gamma_mul_Gamma_eq_betaIntegral + - Complex.differentiable_one_div_Gamma + - Complex.betaIntegral_symm + - Complex.Gamma_mul_Gamma_add_half + - Complex.betaIntegral_convergent + - Complex.Gamma_mul_Gamma_one_sub + - Complex.one_div_Gamma_eq_self_mul_one_div_Gamma_add_one + - Complex.betaIntegral_eval_one_right + - Complex.betaIntegral_eval_nat_add_one_right + - Complex.betaIntegral_convergent_left + - Real.GammaSeq_tendsto_Gamma + - Complex.Gamma_ne_zero + - Complex.Gamma_ne_zero_of_re_pos + - Complex.GammaSeq_eq_approx_Gamma_integral + - Complex.GammaSeq_tendsto_Gamma + - Complex.betaIntegral_scaled + - Complex.GammaSeq_eq_betaIntegral_of_re_pos + - Real.Gamma_mul_Gamma_one_sub + - Complex.GammaSeq_mul + - Complex.approx_Gamma_integral_tendsto_Gamma_integral + - Real.Gamma_mul_Gamma_add_half + - path: Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean + theorems: + - SimpleGraph.incMatrix_apply_mul_incMatrix_apply + - SimpleGraph.sum_incMatrix_apply_of_mem_edgeSet + - SimpleGraph.incMatrix_of_mem_incidenceSet + - SimpleGraph.incMatrix_mul_transpose + - SimpleGraph.incMatrix_apply_eq_one_iff + - SimpleGraph.incMatrix_mul_transpose_apply_of_adj + - SimpleGraph.incMatrix_apply_mul_incMatrix_apply_of_not_adj + - SimpleGraph.incMatrix_mul_transpose_diag + - SimpleGraph.incMatrix_transpose_mul_diag + - SimpleGraph.sum_incMatrix_apply + - SimpleGraph.incMatrix_of_not_mem_incidenceSet + - path: Mathlib/Topology/Category/Profinite/Limits.lean + theorems: + - Profinite.pullback.hom_ext + - Profinite.pullback_fst_eq + - "Profinite.finiteCoproduct.\u03B9_desc_apply" + - Profinite.pullback.condition + - "Profinite.Sigma.\u03B9_comp_toFiniteCoproduct" + - Profinite.finiteCoproduct.hom_ext + - Profinite.pullback_snd_eq + - "Profinite.finiteCoproduct.\u03B9_injective" + - path: Mathlib/Combinatorics/Additive/ETransform.lean + theorems: + - Finset.mulETransformRight_one + - Finset.mulETransformLeft.fst_mul_snd_subset + - Finset.mulETransformLeft.card + - Finset.mulETransformLeft_one + - Finset.mulETransformRight.fst_mul_snd_subset + - Finset.mulDysonETransform_idem + - Finset.mulDysonETransform.subset + - Finset.mulETransformLeft_inv + - Finset.mulETransformRight_inv + - Finset.mulETransformRight.card + - Finset.mulDysonETransform.card + - Finset.mulDysonETransform.smul_finset_snd_subset_fst + - path: Mathlib/CategoryTheory/IsConnected.lean + theorems: + - CategoryTheory.nat_trans_from_is_connected + - CategoryTheory.IsConnected.of_induct + - CategoryTheory.isPreconnected_induction + - CategoryTheory.zigzag_isPreconnected + - CategoryTheory.nonempty_hom_of_preconnected_groupoid + - CategoryTheory.equiv_relation + - CategoryTheory.any_functor_const_on_obj + - CategoryTheory.isPreconnected_of_zigzag + - CategoryTheory.induct_on_objects + - CategoryTheory.IsPreconnected.of_constant_of_preserves_morphisms + - CategoryTheory.constant_of_preserves_morphisms + - CategoryTheory.isConnected_of_equivalent + - path: Mathlib/Data/Nat/Factorial/DoubleFactorial.lean + theorems: + - Nat.doubleFactorial_eq_prod_even + - Nat.doubleFactorial_eq_prod_odd + - Nat.factorial_eq_mul_doubleFactorial + - Nat.doubleFactorial_add_one + - Nat.doubleFactorial_le_factorial + - path: Mathlib/Order/SuccPred/Basic.lean + theorems: + - Order.Ioc_pred_left_of_not_isMin + - exists_succ_iterate_iff_le + - Order.Icc_succ_right + - StrictMono.not_bddAbove_range + - WithBot.succ_unbot + - Order.Ioi_pred_eq_insert_of_not_isMin + - WithTop.pred_untop + - Order.Icc_pred_left + - Order.Ioo_succ_right_of_not_isMax + - Order.Ico_succ_right_eq_insert_of_not_isMax + - Order.le_succ_iterate + - Order.succ_lt_succ_iff_of_not_isMax + - Order.le_succ_bot_iff + - Order.pred_eq_iff_covBy + - Order.succ_le_succ + - Order.pred_succ_iterate_of_not_isMax + - Order.succ_lt_succ_iff + - Order.le_le_succ_iff + - WCovBy.le_succ + - Order.pred_le_iff_eq_or_le + - Order.succ_eq_iInf + - Order.le_succ_iff_eq_or_le + - Order.pred_eq_iSup + - Succ.rec + - WCovBy.pred_le + - Order.Icc_succ_left_of_not_isMax + - Order.succ_le_succ_iff + - Order.pred_iterate_le + - Order.pred_lt_pred_iff + - Order.Ioc_pred_right_of_not_isMin + - Order.Ioc_succ_right + - Order.succ_top + - Order.isMax_iterate_succ_of_eq_of_lt + - Order.Ico_pred_left + - Order.lt_succ_bot_iff + - Order.Ioo_pred_left_of_not_isMin + - Order.Icc_pred_right_of_not_isMin + - SuccOrder.forall_ne_bot_iff + - Order.Ico_pred_right_eq_insert + - Order.pred_eq_pred_iff + - Succ.rec_iff + - Order.Ioo_succ_right_eq_insert_of_not_isMax + - Order.pred_le_pred_iff + - Order.Ioo_pred_right_eq_insert + - Order.succ_eq_succ_iff_of_not_isMax + - Order.pred_le_le_iff + - Order.succ_eq_iff_covBy + - Order.Ico_succ_left_of_not_isMax + - Order.isMax_iterate_succ_of_eq_of_ne + - Order.Ico_succ_right_of_not_isMax + - Order.succ_le_succ_iff_of_not_isMax + - path: Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean + theorems: + - ContinuousMultilinearMap.norm_restr + - ContinuousMultilinearMap.le_opNorm_mul_pow_card_of_le + - ContinuousMultilinearMap.opNorm_zero + - ContinuousMultilinearMap.opNorm_smul_le + - ContinuousMultilinearMap.continuous_eval + - ContinuousMultilinearMap.opNNNorm_pi + - ContinuousMultilinearMap.opNorm_add_le + - ContinuousMultilinearMap.opNorm_zero_iff + - MultilinearMap.restr_norm_le + - ContinuousMultilinearMap.nnnorm_smulRight + - ContinuousMultilinearMap.norm_ofSubsingleton_id_le + - ContinuousMultilinearMap.le_opNNNorm + - ContinuousMultilinearMap.isLeast_opNNNorm + - ContinuousMultilinearMap.norm_mkPiAlgebra + - MultilinearMap.exists_bound_of_continuous + - ContinuousMultilinearMap.norm_mkPiAlgebraFin_zero + - ContinuousMultilinearMap.norm_mkPiAlgebra_le + - ContinuousMultilinearMap.hasSum_eval + - ContinuousMultilinearMap.norm_mkPiRing + - MultilinearMap.norm_map_coord_zero + - ContinuousMultilinearMap.opNorm_neg + - MultilinearMap.mkContinuousLinear_norm_le' + - ContinuousMultilinearMap.norm_mkPiAlgebraFin_le_of_pos + - MultilinearMap.mkContinuousMultilinear_norm_le' + - ContinuousMultilinearMap.norm_ofSubsingleton_id + - ContinuousMultilinearMap.norm_mkPiAlgebraFin_succ_le + - ContinuousMultilinearMap.isLeast_opNorm + - ContinuousMultilinearMap.opNNNorm_prod + - ContinuousMultilinearMap.opNNNorm_le_iff + - ContinuousMultilinearMap.norm_mkPiAlgebraFin + - ContinuousMultilinearMap.norm_compContinuous_linearIsometry_le + - MultilinearMap.norm_image_sub_le_of_bound' + - ContinuousMultilinearMap.norm_ofSubsingleton + - ContinuousMultilinearMap.le_opNorm_mul_pow_of_le + - MultilinearMap.bound_of_shell_of_norm_map_coord_zero + - LinearIsometry.norm_compContinuousMultilinearMap + - ContinuousMultilinearMap.unit_le_opNorm + - ContinuousMultilinearMap.norm_compContinuousLinearMap_le + - ContinuousMultilinearMap.norm_constOfIsEmpty + - ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv + - MultilinearMap.norm_image_sub_le_of_bound + - MultilinearMap.continuous_of_bound + - ContinuousMultilinearMap.norm_mkPiAlgebra_of_empty + - path: Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean + theorems: + - Polynomial.X_pow_sub_one_dvd_prod_cyclotomic + - Polynomial.cyclotomic_dvd_geom_sum_of_dvd + - Polynomial.eq_cyclotomic_iff + - Polynomial.prod_cyclotomic'_eq_X_pow_sub_one + - Polynomial.cyclotomic_eq_X_pow_sub_one_div + - Polynomial.natDegree_cyclotomic + - Polynomial.cyclotomic.eval_apply + - Polynomial.int_cyclotomic_rw + - Polynomial.cyclotomic'_two + - Polynomial.cyclotomic_eq_prod_X_pow_sub_one_pow_moebius + - Polynomial.int_cyclotomic_unique + - Polynomial.cyclotomic'_zero + - Polynomial.X_pow_sub_one_mul_cyclotomic_dvd_X_pow_sub_one_of_dvd + - Polynomial.orderOf_root_cyclotomic_dvd + - Polynomial.cyclotomic.monic + - Polynomial.degree_cyclotomic + - Polynomial.prod_cyclotomic_eq_geom_sum + - Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one + - Polynomial.int_cyclotomic_spec + - Polynomial.cyclotomic_prime_pow_eq_geom_sum + - Polynomial.cyclotomic_eq_prod_X_sub_primitiveRoots + - Polynomial.cyclotomic_prime_mul_X_sub_one + - Polynomial.natDegree_cyclotomic' + - Polynomial.prod_cyclotomic_eq_X_pow_sub_one + - Polynomial.cyclotomic'_eq_X_pow_sub_one_div + - Polynomial.degree_cyclotomic' + - Polynomial.map_cyclotomic_int + - Polynomial.unique_int_coeff_of_cycl + - Polynomial.cyclotomic_one + - Polynomial.cyclotomic.dvd_X_pow_sub_one + - Polynomial.int_coeff_of_cyclotomic' + - Polynomial.roots_of_cyclotomic + - Polynomial.cyclotomic'_splits + - Polynomial.coprime_of_root_cyclotomic + - Polynomial.X_pow_sub_one_mul_prod_cyclotomic_eq_X_pow_sub_one_of_dvd + - Polynomial.X_pow_sub_one_splits + - Polynomial.cyclotomic'_one + - Polynomial.cyclotomic_zero + - Polynomial.degree_cyclotomic_pos + - Polynomial.map_cyclotomic + - Polynomial.cyclotomic_three + - Polynomial.cyclotomic_two + - Polynomial.cyclotomic_prime + - Polynomial.X_pow_sub_one_eq_prod + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean + theorems: + - Matrix.charmatrix_map + - Matrix.aeval_self_charpoly + - "Matrix.charpoly_fromBlocks_zero\u2081\u2082" + - Matrix.charmatrix_apply_eq + - Matrix.charpoly_map + - "Matrix.charpoly_fromBlocks_zero\u2082\u2081" + - Matrix.matPolyEquiv_charmatrix + - Matrix.charpoly_reindex + - Matrix.charmatrix_fromBlocks + - Matrix.charmatrix_reindex + - path: Mathlib/Data/Set/Image.lean + theorems: + - Set.preimage_eq_preimage' + - Option.injective_iff + - Set.image_pair + - Set.exists_mem_image + - Set.preimage_eq_empty + - Set.image_compl_subset + - Set.range_ite_subset + - Set.preimage_eq_empty_iff + - Set.exists_subtype_range_iff + - Set.image_subset + - Set.preimage_eq_univ_iff + - Set.range_subset_singleton + - Set.preimage_const + - Set.range_inr + - Set.image_comp_eq + - Function.Injective.image_injective + - Set.image_symmDiff + - Set.image_eq_range + - Set.exists_subset_range_and_iff + - Set.image_eq_empty + - Nat.mem_range_succ + - Set.image_univ + - Set.range_eq_empty_iff + - Function.Surjective.image_surjective + - Set.image_id + - Subtype.forall_set_subtype + - Function.LeftInverse.image_image + - Set.range_comp + - Set.eq_preimage_iff_image_eq + - Set.image_inter_on + - Set.preimage_eq_iff_eq_image + - Set.preimage_image_preimage + - Set.range_inl + - Set.preimage_singleton_true + - Set.range_eq_iff + - Set.mem_image_iff_of_inverse + - sigma_mk_preimage_image' + - Set.Nontrivial.preimage + - Set.subset_image_iff + - Set.range_comp_subset_range + - Subtype.preimage_coe_eq_empty + - Set.preimage_singleton_false + - Set.image_comm + - Subtype.preimage_val_subset_preimage_val_iff + - Subtype.exists_set_subtype + - Set.exists_range_iff + - Set.image_preimage_eq_iff + - Set.preimage_congr + - sigma_mk_preimage_image_eq_self + - Set.mem_compl_image + - Set.preimage_eq_preimage + - Set.image_singleton + - Function.Injective.preimage_surjective + - Set.image_eq_image + - Set.disjoint_image_image + - Set.image_preimage_eq_range_inter + - Subtype.preimage_coe_eq_preimage_coe_iff + - Set.range_unique + - Set.leftInverse_rangeSplitting + - Set.compl_compl_image + - Set.preimage_inr_image_inl + - Set.insert_image_compl_eq_range + - Set.image_perm + - Set.image_preimage_eq_inter_range + - Set.preimage_injective + - Set.range_inclusion + - Set.powerset_insert + - Set.preimage_inl_range_inr + - Subtype.coe_preimage_self + - Set.subset_image_compl + - Set.image_empty + - Set.preimage_subtype_coe_eq_compl + - Set.image_preimage_inl_union_image_preimage_inr + - Set.image_id' + - Subtype.range_coe + - Set.preimage_iterate_eq + - Set.subset_image_diff + - Set.image_preimage_inter + - Disjoint.of_preimage + - Set.preimage_inl_image_inr + - Set.forall_subtype_range_iff + - Subtype.preimage_coe_nonempty + - Set.image_subset_image_iff + - Set.image_injective + - Subtype.coe_image_subset + - Set.preimage_surjective + - Set.preimage_subset_preimage_iff + - Set.range_ite_subset' + - Function.Surjective.preimage_subset_preimage_iff + - Function.LeftInverse.preimage_preimage + - Subtype.preimage_coe_inter_self + - Set.image_id_eq + - Set.image_iterate_eq + - Set.preimage_range_inter + - Set.exists_eq_const_of_preimage_singleton + - Set.preimage_inr_range_inl + - Set.image_inter_nonempty_iff + - Set.forall_mem_image + - Set.forall_subset_range_iff + - Set.image_mono + - Set.image_inter_preimage + - Set.eq_preimage_subtype_val_iff + - Set.image_surjective + - Set.Nonempty.subset_preimage_const + - Set.image_union + - Set.image_subset_range + - Function.Surjective.nonempty_preimage + - Subtype.preimage_coe_self_inter + - Set.image_insert_eq + - Set.image_univ_of_surjective + - Subtype.coe_image_of_subset + - Set.exists_subset_range_iff + - Set.image_congr + - Set.image_diff_preimage + - Set.image_compl_preimage + - Set.isCompl_range_inl_range_inr + - Set.image_comp + - Set.subsingleton_of_preimage + - Function.Injective.compl_image_eq + - path: Mathlib/RingTheory/Coalgebra.lean + theorems: + - Finsupp.comul_comp_lapply + - Prod.counit_comp_inr + - Prod.comul_comp_snd + - Coalgebra.coassoc_symm_apply + - Finsupp.counit_comp_lsingle + - Finsupp.comul_comp_lsingle + - Prod.comul_comp_inr + - Prod.comul_comp_inl + - Prod.comul_comp_fst + - Prod.counit_comp_inl + - path: Mathlib/Topology/LocallyConstant/Basic.lean + theorems: + - IsLocallyConstant.exists_eq_const + - LocallyConstant.piecewise_apply_left + - IsLocallyConstant.desc + - IsLocallyConstant.comp_continuous + - LocallyConstant.locallyConstant_eq_of_fiber_zero_eq + - LocallyConstant.comap_const + - LocallyConstant.coe_comap + - IsLocallyConstant.range_finite + - LocallyConstant.comap_comp + - IsLocallyConstant.comp + - LocallyConstant.comap_injective + - LocallyConstant.ofIsClopen_fiber_one + - LocallyConstant.piecewise'_apply_right + - LocallyConstant.comap_comap + - LocallyConstant.comap_id + - IsLocallyConstant.apply_eq_of_isPreconnected + - LocallyConstant.exists_eq_const + - LocallyConstant.ofIsClopen_fiber_zero + - LocallyConstant.coe_comap_apply + - LocallyConstant.piecewise'_apply_left + - IsLocallyConstant.tfae + - path: Mathlib/Data/Fin/Basic.lean + theorems: + - Fin.coe_of_injective_castSucc_symm + - Fin.min_val + - Fin.castSucc_lt_succ_iff + - Fin.lt_sub_one_iff + - Fin.nat_cast_self + - Fin.zero_mul' + - Fin.symm_castIso + - Fin.heq_ext_iff + - Fin.add_zero + - Fin.heq_fun_iff + - Fin.zero_eq_one_iff + - Fin.cast_eq_cast + - Fin.castSucc_ne_zero_of_lt + - Fin.cast_nat_eq_last + - Fin.coe_fin_one + - Fin.succ_one_eq_two' + - Fin.succ_castPred_eq_add_one + - Fin.lt_succ_castPred + - Fin.castpred_succ_le_iff + - Fin.rev_le_iff + - Fin.castPred_inj + - Fin.lt_pred_iff + - Fin.exists_eq_add_of_lt + - Fin.le_pred_castSucc_iff + - Fin.coe_succ_lt_iff_lt + - Fin.pred_last + - Fin.neg_nat_cast_eq_one + - Fin.mul_one' + - Fin.lt_castPred_iff + - Fin.succ_ne_last_of_lt + - Fin.castSucc_pred_lt_iff + - Fin.cast_zero + - Fin.castSucc_castPred + - Fin.rev_lt_iff + - Fin.val_add_eq_ite + - Fin.add_one_le_of_lt + - Fin.nonempty_embedding_iff + - Fin.pos_iff_ne_zero' + - Fin.castPred_le_iff + - Fin.castIso_to_equiv + - Fin.coe_of_injective_castLEEmb_symm + - Fin.pred_lt_castPred + - Fin.nat_cast_eq_zero + - Fin.modNat_rev + - Fin.le_zero_iff' + - "Fin.heq_fun\u2082_iff" + - Fin.castSucc_pos' + - Fin.coe_eq_castSucc + - Fin.castPred_le_pred_iff + - Fin.one_mul' + - Fin.pred_castSucc_lt_iff + - Fin.castSucc_pred_lt + - Fin.castPred_lt_iff + - Fin.pred_lt_iff + - Fin.max_val + - Fin.rev_involutive + - Fin.pred_castSucc_lt + - Fin.mk_bit1 + - Fin.succ_le_or_le_castSucc + - Fin.succ_le_castSucc_iff + - Fin.succ_zero_eq_one' + - Fin.lt_castPred_succ_iff + - Fin.exists_eq_add_of_le + - Fin.mul_zero' + - Fin.le_castPred_iff + - Fin.val_bit0 + - Fin.le_rev_iff + - Fin.lt_rev_iff + - Fin.castPred_eq_iff_eq_castSucc + - Fin.castIso_refl + - Fin.lt_succ_castPred_iff + - Fin.rev_eq_iff + - Fin.last_sub + - Fin.le_sub_one_iff + - Fin.castPred_one + - Fin.coe_sub_one + - Fin.pred_lt_castPred_iff + - Fin.coe_neg_one + - Fin.cast_rev + - Fin.castSucc_pred_add_one_eq + - Fin.zero_add + - Fin.one_eq_zero_iff + - Fin.le_castSucc_pred_iff + - Fin.le_val_last + - Fin.neg_last + - Fin.pred_one' + - Fin.succ_castPred_le_iff + - Fin.lt_castPred_succ + - Fin.eq_one_of_neq_zero + - Fin.pred_le_iff + - Fin.rev_pred + - Fin.castSucc_eq_zero_iff' + - path: Mathlib/Analysis/Asymptotics/Asymptotics.lean + theorems: + - Asymptotics.isBigOWith_zero_right_iff + - Asymptotics.IsLittleO.tendsto_zero_of_tendsto + - Asymptotics.IsLittleO.smul_isBigO + - Asymptotics.isBigOWith_inv + - Asymptotics.IsBigO.smul_isLittleO + - Homeomorph.isBigO_congr + - Asymptotics.isLittleO_zero + - Asymptotics.isBigO_iff' + - Asymptotics.IsBigOWith.isBigO + - Asymptotics.isLittleO_norm_pow_id + - Asymptotics.IsLittleO.mul_isBigO + - Asymptotics.IsBigO.exists_mem_basis + - Asymptotics.IsLittleO.trans_isBigOWith + - Asymptotics.div_isBoundedUnder_of_isBigO + - Asymptotics.isBigO_const_iff + - Asymptotics.isBigOWith_of_eq_mul + - Asymptotics.isBigOWith_insert + - Asymptotics.isLittleO_one_left_iff + - Asymptotics.isBigOWith_zero' + - Asymptotics.isBigO_one_iff + - Asymptotics.IsBigOWith.trans_isLittleO + - Homeomorph.isLittleO_congr + - Asymptotics.isLittleO_const_left + - Asymptotics.isLittleO_principal + - Asymptotics.isBigOWith_pi + - Asymptotics.IsLittleO.add_add + - Asymptotics.isBigO_iff_isBigOWith + - Asymptotics.isBigO_const_const_iff + - PartialHomeomorph.isLittleO_congr + - Asymptotics.IsBigOWith.sub_isLittleO + - Asymptotics.IsBigOWith.mul + - Asymptotics.bound_of_isBigO_cofinite + - Asymptotics.isBigO_neg_right + - Asymptotics.IsBigO.sum + - Asymptotics.isBigOWith_neg_left + - Asymptotics.IsBigOWith.right_le_add_of_lt_one + - Asymptotics.isLittleO_top + - Asymptotics.IsBigOWith.eq_zero_imp + - Asymptotics.IsLittleO.sub + - Asymptotics.isBigO_iff_eventually + - Asymptotics.isLittleO_one_iff + - Asymptotics.isBigOWith_of_le + - Asymptotics.IsBigOWith.sum + - Asymptotics.isLittleO_irrefl' + - Asymptotics.IsLittleO.eventuallyLE + - Asymptotics.isBigO_iff_isBoundedUnder_le_div + - Asymptotics.IsLittleO.symm + - Asymptotics.isLittleO_of_subsingleton + - Asymptotics.IsBigOWith.eventually_mul_div_cancel + - Asymptotics.isBigO_norm_right + - Asymptotics.isLittleO_pi + - Asymptotics.isLittleO_iff_forall_isBigOWith + - Asymptotics.bound_of_isBigO_nat_atTop + - Asymptotics.isLittleO_pow_pow + - Asymptotics.IsBigOWith.add + - Asymptotics.isBigOWith_norm_right + - Asymptotics.isBigO_iff'' + - Asymptotics.isBigO_snd_prod' + - Asymptotics.isBigO_iff + - Asymptotics.isLittleO_pow_sub_pow_sub + - Asymptotics.isBigO_norm_left + - Asymptotics.isBigO_const_left_iff_pos_le_norm + - Asymptotics.IsBigOWith.pow' + - Asymptotics.IsBigO.of_bound' + - Asymptotics.isLittleO_iff_nat_mul_le_aux + - Asymptotics.isBigOWith_self_const_mul' + - Asymptotics.IsLittleO.sum + - Asymptotics.isBigO_atTop_iff_eventually_exists + - Asymptotics.isBigO_nat_atTop_iff + - Asymptotics.isBigO_atTop_iff_eventually_exists_pos + - Asymptotics.isBigOWith_zero + - Asymptotics.IsBigO.mul_isLittleO + - Asymptotics.isBigO_const_smul_left + - Asymptotics.IsBigOWith.prod_left_same + - Asymptotics.isLittleO_neg_right + - Asymptotics.isLittleO_const_const_iff + - Asymptotics.IsBigOWith.of_pow + - Asymptotics.isBigO_iff_div_isBoundedUnder + - Asymptotics.isLittleO_iff_exists_eq_mul + - Asymptotics.isLittleO_const_left_of_ne + - Asymptotics.isBigOWith_const_const + - Asymptotics.isLittleO_const_smul_right + - Asymptotics.isBigO_one_nat_atTop_iff + - Asymptotics.IsBigOWith.inv_rev + - Asymptotics.isLittleO_const_smul_left + - Asymptotics.isBigO_pi + - Asymptotics.isLittleO_neg_left + - Asymptotics.IsBigO.sub + - Asymptotics.isBigO_cofinite_iff + - Asymptotics.isLittleO_map + - Asymptotics.isBigOWith_map + - Asymptotics.isLittleO_norm_left + - Asymptotics.isBigOWith_principal + - Asymptotics.isLittleO_congr + - Asymptotics.isLittleO_pow_id + - Asymptotics.isBigOWith_top + - Asymptotics.isBigO_iff_exists_eq_mul + - Asymptotics.IsBigOWith.trans + - Asymptotics.IsBigOWith.pow + - Asymptotics.IsBigOWith.smul + - Asymptotics.isBigO_top + - Asymptotics.IsLittleO.pow + - Asymptotics.IsBigOWith.sub + - Asymptotics.isBigO_map + - Asymptotics.IsBigOWith.weaken + - PartialHomeomorph.isBigO_congr + - Asymptotics.isBigO_principal + - Asymptotics.isBigOWith_iff_exists_eq_mul + - Asymptotics.isBigOWith_const_one + - Asymptotics.isBigOWith_norm_left + - Asymptotics.isBigOWith_congr + - Asymptotics.isBigO_neg_left + - PartialHomeomorph.isBigOWith_congr + - Asymptotics.isBigO_congr + - Asymptotics.isBigO_const_smul_right + - Asymptotics.isBigO_fst_prod' + - Asymptotics.isLittleO_iff + - Asymptotics.IsLittleO.tendsto_inv_smul_nhds_zero + - Asymptotics.IsBigO.of_pow + - Asymptotics.isBigOWith_neg_right + - Asymptotics.isLittleO_norm_right + - Asymptotics.IsBigO.smul + - Asymptotics.isBigOWith_iff + - Asymptotics.IsLittleO.tendsto_div_nhds_zero + - Asymptotics.IsBigOWith.right_le_sub_of_lt_one + - Asymptotics.IsBigO.eq_zero_of_norm_pow_within + - Asymptotics.IsBigO.eq_zero_of_norm_pow + - path: Mathlib/Dynamics/FixedPoints/Basic.lean + theorems: + - Function.IsFixedPt.perm_pow + - Function.Commute.invOn_fixedPoints_comp + - Function.Commute.left_bijOn_fixedPoints_comp + - Function.mem_fixedPoints_iff + - Function.fixedPoints_id + - Function.Commute.right_bijOn_fixedPoints_comp + - Function.IsFixedPt.apply + - Function.IsFixedPt.preimage_iterate + - path: Mathlib/Data/Fin/Tuple/Sort.lean + theorems: + - Tuple.graph.card + - Tuple.eq_sort_iff' + - Tuple.sort_eq_refl_iff_monotone + - Tuple.antitone_pair_of_not_sorted' + - Tuple.comp_perm_comp_sort_eq_comp_sort + - Tuple.monotone_proj + - Tuple.lt_card_le_iff_apply_le_of_monotone + - Tuple.self_comp_sort + - Tuple.eq_sort_iff + - Tuple.monotone_sort + - path: Mathlib/Data/Option/Defs.lean + theorems: + - Option.mem_toList + - Option.mem_some_iff + - Option.elim'_eq_elim + - path: Mathlib/CategoryTheory/Limits/Types.lean + theorems: + - CategoryTheory.Limits.Types.FilteredColimit.isColimit_eq_iff + - CategoryTheory.Limits.Types.Small.limitCone_pt_ext + - CategoryTheory.Limits.Types.colimit_sound + - CategoryTheory.Limits.Types.Quot.desc_colimitCocone + - CategoryTheory.Limits.Types.FilteredColimit.rel_eq_eqvGen_quot_rel + - CategoryTheory.Limits.Types.FilteredColimit.eqvGen_quot_rel_of_rel + - CategoryTheory.Limits.Types.Quot.desc_toCocone_desc + - CategoryTheory.Limits.Types.FilteredColimit.rel_equiv + - CategoryTheory.Limits.Types.isLimit_iff + - CategoryTheory.Limits.Types.colimitEquivQuot_apply + - CategoryTheory.Limits.Types.FilteredColimit.rel_of_quot_rel + - CategoryTheory.Limits.Types.colimit_eq + - "CategoryTheory.Limits.Types.Limit.\u03C0_mk" + - CategoryTheory.Limits.Types.isColimit_iff_bijective_desc + - CategoryTheory.Limits.Types.colimit_sound' + - CategoryTheory.Limits.Types.jointly_surjective_of_isColimit + - CategoryTheory.Limits.Types.FilteredColimit.colimit_eq_iff_aux + - CategoryTheory.Limits.Types.isLimitEquivSections_symm_apply + - CategoryTheory.Limits.Types.limit_ext + - CategoryTheory.Limits.Types.isLimit_iff_bijective_sectionOfCone + - path: Mathlib/RingTheory/PowerSeries/Basic.lean + theorems: + - PowerSeries.monomial_zero_eq_C_apply + - PowerSeries.coeff_prod + - PowerSeries.rescale_zero_apply + - PowerSeries.coeff_zero_X_mul + - PowerSeries.map_C + - Polynomial.coe_eq_one_iff + - PowerSeries.coeff_one_X + - PowerSeries.coeff_X + - PowerSeries.monomial_eq_mk + - PowerSeries.X_dvd_iff + - PowerSeries.ext + - Polynomial.coe_add + - PowerSeries.coeff_X_pow_self + - PowerSeries.X_ne_zero + - PowerSeries.coeff_mul_X_pow' + - PowerSeries.coeff_X_pow_mul' + - PowerSeries.rescale_one + - PowerSeries.X_prime + - PowerSeries.rescale_injective + - PowerSeries.monomial_zero_eq_C + - PowerSeries.smul_eq_C_mul + - PowerSeries.coeff_ne_zero_C + - PowerSeries.map_X + - PowerSeries.eq_shift_mul_X_add_const + - Polynomial.coe_mul + - Polynomial.coe_bit1 + - PowerSeries.coeff_zero_C + - PowerSeries.coeff_zero_eq_constantCoeff + - Polynomial.coe_C + - PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero + - PowerSeries.coeff_X_pow + - PowerSeries.eq_X_mul_shift_add_const + - PowerSeries.rescale_mk + - Polynomial.coe_injective + - PowerSeries.coeff_zero_X + - "Polynomial.eval\u2082_C_X_eq_coe" + - PowerSeries.coeff_C_mul_X_pow + - PowerSeries.rescale_zero + - PowerSeries.coeff_monomial + - PowerSeries.coeff_mul_X_pow + - Polynomial.coe_eq_zero_iff + - PowerSeries.coeff_X_pow_mul + - PowerSeries.rescale_neg_one_X + - PowerSeries.coeff_zero_eq_constantCoeff_apply + - PowerSeries.span_X_isPrime + - PowerSeries.rescale_mul + - PowerSeries.coeff_C + - Polynomial.coe_monomial + - PowerSeries.coeff_mul + - PowerSeries.coeff_zero_mul_X + - PowerSeries.X_pow_dvd_iff + - PowerSeries.coeff_succ_X_mul + - PowerSeries.coeff_def + - Polynomial.coe_one + - PowerSeries.rescale_X + - PowerSeries.rescale_rescale + - PowerSeries.coeff_succ_mul_X + - path: Mathlib/RingTheory/PrincipalIdealDomain.lean + theorems: + - Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem + - IsBezout.gcd_eq_sum + - Submodule.IsPrincipal.associated_generator_span_self + - nonPrincipals_zorn + - gcd_dvd_iff_exists + - Ideal.IsPrincipal.of_comap + - IsBezout.gcd_dvd_left + - PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset + - PrincipalIdealRing.factors_spec + - exists_associated_pow_of_mul_eq_pow' + - Submodule.IsPrincipal.of_comap + - Irreducible.coprime_iff_not_dvd + - Submodule.IsPrincipal.generator_map_dvd_of_mem + - span_gcd + - IsPrime.to_maximal_ideal + - IsBezout.span_gcd_eq_span_gcd + - IsBezout.dvd_gcd + - IsPrincipalIdealRing.of_prime + - PrincipalIdealRing.isMaximal_of_irreducible + - Submodule.IsPrincipal.mem_iff_generator_dvd + - Submodule.IsPrincipal.mem_iff_eq_smul_generator + - IsRelPrime.isCoprime + - nonPrincipals_eq_empty_iff + - Submodule.IsPrincipal.prime_generator_of_isPrime + - Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero + - exists_gcd_eq_mul_add_mul + - Submodule.IsPrincipal.generator_mem + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean + theorems: + - MeasureTheory.condexpInd_of_measurable + - MeasureTheory.condexpIndL1Fin_smul' + - MeasureTheory.condexpIndL1_disjoint_union + - MeasureTheory.condexpInd_disjoint_union + - MeasureTheory.condexpIndL1Fin_add + - MeasureTheory.condexpIndL1_smul + - MeasureTheory.condexpIndL1_of_not_measurableSet + - MeasureTheory.condexpL1CLM_indicatorConst + - MeasureTheory.condexpIndL1_of_measurableSet_of_measure_ne_top + - MeasureTheory.condexpInd_empty + - MeasureTheory.q + - MeasureTheory.condexpL1_of_aestronglyMeasurable' + - MeasureTheory.condexpIndL1Fin_smul + - MeasureTheory.condexpInd_nonneg + - MeasureTheory.condexpIndL1_add + - MeasureTheory.condexpL1CLM_smul + - MeasureTheory.condexpL1CLM_lpMeas + - MeasureTheory.condexpIndL1_smul' + - MeasureTheory.condexpL1_mono + - MeasureTheory.aestronglyMeasurable'_condexpL1 + - MeasureTheory.set_integral_condexpL1 + - MeasureTheory.norm_condexpIndL1_le + - MeasureTheory.set_integral_condexpL1CLM + - MeasureTheory.aestronglyMeasurable'_condexpL1CLM + - MeasureTheory.condexpIndL1_of_measure_eq_top + - MeasureTheory.norm_condexpIndL1Fin_le + - MeasureTheory.condexpIndL1Fin_disjoint_union + - MeasureTheory.condexpL1_smul + - MeasureTheory.set_integral_condexpL1CLM_of_measure_ne_top + - MeasureTheory.condexpInd_ae_eq_condexpIndSMul + - path: Mathlib/Algebra/Tropical/Basic.lean + theorems: + - Tropical.add_eq_left + - Tropical.add_eq_left_iff + - Tropical.add_eq_zero_iff + - Tropical.add_pow + - Tropical.add_eq_right_iff + - Tropical.add_eq_right + - Tropical.mul_eq_zero_iff + - Tropical.succ_nsmul + - Tropical.add_eq_iff + - path: Mathlib/AlgebraicGeometry/Spec.lean + theorems: + - AlgebraicGeometry.Spec.sheafedSpaceMap_id + - AlgebraicGeometry.StructureSheaf.toPushforwardStalk_comp + - AlgebraicGeometry.Spec.locallyRingedSpaceMap_comp + - AlgebraicGeometry.Spec_map_localization_isIso + - AlgebraicGeometry.stalkMap_toStalk + - AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux + - AlgebraicGeometry.Spec.basicOpen_hom_ext + - AlgebraicGeometry.localRingHom_comp_stalkIso + - "AlgebraicGeometry.Spec_\u0393_naturality" + - AlgebraicGeometry.Spec.locallyRingedSpaceMap_id + - path: Mathlib/Algebra/Star/Unitary.lean + theorems: + - unitary.map_mem + - IsUnit.mem_unitary_of_star_mul_self + - unitary.toUnits_comp_map + - unitary.star_mem + - unitary.coe_zpow + - unitary.coe_div + - path: Mathlib/Probability/Kernel/Disintegration.lean + theorems: + - MeasureTheory.AEStronglyMeasurable.ae_integrable_condKernel_iff + - ProbabilityTheory.lintegral_condKernelReal_univ + - ProbabilityTheory.set_lintegral_condKernel_univ_right + - ProbabilityTheory.lintegral_condKernel_mem + - MeasureTheory.AEStronglyMeasurable.integral_condKernel + - ProbabilityTheory.measure_eq_compProd_real + - MeasureTheory.Integrable.integral_norm_condKernel + - ProbabilityTheory.kernel.const_eq_compProd + - ProbabilityTheory.lintegral_condKernelReal + - ProbabilityTheory.lintegral_condKernelReal_mem + - ProbabilityTheory.set_lintegral_condKernelReal_Iic + - ProbabilityTheory.kernel.const_unit_eq_compProd + - ProbabilityTheory.eq_condKernel_of_measure_eq_compProd + - ProbabilityTheory.lintegral_condKernel + - ProbabilityTheory.set_lintegral_condKernelReal_prod + - ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real + - ProbabilityTheory.set_lintegral_condKernel_eq_measure_prod + - ProbabilityTheory.measure_eq_compProd + - ProbabilityTheory.ae_condKernelReal_eq_one + - ProbabilityTheory.set_integral_condKernel + - ProbabilityTheory.exists_cond_kernel + - MeasureTheory.Integrable.norm_integral_condKernel + - ProbabilityTheory.set_lintegral_condKernelReal_univ + - ProbabilityTheory.eq_condKernel_of_measure_eq_compProd' + - ProbabilityTheory.integral_condKernel + - ProbabilityTheory.condKernel_def + - ProbabilityTheory.set_integral_condKernel_univ_left + - ProbabilityTheory.kernel.const_eq_compProd_real + - ProbabilityTheory.set_lintegral_condKernel_univ_left + - ProbabilityTheory.set_lintegral_condKernel + - ProbabilityTheory.condKernel_apply_of_ne_zero_of_measurableSet + - ProbabilityTheory.condKernel_apply_of_ne_zero + - MeasureTheory.Integrable.condKernel_ae + - path: Mathlib/Computability/TMToPartrec.lean + theorems: + - Turing.PartrecToTM2.ret_supports + - Turing.ToPartrec.stepNormal_then + - Turing.PartrecToTM2.copy_ok + - Turing.PartrecToTM2.codeSupp_comp + - Turing.PartrecToTM2.pred_ok + - Turing.PartrecToTM2.unrev_ok + - Turing.PartrecToTM2.supports_biUnion + - Turing.PartrecToTM2.clear_ok + - Turing.ToPartrec.Code.head_eval + - Turing.ToPartrec.Cont.then_eval + - "Turing.PartrecToTM2.move\u2082_ok" + - Turing.PartrecToTM2.head_stack_ok + - Turing.ToPartrec.Code.succ_eval + - Turing.PartrecToTM2.K'.elim_update_stack + - Turing.ToPartrec.cont_eval_fix + - Turing.PartrecToTM2.head_main_ok + - "Turing.PartrecToTM2.contSupp_cons\u2081" + - Turing.ToPartrec.Code.nil_eval + - Turing.PartrecToTM2.contSupp_fix + - Turing.PartrecToTM2.tr_ret_respects + - Turing.ToPartrec.Code.tail_eval + - Turing.PartrecToTM2.K'.elim_update_rev + - Turing.ToPartrec.Code.id_eval + - Turing.ToPartrec.Code.comp_eval + - "Turing.PartrecToTM2.trStmts\u2081_supports'" + - Turing.ToPartrec.Code.fix_eval + - Turing.PartrecToTM2.head_supports + - Turing.ToPartrec.Code.pred_eval + - Turing.PartrecToTM2.splitAtPred_eq + - Turing.PartrecToTM2.codeSupp_fix + - Turing.ToPartrec.Code.Ok.zero + - Turing.PartrecToTM2.trNat_zero + - Turing.PartrecToTM2.K'.elim_update_aux + - Turing.PartrecToTM2.supports_insert + - Turing.PartrecToTM2.codeSupp'_self + - Turing.ToPartrec.Code.case_eval + - Turing.ToPartrec.code_is_ok + - "Turing.PartrecToTM2.trStmts\u2081_self" + - Turing.PartrecToTM2.supports_singleton + - "Turing.PartrecToTM2.trStmts\u2081_trans" + - Turing.ToPartrec.Code.zero'_eval + - Turing.ToPartrec.Code.zero_eval + - "Turing.PartrecToTM2.trStmts\u2081_supports" + - Turing.ToPartrec.stepNormal.is_ret + - Turing.PartrecToTM2.contSupp_supports + - Turing.PartrecToTM2.tr_eval + - "Turing.PartrecToTM2.trList_ne_cons\u2097" + - Turing.ToPartrec.stepRet_then + - Turing.ToPartrec.stepRet_eval + - Turing.ToPartrec.Code.exists_code.comp + - Turing.ToPartrec.Code.cons_eval + - Turing.ToPartrec.Code.exists_code + - Turing.PartrecToTM2.codeSupp'_supports + - Turing.PartrecToTM2.supports_union + - Turing.PartrecToTM2.succ_ok + - Turing.PartrecToTM2.codeSupp_cons + - path: Mathlib/Topology/Algebra/Group/Basic.lean + theorems: + - subset_mul_closure_one + - continuousInv_inf + - "QuotientGroup.continuous_smul\u2081" + - QuotientGroup.isClosedMap_coe + - IsOpen.div_left + - IsOpen.closure_mul + - GroupTopology.continuous_mul' + - TopologicalGroup.ext + - GroupTopology.coinduced_continuous + - IsOpen.mul_right + - compl_mul_closure_one_eq + - tendsto_inv_nhdsWithin_Ici_inv + - IsClosed.mul_closure_one_eq + - DenseRange.topologicalClosure_map_subgroup + - exists_disjoint_smul_of_isCompact + - tendsto_inv_nhdsWithin_Iio_inv + - mul_singleton_mem_nhds + - mul_singleton_mem_nhds_of_nhds_one + - inv_mem_nhds_one + - tendsto_inv_nhdsWithin_Iic_inv + - singleton_mul_mem_nhds_of_nhds_one + - IsClosed.smul_left_of_isCompact + - IsCompact.mul_closure_one_eq_closure + - compact_open_separated_mul_left + - group_inseparable_iff + - MulAction.isClosedMap_quotient + - ContinuousInv.of_nhds_one + - Homeomorph.mulRight_symm + - compact_open_separated_mul_right + - map_mul_right_nhds_one + - "continuous_of_continuousAt_one\u2082" + - isClosed_setOf_map_inv + - Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite + - Filter.tendsto_const_div_iff + - Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite + - smul_mem_nhds + - GroupTopology.continuous_inv' + - Filter.tendsto_sub_const_iff + - inv_mem_connectedComponent_one + - exists_closed_nhds_one_inv_eq_mul_subset + - TopologicalGroup.exists_antitone_basis_nhds_one + - IsOpen.div_right + - continuousInv_iInf + - TopologicalGroup.of_nhds_one + - continuous_zpow + - nhds_mul + - IsOpen.mul_closure + - compact_covered_by_mul_left_translates + - tendsto_inv_nhdsWithin_Ici + - TopologicalGroup.t2Space_of_one_sep + - IsOpen.div_closure + - IsOpen.smul_left + - Homeomorph.mulLeft_symm + - GroupTopology.toTopologicalSpace_injective + - exists_nhds_split_inv + - IsCompact.inv + - map_mul_left_nhds_one + - IsOpen.closure_div + - singleton_mul_mem_nhds + - topologicalGroup_inf + - IsCompact.locallyCompactSpace_of_mem_nhds_of_group + - tendsto_inv_nhdsWithin_Ioi_inv + - tendsto_inv_nhdsWithin_Ioi + - discreteTopology_of_isOpen_singleton_one + - TopologicalGroup.of_comm_of_nhds_one + - TopologicalGroup.of_nhds_one' + - mem_closure_iff_nhds_one + - nhds_translation_div + - Filter.tendsto_div_const_iff + - Filter.HasBasis.nhds_of_one + - compl_mul_closure_one_eq_iff + - tendsto_inv_nhdsWithin_Iio + - TopologicalGroup.t1Space + - IsClosed.mul_right_of_isCompact + - tendsto_div_nhds_one_iff + - Subgroup.is_normal_topologicalClosure + - eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_group + - mul_mem_connectedComponent_one + - path: .lake/packages/std/Std/Data/Char.lean + theorems: + - String.csize_eq + - String.csize_le_4 + - String.csize_pos + - path: Mathlib/Data/Finsupp/Defs.lean + theorems: + - Finsupp.support_update_ne_zero + - Finsupp.single_apply_ne_zero + - Finsupp.support_update_zero + - Finsupp.single_apply_left + - Finsupp.erase_zero + - Finsupp.embDomain_injective + - Finsupp.zipWith_single_single + - Finsupp.single_swap + - Finsupp.embDomain_single + - Finsupp.equivFunOnFinite_symm_single + - Finsupp.eq_single_iff + - Finsupp.single_eq_single_iff + - Finsupp.update_eq_sub_add_single + - Finsupp.support_update + - Finsupp.single_injective + - Finsupp.coe_eq_zero + - Finsupp.erase_apply + - Finsupp.erase_update_eq_erase + - Finsupp.unique_ext + - Finsupp.update_erase_eq_update + - Finsupp.erase_same + - Finsupp.equivFunOnFinite_single + - Finsupp.support_zipWith + - Finsupp.erase_add + - Finsupp.mulHom_ext + - Finsupp.card_support_eq_one + - Finsupp.coe_update + - Finsupp.single_of_single_apply + - Finsupp.support_erase + - Finsupp.single_zero + - Finsupp.support_subset_iff + - Finsupp.support_nonempty_iff + - Finsupp.embDomain_mapRange + - Finsupp.induction + - Finsupp.erase_eq_sub_single + - "Finsupp.induction\u2082" + - Finsupp.support_update_subset + - Finsupp.support_sub + - Finsupp.card_support_le_one + - Finsupp.erase_idem + - Finsupp.single_eq_same + - Finsupp.card_support_eq_zero + - Finsupp.addHom_ext + - Finsupp.support_single_ne_bot + - Finsupp.single_apply_eq_zero + - Finsupp.embDomain_notin_range + - Finsupp.single_add_erase + - Finsupp.mapRange_neg + - Finsupp.support_single_subset + - Finsupp.mapRange_zero + - Finsupp.single_eq_update + - Finsupp.erase_ne + - Finsupp.single_eq_set_indicator + - Finsupp.ext_iff' + - Finsupp.update_eq_erase_add_single + - Finsupp.single_of_embDomain_single + - Finsupp.single_add_single_eq_single_add_single + - Finsupp.support_mapRange_of_injective + - Finsupp.erase_add_single + - Finsupp.support_add_eq + - Finsupp.support_onFinset + - AddEquiv.finsuppUnique_symm + - Finsupp.unique_single_eq_iff + - Finsupp.zero_update + - Finsupp.update_eq_single_add_erase + - Finsupp.support_subset_singleton' + - Finsupp.card_support_le_one' + - Finsupp.single_eq_zero + - Finsupp.erase_update_of_ne + - Finsupp.mapRange_single + - Finsupp.embDomain_apply + - Finsupp.single_apply + - Finsupp.single_apply_mem + - Finsupp.mem_support_single + - Finsupp.erase_of_not_mem_support + - Finsupp.mapRange_add + - Finsupp.mapRange_sub + - Finsupp.erase_single + - Finsupp.single_eq_of_ne + - Finsupp.mem_support_onFinset + - Finsupp.card_support_eq_one' + - path: Mathlib/MeasureTheory/SetSemiring.lean + theorems: + - MeasureTheory.IsSetSemiring.pairwiseDisjoint_diffFinset + - "MeasureTheory.IsSetSemiring.disjoint_diffFinset\u2080" + - "MeasureTheory.IsSetSemiring.diffFinset\u2080_subset" + - "MeasureTheory.IsSetSemiring.sUnion_union_diffFinset\u2080_of_subset" + - "MeasureTheory.IsSetSemiring.pairwiseDisjoint_union_diffFinset\u2080" + - MeasureTheory.IsSetSemiring.sUnion_diffFinset + - MeasureTheory.IsSetSemiring.disjoint_sUnion_diffFinset + - "MeasureTheory.IsSetSemiring.sUnion_diffFinset\u2080_subset" + - MeasureTheory.IsSetSemiring.not_mem_diffFinset + - MeasureTheory.IsSetRing.inter_mem + - "MeasureTheory.IsSetSemiring.sUnion_union_sUnion_diffFinset\u2080_of_subset" + - MeasureTheory.IsSetSemiring.empty_not_mem_diffFinset + - MeasureTheory.IsSetRing.biInter_mem + - "MeasureTheory.IsSetSemiring.diff_sUnion_eq_sUnion_diffFinset\u2080" + - MeasureTheory.IsSetRing.biUnion_mem + - MeasureTheory.IsSetSemiring.sUnion_insert_diffFinset + - "MeasureTheory.IsSetSemiring.disjoint_sUnion_diffFinset\u2080" + - MeasureTheory.IsSetRing.isSetSemiring + - "MeasureTheory.IsSetSemiring.pairwiseDisjoint_diffFinset\u2080" + - MeasureTheory.IsSetSemiring.diffFinset_subset + - MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq + - "MeasureTheory.IsSetSemiring.empty_not_mem_diffFinset\u2080" + - MeasureTheory.IsSetSemiring.pairwiseDisjoint_insert_diffFinset + - path: Mathlib/Topology/Order/Basic.lean + theorems: + - Ioc_subset_closure_interior + - tendsto_Ioi_atBot + - TFAE_mem_nhdsWithin_Iic + - IsOpen.exists_Ioo_subset + - Filter.Tendsto.add_atTop + - Filter.Tendsto.atBot_add + - nhdsWithin_Iio_eq_bot_iff + - closure_Ioo + - frontier_Iio' + - Monotone.map_csSup_of_continuousAt + - IsLUB.exists_seq_strictMono_tendsto_of_not_mem + - tendsto_nhds_top_mono + - nhds_top_order + - pi_Iio_mem_nhds + - nhdsWithin_Ici_eq'' + - Monotone.map_iInf_of_continuousAt' + - countable_setOf_isolated_right + - mem_nhds_iff_exists_Ioo_subset' + - exists_seq_strictAnti_strictMono_tendsto + - pi_Ico_mem_nhds + - Icc_mem_nhds_iff + - Dense.exists_countable_dense_subset_no_bot_top + - map_coe_atBot_of_Ioo_subset + - nhdsWithin_Ioi_neBot' + - exists_Ico_subset_of_mem_nhds' + - interior_Ici' + - tendsto_comp_coe_Ioi_atBot + - exists_seq_strictAnti_tendsto' + - mem_nhdsWithin_Iic_iff_exists_Ioc_subset' + - tendsto_comp_coe_Ioo_atBot + - Monotone.map_ciSup_of_continuousAt + - frontier_Icc + - Monotone.map_iSup_of_continuousAt + - mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset + - nhdsWithin_Iic_eq' + - mem_nhdsWithin_Iic_iff_exists_Icc_subset + - SecondCountableTopology.of_separableSpace_orderTopology + - closure_Ioi' + - frontier_Ico + - induced_topology_le_preorder + - exists_seq_tendsto_sSup + - exists_seq_strictMono_tendsto' + - frontier_Ici' + - mem_nhdsWithin_Ici_iff_exists_Ico_subset' + - countable_of_isolated_right' + - comap_coe_nhdsWithin_Ioi_of_Ioo_subset + - nhds_basis_Ioo_pos + - StrictMono.induced_topology_eq_preorder + - nhdsWithin_Ioi_eq_bot_iff + - nhdsWithin_Iic_basis' + - tendsto_order_unbounded + - tendsto_order + - map_coe_atTop_of_Ioo_subset + - Antitone.map_iInf_of_continuousAt' + - Ioc_mem_nhds_iff + - Ico_subset_closure_interior + - countable_image_lt_image_Ioi + - nhds_basis_zero_abs_sub_lt + - LinearOrderedAddCommGroup.tendsto_nhds + - nhds_top_basis + - TFAE_mem_nhdsWithin_Ici + - frontier_Iic' + - Monotone.tendsto_nhdsWithin_Iio + - exists_Icc_mem_subset_of_mem_nhdsWithin_Iic + - Filter.Tendsto.atTop_add + - IsLUB.exists_seq_monotone_tendsto + - pi_Ioo_mem_nhds + - induced_topology_eq_preorder + - Dense.topology_eq_generateFrom + - countable_setOf_covBy_right + - dense_of_exists_between + - isOpen_iff_generate_intervals + - isLUB_of_mem_closure + - mem_nhdsWithin_Iio_iff_exists_Ico_subset + - IsLUB.frequently_mem + - frontier_Ioi' + - Antitone.map_iSup_of_continuousAt' + - TFAE_mem_nhdsWithin_Ioi + - mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset + - exists_Icc_mem_subset_of_mem_nhdsWithin_Ici + - nhds_order_unbounded + - nhdsWithin_Ici_eq' + - interior_Ioc + - frontier_Ioc + - countable_of_isolated_left' + - closure_Ioc + - interior_Icc + - atBot_le_nhds_bot + - tendsto_Ioo_atTop + - nhdsWithin_Iio_neBot' + - mem_nhdsWithin_Ioi_iff_exists_Ioc_subset + - Ico_mem_nhds_iff + - Monotone.map_sSup_of_continuousAt + - tendsto_comp_coe_Ioo_atTop + - nhds_basis_abs_sub_lt + - nhds_eq_order + - tendsto_comp_coe_Iio_atTop + - exists_countable_dense_no_bot_top + - tendsto_Ioo_atBot + - pi_Ioc_mem_nhds + - Monotone.map_iSup_of_continuousAt' + - exists_seq_strictMono_tendsto + - countable_setOf_covBy_left + - frontier_Ioo + - nhds_basis_Ioo' + - tendsto_Iio_atTop + - TFAE_mem_nhdsWithin_Iio + - orderTopology_of_nhds_abs + - closure_Ico + - comap_coe_nhdsWithin_Iio_of_Ioo_subset + - IsLUB.mem_upperBounds_of_tendsto + - nhds_bot_order + - exists_Icc_mem_subset_of_mem_nhds + - path: Mathlib/Topology/FiberBundle/Basic.lean + theorems: + - FiberBundleCore.mem_localTrivAt_baseSet + - FiberPrebundle.isOpen_target_of_mem_pretrivializationAtlas_inter + - FiberBundleCore.mem_trivChange_source + - FiberBundleCore.open_source' + - FiberBundleCore.localTrivAt_apply + - FiberPrebundle.continuous_symm_of_mem_pretrivializationAtlas + - FiberPrebundle.continuous_totalSpaceMk + - FiberBundle.exists_trivialization_Icc_subset + - FiberBundleCore.continuous_const_section + - FiberBundle.totalSpaceMk_closedEmbedding + - FiberPrebundle.continuous_proj + - FiberPrebundle.continuousOn_of_comp_right + - FiberBundleCore.mem_localTrivAsPartialEquiv_target + - FiberPrebundle.isOpen_source + - FiberBundleCore.mk_mem_localTrivAt_source + - FiberBundleCore.localTrivAsPartialEquiv_trans + - FiberPrebundle.inducing_totalSpaceMk_of_inducing_comp + - FiberPrebundle.mem_pretrivializationAt_source + - path: Mathlib/Topology/Connected/PathConnected.lean + theorems: + - Path.truncate_zero_one + - locPathConnected_of_isOpen + - Path.reparam_id + - pathComponent_subset_component + - Path.truncate_zero_zero + - JoinedIn.trans + - Path.trans_apply + - IsPathConnected.exists_path_through_family' + - Path.trans_pi_eq_pi_trans + - Path.truncate_range + - Path.refl_symm + - Path.map_id + - Path.trans_range + - Path.refl_reparam + - Path.refl_trans_refl + - IsPathConnected.image' + - JoinedIn.joined_subtype + - Path.symm_symm + - IsPathConnected.exists_path_through_family + - Path.ext + - PathConnectedSpace.exists_path_through_family + - joinedIn_iff_joined + - Path.trans_prod_eq_prod_trans + - pathComponent_congr + - IsPathConnected.preimage_coe + - pathConnectedSpace_iff_zerothHomotopy + - isPathConnected_iff_eq + - Path.continuous_uncurry_extend_of_continuous_family + - IsPathConnected.isConnected + - IsPathConnected.union + - joinedIn_univ + - isPathConnected_singleton + - Continuous.path_trans + - Path.extend_zero + - Path.extend_one + - Function.Surjective.pathConnectedSpace + - Path.truncate_const_continuous_family + - Path.refl_range + - locPathConnected_of_bases + - isPathConnected_iff_pathConnectedSpace + - pathConnectedSpace_iff_univ + - Path.trans_continuous_family + - JoinedIn.mem + - Path.truncate_self + - Path.trans_symm + - pathComponentIn_univ + - Path.range_reparam + - pathConnectedSpace_iff_eq + - Specializes.joinedIn + - Inducing.isPathConnected_iff + - isPathConnected_range + - pathConnectedSpace_iff_connectedSpace + - IsOpen.isConnected_iff_isPathConnected + - Homeomorph.isPathConnected_preimage + - Path.map_map + - Path.truncate_one_one + - Path.map_coe + - JoinedIn.symm + - Path.symm_range + - path: Mathlib/Algebra/Algebra/Spectrum.lean + theorems: + - spectrum.star_mem_resolventSet_iff + - spectrum.zero_not_mem_iff + - spectrum.resolventSet_of_subsingleton + - spectrum.zero_mem_resolventSet_of_unit + - spectrum.add_mem_iff + - spectrum.unit_mem_mul_iff_mem_swap_mul + - spectrum.sub_singleton_eq + - spectrum.one_eq + - spectrum.mem_resolventSet_of_left_right_inverse + - SpectrumRestricts.image + - spectrum.map_inv + - spectrum.singleton_sub_eq + - SpectrumRestricts.comp + - spectrum.algebraMap_mem_iff + - spectrum.inv_mem_resolventSet + - spectrum.units_smul_resolvent_self + - spectrum.singleton_add_eq + - spectrum.nonzero_mul_eq_swap_mul + - spectrum.zero_eq + - spectrum.smul_eq_smul + - spectrum.units_smul_resolvent + - spectrum.scalar_eq + - AlgHom.mem_resolventSet_apply + - AlgHom.apply_mem_spectrum + - spectrum.not_mem_iff + - spectrum.zero_mem_iff + - SpectrumRestricts.algebraMap_image + - AlgEquiv.spectrum_eq + - spectrum.neg_eq + - spectrum.smul_mem_smul_iff + - SpectrumRestricts.of_subset_range_algebraMap + - spectrum.of_subsingleton + - path: Mathlib/Data/Real/EReal.lean + theorems: + - EReal.preimage_coe_Ioo_top + - EReal.toReal_le_toReal + - EReal.toReal_mul + - EReal.coe_coe_sign + - EReal.coe_ennreal_pos + - EReal.image_coe_Ioi + - "EReal.induction\u2082" + - EReal.eq_bot_iff_forall_lt + - EReal.coe_ennreal_top_mul + - EReal.toReal_neg + - EReal.le_coe_toReal + - EReal.abs_neg + - EReal.preimage_coe_Iio_top + - EReal.bot_mul_of_pos + - EReal.preimage_coe_Ioo_bot + - EReal.abs_zero + - EReal.coe_ennreal_add + - EReal.range_coe_eq_Ioo + - EReal.image_coe_Ico + - EReal.sign_mul_abs + - EReal.preimage_coe_Icc + - EReal.toReal_sub + - EReal.bot_lt_add_iff + - EReal.add_lt_add_of_lt_of_le' + - EReal.sign_neg + - EReal.toReal_add + - EReal.top_mul_of_neg + - EReal.image_coe_Ici + - EReal.le_iff_sign + - EReal.coe_ennreal_mul + - EReal.sign_eq_and_abs_eq_iff_eq + - EReal.preimage_coe_Ico + - EReal.preimage_coe_Ioc + - EReal.eq_top_iff_forall_lt + - EReal.addLECancellable_coe + - EReal.neg_le + - EReal.preimage_coe_Ioi + - EReal.range_coe + - EReal.preimage_coe_Iic + - EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal + - EReal.mul_comm + - EReal.bot_mul_of_neg + - EReal.preimage_coe_Ioi_bot + - EReal.image_coe_Ioo + - EReal.neg_lt_iff_neg_lt + - EReal.coe_ennreal_eq_zero + - EReal.sign_coe + - EReal.abs_eq_zero_iff + - EReal.coe_ennreal_eq_one + - EReal.preimage_coe_Ioo + - EReal.abs_mul + - EReal.coe_abs + - EReal.exists_rat_btwn_of_lt + - EReal.sign_mul + - EReal.image_coe_Ioc + - EReal.image_coe_Iic + - EReal.abs_mul_sign + - EReal.le_neg_of_le_neg + - EReal.image_coe_Icc + - EReal.coe_toReal_le + - EReal.preimage_coe_Ico_top + - EReal.preimage_coe_Ici + - EReal.preimage_coe_Ioo_bot_top + - EReal.preimage_coe_Ioc_bot + - EReal.preimage_coe_Iio + - EReal.neg_mul + - EReal.add_lt_add + - EReal.add_lt_top + - EReal.image_coe_Iio + - EReal.top_mul_of_pos + - EReal.coe_toReal + - path: Mathlib/Data/Nat/Choose/Basic.lean + theorems: + - Nat.add_choose_mul_factorial_mul_factorial + - Nat.choose_eq_zero_of_lt + - Nat.choose_eq_descFactorial_div_factorial + - Nat.factorial_mul_factorial_dvd_factorial + - Nat.multichoose_zero_succ + - Nat.ascFactorial_eq_factorial_mul_choose + - Nat.triangle_succ + - Nat.choose_pos + - Nat.choose_succ_self_right + - Nat.choose_mul_factorial_mul_factorial + - Nat.choose_mul + - Nat.ascFactorial_eq_factorial_mul_choose' + - Nat.choose_zero_right + - Nat.choose_symm + - Nat.multichoose_zero_right + - Nat.multichoose_one + - Nat.choose_le_succ_of_lt_half_left + - Nat.choose_mul_succ_eq + - Nat.choose_eq_asc_factorial_div_factorial + - Nat.choose_eq_factorial_div_factorial + - Nat.factorial_mul_factorial_dvd_factorial_add + - Nat.choose_eq_asc_factorial_div_factorial' + - Nat.add_choose + - Nat.choose_le_succ + - Nat.choose_two_right + - Nat.choose_symm_of_eq_add + - Nat.succ_mul_choose_eq + - Nat.choose_symm_half + - Nat.choose_succ_right_eq + - Nat.descFactorial_eq_factorial_mul_choose + - Nat.choose_le_middle + - Nat.multichoose_succ_succ + - Nat.multichoose_one_right + - Nat.multichoose_two + - Nat.choose_self + - Nat.choose_le_add + - Nat.multichoose_eq + - path: Mathlib/Analysis/BoxIntegral/Integrability.lean + theorems: + - MeasureTheory.SimpleFunc.hasBoxIntegral + - MeasureTheory.IntegrableOn.hasBoxIntegral + - BoxIntegral.HasIntegral.of_aeEq_zero + - BoxIntegral.HasIntegral.congr_ae + - MeasureTheory.ContinuousOn.hasBoxIntegral + - BoxIntegral.hasIntegralIndicatorConst + - path: Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean + theorems: + - CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_fst + - CategoryTheory.Limits.coprod.map_desc + - CategoryTheory.Limits.prod.diag_map_fst_snd + - CategoryTheory.Limits.prod.lift_map + - CategoryTheory.Limits.prod.map_id_id + - CategoryTheory.Limits.coprod.desc_comp_inl_comp_inr + - CategoryTheory.Limits.coprod.pentagon + - CategoryTheory.Limits.coprod.associator_naturality + - CategoryTheory.Limits.coprod.map_map + - CategoryTheory.Limits.BinaryCofan.isColimit_iff_isIso_inl + - CategoryTheory.Limits.coprod.map_id_comp + - CategoryTheory.Limits.coprod.triangle + - CategoryTheory.Limits.prod.pentagon + - CategoryTheory.Limits.coprod.diag_comp + - CategoryTheory.Limits.coprodComparison_natural + - CategoryTheory.Limits.coprod.map_comp_id + - CategoryTheory.Limits.prod.map_comp_id + - CategoryTheory.Limits.prod.associator_naturality + - CategoryTheory.Limits.inv_prodComparison_map_fst + - CategoryTheory.Limits.coprod.map_swap + - CategoryTheory.Limits.prod.lift_fst_comp_snd_comp + - CategoryTheory.Limits.prod.map_swap + - CategoryTheory.Limits.coprod.map_id_id + - CategoryTheory.Limits.coprod.map_comp_inl_inr_codiag + - CategoryTheory.Limits.prod.map_id_comp + - CategoryTheory.Limits.coprod.desc_comp_assoc + - CategoryTheory.Limits.map_inr_inv_coprodComparison + - CategoryTheory.Limits.prod.triangle + - CategoryTheory.Limits.coprod.desc_inl_inr + - CategoryTheory.Limits.prodComparison_natural + - CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_snd + - CategoryTheory.Limits.prod.map_map + - CategoryTheory.Limits.coprod.map_codiag + - CategoryTheory.Limits.coprodComparison_inv_natural + - CategoryTheory.Limits.coprod.desc_comp + - CategoryTheory.Limits.prod.lift_fst_snd + - CategoryTheory.Limits.prod.diag_map_fst_snd_comp + - CategoryTheory.Limits.inv_prodComparison_map_snd + - CategoryTheory.Limits.prodComparison_inv_natural + - CategoryTheory.Limits.prod.diag_map + - CategoryTheory.Limits.map_inl_inv_coprodComparison + - CategoryTheory.Limits.braid_natural + - CategoryTheory.Limits.prod_rightUnitor_inv_naturality + - CategoryTheory.Limits.coprod.map_inl_inr_codiag + - CategoryTheory.Limits.prod.comp_diag + - CategoryTheory.Limits.prod.leftUnitor_inv_naturality + - CategoryTheory.Limits.BinaryCofan.isColimit_iff_isIso_inr + - path: Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean + theorems: + - "Polynomial.eval\u2082_one_cyclotomic_prime_pow" + - Polynomial.eval_one_cyclotomic_not_prime_pow + - Polynomial.sub_one_pow_totient_lt_cyclotomic_eval + - Polynomial.cyclotomic_eval_lt_add_one_pow_totient + - "Polynomial.eval\u2082_one_cyclotomic_prime" + - Polynomial.sub_one_pow_totient_le_cyclotomic_eval + - Polynomial.eval_one_cyclotomic_prime_pow + - Polynomial.cyclotomic_eval_le_add_one_pow_totient + - Polynomial.eval_one_cyclotomic_prime + - Polynomial.sub_one_pow_totient_lt_natAbs_cyclotomic_eval + - Polynomial.cyclotomic_neg_one_pos + - Polynomial.cyclotomic_pos + - path: Mathlib/Combinatorics/SimpleGraph/Connectivity.lean + theorems: + - SimpleGraph.Walk.IsCircuit.rotate + - SimpleGraph.isBridge_iff_adj_and_forall_walk_mem_edges + - SimpleGraph.Path.loop_eq + - SimpleGraph.Walk.toSubgraph_cons_nil_eq_subgraphOfAdj + - SimpleGraph.Walk.mem_verts_toSubgraph + - SimpleGraph.Walk.transfer_append + - SimpleGraph.Walk.IsPath.of_map + - SimpleGraph.Walk.support_takeUntil_subset + - SimpleGraph.Walk.mem_support_iff + - SimpleGraph.Walk.length_transfer + - SimpleGraph.Walk.length_bypass_le + - SimpleGraph.Walk.length_tail_add_one + - SimpleGraph.Walk.isCycle_copy + - SimpleGraph.Walk.takeUntil_copy + - SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts + - SimpleGraph.Walk.coe_support + - SimpleGraph.Walk.map_map + - SimpleGraph.Walk.IsTrail.of_append_left + - SimpleGraph.Walk.snd_mem_support_of_mem_edges + - SimpleGraph.Walk.concatRec_concat + - SimpleGraph.Walk.adj_getVert_succ + - SimpleGraph.Walk.support_nonempty + - SimpleGraph.Walk.map_isCycle_iff_of_injective + - SimpleGraph.Walk.mem_support_append_iff + - SimpleGraph.Path.not_mem_edges_of_loop + - SimpleGraph.Walk.support_rotate + - SimpleGraph.Walk.count_support_takeUntil_eq_one + - SimpleGraph.Walk.map_copy + - SimpleGraph.Walk.IsTrail.rotate + - SimpleGraph.Walk.darts_map + - SimpleGraph.Walk.count_edges_takeUntil_le_one + - SimpleGraph.Walk.IsPath.reverse + - SimpleGraph.Walk.reverse_reverse + - SimpleGraph.Walk.append_nil + - SimpleGraph.top_preconnected + - SimpleGraph.Walk.IsTrail.dropUntil + - SimpleGraph.reachable_iff_reflTransGen + - SimpleGraph.set_walk_length_toFinset_eq + - SimpleGraph.reachable_iff_exists_finsetWalkLength_nonempty + - SimpleGraph.Walk.toSubgraph_rotate + - SimpleGraph.Walk.getVert_of_length_le + - SimpleGraph.Walk.edges_reverse + - SimpleGraph.Walk.support_eq_cons + - SimpleGraph.Walk.cons_isTrail_iff + - SimpleGraph.Walk.darts_nodup_of_support_nodup + - SimpleGraph.Walk.reverseAux_eq_reverse_append + - SimpleGraph.Walk.darts_bypass_subset + - SimpleGraph.Walk.exists_boundary_dart + - SimpleGraph.Walk.reverse_append + - SimpleGraph.Walk.reverse_transfer + - SimpleGraph.Walk.map_isPath_of_injective + - SimpleGraph.Walk.map_snd_darts + - SimpleGraph.Walk.map_eq_of_eq + - SimpleGraph.Walk.cons_map_snd_darts + - SimpleGraph.Walk.edges_concat + - SimpleGraph.Walk.exists_cons_eq_concat + - SimpleGraph.Walk.darts_takeUntil_subset + - SimpleGraph.Walk.fst_mem_support_of_mem_edges + - SimpleGraph.Walk.append_reverseAux + - SimpleGraph.Walk.support_dropUntil_subset + - SimpleGraph.Walk.IsCycle.ne_bot + - SimpleGraph.Walk.mem_support_nil_iff + - SimpleGraph.Walk.nil_iff_support_eq + - SimpleGraph.Walk.mem_support_iff_exists_append + - SimpleGraph.Walk.end_mem_support + - SimpleGraph.Walk.chain_dartAdj_darts + - SimpleGraph.Walk.toSubgraph_append + - SimpleGraph.Walk.cons_isCycle_iff + - SimpleGraph.Walk.reverseAux_append + - SimpleGraph.Walk.darts_dropUntil_subset + - SimpleGraph.Walk.IsPath.dropUntil + - SimpleGraph.Walk.IsTrail.takeUntil + - SimpleGraph.Walk.concat_append + - SimpleGraph.Path.mk'_mem_edges_singleton + - SimpleGraph.Walk.length_reverse + - SimpleGraph.Walk.mem_darts_reverse + - SimpleGraph.Walk.transfer_transfer + - SimpleGraph.Walk.support_transfer + - SimpleGraph.Walk.IsPath.length_lt + - SimpleGraph.Walk.transfer_eq_map_of_le + - SimpleGraph.Walk.toSubgraph_reverse + - SimpleGraph.Walk.isTrail_copy + - SimpleGraph.Walk.IsTrail.of_cons + - SimpleGraph.Walk.length_append + - SimpleGraph.Walk.toSubgraph_map + - SimpleGraph.Walk.reverse_map + - SimpleGraph.Walk.isPath_copy + - SimpleGraph.Walk.map_toDeleteEdges_eq + - SimpleGraph.Path.map_injective + - SimpleGraph.Reachable.elim_path + - SimpleGraph.Walk.IsPath.takeUntil + - SimpleGraph.Walk.IsPath.transfer + - SimpleGraph.Walk.reverse_isTrail_iff + - SimpleGraph.Walk.map_fst_darts + - SimpleGraph.ConnectedComponent.supp_injective + - SimpleGraph.Walk.copy_cons + - SimpleGraph.Walk.darts_reverse + - SimpleGraph.Walk.nil_copy + - SimpleGraph.Walk.toDeleteEdges_copy + - SimpleGraph.Walk.rotate_darts + - SimpleGraph.Walk.mem_edges_toSubgraph + - SimpleGraph.Walk.reverse_concat + - SimpleGraph.ConnectedComponent.iso_image_comp_eq_map_iff_eq_comp + - SimpleGraph.ConnectedComponent.map_id + - SimpleGraph.Walk.length_support + - SimpleGraph.Walk.support_bypass_subset + - SimpleGraph.Walk.IsTrail.reverse + - SimpleGraph.Walk.finite_neighborSet_toSubgraph + - SimpleGraph.Walk.end_mem_tail_support_of_ne + - SimpleGraph.Walk.subset_support_append_right + - SimpleGraph.Walk.length_reverseAux + - SimpleGraph.Walk.take_spec + - SimpleGraph.Walk.end_mem_verts_toSubgraph + - SimpleGraph.isBridge_iff_adj_and_forall_cycle_not_mem + - SimpleGraph.Walk.reverse_cons + - SimpleGraph.Path.cons_isCycle + - SimpleGraph.set_walk_self_length_zero_eq + - SimpleGraph.Walk.cons_copy + - SimpleGraph.Walk.support_copy + - SimpleGraph.Walk.transfer_self + - SimpleGraph.Walk.map_isTrail_iff_of_injective + - SimpleGraph.Walk.dart_snd_mem_support_of_mem_darts + - SimpleGraph.Walk.map_fst_darts_append + - SimpleGraph.Iso.connectedComponentEquiv_refl + - SimpleGraph.ConnectedComponent.iso_inv_image_comp_eq_iff_eq_map + - SimpleGraph.Walk.edges_nodup_of_support_nodup + - SimpleGraph.set_walk_length_zero_eq_of_ne + - SimpleGraph.Walk.copy_nil + - SimpleGraph.Walk.bypass_isPath + - SimpleGraph.Walk.edges_map + - SimpleGraph.Walk.support_ne_nil + - SimpleGraph.Walk.map_eq_nil_iff + - SimpleGraph.Walk.cons_support_tail + - SimpleGraph.set_walk_length_succ_eq + - SimpleGraph.Walk.map_injective_of_injective + - SimpleGraph.coe_finsetWalkLength_eq + - SimpleGraph.Walk.edges_subset_edgeSet + - SimpleGraph.ConnectedComponent.map_comp + - SimpleGraph.adj_and_reachable_delete_edges_iff_exists_cycle + - SimpleGraph.Walk.support_map + - SimpleGraph.Walk.bypass_copy + - SimpleGraph.Walk.darts_concat + - SimpleGraph.Walk.copy_copy + - SimpleGraph.Walk.length_copy + - SimpleGraph.Walk.concat_ne_nil + - SimpleGraph.Iso.connectedComponentEquiv_symm + - SimpleGraph.Walk.not_nil_iff + - SimpleGraph.Walk.edges_append + - SimpleGraph.Walk.reverse_copy + - SimpleGraph.connected_iff_exists_forall_reachable + - SimpleGraph.Walk.tail_support_append + - SimpleGraph.reachable_deleteEdges_iff_exists_cycle.aux + - SimpleGraph.Walk.dropUntil_copy + - SimpleGraph.Walk.IsTrail.of_append_right + - SimpleGraph.Walk.IsCycle.transfer + - SimpleGraph.Walk.append_copy_copy + - SimpleGraph.Iso.connectedComponentEquiv_trans + - SimpleGraph.Walk.isCircuit_copy + - SimpleGraph.Iso.symm_apply_reachable + - SimpleGraph.Walk.length_map + - SimpleGraph.Walk.coe_support_append' + - SimpleGraph.Walk.IsPath.nil + - SimpleGraph.Walk.IsPath.of_append_left + - SimpleGraph.Walk.length_takeUntil_le + - SimpleGraph.Walk.subset_support_append_left + - SimpleGraph.Walk.map_append + - SimpleGraph.Walk.isPath_reverse_iff + - SimpleGraph.Walk.length_darts + - SimpleGraph.Walk.coe_support_append + - SimpleGraph.Walk.length_edges + - SimpleGraph.Walk.start_mem_verts_toSubgraph + - SimpleGraph.Walk.getVert_zero + - SimpleGraph.Walk.length_eq_zero_iff + - SimpleGraph.Preconnected.set_univ_walk_nonempty + - SimpleGraph.Walk.darts_append + - SimpleGraph.Walk.cons_isPath_iff + - SimpleGraph.Walk.IsCycle.rotate + - SimpleGraph.Walk.length_dropUntil_le + - SimpleGraph.Walk.IsTrail.nil + - SimpleGraph.Walk.IsPath.tail + - SimpleGraph.Walk.mem_tail_support_append_iff + - SimpleGraph.Walk.IsPath.of_append_right + - SimpleGraph.Walk.exists_length_eq_zero_iff + - SimpleGraph.Walk.support_reverse + - SimpleGraph.Walk.IsPath.of_cons + - SimpleGraph.Walk.edges_transfer + - SimpleGraph.Walk.darts_copy + - SimpleGraph.Walk.map_id + - SimpleGraph.Walk.toSubgraph_le_induce_support + - SimpleGraph.card_set_walk_length_eq + - SimpleGraph.Walk.nil_iff_length_eq + - SimpleGraph.Walk.append_assoc + - SimpleGraph.Walk.start_mem_support + - SimpleGraph.Walk.chain'_dartAdj_darts + - SimpleGraph.Walk.isPath_iff_eq_nil + - SimpleGraph.Walk.concat_inj + - SimpleGraph.Walk.support_append + - path: Mathlib/Order/LocallyFinite.lean + theorems: + - Finset.map_subtype_embedding_Iic + - Multiset.mem_Ioi + - Set.toFinset_Ioc + - Set.toFinset_Icc + - Multiset.mem_Iio + - Ico_toDual + - Multiset.mem_Icc + - Ico_ofDual + - WithTop.aux + - Multiset.mem_Iic + - Set.toFinset_Iic + - Finset.map_subtype_embedding_Ioi + - Icc_toDual + - Ioo_ofDual + - Finset.map_subtype_embedding_Ico + - Set.toFinset_Iio + - Set.finite_iff_bddBelow_bddAbove + - Finset.map_subtype_embedding_Ioo + - Multiset.mem_Ici + - Finset.map_subtype_embedding_Ioc + - Multiset.mem_Ico + - Icc_ofDual + - Set.toFinset_Ico + - Set.toFinset_Ici + - Set.toFinset_Ioi + - Multiset.mem_Ioc + - Ioc_toDual + - Ioo_toDual + - Set.toFinset_Ioo + - Ioc_ofDual + - Set.finite_iff_bddAbove + - Finset.map_subtype_embedding_Iio + - Finset.map_subtype_embedding_Icc + - path: Mathlib/Combinatorics/Composition.lean + theorems: + - Composition.sigma_eq_iff_blocks_eq + - CompositionAsSet.blocks_partial_sum + - Composition.ones_sizeUpTo + - List.get_splitWrtCompositionAux + - Composition.sizeUpTo_ofLength_le + - CompositionAsSet.toComposition_length + - Composition.mem_range_embedding + - Composition.sizeUpTo_le + - Composition.length_le + - Composition.sizeUpTo_succ + - Composition.disjoint_range + - Composition.orderEmbOfFin_boundaries + - Composition.index_exists + - Composition.ne_single_iff + - Composition.length_pos_of_pos + - CompositionAsSet.blocks_sum + - Composition.sizeUpTo_zero + - Composition.sizeUpTo_strict_mono + - CompositionAsSet.length_lt_card_boundaries + - Composition.blocksFun_congr + - Composition.eq_ones_iff_le_length + - Composition.eq_single_iff_length + - Composition.toCompositionAsSet_length + - Composition.ones_blocksFun + - Composition.ne_ones_iff + - CompositionAsSet.toComposition_boundaries + - Composition.mem_range_embedding_iff' + - Composition.single_embedding + - List.splitWrtComposition_join + - Composition.toCompositionAsSet_blocks + - Composition.boundary_zero + - Composition.index_embedding + - List.length_splitWrtCompositionAux + - List.join_splitWrtCompositionAux + - composition_card + - Composition.sum_blocksFun + - Composition.embedding_comp_inv + - CompositionAsSet.boundary_zero + - List.length_pos_of_mem_splitWrtComposition + - Composition.single_blocksFun + - Composition.card_boundaries_eq_succ_length + - List.sum_take_map_length_splitWrtComposition + - compositionAsSet_card + - Composition.eq_ones_iff_length + - Composition.mem_range_embedding_iff + - Composition.invEmbedding_comp + - Composition.sizeUpTo_index_le + - Composition.eq_ones_iff + - List.splitWrtCompositionAux_cons + - Composition.boundary_last + - Composition.ones_embedding + - path: Mathlib/Logic/Function/Basic.lean + theorems: + - Function.update_idem + - Function.hfunext + - Function.factorsThrough_iff + - Function.LeftInverse.comp + - Function.LeftInverse.eq_rec_eq + - Function.Bijective.comp_right + - Function.invFun_eq + - Function.LeftInverse.eq_rightInverse + - Function.LeftInverse.cast_eq + - Function.Bijective.existsUnique_iff + - Function.Injective.beq_eq + - cast_bijective + - Function.Injective.dite + - forall_existsUnique_iff' + - forall_existsUnique_iff + - Symmetric.forall_existsUnique_iff + - Symmetric.forall_existsUnique_iff' + - Function.update_apply + - eq_mp_bijective + - Function.update_eq_self_iff + - "Function.apply_update\u2082" + - Function.invFun_eq_of_injective_of_rightInverse + - Function.exists_update_iff + - Function.extend_apply' + - Function.forall_update_iff + - Function.Involutive.ite_not + - Function.update_comm + - Function.partialInv_of_injective + - Function.update_injective + - Function.eq_update_self_iff + - Function.FactorsThrough.extend_apply + - eq_mpr_bijective + - Function.Injective.of_comp_right + - Function.extend_injective + - Function.funext_iff_of_subsingleton + - Function.not_surjective_Type + - Function.pred_update + - Function.apply_update + - Function.symmetric_apply_eq_iff + - path: Mathlib/Data/Sigma/Basic.lean + theorems: + - Function.Surjective.sigma_map + - Function.eq_of_sigmaMk_comp + - Sigma.ext_iff + - Sigma.mk.inj_iff + - Function.Injective.sigma_map + - PSigma.ext_iff + - Sigma.fst_injective + - path: Mathlib/Topology/Sequences.lean + theorems: + - FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto + - tendsto_nhds_iff_seq_tendsto + - IsSeqCompact.totallyBounded + - mem_closure_iff_seq_limit + - UniformSpace.compactSpace_iff_seqCompactSpace + - IsSeqCompact.isComplete + - path: Mathlib/MeasureTheory/MeasurableSpace/Defs.lean + theorems: + - MeasurableSet.sUnion + - MeasurableSpace.generateFrom_singleton_univ + - MeasurableSpace.forall_generateFrom_mem_iff_mem_iff + - MeasurableSet.inter + - MeasurableSet.union + - MeasurableSpace.generateFrom_induction + - MeasurableSet.iUnion + - MeasurableSpace.measurableSet_sInf + - MeasurableSpace.measurableSet_iSup + - MeasurableSet.iInter + - MeasurableSpace.measurableSet_sSup + - MeasurableSet.ite' + - MeasurableSpace.generateFrom_empty + - MeasurableSpace.measurableSet_injective + - Set.Countable.measurableSet + - MeasurableSpace.measurableSet_iInf + - MeasurableSet.congr + - MeasurableSpace.generateFrom_insert_empty + - MeasurableSet.cond + - MeasurableSet.himp + - MeasurableSet.biInter + - MeasurableSet.const + - MeasurableSpace.generateFrom_singleton_empty + - MeasurableSet.biUnion + - measurableSet_insert + - MeasurableSet.sInter + - path: Mathlib/Logic/Embedding/Basic.lean + theorems: + - Equiv.toEmbedding_injective + - Function.Embedding.setValue_eq + - Function.Embedding.equiv_toEmbedding_trans_symm_toEmbedding + - Function.Embedding.equiv_symm_toEmbedding_trans_toEmbedding + - path: Mathlib/Data/Polynomial/Monic.lean + theorems: + - Polynomial.Monic.add_of_left + - Polynomial.monic_X_pow_add_C + - Polynomial.Monic.not_dvd_of_natDegree_lt + - Polynomial.Monic.natDegree_mul' + - Polynomial.Monic.natDegree_pow + - Polynomial.isUnit_leadingCoeff_mul_left_eq_zero_iff + - Polynomial.Monic.mul_natDegree_lt_iff + - Polynomial.degree_smul_of_smul_regular + - Polynomial.monic_X_pow_sub + - Polynomial.monic_X_pow_sub_C + - Polynomial.Monic.sub_of_right + - Polynomial.monic_zero_iff_subsingleton' + - Polynomial.Monic.of_mul_monic_left + - Polynomial.Monic.nextCoeff_mul + - Polynomial.Monic.pow + - Polynomial.Monic.add_of_right + - Polynomial.Monic.mul_left_ne_zero + - Polynomial.Monic.of_mul_monic_right + - Polynomial.Monic.natDegree_mul + - Polynomial.natDegree_smul_of_smul_regular + - Polynomial.leadingCoeff_smul_of_smul_regular + - Polynomial.monic_of_injective + - Polynomial.monic_of_isUnit_leadingCoeff_inv_smul + - Polynomial.monic_mul_C_of_leadingCoeff_mul_eq_one + - Polynomial.Monic.as_sum + - Polynomial.Monic.mul_right_eq_zero_iff + - Polynomial.Monic.natDegree_mul_comm + - Polynomial.Monic.mul_right_ne_zero + - Polynomial.Monic.mul_left_eq_zero_iff + - Polynomial.degree_map_eq_of_injective + - Polynomial.Monic.eq_one_of_isUnit + - Polynomial.Monic.natDegree_map + - Polynomial.monic_C_mul_of_mul_leadingCoeff_eq_one + - Polynomial.Monic.nextCoeff_multiset_prod + - Polynomial.Monic.mul + - Polynomial.natDegree_pow_X_add_C + - Polynomial.isUnit_leadingCoeff_mul_right_eq_zero_iff + - Polynomial.leadingCoeff_of_injective + - Polynomial.Monic.degree_mul_comm + - Polynomial.Monic.sub_of_left + - Polynomial.Monic.map + - Polynomial.nextCoeff_map + - Polynomial.Monic.natDegree_eq_zero_iff_eq_one + - Polynomial.eq_of_monic_of_associated + - Polynomial.not_isUnit_X_pow_sub_one + - Polynomial.monic_of_degree_le + - Polynomial.Monic.eq_one_of_map_eq_one + - Polynomial.monic_X_sub_C + - Polynomial.Monic.nextCoeff_pow + - Polynomial.Monic.degree_map + - Polynomial.Monic.degree_le_zero_iff_eq_one + - Polynomial.leadingCoeff_map' + - Polynomial.monic_multiset_prod_of_monic + - Polynomial.Monic.isRegular + - Polynomial.ne_zero_of_ne_zero_of_monic + - Polynomial.monic_X_pow_add + - path: Mathlib/NumberTheory/NumberField/Embeddings.lean + theorems: + - NumberField.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq + - NumberField.InfinitePlace.isReal_iff + - NumberField.InfinitePlace.even_finrank_of_not_isUnramified + - NumberField.InfinitePlace.isUnramified_iff_stabilizer_eq_bot + - NumberField.InfinitePlace.IsUnramified.comap_algHom + - NumberField.InfinitePlace.card_isUnramified_compl + - NumberField.ComplexEmbedding.IsReal.comp + - NumberField.InfinitePlace.card_add_two_mul_card_eq_rank + - NumberField.ComplexEmbedding.place_conjugate + - NumberField.InfinitePlace.one_le_of_lt_one + - NumberField.ComplexEmbedding.isReal_comp_iff + - NumberField.InfinitePlace.eq_iff_eq + - NumberField.InfinitePlace.not_isUnramified_iff + - NumberField.InfinitePlace.isUnramified_iff_card_stabilizer_eq_one + - NumberField.InfinitePlace.norm_embedding_eq + - NumberField.InfinitePlace.le_iff_le + - NumberField.InfinitePlace.comap_surjective + - NumberField.InfinitePlace.nat_card_stabilizer_eq_one_or_two + - NumberField.InfinitePlace.mult_comap_le + - NumberField.InfinitePlace.even_nat_card_aut_of_not_isUnramified + - NumberField.InfinitePlace.isReal_or_isComplex + - IsUnramifiedAtInfinitePlaces.bot + - NumberField.InfinitePlace.IsReal.comap + - NumberField.InfinitePlace.IsUnramified.of_restrictScalars + - NumberField.ComplexEmbedding.IsConj.isReal_comp + - NumberField.adjoin_eq_top_of_infinitePlace_lt + - NumberField.InfinitePlace.isComplex_smul_iff + - NumberField.Embeddings.finite_of_norm_le + - NumberField.InfinitePlace.isComplex_iff + - NumberField.InfinitePlace.sum_mult_eq + - NumberField.Embeddings.pow_eq_one_of_norm_eq_one + - NumberField.InfinitePlace.card_isUnramified + - NumberField.Embeddings.range_eval_eq_rootSet_minpoly + - NumberField.InfinitePlace.mk_conjugate_eq + - NumberField.InfinitePlace.nrComplexPlaces_eq_zero_of_finrank_eq_one + - NumberField.ComplexEmbedding.IsConj.isUnramified_mk_iff + - NumberField.InfinitePlace.card_complex_embeddings + - NumberField.InfinitePlace.IsUnramified.stabilizer_eq_bot + - NumberField.InfinitePlace.embedding_mk_eq_of_isReal + - NumberField.InfinitePlace.isReal_comap_iff + - NumberField.InfinitePlace.embedding_mk_eq + - NumberField.InfinitePlace.mk_eq_iff + - NumberField.InfinitePlace.prod_eq_abs_norm + - IsPrimitiveRoot.nrRealPlaces_eq_zero_of_two_lt + - NumberField.InfinitePlace.isUnramified_smul_iff + - NumberField.InfinitePlace.not_isReal_of_mk_isComplex + - NumberField.Embeddings.coeff_bdd_of_norm_le + - NumberField.InfinitePlace.card_filter_mk_eq + - NumberField.is_primitive_element_of_infinitePlace_lt + - NumberField.InfinitePlace.not_isComplex_iff_isReal + - NumberField.ComplexEmbedding.IsConj.symm + - NumberField.InfinitePlace.even_card_aut_of_not_isUnramifiedIn + - NumberField.ComplexEmbedding.IsReal.coe_embedding_apply + - NumberField.InfinitePlace.mem_orbit_iff + - NumberField.InfinitePlace.card_stabilizer + - NumberField.InfinitePlace.not_isUnramified_iff_card_stabilizer_eq_two + - NumberField.ComplexEmbedding.IsConj.coe_stabilzer_mk + - NumberField.InfinitePlace.not_isReal_iff_isComplex + - NumberField.InfinitePlace.isUnramified_iff + - NumberField.Embeddings.card + - NumberField.InfinitePlace.isUnramified_iff_mult_le + - NumberField.InfinitePlace.even_finrank_of_not_isUnramifiedIn + - NumberField.InfinitePlace.nrRealPlaces_eq_one_of_finrank_eq_one + - IsUnramifiedAtInfinitePlaces.card_infinitePlace + - NumberField.InfinitePlace.isUnramifiedIn_comap + - NumberField.InfinitePlace.mem_stabilizer_mk_iff + - path: Mathlib/Data/Complex/Order.lean + theorems: + - Complex.not_lt_iff + - Complex.monotone_ofReal + - Complex.neg_re_eq_abs + - Complex.re_eq_neg_abs + - Complex.real_lt_real + - Complex.eq_re_of_ofReal_le + - Complex.real_le_real + - Complex.not_le_iff + - Complex.re_eq_abs + - path: Mathlib/Analysis/Complex/AbsMax.lean + theorems: + - Complex.eventually_eq_of_isLocalMax_norm + - Complex.norm_eqOn_of_isPreconnected_of_isMaxOn + - Complex.eventually_eq_or_eq_zero_of_isLocalMin_norm + - Complex.isOpen_setOf_mem_nhds_and_isMaxOn_norm + - Complex.eqOn_of_isPreconnected_of_isMaxOn_norm + - "Complex.norm_max_aux\u2082" + - Complex.norm_le_of_forall_mem_frontier_norm_le + - Complex.norm_eqOn_closedBall_of_isMaxOn + - "Complex.norm_max_aux\u2083" + - Complex.norm_eventually_eq_of_isLocalMax + - "Complex.norm_max_aux\u2081" + - Complex.eqOn_closure_of_eqOn_frontier + - Complex.eq_of_isMaxOn_of_ball_subset + - path: Mathlib/Topology/UniformSpace/Matrix.lean + theorems: + - Matrix.uniformContinuous + - Matrix.uniformity + - path: Mathlib/Analysis/InnerProductSpace/Rayleigh.lean + theorems: + - ContinuousLinearMap.rayleigh_smul + - IsSelfAdjoint.linearly_dependent_of_isLocalExtrOn + - ContinuousLinearMap.iSup_rayleigh_eq_iSup_rayleigh_sphere + - IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn + - LinearMap.IsSymmetric.hasStrictFDerivAt_reApplyInnerSelf + - IsSelfAdjoint.hasEigenvector_of_isMinOn + - IsSelfAdjoint.hasEigenvector_of_isLocalExtrOn + - ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere + - ContinuousLinearMap.iInf_rayleigh_eq_iInf_rayleigh_sphere + - LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional + - IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn_real + - LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional + - path: Mathlib/LinearAlgebra/Eigenspace/Basic.lean + theorems: + - Module.End.mem_spectrum_of_hasEigenvalue + - Module.End.pos_finrank_generalizedEigenspace_of_hasEigenvalue + - Module.End.eigenspace_zero + - Module.End.generalizedEigenspace_restrict + - Module.End.mem_maximalGeneralizedEigenspace + - Module.End.exp_ne_zero_of_hasGeneralizedEigenvalue + - Module.End.iSup_generalizedEigenspace_le_smul + - Module.End.eigenspace_restrict_eq_bot + - Module.End.maximalGeneralizedEigenspace_eq + - Module.End.eigenspace_restrict_le_eigenspace + - Module.End.hasEigenvalue_iff_mem_spectrum + - Module.End.map_smul_of_iInf_generalizedEigenspace_ne_bot + - Module.End.generalized_eigenvec_disjoint_range_ker + - Submodule.inf_generalizedEigenspace + - Module.End.hasEigenvalue_of_hasGeneralizedEigenvalue + - Module.End.isNilpotent_restrict_iSup_sub_algebraMap + - Module.End.generalizedEigenspace_zero + - Module.End.eigenspace_div + - Module.End.map_add_of_iInf_generalizedEigenspace_ne_bot_of_commute + - Module.End.isNilpotent_restrict_sub_algebraMap + - Module.End.mapsTo_generalizedEigenspace_of_comm + - Module.End.independent_generalizedEigenspace + - Module.End.map_generalizedEigenrange_le + - Module.End.disjoint_iSup_generalizedEigenspace + - Module.End.mapsTo_iSup_generalizedEigenspace_of_comm + - Module.End.mem_eigenspace_iff + - Module.End.disjoint_generalizedEigenspace + - Module.End.hasEigenvalue_of_hasEigenvector + - Module.End.injOn_generalizedEigenspace + - Module.End.hasGeneralizedEigenvalue_of_hasGeneralizedEigenvalue_of_le + - Module.End.HasEigenvector.pow_apply + - path: Mathlib/RingTheory/SimpleModule.lean + theorems: + - isSimpleModule_self_iff_isUnit + - covBy_iff_quot_is_simple + - IsSemisimpleRing.ideal_eq_span_idempotent + - JordanHolderModule.second_iso + - isSimpleModule_iff_isAtom + - IsSimpleModule.annihilator_isMaximal + - isSimpleModule_iff_toSpanSingleton_surjective + - IsSemisimpleModule.annihilator_isRadical + - IsSemisimpleModule.sup + - IsSimpleModule.nontrivial + - isSemisimpleModule_biSup_of_isSemisimpleModule_submodule + - IsSimpleModule.toSpanSingleton_surjective + - IsSemisimpleModule.of_sSup_simples_eq_top + - LinearMap.isCoatom_ker_of_surjective + - isSimpleModule_iff_quot_maximal + - isSemisimpleModule_of_isSemisimpleModule_submodule' + - LinearMap.surjective_or_eq_zero + - IsSemisimpleModule.sSup_simples_eq_top + - LinearMap.injective_or_eq_zero + - IsSimpleModule.ker_toSpanSingleton_isMaximal + - isSemisimpleModule_of_isSemisimpleModule_submodule + - isSimpleModule_iff_isCoatom + - RingHom.isSemisimpleRing_of_surjective + - path: Mathlib/Data/Polynomial/Basic.lean + theorems: + - Polynomial.erase_same + - Polynomial.X_pow_eq_monomial + - Polynomial.monomial_one_right_eq_X_pow + - Polynomial.support_X_pow + - Polynomial.toFinsupp_C_mul_X_pow + - Polynomial.support_eq_empty + - Polynomial.mul_eq_sum_sum + - Polynomial.ofFinsupp_pow + - Polynomial.coeff_erase + - Polynomial.C_mul_X_eq_monomial + - Polynomial.sum_zero_index + - Polynomial.toFinsupp_pow + - Polynomial.monomial_mul_monomial + - Polynomial.ofFinsupp_neg + - Polynomial.ofFinsupp_mul + - Polynomial.rat_smul_eq_C_mul + - Polynomial.ofFinsupp_sub + - Polynomial.support_X + - Polynomial.not_mem_support_iff + - Polynomial.support_update_ne_zero + - Polynomial.coeff_monomial_succ + - Polynomial.eta + - Polynomial.card_support_eq_zero + - Polynomial.monomial_neg + - Polynomial.X_pow_mul + - Polynomial.support_C_mul_X + - Polynomial.X_mul_monomial + - Polynomial.toFinsupp_erase + - Polynomial.support_erase + - Polynomial.erase_monomial + - Polynomial.forall_eq_iff_forall_eq + - Polynomial.X_pow_mul_assoc + - Polynomial.ofFinsupp_eq_one + - Polynomial.sum_add' + - Polynomial.coeff_monomial + - Polynomial.update_zero_eq_erase + - Polynomial.X_pow_mul_monomial + - Polynomial.support_neg + - Polynomial.monomial_pow + - Polynomial.monomial_mul_X_pow + - Polynomial.coeff_C_ne_zero + - Polynomial.C_mul_X_pow_eq_monomial + - Polynomial.binomial_eq_binomial + - Polynomial.support_update + - Polynomial.coeff_C + - Polynomial.C_mul_monomial + - Polynomial.coeff_injective + - Polynomial.coeff_ofNat_succ + - Polynomial.coeff_one_zero + - Polynomial.sum_C_mul_X_pow_eq + - Polynomial.ofFinsupp_erase + - Polynomial.monomial_left_inj + - Polynomial.coeff_update_same + - Polynomial.smul_monomial + - Polynomial.support_C_mul_X' + - Polynomial.toFinsupp_eq_zero + - Polynomial.monomial_add_erase + - Polynomial.X_mul + - Polynomial.coeff_X_of_ne_one + - Polynomial.support_monomial' + - Polynomial.support_toFinsupp + - Polynomial.ext_iff + - Polynomial.erase_ne + - Polynomial.support_ofFinsupp + - Polynomial.toFinsupp_eq_one + - Polynomial.toFinsupp_neg + - Polynomial.support_add + - Polynomial.mem_support_iff + - Polynomial.coeff_one + - Polynomial.toFinsupp_mul + - Polynomial.toFinsupp_C_mul_X + - Polynomial.support_C_mul_X_pow + - Polynomial.ofFinsupp_single + - Polynomial.monomial_mul_X + - Polynomial.erase_zero + - Polynomial.toFinsupp_X_pow + - Polynomial.smul_X_eq_monomial + - Polynomial.addHom_ext + - Polynomial.monomial_mul_C + - Polynomial.coeff_ofFinsupp + - Polynomial.coeff_C_succ + - Polynomial.support_update_zero + - Polynomial.subsingleton_iff_subsingleton + - Polynomial.sum_add_index + - Polynomial.X_ne_zero + - Polynomial.C_eq_int_cast + - Polynomial.C_0 + - Polynomial.support_X_empty + - Polynomial.ofFinsupp_eq_zero + - Polynomial.coeff_nat_cast_ite + - Polynomial.eq_zero_of_eq_zero + - Polynomial.toFinsupp_apply + - Polynomial.ofFinsupp_add + - Polynomial.addSubmonoid_closure_setOf_eq_monomial + - Polynomial.C_bit1 + - Polynomial.support_C_mul_X_pow' + - Polynomial.coeff_sub + - Polynomial.monomial_sub + - Polynomial.coeff_update_ne + - Polynomial.toFinsupp_monomial + - Polynomial.coeff_neg + - Polynomial.coeff_update + - path: Mathlib/Algebra/Algebra/Bilinear.lean + theorems: + - LinearMap.mulRight_one + - LinearMap.mulLeft_mul + - LinearMap.pow_mulRight + - LinearMap.mulLeft_eq_zero_iff + - LinearMap.mulRight_mul + - LinearMap.commute_mulLeft_right + - LinearMap.pow_mulLeft + - LinearMap.mulRight_injective + - LinearMap.mulRight_eq_zero_iff + - LinearMap.mulLeft_one + - Algebra.lmul_injective + - Algebra.lmul_isUnit_iff + - LinearMap.mulLeft_injective + - LinearMap.mul_injective + - path: Mathlib/Data/ZMod/Basic.lean + theorems: + - ZMod.natAbs_valMinAbs_add_le + - ZMod.isUnit_iff_coprime + - ZMod.natAbs_valMinAbs_le + - ZMod.nat_coe_zmod_eq_iff + - ZMod.valMinAbs_mul_two_eq_iff + - ZMod.val_mul + - ZMod.int_coe_zmod_eq_iff + - ZMod.cast_one + - ZMod.cast_add_eq_ite + - ZMod.int_coe_ringEquivCongr + - ZMod.int_cast_zmod_eq_zero_iff_dvd + - ZMod.ringEquivCongr_refl_apply + - ZMod.natAbs_mod_two + - RingHom.ext_zmod + - ZMod.addOrderOf_coe + - ZMod.mul_inv_cancel_aux + - ZMod.ringEquivCongr_val + - ZMod.val_unit' + - ZMod.valMinAbs_def_pos + - ZMod.val_eq_zero + - ZMod.inv_mul_of_unit + - ZMod.val_add_of_le + - ZMod.eq_iff_modEq_nat + - ZMod.natAbs_valMinAbs_neg + - ZMod.int_cast_zmod_cast + - ZMod.val_add_of_lt + - ZMod.val_coe_unit_coprime + - ZMod.cast_zero + - ZMod.ringEquivCongr_ringEquivCongr_apply + - ZMod.subsingleton_iff + - Prod.fst_zmod_cast + - ZMod.val_injective + - ZMod.coe_mul_inv_eq_one + - ZMod.neg_eq_self_iff + - ZMod.inv_coe_unit + - ZMod.add_self_eq_zero_iff_eq_zero + - ZMod.eq_one_of_isUnit_natCast + - ZMod.val_add + - ZMod.nat_cast_mod + - ZMod.valMinAbs_natCast_eq_self + - ZMod.neg_val + - ZMod.ringEquivCongr_refl + - ZMod.cast_cast_zmod_of_le + - ZMod.ringHom_eq_of_ker_eq + - ZMod.val_add_val_of_le + - ZMod.isUnit_prime_iff_not_dvd + - ZMod.val_eq_ite_valMinAbs + - ZMod.val_lt + - Prod.snd_zmod_cast + - ZMod.val_add_le + - ZMod.valMinAbs_neg_of_ne_half + - ZMod.mul_inv_eq_gcd + - ZMod.valMinAbs_eq_zero + - ZMod.cast_mul + - ZMod.valMinAbs_spec + - ZMod.valMinAbs_nonneg_iff + - ZMod.ne_neg_self + - ZMod.ker_int_castAddHom + - Nat.le_div_two_iff_mul_two_le + - ZMod.valMinAbs_mem_Ioc + - ZMod.val_nat_cast_of_lt + - ZMod.nat_cast_toNat + - ZMod.neg_eq_self_mod_two + - ZMod.valMinAbs_natCast_of_half_lt + - ZMod.ringEquivCongr_symm + - ZMod.ringHom_map_cast + - ZMod.val_neg_one + - ZMod.nat_cast_comp_val + - ZMod.valMinAbs_zero + - ZMod.val_nat_cast + - ZMod.val_cast_eq_val_of_lt + - Nat.range_mul_add + - ZMod.val_one_eq_one_mod + - ZMod.cast_sub_one + - ZMod.inv_zero + - ZMod.val_neg_of_ne_zero + - ZMod.nat_cast_zmod_eq_zero_iff_dvd + - ZMod.castHom_bijective + - ZMod.cast_add + - ZMod.ringEquivCongr_trans + - ZMod.cast_eq_val + - ZMod.coe_int_cast + - ZMod.val_int_cast + - ZMod.cast_neg_one + - ZMod.nat_cast_zmod_val + - ZMod.valMinAbs_natCast_of_le_half + - ZMod.valMinAbs_natAbs_eq_min + - ZMod.natAbs_min_of_le_div_two + - ZMod.int_cast_eq_int_cast_iff_dvd_sub + - ZMod.addOrderOf_coe' + - ZMod.ker_int_castRingHom + - ZMod.nat_cast_eq_nat_cast_iff + - ZMod.prime_ne_zero + - ZMod.neg_val' + - ZMod.ringChar_zmod_n + - ZMod.val_one + - ZMod.cast_zmod_eq_zero_iff_of_le + - ZMod.val_mul_le + - ZMod.cast_injective_of_le + - ZMod.val_cast_of_lt + - ZMod.castHom_injective + - ZMod.mul_inv_of_unit + - ZMod.val_sub + - ZMod.int_cast_comp_cast + - ZMod.nat_cast_natAbs_valMinAbs + - ZMod.coe_valMinAbs + - path: Mathlib/Data/Nat/Choose/Factorization.lean + theorems: + - Nat.prod_pow_factorization_choose + - Nat.factorization_choose_le_log + - Nat.factorization_choose_le_one + - Nat.factorization_choose_eq_zero_of_lt + - Nat.factorization_choose_of_lt_three_mul + - Nat.factorization_factorial_eq_zero_of_lt + - Nat.prod_pow_factorization_centralBinom + - Nat.factorization_centralBinom_of_two_mul_self_lt_three_mul + - path: Mathlib/GroupTheory/Perm/Basic.lean + theorems: + - Equiv.swap_mul_eq_iff + - Equiv.swap_mul_swap_mul_swap + - Equiv.pow_mulLeft + - Set.MapsTo.perm_pow + - Equiv.swap_smul_self_smul + - Equiv.Perm.inv_aux + - Set.SurjOn.perm_pow + - Equiv.swap_mul_eq_mul_swap + - Equiv.pow_addLeft + - Equiv.mul_swap_eq_iff + - Equiv.zpow_addRight + - Equiv.pow_addRight + - Equiv.Perm.subtypePerm_zpow + - Equiv.swap_apply_apply + - Equiv.Perm.subtypeCongrHom_injective + - Equiv.Perm.sigmaCongrRightHom_injective + - Equiv.Perm.sumCongrHom_injective + - Equiv.Perm.subtypePerm_pow + - Equiv.Perm.ofSubtype_subtypePerm + - Equiv.pow_mulRight + - Equiv.mul_swap_mul_self + - Equiv.Perm.zpow_apply_comm + - Set.BijOn.perm_pow + - Equiv.zpow_mulRight + - Equiv.Perm.mem_iff_ofSubtype_apply_mem + - Equiv.Perm.zpow_aux + - path: Mathlib/Dynamics/Newton.lean + theorems: + - Polynomial.isFixedPt_newtonMap_of_isUnit_iff + - Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent + - Polynomial.exists_unique_nilpotent_sub_and_aeval_eq_zero + - Polynomial.isFixedPt_newtonMap_of_aeval_eq_zero + - Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap + - Polynomial.newtonMap_apply_of_isUnit + - Polynomial.newtonMap_apply_of_not_isUnit + - path: Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean + theorems: + - Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree + - Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree_le + - Polynomial.IsEisensteinAt.coeff_mem + - Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_aeval_zero_of_monic_mem_map + - "Polynomial.dvd_pow_natDegree_of_eval\u2082_eq_zero" + - Polynomial.scaleRoots.isWeaklyEisensteinAt + - Polynomial.IsWeaklyEisensteinAt.pow_natDegree_le_of_root_of_monic_mem + - path: Mathlib/RingTheory/MvPolynomial/Homogeneous.lean + theorems: + - MvPolynomial.IsHomogeneous.funext_of_le_card + - MvPolynomial.homogeneousSubmodule_mul + - MvPolynomial.IsHomogeneous.finSuccEquiv_coeff_isHomogeneous + - MvPolynomial.homogeneousComponent_eq_zero' + - "MvPolynomial.IsHomogeneous.eval\u2082" + - MvPolynomial.IsHomogeneous.exists_eval_ne_zero_of_coeff_finSuccEquiv_ne_zero_aux + - MvPolynomial.IsHomogeneous.pow + - MvPolynomial.IsHomogeneous.C_mul + - MvPolynomial.IsHomogeneous.funext + - MvPolynomial.isHomogeneous_of_totalDegree_zero + - MvPolynomial.isHomogeneous_X_pow + - MvPolynomial.IsHomogeneous.inj_right + - MvPolynomial.IsHomogeneous.coeff_eq_zero + - MvPolynomial.isHomogeneous_C + - MvPolynomial.homogeneousComponent_homogeneous_polynomial + - MvPolynomial.homogeneousSubmodule_eq_finsupp_supported + - MvPolynomial.IsHomogeneous.rename_isHomogeneous + - MvPolynomial.sum_homogeneousComponent + - MvPolynomial.IsHomogeneous.map + - MvPolynomial.IsHomogeneous.totalDegree + - MvPolynomial.homogeneousComponent_eq_zero + - MvPolynomial.isHomogeneous_monomial + - MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero_of_le_card + - MvPolynomial.IsHomogeneous.totalDegree_le + - MvPolynomial.IsHomogeneous.coeff_isHomogeneous_of_optionEquivLeft_symm + - MvPolynomial.IsHomogeneous.prod + - MvPolynomial.homogeneousComponent_isHomogeneous + - MvPolynomial.isHomogeneous_X + - MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero + - MvPolynomial.homogeneousComponent_C_mul + - MvPolynomial.IsHomogeneous.exists_eval_ne_zero_of_totalDegree_le_card_aux + - MvPolynomial.IsHomogeneous.rename_isHomogeneous_iff + - path: Mathlib/Algebra/Group/Equiv/Basic.lean + theorems: + - MulEquiv.comp_left_injective + - MulEquiv.comp_right_injective + - MulEquiv.coe_monoidHom_comp_coe_monoidHom_symm + - MulEquiv.coe_monoidHom_symm_comp_coe_monoidHom + - path: Mathlib/Algebra/Function/Indicator.lean + theorems: + - Set.apply_mulIndicator_symmDiff + - Set.mulIndicator_mul_eq_left + - Set.indicator_mul_left + - Set.indicator_mul_right + - Set.mulIndicator_preimage_of_not_mem + - Set.mulIndicator_const_preimage + - Set.mulIndicator_inter_mulSupport + - Set.mulIndicator_one + - Set.mulIndicator_eq_self_of_superset + - Set.inter_indicator_mul + - Set.mem_range_mulIndicator + - Set.mulIndicator_union_of_not_mem_inter + - Set.indicator_eq_one_iff_mem + - Set.mulIndicator_compl' + - Set.mulIndicator_rel_mulIndicator + - MonoidHom.map_mulIndicator + - Set.mulIndicator_mulIndicator + - Set.mulIndicator_diff' + - Set.inter_indicator_one + - Set.mulIndicator_compl_mul_self_apply + - Set.mulIndicator_diff + - Set.mulIndicator_eq_one + - Set.mulIndicator_self_mul_compl_apply + - Set.mulIndicator_mul_eq_right + - Set.mulIndicator_congr + - Set.mulIndicator_const_preimage_eq_union + - Set.mulIndicator_mul + - Set.mulIndicator_apply + - Set.mulIndicator_eq_self + - Set.mulIndicator_image + - Set.indicator_one_inj + - Set.mulIndicator_comp_right + - Set.mulIndicator_comp_of_one + - Set.indicator_eq_zero_iff_not_mem + - Set.mulIndicator_apply_ne_one + - Set.indicator_prod_one + - Set.mulSupport_mulIndicator + - Set.mulIndicator_union_mul_inter_apply + - Set.mulIndicator_mul_compl_eq_piecewise + - Set.comp_mulIndicator + - Set.mulIndicator_one_preimage + - Set.mulIndicator_eq_one_or_self + - Set.mulIndicator_apply_eq_self + - Set.indicator_mul + - path: Mathlib/RingTheory/Polynomial/Bernstein.lean + theorems: + - bernsteinPolynomial.iterate_derivative_at_1_eq_zero_of_lt + - bernsteinPolynomial.derivative_zero + - bernsteinPolynomial.linearIndependent_aux + - bernsteinPolynomial.derivative_succ + - bernsteinPolynomial.map + - bernsteinPolynomial.flip + - bernsteinPolynomial.sum + - bernsteinPolynomial.iterate_derivative_at_1 + - bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt + - bernsteinPolynomial.derivative_succ_aux + - bernsteinPolynomial.iterate_derivative_at_0_ne_zero + - bernsteinPolynomial.eval_at_1 + - bernsteinPolynomial.iterate_derivative_at_0 + - bernsteinPolynomial.variance + - bernsteinPolynomial.sum_smul + - bernsteinPolynomial.iterate_derivative_at_1_ne_zero + - bernsteinPolynomial.eval_at_0 + - bernsteinPolynomial.flip' + - bernsteinPolynomial.eq_zero_of_lt + - path: Mathlib/Computability/AkraBazzi/AkraBazzi.lean + theorems: + - AkraBazziRecurrence.eventually_deriv_one_add_smoothingFn + - AkraBazziRecurrence.eventually_bi_mul_le_r + - AkraBazziRecurrence.eventually_one_add_smoothingFn_r_pos + - AkraBazziRecurrence.strictAnti_sumCoeffsExp + - AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T + - AkraBazziRecurrence.eventually_deriv_one_sub_smoothingFn + - AkraBazziRecurrence.differentiableAt_smoothingFn + - AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge + - AkraBazziRecurrence.eventually_asympBound_pos + - AkraBazziRecurrence.isBigO_symm_asympBound + - AkraBazziRecurrence.isBigO_apply_r_sub_b + - AkraBazziRecurrence.exists_eventually_r_le_const_mul + - AkraBazziRecurrence.eventually_r_pos + - AkraBazziRecurrence.exists_eventually_const_mul_le_r + - AkraBazziRecurrence.dist_r_b' + - AkraBazziRecurrence.bi_min_div_two_pos + - AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_sub_smoothingFn + - AkraBazziRecurrence.eventually_one_add_smoothingFn_pos + - AkraBazziRecurrence.tendsto_atTop_r + - AkraBazziRecurrence.isTheta_smoothingFn_sub_self + - AkraBazziRecurrence.one_mem_range_sumCoeffsExp + - AkraBazziRecurrence.eventually_asympBound_r_pos + - AkraBazziRecurrence.one_add_smoothingFn_le_two + - AkraBazziRecurrence.eventually_r_ge + - AkraBazziRecurrence.eventually_one_sub_smoothingFn_r_pos + - AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_sub_smoothingFn + - AkraBazziRecurrence.eventually_one_sub_smoothingFn_nonneg + - AkraBazziRecurrence.isLittleO_self_div_log_id + - AkraBazziRecurrence.isBigO_asympBound + - AkraBazziRecurrence.deriv_smoothingFn + - AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_sub_smoothingFn + - AkraBazziRecurrence.growsPolynomially_deriv_rpow_p_mul_one_sub_smoothingFn + - AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn + - AkraBazziRecurrence.strictAntiOn_smoothingFn + - AkraBazziRecurrence.eventually_r_le_b + - AkraBazziRecurrence.sumCoeffsExp_p_eq_one + - AkraBazziRecurrence.isEquivalent_smoothingFn_sub_self + - AkraBazziRecurrence.continuous_sumCoeffsExp + - AkraBazziRecurrence.T_isBigO_smoothingFn_mul_asympBound + - AkraBazziRecurrence.eventually_one_sub_smoothingFn_gt_const_real + - AkraBazziRecurrence.T_pos + - AkraBazziRecurrence.eventually_atTop_sumTransform_le + - AkraBazziRecurrence.eventually_one_add_smoothingFn_nonneg + - AkraBazziRecurrence.bi_min_div_two_lt_one + - AkraBazziRecurrence.tendsto_zero_sumCoeffsExp + - AkraBazziRecurrence.asympBound_pos + - AkraBazziRecurrence.isLittleO_deriv_one_add_smoothingFn + - AkraBazziRecurrence.eventually_b_le_r + - AkraBazziRecurrence.rpow_p_mul_one_sub_smoothingFn_le + - AkraBazziRecurrence.isLittleO_deriv_smoothingFn + - AkraBazziRecurrence.asympBound_def' + - AkraBazziRecurrence.isLittleO_smoothingFn_one + - AkraBazziRecurrence.growsPolynomially_deriv_rpow_p_mul_one_add_smoothingFn + - AkraBazziRecurrence.eventually_atTop_sumTransform_ge + - AkraBazziRecurrence.tendsto_atTop_sumCoeffsExp + - AkraBazziRecurrence.isLittleO_deriv_one_sub_smoothingFn + - AkraBazziRecurrence.isTheta_deriv_rpow_p_mul_one_add_smoothingFn + - AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_add_smoothingFn + - AkraBazziRecurrence.eventually_log_b_mul_pos + - AkraBazziRecurrence.strictMonoOn_one_sub_smoothingFn + - path: Mathlib/RingTheory/UniqueFactorizationDomain.lean + theorems: + - UniqueFactorizationMonoid.mem_normalizedFactors_iff + - Associates.dvd_count_pow + - UniqueFactorizationMonoid.mem_normalizedFactors_eq_of_associated + - UniqueFactorizationMonoid.le_multiplicity_iff_replicate_le_normalizedFactors + - Associates.dvd_of_mem_factors + - UniqueFactorizationMonoid.factors_pow_count_prod + - Associates.eq_pow_of_mul_eq_pow + - Associates.prod_le_prod_iff_le + - UniqueFactorizationMonoid.associated_iff_normalizedFactors_eq_normalizedFactors + - irreducible_iff_prime_of_exists_prime_factors + - Associates.prime_pow_dvd_iff_le + - Associates.exists_prime_dvd_of_not_inf_one + - Associates.prod_add + - Associates.count_ne_zero_iff_dvd + - Associates.mem_factors'_iff_dvd + - Associates.prod_le + - Associates.FactorSet.unique + - Associates.mem_factorSet_some + - Associates.factors_mul + - Associates.count_of_coprime + - UniqueFactorizationMonoid.normalizedFactors_pos + - UniqueFactorizationMonoid.dvd_iff_normalizedFactors_le_normalizedFactors + - associated_of_factorization_eq + - Associates.FactorSet.coe_add + - UniqueFactorizationMonoid.exists_reduced_factors + - Associates.mem_factors'_of_dvd + - Associates.dvd_count_of_dvd_count_mul + - Associates.mem_factorSet_top + - Associates.factors_mono + - UniqueFactorizationMonoid.prime_pow_coprime_prod_of_coprime_insert + - UniqueFactorizationMonoid.normalizedFactors_multiset_prod + - Associates.factors_eq_some_iff_ne_zero + - UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors + - factorization_mul + - Associates.count_factors_eq_find_of_dvd_pow + - UniqueFactorizationMonoid.normalizedFactors_prod_eq + - MulEquiv.uniqueFactorizationMonoid + - UniqueFactorizationMonoid.of_exists_unique_irreducible_factors + - UniqueFactorizationMonoid.pow_right_injective + - Associates.count_mul + - UniqueFactorizationMonoid.factors_prod + - UniqueFactorizationMonoid.multiplicative_prime_power + - factorization_zero + - Associates.eq_factors_of_eq_counts + - UniqueFactorizationMonoid.factors_zero + - factorization_eq_count + - Associates.factors_mk + - UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors + - UniqueFactorizationMonoid.factors_one + - UniqueFactorizationMonoid.factors_eq_normalizedFactors + - multiplicity.finite_of_not_isUnit + - UniqueFactorizationMonoid.normalizedFactors_irreducible + - factorization_pow + - support_factorization + - Associates.count_zero + - Associates.mem_factors_iff_dvd + - Associates.count_some + - UniqueFactorizationMonoid.normalizedFactors_zero + - UniqueFactorizationMonoid.normalizedFactors_mul + - Irreducible.normalizedFactors_pow + - Associates.prod_mono + - Associates.factors_one + - UniqueFactorizationMonoid.multiplicative_of_coprime + - UniqueFactorizationMonoid.exists_associated_prime_pow_of_unique_normalized_factor + - Associates.eq_of_prod_eq_prod + - Associates.count_pow + - Associates.sup_mul_inf + - Associates.eq_pow_count_factors_of_dvd_pow + - Associates.FactorSet.sup_add_inf_eq_add + - UniqueFactorizationMonoid.prime_of_factor + - Associates.pow_factors + - WfDvdMonoid.not_unit_iff_exists_factors_eq + - Associates.reducible_not_mem_factorSet + - Associates.factors_subsingleton + - UniqueFactorizationMonoid.exists_prime_iff + - Associates.mem_factors_of_dvd + - UniqueFactorizationMonoid.exists_mem_factors_of_dvd + - Associates.coprime_iff_inf_one + - Associates.eq_pow_find_of_dvd_irreducible_pow + - UniqueFactorizationMonoid.exists_mem_normalizedFactors + - UniqueFactorizationMonoid.factors_pow + - Associates.count_mul_of_coprime + - UniqueFactorizationMonoid.ne_zero_of_mem_factors + - prime_factors_irreducible + - UniqueFactorizationMonoid.induction_on_coprime + - Associates.dvd_of_mem_factors' + - Ideal.IsPrime.exists_mem_prime_of_ne_bot + - Associates.factors_prime_pow + - Associates.unique' + - Associates.count_self + - irreducible_iff_prime_of_exists_unique_irreducible_factors + - Associates.is_pow_of_dvd_count + - UniqueFactorizationMonoid.normalizedFactors_prod_of_prime + - UniqueFactorizationMonoid.induction_on_prime_power + - UniqueFactorizationMonoid.count_normalizedFactors_eq + - Associates.count_eq_zero_of_ne + - UniqueFactorizationMonoid.factors_mul + - factorization_one + - UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd + - UniqueFactorizationMonoid.count_normalizedFactors_eq' + - UniqueFactorizationMonoid.exists_prime_factors + - UniqueFactorizationMonoid.induction_on_prime + - Associates.count_le_count_of_factors_le + - UniqueFactorizationMonoid.exists_mem_factors + - UniqueFactorizationMonoid.factors_pos + - UniqueFactorizationMonoid.normalizedFactors_eq_of_dvd + - WfDvdMonoid.of_exists_prime_factors + - prime_factors_unique + - Associates.eq_of_factors_eq_factors + - Associates.factors_self + - Associates.count_mul_of_coprime' + - UniqueFactorizationMonoid.normalizedFactors_of_irreducible_pow + - UniqueFactorizationMonoid.normalize_normalized_factor + - Associates.factors_prod + - Associates.factors'_cong + - WfDvdMonoid.iff_wellFounded_associates + - UniqueFactorizationMonoid.normalizedFactors_one + - WfDvdMonoid.of_wellFounded_associates + - UniqueFactorizationMonoid.normalizedFactors_pow + - UniqueFactorizationMonoid.normalizedFactors_prod + - Associates.factors_le + - WfDvdMonoid.max_power_factor' + - Associates.factors_eq_none_iff_zero + - WfDvdMonoid.exists_factors + - path: Mathlib/Data/Polynomial/Reverse.lean + theorems: + - Polynomial.reverse_mul_X_pow + - Polynomial.reflect_monomial + - Polynomial.revAtFun_invol + - Polynomial.reflect_C_mul + - Polynomial.reflect_support + - Polynomial.revAtFun_inj + - Polynomial.reflect_C + - Polynomial.revAt_zero + - Polynomial.reflect_sub + - Polynomial.reverse_mul_of_domain + - Polynomial.coeff_reverse + - Polynomial.reverse_add_C + - Polynomial.reflect_mul_induction + - Polynomial.reflect_one_X + - Polynomial.reverse_X_pow_mul + - "Polynomial.eval\u2082_reflect_eq_zero_iff" + - Polynomial.reverse_natDegree + - Polynomial.natDegree_eq_reverse_natDegree_add_natTrailingDegree + - Polynomial.coeff_reflect + - Polynomial.reverse_leadingCoeff + - Polynomial.reflect_C_mul_X_pow + - Polynomial.reverse_X_mul + - Polynomial.revAt_eq_self_of_lt + - Polynomial.reverse_eq_zero + - Polynomial.reverse_C + - Polynomial.reverse_natTrailingDegree + - Polynomial.reverse_mul_X + - Polynomial.revAt_add + - Polynomial.trailingCoeff_mul + - Polynomial.reflect_add + - Polynomial.reflect_eq_zero_iff + - Polynomial.coeff_zero_reverse + - Polynomial.reverse_natDegree_le + - Polynomial.reverse_C_add + - "Polynomial.eval\u2082_reflect_mul_pow" + - Polynomial.reflect_neg + - Polynomial.reverse_trailingCoeff + - path: Mathlib/LinearAlgebra/Finsupp.lean + theorems: + - Finsupp.total_apply_of_mem_supported + - Finsupp.finsuppProdLEquiv_apply + - Finsupp.supported_union + - Submodule.exists_finset_of_mem_iSup + - mem_span_set + - Finsupp.lmapDomain_disjoint_ker + - Finsupp.total_equivMapDomain + - Finsupp.supported_empty + - Finsupp.mem_supported_support + - Finsupp.mem_span_range_iff_exists_finsupp + - span_eq_iUnion_nat + - Finsupp.total_surjective + - Finsupp.total_onFinset + - mem_span_finset + - Finsupp.total_eq_fintype_total_apply + - Finsupp.lapply_comp_lsingle_same + - Finsupp.domLCongr_single + - Fintype.total_apply_single + - LinearMap.splittingOfFinsuppSurjective_splits + - Finsupp.supported_eq_span_single + - Finsupp.total_single + - Submodule.mem_sSup_iff_exists_finset + - Finsupp.lmapDomain_total + - Finsupp.lcongr_symm + - Finsupp.supported_iUnion + - Finsupp.total_comapDomain + - Finsupp.lsingle_range_le_ker_lapply + - Finsupp.total_option + - Finsupp.total_unique + - Finsupp.total_total + - Finsupp.span_eq_range_total + - Span.finsupp_total_repr + - Finsupp.iSup_lsingle_range + - Finsupp.lcongr_symm_single + - Finsupp.supported_iInter + - Finsupp.disjoint_supported_supported + - Finsupp.lmapDomain_supported + - Finsupp.LinearEquiv.finsuppUnique_symm_apply + - Finsupp.span_image_eq_map_total + - Finsupp.supported_comap_lmapDomain + - mem_span_set' + - Finsupp.range_total + - Finsupp.restrictDom_comp_subtype + - Finsupp.lsum_comp_lsingle + - Finsupp.mem_span_image_iff_total + - Finsupp.total_fin_zero + - Finsupp.span_single_image + - LinearMap.splittingOfFunOnFintypeSurjective_splits + - Finsupp.disjoint_supported_supported_iff + - Finsupp.total_comp + - top_le_span_range_iff_forall_exists_fun + - Fintype.range_total + - Finsupp.apply_total + - Finsupp.supported_inter + - Finsupp.iInf_ker_lapply_le_bot + - Finsupp.mem_supported' + - Finsupp.total_zero_apply + - mem_span_range_iff_exists_fun + - Finsupp.sum_smul_index_linearMap' + - Finsupp.total_embDomain + - Finsupp.finsuppProdLEquiv_symm_apply + - Finsupp.disjoint_lsingle_lsingle + - Finsupp.lapply_comp_lsingle_of_ne + - path: Mathlib/Algebra/GroupWithZero/Defs.lean + theorems: + - zero_eq_mul_self + - "mul_inv_cancel_left\u2080" + - "mul_inv_cancel_right\u2080" + - mul_self_eq_zero + - zero_eq_mul + - path: Mathlib/Data/List/Forall2.lean + theorems: + - List.rel_filter + - "List.forall\u2082_nil_right_iff" + - "List.forall\u2082_take_append" + - "List.Forall\u2082.imp" + - "List.forall\u2082_and_left" + - "List.forall\u2082_iff_zip" + - "List.forall\u2082_map_left_iff" + - List.rel_mem + - "List.forall\u2082_take" + - "List.forall\u2082_same" + - "List.forall\u2082_zip" + - "List.forall\u2082_nil_left_iff" + - "List.forall\u2082_eq_eq_eq" + - "List.forall\u2082_drop_append" + - List.rel_reverse + - "List.sublistForall\u2082_iff" + - "List.forall\u2082_drop" + - "List.forall\u2082_reverse_iff" + - List.rel_filterMap + - "List.forall\u2082_map_right_iff" + - path: Mathlib/Data/Finset/Slice.lean + theorems: + - Finset.subset_powersetCard_univ_iff + - Set.sized_iUnion + - "Set.sized_iUnion\u2082" + - Set.sized_singleton + - Set.Sized.card_le + - Set.sized_empty + - Finset.sum_card_slice + - path: Mathlib/MeasureTheory/Covering/OneDim.lean + theorems: + - Real.Icc_mem_vitaliFamily_at_right + - Real.tendsto_Icc_vitaliFamily_left + - Real.Icc_mem_vitaliFamily_at_left + - Real.tendsto_Icc_vitaliFamily_right + - path: Mathlib/LinearAlgebra/FreeModule/PID.lean + theorems: + - LinearIndependent.restrict_scalars_algebras + - generator_maximal_submoduleImage_dvd + - Ideal.exists_smith_normal_form + - Submodule.basisOfPid_bot + - Submodule.exists_smith_normal_form_of_le + - Basis.SmithNormalForm.repr_apply_embedding_eq_repr_smul + - eq_bot_of_generator_maximal_map_eq_zero + - Submodule.nonempty_basis_of_pid + - Basis.SmithNormalForm.coord_apply_embedding_eq_smul_coord + - dvd_generator_iff + - Ideal.smithCoeffs_ne_zero + - Basis.SmithNormalForm.toMatrix_restrict_eq_toMatrix + - Submodule.basis_of_pid_aux + - Basis.SmithNormalForm.repr_comp_embedding_eq_smul + - Module.free_of_finite_type_torsion_free + - Basis.SmithNormalForm.repr_eq_zero_of_nmem_range + - eq_bot_of_generator_maximal_submoduleImage_eq_zero + - path: Mathlib/Algebra/Order/Monovary.lean + theorems: + - antivary_iff_forall_smul_nonpos + - monovaryOn_iff_smul_rearrangement + - antivaryOn_iff_smul_rearrangement + - monovaryOn_iff_forall_mul_nonneg + - monovary_inv_left + - antivary_inv + - monovaryOn_inv_right + - monovary_inv_right + - antivary_iff_smul_rearrangement + - antivaryOn_iff_mul_rearrangement + - "monovaryOn_inv\u2080" + - "monovaryOn_inv_right\u2080" + - antivaryOn_inv + - monovary_iff_forall_smul_nonneg + - "monovary_inv_right\u2080" + - "antivary_inv\u2080" + - "antivaryOn_inv_right\u2080" + - "monovary_inv\u2080" + - antivary_iff_mul_rearrangement + - monovaryOn_iff_forall_smul_nonneg + - antivary_inv_left + - monovaryOn_inv + - antivaryOn_inv_left + - monovary_iff_smul_rearrangement + - "antivary_inv_right\u2080" + - antivary_inv_right + - monovaryOn_inv_left + - antivaryOn_inv_right + - monovary_inv + - monovary_iff_forall_mul_nonneg + - antivaryOn_iff_forall_smul_nonpos + - monovary_iff_mul_rearrangement + - path: Mathlib/Data/ENNReal/Basic.lean + theorems: + - ENNReal.iUnion_Icc_coe_nat + - ENNReal.ofReal_one + - ENNReal.coe_nnreal_eq + - Set.OrdConnected.image_ennreal_ofReal + - ENNReal.toReal_eq_toReal_iff' + - ENNReal.lt_iff_exists_add_pos_lt + - ENNReal.ofReal_coe_nat + - ENNReal.iUnion_Iio_coe_nat + - ENNReal.toReal_eq_one_iff + - ENNReal.toReal_eq_zero_iff + - ENNReal.toReal_le_coe_of_le_coe + - ENNReal.iInter_Ici_coe_nat + - ENNReal.iUnion_Ico_coe_nat + - ENNReal.exists_ne_top + - ENNReal.coe_toNNReal_le_self + - ENNReal.ofReal_zero + - ENNReal.toReal_eq_toReal_iff + - ENNReal.le_of_forall_pos_le_add + - ENNReal.iInter_Ioi_coe_nat + - Set.OrdConnected.image_coe_nnreal_ennreal + - ENNReal.ofReal_toReal + - ENNReal.iUnion_Ioc_coe_nat + - ENNReal.ofReal_toReal_eq_iff + - ENNReal.toNNReal_eq_one_iff + - ENNReal.coe_mem_upperBounds + - ENNReal.abs_toReal + - ENNReal.iUnion_Ioo_coe_nat + - ENNReal.exists_nat_gt + - ENNReal.toReal_nat + - ENNReal.toNNReal_eq_toNNReal_iff' + - ENNReal.toNNReal_nat + - ENNReal.toReal_ofReal_eq_iff + - ENNReal.le_of_top_imp_top_of_toNNReal_le + - ENNReal.lt_iff_exists_rat_btwn + - ENNReal.iInf_ne_top + - path: Mathlib/MeasureTheory/Integral/Layercake.lean + theorems: + - MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul + - MeasureTheory.Integrable.integral_eq_integral_meas_lt + - MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul + - MeasureTheory.Integrable.integral_eq_integral_meas_le + - MeasureTheory.lintegral_rpow_eq_lintegral_meas_le_mul + - MeasureTheory.countable_meas_le_ne_meas_lt + - MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite + - MeasureTheory.lintegral_eq_lintegral_meas_lt + - MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable + - MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul + - MeasureTheory.Integrable.integral_eq_integral_Ioc_meas_le + - MeasureTheory.lintegral_eq_lintegral_meas_le + - path: Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean + theorems: + - Matrix.inv_inv_inv + - Matrix.nonsing_inv_mul_cancel_left + - Matrix.vecMul_surjective_iff_exists_left_inverse + - Matrix.det_invOf + - Matrix.linearIndependent_cols_iff_isUnit + - Matrix.invOf_diagonal_eq + - Matrix.linearIndependent_rows_iff_isUnit + - Matrix.inv_mul_eq_iff_eq_mul_of_invertible + - Matrix.det_smul_inv_mulVec_eq_cramer + - Matrix.nonsing_inv_mul + - Matrix.invOf_submatrix_equiv_eq + - Matrix.isUnit_iff_isUnit_det + - Matrix.isUnit_nonsing_inv_det_iff + - Matrix.mul_inv_rev + - Matrix.left_inv_eq_left_inv + - Matrix.list_prod_inv_reverse + - Matrix.invOf_eq + - Matrix.transpose_nonsing_inv + - Matrix.inv_diagonal + - Matrix.nonsing_inv_eq_ring_inverse + - Matrix.mul_nonsing_inv_cancel_left + - Matrix.conjTranspose_nonsing_inv + - Matrix.isUnit_det_transpose + - Matrix.vecMul_injective_iff_isUnit + - Matrix.det_nonsing_inv + - Matrix.right_inv_eq_right_inv + - Matrix.isUnit_diagonal + - Matrix.right_inv_eq_left_inv + - Matrix.isUnit_submatrix_equiv + - Matrix.invOf_eq_nonsing_inv + - Matrix.unitOfDetInvertible_eq_nonsingInvUnit + - Matrix.mul_inv_eq_iff_eq_mul_of_invertible + - Matrix.det_nonsing_inv_mul_det + - Matrix.inv_inj + - Matrix.mulVec_surjective_iff_isUnit + - Matrix.nonsing_inv_cancel_or_zero + - Matrix.inv_smul + - Matrix.inv_zero + - Matrix.mul_eq_one_comm + - Matrix.nonsing_inv_apply + - Matrix.vecMul_surjective_iff_isUnit + - Matrix.inv_adjugate + - Matrix.nonsing_inv_nonsing_inv + - Matrix.det_conj + - Matrix.det_conj' + - Matrix.inv_submatrix_equiv + - Matrix.mul_nonsing_inv_cancel_right + - Matrix.coe_units_inv + - Matrix.mulVec_injective_iff_isUnit + - Matrix.inv_inv_of_invertible + - Matrix.nonsing_inv_apply_not_isUnit + - Matrix.mul_nonsing_inv + - Matrix.mulVec_surjective_iff_exists_right_inverse + - Matrix.inv_smul' + - path: Mathlib/LinearAlgebra/Determinant.lean + theorems: + - LinearMap.det_toLin' + - Basis.det_ne_zero + - Pi.basisFun_det + - LinearEquiv.det_conj + - LinearMap.det_cases + - LinearMap.det_toMatrix_eq_det_toMatrix + - LinearEquiv.coe_ofIsUnitDet + - LinearMap.det_eq_one_of_finrank_eq_zero + - LinearMap.detAux_def' + - LinearEquiv.det_coe_symm + - LinearMap.coe_det + - LinearMap.det_zero + - Basis.det_smul_mk_coord_eq_det_update + - LinearMap.det_zero' + - is_basis_iff_det + - LinearMap.associated_det_comp_equiv + - LinearMap.det_conj + - Matrix.det_conj_of_mul_eq_one + - AlternatingMap.map_basis_eq_zero_iff + - Basis.det_reindex_symm + - LinearMap.det_toMatrix + - Matrix.det_comm' + - LinearMap.det_toLin + - Basis.det_unitsSMul_self + - Basis.det_map + - Basis.det_reindex + - LinearMap.bot_lt_ker_of_det_eq_zero + - LinearMap.range_lt_top_of_det_eq_zero + - LinearMap.detAux_def'' + - Basis.det_comp_basis + - LinearMap.isUnit_det + - Basis.det_self + - LinearMap.associated_det_of_eq_comp + - LinearEquiv.isUnit_det + - LinearMap.det_eq_one_of_subsingleton + - LinearMap.det_eq_det_toMatrix_of_finset + - LinearEquiv.det_symm_mul_det + - Basis.det_isEmpty + - Matrix.det_comm + - LinearEquiv.det_mul_det_symm + - LinearMap.finiteDimensional_of_det_ne_one + - LinearMap.det_toMatrix' + - LinearMap.det_smul + - Basis.det_comp + - AlternatingMap.eq_smul_basis_det + - path: Mathlib/GroupTheory/Subgroup/Basic.lean + theorems: + - Subgroup.mem_sup' + - Subgroup.closure_singleton_one + - Subgroup.exists_ne_one_of_nontrivial + - Group.conjugates_subset_normal + - Subgroup.comap_normalizer_eq_of_surjective + - IsConj.eq_of_left_mem_center + - Subgroup.subgroupOf_sup + - Subgroup.normalClosure_le_normal + - Subgroup.subgroupOf_map_subtype + - Subgroup.Normal.map + - SubgroupClass.coeSubtype + - Group.conj_mem_conjugatesOfSet + - zpow_mem + - Subgroup.mulSingle_mem_pi + - Subgroup.le_pi_iff + - Subgroup.comap_injective_isCommutative + - MonoidHom.closure_preimage_le + - Subgroup.prod_eq_bot_iff + - Subgroup.normalClosure_eq_iInf + - normalizerCondition_iff_only_full_group_self_normalizing + - Subgroup.equivMapOfInjective_coe_mulEquiv + - Subgroup.codisjoint_subgroupOf_sup + - Subgroup.map_top_of_surjective + - Subgroup.subgroupOf_normalizer_eq + - Subgroup.pi_eq_bot_iff + - Subgroup.mem_sSup_of_directedOn + - Subgroup.comap_lt_comap_of_surjective + - Subgroup.map_le_map_iff' + - Subgroup.mem_map + - Subgroup.commute_of_normal_of_disjoint + - Subgroup.comap_normalClosure + - Subgroup.comap_sup_eq_of_le_range + - Subgroup.characteristic_iff_map_eq + - Subgroup.subgroupOf_eq_top + - Subgroup.comap_map_eq_self + - Subgroup.map_le_map_iff_of_injective + - exists_inv_mem_iff_exists_mem + - Subgroup.ne_bot_iff_exists_ne_one + - Subgroup.le_normalizer_of_normal + - Subgroup.coe_iInf + - MonoidHom.range_eq_map + - Subgroup.SubgroupNormal.mem_comm + - Subgroup.nontrivial_iff_exists_ne_one + - Subgroup.nontrivial_iff_ne_bot + - Subgroup.mem_closure_singleton + - IsConj.normalClosure_eq_top_of + - Subgroup.mul_injective_of_disjoint + - MonoidHom.range_top_iff_surjective + - Subgroup.map_subtype_le_map_subtype + - MonoidHom.rangeRestrict_injective_iff + - Subgroup.characteristic_iff_le_map + - Subgroup.map_one_eq_bot + - Subgroup.coe_eq_univ + - Subgroup.map_normalClosure + - Subgroup.comap_id + - Subgroup.comap_map_eq + - Subgroup.bot_prod_bot + - Subgroup.coe_eq_singleton + - Subgroup.map_eq_range_iff + - SubgroupClass.coe_inclusion + - Subgroup.closure_closure_coe_preimage + - Subgroup.map_inf_eq + - Subgroup.le_normalizer_map + - Subgroup.inf_subgroupOf_left + - SubgroupClass.subset_union + - Subgroup.le_prod_iff + - Subgroup.subgroupOf_eq_bot + - Subgroup.subgroupOf_inj + - Subgroup.pi_empty + - MonoidHom.ker_toHomUnits + - Subgroup.mem_normalizer_iff'' + - Subgroup.subsingleton_iff + - Subgroup.bot_or_nontrivial + - Subgroup.mk_eq_one + - SubgroupClass.subtype_comp_inclusion + - abs_mem_iff + - Subgroup.map_comap_eq_self + - Subgroup.toSubmonoid_injective + - Subgroup.pi_bot + - MonoidHom.map_range + - Subgroup.map_le_map_iff + - Subgroup.map_symm_eq_iff_map_eq + - Subgroup.normalCore_le + - Subgroup.bot_or_exists_ne_one + - Subgroup.closure_preimage_eq_top + - Subgroup.closure_le_normalClosure + - Subgroup.prod_le_iff + - Subgroup.normalizer_eq_top + - Subgroup.map_comap_eq + - MonoidHom.ker_prodMap + - SubgroupClass.inclusion_inclusion + - MonoidHom.liftOfRightInverseAux_comp_apply + - Subgroup.map_equiv_normalizer_eq + - ConjClasses.mk_bijOn + - Subgroup.iSup_eq_closure + - MonoidHom.eq_liftOfRightInverse + - Subgroup.prod_top + - Subgroup.mem_sup + - SubgroupClass.inclusion_right + - Subgroup.sup_eq_closure + - MonoidHom.ker_eq_bot_iff + - Subgroup.closure_induction' + - SubgroupClass.inclusion_self + - MonoidHom.subgroupOf_range_eq_of_le + - Subgroup.characteristic_iff_map_le + - MonoidHom.range_le_ker_iff + - Subgroup.comap_injective + - mul_mem_cancel_left + - Subgroup.subtype_range + - Subgroup.coe_inclusion + - Subgroup.map_injective_of_ker_le + - Subgroup.pi_top + - Subgroup.le_normalizer_comap + - Subgroup.map_eq_map_iff + - Subgroup.top_prod + - Subgroup.le_normalizer + - Subgroup.mem_normalizer_iff' + - MonoidHom.restrict_range + - Group.mem_conjugatesOfSet_iff + - Subgroup.map_eq_bot_iff_of_injective + - Subgroup.mem_map_equiv + - Subgroup.mem_centralizer_iff_commutator_eq_one + - Subgroup.mem_center_iff + - Subgroup.disjoint_def + - MonoidHom.eq_iff + - Subgroup.comap_normalizer_eq_of_injective_of_le_range + - div_mem + - Subgroup.disjoint_iff_mul_eq_one + - Subgroup.coe_iSup_of_directed + - Subgroup.mem_closure_pair + - Subgroup.center_le_normalizer + - Subgroup.mem_iInf + - Subgroup.mem_iSup_of_directed + - path: Mathlib/Topology/Algebra/InfiniteSum/Group.lean + theorems: + - summable_const_iff + - HasSum.hasSum_iff_compl + - Summable.trans_sub + - Summable.countable_support + - tsum_eq_add_tsum_ite + - Summable.summable_of_eq_zero_or_self + - tsum_neg + - Finset.hasSum_compl_iff + - summable_iff_cauchySeq_finset + - sum_add_tsum_subtype_compl + - Summable.vanishing + - HasSum.sub + - HasSum.update + - summable_iff_of_summable_sub + - Summable.tsum_vanishing + - summable_iff_vanishing + - hasSum_ite_sub_hasSum + - HasSum.hasSum_compl_iff + - Summable.comp_injective + - cauchySeq_finset_iff_vanishing + - summable_iff_tsum_vanishing + - HasSum.neg + - cauchySeq_finset_iff_tsum_vanishing + - tendsto_tsum_compl_atTop_zero + - tsum_const + - Summable.tendsto_cofinite_zero + - Summable.of_neg + - path: Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean + theorems: + - "TensorAlgebra.\u03B9_range_disjoint_one" + - TensorAlgebra.induction + - "FreeAlgebra.toTensor_\u03B9" + - "TensorAlgebra.lift_\u03B9_apply" + - "TensorAlgebra.\u03B9_leftInverse" + - TensorAlgebra.algebraMap_leftInverse + - "TensorAlgebra.\u03B9_eq_algebraMap_iff" + - TensorAlgebra.hom_ext + - "TensorAlgebra.lift_comp_\u03B9" + - "TensorAlgebra.\u03B9_eq_zero_iff" + - "TensorAlgebra.\u03B9_comp_lift" + - "TensorAlgebra.\u03B9_ne_one" + - TensorAlgebra.lift_unique + - path: Mathlib/CategoryTheory/Types.lean + theorems: + - CategoryTheory.epi_iff_surjective + - CategoryTheory.mono_iff_injective + - CategoryTheory.FunctorToTypes.eqToHom_map_comp_apply + - CategoryTheory.homOfElement_eq_iff + - CategoryTheory.FunctorToTypes.map_comp_apply + - CategoryTheory.types_ext + - CategoryTheory.FunctorToTypes.map_id_apply + - path: Mathlib/Order/Filter/Basic.lean + theorems: + - Filter.biInter_mem + - Filter.disjoint_comap_iff + - Filter.mem_iInf_of_directed + - Filter.comap_pure + - Filter.map_biInf_eq + - Filter.mem_iInf + - Filter.frequently_imp_distrib_right + - Filter.map_inf' + - Filter.EventuallyLE.mul_nonneg + - Filter.eventually_const + - Pairwise.exists_mem_filter_of_disjoint + - Filter.frequently_or_distrib + - Filter.tendsto_principal + - Filter.pure_bind + - Filter.bind_mono + - Filter.comap_sup + - Filter.tendsto_map'_iff + - Filter.tendsto_iSup + - Filter.iInf_sets_eq_finite' + - Filter.not_disjoint_self_iff + - Filter.frequently_imp_distrib + - Filter.comap_surjective_eq_bot + - Filter.tendsto_comap_iff + - Filter.range_mem_map + - Filter.tendsto_iInf + - Filter.comap_comm + - Filter.map_le_map_iff + - Filter.eventually_all_finite + - Filter.principal_singleton + - Filter.frequently_or_distrib_right + - Filter.comap_sSup + - Filter.Eventually.and_frequently + - Filter.compl_mem_kernMap + - Filter.frequently_iff_neBot + - Filter.EventuallyLE.mul_le_mul' + - Filter.comap_eq_of_inverse + - Filter.sInter_mem + - Filter.isCompl_principal + - Filter.Frequently.exists + - Filter.sup_neBot + - Filter.mem_iInf_of_finite + - Filter.frequently_false + - Filter.EventuallyEq.sub_eq + - Filter.inf_principal_eq_bot + - Filter.mem_iSup + - Filter.sInter_comap_sets + - Set.Finite.eventuallyLE_iUnion + - Filter.comap_eval_neBot_iff' + - Filter.map_top + - Set.PairwiseDisjoint.exists_mem_filter + - Filter.eq_biInf_of_mem_iff_exists_mem + - Filter.mem_iInf_of_iInter + - Filter.mem_inf_principal' + - Filter.image_mem_map_iff + - Filter.comap_eval_neBot_iff + - Filter.map_comap + - Filter.eventually_or_distrib_right + - Filter.map_eq_bot_iff + - Filter.principal_univ + - Filter.frequently_iff + - Filter.frequently_iSup + - Filter.map_eq_of_inverse + - Filter.tendsto_pure + - Filter.mem_iInf' + - Filter.push_pull + - Filter.neBot_of_comap + - Filter.mem_of_eq_bot + - Filter.not_frequently + - Filter.not_le + - Filter.eventuallyLE_antisymm_iff + - Filter.kernMap_principal + - Filter.mem_comap_iff_compl + - Filter.map_comm + - Filter.compl_mem_comk + - Filter.comap_le_iff_le_kernMap + - Filter.mem_generate_iff + - Filter.frequently_const + - Filter.comap_map + - Filter.seq_pure + - Filter.eventuallyEq_iff_sub + - Filter.principal_eq_iff_eq + - Filter.iInf_neBot_of_directed' + - Filter.le_pure_iff + - Filter.comap_inf_principal_range + - Filter.frequently_sup + - Filter.mem_comap_iff + - Filter.tendsto_congr' + - Filter.biInf_sets_eq + - Filter.le_map_of_right_inverse + - Filter.frequently_iff_forall_eventually_exists_and + - Filter.tendsto_inf + - Filter.pure_seq_eq_map + - Filter.inter_eventuallyEq_left + - Filter.comap_inf_principal_neBot_of_image_mem + - Filter.not_tendsto_iff_exists_frequently_nmem + - Filter.eventually_or_distrib_left + - Filter.iSup_principal + - Filter.disjoint_iff + - Filter.tendsto_of_isEmpty + - Filter.EventuallyLE.mul_le_mul + - Filter.eventually_all + - Filter.frequently_and_distrib_right + - Filter.iInf_sets_eq + - Filter.tendsto_comap'_iff + - Filter.frequently_true_iff_neBot + - Filter.inf_principal + - Filter.disjoint_comap + - Filter.le_comap_top + - Filter.EventuallyEq.prod_mk + - Filter.iSup_inf_principal + - Filter.set_eventuallyLE_iff_inf_principal_le + - Filter.eq_top_of_neBot + - Filter.image_mem_of_mem_comap + - Filter.map_comap_setCoe_val + - Filter.tendsto_principal_principal + - Filter.eventually_congr + - Filter.push_pull' + - Filter.seq_assoc + - Filter.iSup_neBot + - Filter.eventually_imp_distrib_right + - Filter.mem_seq_iff + - Filter.ext_iff + - Filter.image_coe_mem_of_mem_comap + - Filter.principal_bind + - Filter.eventuallyEq_univ + - Filter.frequently_bot + - Filter.EventuallyLE.sup_le + - Filter.generate_eq_biInf + - Filter.mem_top + - Filter.iInf_principal_finset + - Filter.Tendsto.of_tendsto_comp + - Filter.map_inf + - Filter.comap_bot + - Filter.mem_biInf_of_directed + - Filter.comap_neBot_iff_frequently + - Filter.subtype_coe_map_comap + - Filter.sup_principal + - Filter.frequently_sSup + - Filter.frequently_top + - Filter.iInf_sets_induct + - Filter.disjoint_map + - Filter.map_inf_principal_preimage + - Filter.mem_kernMap_iff_compl + - Filter.map_comap_of_mem + - Filter.Frequently.and_eventually + - Filter.principal_le_iff + - Filter.comap_fst_neBot_iff + - Filter.comap_snd_neBot_iff + - Filter.map_neBot_iff + - Filter.mem_comap_prod_mk + - Filter.map_iInf_eq + - Filter.frequently_comap + - Filter.map_comap_of_surjective + - Filter.EventuallyLE.sup + - Filter.iInf_sets_eq_finite + - Filter.bind_inf_principal + - Filter.frequently_or_distrib_left + - Filter.set_eventuallyEq_iff_inf_principal + - Filter.frequently_principal + - Filter.map_equiv_symm + - Filter.principal_eq_map_coe_top + - Filter.tendsto_sup + - Filter.Eventually.choice + - Filter.principal_mono + - Filter.mem_inf_principal + - Filter.iInf_principal + - Filter.frequently_imp_distrib_left + - Filter.Tendsto.if + - Filter.inf_principal_eq_bot_iff_comap + - Filter.tendsto_bot + - Filter.eventuallyEq_empty + - Filter.exists_mem_and_iff + - Filter.map_eq_map_iff_of_injOn + - Filter.mem_iInf_finset + - Filter.mem_comap' + - Filter.inter_eventuallyEq_right + - Filter.compl_mem_comap + - Set.Finite.eventuallyLE_iInter + - Filter.inf_eq_bot_iff + - path: Mathlib/LinearAlgebra/Orientation.lean + theorems: + - Orientation.map_of_isEmpty + - Orientation.map_refl + - Orientation.map_eq_iff_det_pos + - Basis.det_adjustToOrientation + - Basis.orientation_comp_linearEquiv_eq_iff_det_pos + - Orientation.eq_or_eq_neg_of_isEmpty + - Basis.orientation_map + - Basis.map_orientation_eq_det_inv_smul + - Basis.orientation_neg_single + - Orientation.reindex_refl + - Basis.orientation_isEmpty + - Basis.orientation_adjustToOrientation + - Orientation.eq_or_eq_neg + - Orientation.map_eq_neg_iff_det_neg + - Basis.orientation_eq_or_eq_neg + - Basis.orientation_comp_linearEquiv_eq_neg_iff_det_neg + - Basis.abs_det_adjustToOrientation + - Basis.orientation_reindex + - Basis.adjustToOrientation_apply_eq_or_eq_neg + - Basis.orientation_eq_iff_det_pos + - path: Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean + theorems: + - CategoryTheory.GrothendieckTopology.Plus.toPlus_mk + - CategoryTheory.GrothendieckTopology.Plus.isSheaf_plus_plus + - CategoryTheory.GrothendieckTopology.Plus.sep + - CategoryTheory.Meq.equiv_symm_eq_apply + - CategoryTheory.GrothendieckTopology.Plus.toPlus_apply + - CategoryTheory.GrothendieckTopology.sheafifyMap_comp + - CategoryTheory.GrothendieckTopology.Plus.inj_of_sep + - CategoryTheory.GrothendieckTopology.Plus.exists_of_sep + - CategoryTheory.GrothendieckTopology.Plus.res_mk_eq_mk_pullback + - CategoryTheory.GrothendieckTopology.Plus.isSheaf_of_sep + - CategoryTheory.GrothendieckTopology.sheafifyMap_id + - CategoryTheory.GrothendieckTopology.sheafifyMap_sheafifyLift + - CategoryTheory.GrothendieckTopology.Plus.eq_mk_iff_exists + - CategoryTheory.GrothendieckTopology.isoSheafify_inv + - CategoryTheory.GrothendieckTopology.Plus.toPlus_eq_mk + - CategoryTheory.GrothendieckTopology.sheafify_hom_ext + - CategoryTheory.GrothendieckTopology.toSheafify_sheafifyLift + - CategoryTheory.GrothendieckTopology.isIso_toSheafify + - CategoryTheory.GrothendieckTopology.sheafifyLift_unique + - CategoryTheory.Sheaf.Hom.mono_iff_presheaf_mono + - CategoryTheory.GrothendieckTopology.toSheafify_naturality + - CategoryTheory.GrothendieckTopology.Plus.exists_rep + - path: Mathlib/Analysis/Convex/Deriv.lean + theorems: + - StrictMonoOn.strictConvexOn_of_deriv + - convexOn_of_hasDerivWithinAt2_nonneg + - concaveOn_of_hasDerivWithinAt2_nonpos + - StrictMonoOn.exists_slope_lt_deriv + - StrictMonoOn.exists_deriv_lt_slope + - MonotoneOn.convexOn_of_deriv + - concaveOn_of_deriv2_nonpos + - strictConvexOn_of_deriv2_pos + - StrictAntiOn.strictConcaveOn_of_deriv + - StrictMonoOn.exists_deriv_lt_slope_aux + - StrictMonoOn.exists_slope_lt_deriv_aux + - convexOn_of_deriv2_nonneg + - strictConcaveOn_of_deriv2_neg + - AntitoneOn.concaveOn_of_deriv + - path: Mathlib/FieldTheory/Finite/Basic.lean + theorems: + - FiniteField.sum_subgroup_pow_eq_zero + - Nat.sq_add_sq_modEq + - ZMod.frobenius_zmod + - FiniteField.exists_root_sum_quadratic + - CharP.sq_add_sq + - FiniteField.sum_subgroup_units + - FiniteField.card_cast_subgroup_card_ne_zero + - FiniteField.isSquare_iff + - ZMod.card_units + - FiniteField.sum_pow_lt_card_sub_one + - FiniteField.pow_card + - FiniteField.expand_card + - FiniteField.cast_card_eq_zero + - FiniteField.unit_isSquare_iff + - FiniteField.pow_card_sub_one_eq_one + - FiniteField.frobenius_pow + - FiniteField.card_image_polynomial_eval + - Int.ModEq.pow_card_sub_one_eq_one + - FiniteField.prod_univ_units_id_eq_neg_one + - ZMod.sq_add_sq + - FiniteField.X_pow_card_sub_X_natDegree_eq + - FiniteField.card + - ZMod.units_pow_card_sub_one_eq_one + - FiniteField.sum_pow_units + - ZMod.expand_card + - FiniteField.roots_X_pow_card_sub_X + - ZMod.pow_totient + - ZMod.pow_card + - card_eq_pow_finrank + - FiniteField.pow_card_pow + - ZMod.pow_card_pow + - FiniteField.forall_pow_eq_one_iff + - FiniteField.pow_dichotomy + - Nat.ModEq.pow_totient + - FiniteField.exists_nonsquare + - FiniteField.sum_subgroup_units_eq_zero + - Nat.sq_add_sq_zmodEq + - FiniteField.even_card_iff_char_two + - path: Mathlib/Analysis/SumOverResidueClass.lean + theorems: + - not_summable_indicator_mod_of_antitone_of_neg + - summable_indicator_mod_iff_summable_indicator_mod + - not_summable_of_antitone_of_neg + - Finset.sum_indicator_mod + - summable_indicator_mod_iff_summable + - summable_indicator_mod_iff + - path: Mathlib/Analysis/Normed/Group/AddCircle.lean + theorems: + - AddCircle.norm_eq_of_zero + - AddCircle.le_add_order_smul_norm_of_isOfFinAddOrder + - AddCircle.norm_le_half_period + - UnitAddCircle.norm_eq + - AddCircle.coe_real_preimage_closedBall_period_zero + - AddCircle.coe_real_preimage_closedBall_inter_eq + - AddCircle.norm_eq' + - AddCircle.norm_half_period_eq + - AddCircle.coe_real_preimage_closedBall_eq_iUnion + - AddCircle.norm_coe_mul + - AddCircle.exists_norm_eq_of_isOfFinAddOrder + - AddCircle.norm_neg_period + - AddCircle.closedBall_eq_univ_of_half_period_le + - AddCircle.norm_eq + - AddCircle.norm_div_nat_cast + - path: Mathlib/Combinatorics/SimpleGraph/Turan.lean + theorems: + - SimpleGraph.turanGraph_zero + - SimpleGraph.not_cliqueFree_of_isTuranMaximal + - SimpleGraph.turanGraph_cliqueFree + - SimpleGraph.turanGraph_eq_top + - SimpleGraph.IsTuranMaximal.le_iff_eq + - path: .lake/packages/std/Std/Data/UInt.lean + theorems: + - UInt8.toUInt32_toNat + - UInt8.toUInt16_toNat + - USize.toUInt64_toNat + - UInt16.toUInt64_toNat + - UInt16.toUInt32_toNat + - UInt32.toUInt64_toNat + - USize.size_eq + - USize.le_size + - UInt8.toUInt64_toNat + - USize.size_le + - USize.toNat_lt + - path: Mathlib/MeasureTheory/MeasurableSpace/Basic.lean + theorems: + - MeasurableSet.image_inclusion' + - measurable_to_countable + - measurableSet_pi_of_nonempty + - measurable_indicator_const_iff + - MeasurableEquiv.measurableSet_preimage + - MeasurableSet.subtype_image + - MeasurableSet.pi + - MeasurableSpace.map_const + - exists_measurable_piecewise + - measurableSet_range_inr + - measurable_to_prop + - measurable_update' + - Measurable.piecewise + - MeasurableSet.tProd + - Filter.principal_isMeasurablyGenerated_iff + - MeasurableEquiv.coe_sumPiEquivProdPi_symm + - MeasurableSpace.generateFrom_singleton + - MeasurableEquiv.measurable_comp_iff + - MeasurableEmbedding.measurableSet_image + - MeasurableSet.measurableSet_blimsup + - measurable_eq_mp + - measurableSet_setOf + - MeasurableSet.measurableSet_liminf + - measurable_const' + - measurable_updateFinset + - Measurable.iff + - MeasurableSet.measurableSet_bliminf + - MeasurableEmbedding.measurable_rangeSplitting + - measurable_iUnionLift + - measurableSet_prod_of_nonempty + - measurable_findGreatest + - measurable_piCongrLeft + - measurableSet_pi + - measurableSet_inl_image + - MeasurableEmbedding.comp + - Measurable.forall + - measurable_to_bool + - measurable_uniqueElim + - MeasurableEquiv.coe_sumPiEquivProdPi + - MeasurableEquiv.preimage_image + - measurable_pi_iff + - MeasurableEquiv.coe_piCongrLeft + - MeasurableEmbedding.measurable_extend + - MeasurableEquiv.toEquiv_injective + - measurable_tProd_mk + - measurable_piEquivPiSubtypeProd_symm + - MeasurableEmbedding.measurableSet_range + - MeasurableEmbedding.id + - MeasurableEquiv.image_symm + - MeasurableEquiv.measurableSet_image + - measurableSet_prod + - MeasurableEquiv.image_preimage + - exists_measurable_piecewise_nat + - MeasurableSpace.comap_const + - Measurable.measurable_of_countable_ne + - Measurable.dite + - MeasurableSet.measurableSet_limsup + - measurable_from_prod_countable + - MeasurableEmbedding.of_measurable_inverse_on_range + - measurable_liftCover + - MeasurableEquiv.eq_image_iff_symm_image_eq + - MeasurableEmbedding.measurableSet_preimage + - Measurable.prod + - Measurable.exists + - measurableSet_inr_image + - MeasurableEmbedding.measurable_comp_iff + - measurable_find + - measurable_tProd_elim + - MeasurableSet.of_union_cover + - path: Mathlib/Data/Polynomial/Degree/Definitions.lean + theorems: + - Polynomial.degree_of_subsingleton + - Polynomial.degree_X_add_C + - Polynomial.degree_linear_lt_degree_C_mul_X_sq + - Polynomial.degree_le_degree + - Polynomial.degree_sub_C + - Polynomial.as_sum_range_C_mul_X_pow + - Polynomial.degree_cubic + - Polynomial.leadingCoeff_X_pow_add_C + - Polynomial.natDegree_X_pow_sub_C + - Polynomial.leadingCoeff_monomial + - Polynomial.natDegree_monomial + - Polynomial.leadingCoeff_add_of_degree_lt' + - Polynomial.monomial_natDegree_leadingCoeff_eq_self + - Polynomial.card_supp_le_succ_natDegree + - Polynomial.degree_smul_le + - Polynomial.degree_monomial + - Polynomial.natDegree_mul' + - Polynomial.natDegree_neg + - Polynomial.natDegree_int_cast + - Polynomial.natDegree_X_mul + - Polynomial.leadingCoeff_X_sub_C + - Polynomial.degree_sub_lt + - Polynomial.nextCoeff_X_sub_C + - Polynomial.natDegree_sub_C + - Polynomial.degree_quadratic_lt_degree_C_mul_X_cb + - Polynomial.degree_nat_cast_le + - Polynomial.natDegree_mul_le + - Polynomial.natDegree_lt_natDegree_iff + - Polynomial.degree_C_le + - Polynomial.leadingCoeff_linear + - Polynomial.natDegree_le_pred + - Polynomial.natDegree_linear + - Polynomial.eq_one_of_monic_natDegree_zero + - Polynomial.leadingCoeff_neg + - Polynomial.degree_eq_iff_natDegree_eq + - Polynomial.degree_C_mul_X_pow + - Polynomial.monic_of_natDegree_le_of_coeff_eq_one + - Polynomial.degree_sum_le + - Polynomial.natDegree_eq_zero + - Polynomial.natDegree_pow_le + - Polynomial.natDegree_of_subsingleton + - Polynomial.X_pow_add_C_ne_zero + - Polynomial.degree_pow' + - Polynomial.degree_add_le + - Polynomial.natDegree_eq_of_natDegree_add_eq_zero + - Polynomial.degree_update_le + - Polynomial.degree_C_mul_X_le + - Polynomial.natDegree_eq_of_natDegree_add_lt_left + - Polynomial.ne_zero_of_degree_ge_degree + - Polynomial.leadingCoeff_add_of_degree_lt + - Polynomial.le_natDegree_of_ne_zero + - Polynomial.natDegree_X_sub_C + - Polynomial.natDegree_mul_X + - Polynomial.degree_mul_le + - Polynomial.degree_quadratic + - Polynomial.le_degree_of_ne_zero + - Polynomial.eq_X_add_C_of_degree_le_one + - Polynomial.leadingCoeff_X_add_C + - Polynomial.natDegree_eq_of_degree_eq_some + - Polynomial.natDegree_monomial_le + - Polynomial.leadingCoeff_sub_of_degree_lt + - Polynomial.natDegree_add_C + - Polynomial.degree_lt_degree + - Polynomial.leadingCoeff_mul + - Polynomial.degree_pow_le + - Polynomial.leadingCoeff_sub_of_degree_lt' + - Polynomial.nextCoeff_X_add_C + - Polynomial.leadingCoeff_C_mul_X_pow + - Polynomial.natDegree_add_le + - Polynomial.degree_mul + - Polynomial.nextCoeff_eq_zero + - Polynomial.le_natDegree_of_coe_le_degree + - Polynomial.nextCoeff_C_eq_zero + - Polynomial.leadingCoeff_sub_of_degree_eq + - Polynomial.natDegree_sub_le + - Polynomial.eq_C_of_degree_le_zero + - Polynomial.X_pow_sub_C_ne_zero + - Polynomial.Monic.eq_X_add_C + - Polynomial.as_sum_support_C_mul_X_pow + - Polynomial.degree_C_mul_X_pow_le + - Polynomial.Monic.degree_mul + - Polynomial.natDegree_mul_X_pow + - Polynomial.degree_one_le + - Polynomial.degree_monomial_le + - Polynomial.leadingCoeff_X + - Polynomial.ext_iff_natDegree_le + - Polynomial.degree_sub_le + - Polynomial.card_support_C_mul_X_pow_le_one + - Polynomial.leadingCoeff_X_pow + - Polynomial.nextCoeff_ne_zero + - Polynomial.coeff_pow_mul_natDegree + - Polynomial.coeff_eq_zero_of_natDegree_lt + - Polynomial.leadingCoeff_X_pow_sub_C + - Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot + - Polynomial.Monic.ne_zero_of_ne + - Polynomial.leadingCoeff_C_mul_X + - Polynomial.degree_lt_iff_coeff_zero + - Polynomial.degree_C + - Polynomial.leadingCoeff_monic_mul + - Polynomial.natDegree_X_pow_le + - Polynomial.degree_sub_eq_left_of_degree_lt + - Polynomial.degree_quadratic_le + - Polynomial.natDegree_eq_of_degree_eq + - Polynomial.leadingCoeff_add_of_degree_eq + - Polynomial.degree_lt_degree_mul_X + - Polynomial.degree_pow_le_of_le + - Polynomial.not_isUnit_X + - Polynomial.leadingCoeff_cubic + - Polynomial.natDegree_C + - Polynomial.degree_mul' + - Polynomial.degree_eq_iff_natDegree_eq_of_pos + - Polynomial.degree_X_pow_sub_C + - Polynomial.natDegree_eq_zero_iff_degree_le_zero + - Polynomial.sum_over_range' + - Polynomial.leadingCoeff_mul_monic + - Polynomial.degree_C_lt_degree_C_mul_X + - Polynomial.degree_neg + - Polynomial.ite_le_natDegree_coeff + - Polynomial.leadingCoeff_pow_X_add_C + - Polynomial.C_mul_X_pow_eq_self + - Polynomial.natDegree_mem_support_of_nonzero + - Polynomial.nextCoeff_of_natDegree_pos + - Polynomial.degree_X_pow + - Polynomial.degree_add_eq_of_leadingCoeff_add_ne_zero + - Polynomial.degree_add_eq_left_of_degree_lt + - Polynomial.degree_add_eq_right_of_degree_lt + - Polynomial.eq_C_coeff_zero_iff_natDegree_eq_zero + - Polynomial.coeff_mul_X_sub_C + - Polynomial.natDegree_pow' + - Polynomial.degree_eq_natDegree + - Polynomial.degree_mul_X_pow + - Polynomial.sum_fin + - Polynomial.degree_int_cast_le + - Polynomial.supDegree_eq_natDegree + - Polynomial.degree_cubic_le + - Polynomial.degree_erase_le + - Polynomial.nonempty_support_iff + - Polynomial.natDegree_C_mul_X + - Polynomial.degree_erase_lt + - Polynomial.degree_X_pow_add_C + - Polynomial.degree_sub_eq_right_of_degree_lt + - Polynomial.leadingCoeff_pow' + - Polynomial.natDegree_C_add + - Polynomial.natDegree_X_pow_mul + - Polynomial.natDegree_lt_natDegree + - Polynomial.degree_X_pow_le + - Polynomial.degree_C_mul_X + - Polynomial.degree_le_iff_coeff_zero + - Polynomial.leadingCoeff_ne_zero + - Polynomial.natDegree_nat_cast + - Polynomial.coeff_mul_degree_add_degree + - Polynomial.coeff_mul_add_eq_of_natDegree_le + - Polynomial.Monic.ne_zero + - Polynomial.natDegree_X_pow_add_C + - Polynomial.zero_le_degree_iff + - Polynomial.leadingCoeff_mul' + - Polynomial.degree_mul_X + - path: Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean + theorems: + - "ChainComplex.isoHomology\u03B9\u2080_inv_naturality" + - HomologicalComplex.homologyMap_comp + - "HomologicalComplex.isIso_homology\u03B9" + - HomologicalComplex.homologyMap_neg + - "HomologicalComplex.toCycles_comp_homology\u03C0" + - HomologicalComplex.isIso_pOpcycles + - "HomologicalComplex.homologyIsoSc'_inv_\u03B9" + - HomologicalComplex.liftCycles_comp_cyclesMap + - ChainComplex.isIso_descOpcycles_iff + - HomologicalComplex.descOpcycles_comp + - HomologicalComplex.isIso_iCycles + - HomologicalComplex.cyclesMap_comp + - HomologicalComplex.opcyclesMap_comp_descOpcycles + - HomologicalComplex.homologyMap_add + - "CochainComplex.isoHomology\u03C0\u2080_inv_naturality" + - HomologicalComplex.opcyclesMap_comp + - HomologicalComplex.opcyclesIsoSc'_inv_fromOpcycles + - HomologicalComplex.fromOpcycles_d + - "HomologicalComplex.\u03C0_homologyIsoSc'_inv" + - HomologicalComplex.cyclesIsoSc'_inv_iCycles + - CochainComplex.isIso_liftCycles_iff + - HomologicalComplex.cyclesIsoSc'_hom_iCycles + - HomologicalComplex.toCycles_cyclesIsoSc'_hom + - HomologicalComplex.exactAt_iff + - "HomologicalComplex.liftCycles_homology\u03C0_eq_zero_of_boundary" + - HomologicalComplex.d_toCycles + - "HomologicalComplex.isIso_homology\u03C0" + - "HomologicalComplex.homology\u03B9_comp_fromOpcycles" + - HomologicalComplex.exactAt_iff_isZero_homology + - HomologicalComplex.iCycles_d + - "HomologicalComplex.homologyIsoSc'_hom_\u03B9" + - HomologicalComplex.liftCycles_i + - HomologicalComplex.comp_liftCycles + - HomologicalComplex.homologyMap_sub + - HomologicalComplex.p_descOpcycles + - HomologicalComplex.pOpcycles_opcyclesIsoSc'_hom + - "HomologicalComplex.\u03C0_homologyIsoSc'_hom" + - HomologicalComplex.pOpcycles_opcyclesIsoSc'_inv + - "HomologicalComplex.homology\u03B9_descOpcycles_eq_zero_of_boundary" + - path: Mathlib/Analysis/Calculus/BumpFunction/Basic.lean + theorems: + - ContDiffBump.one_lt_rOut_div_rIn + - ContDiffBump.support_eq + - ContDiffBump.hasCompactSupport + - ContDiffBump.tsupport_eq + - ContDiffBump.sub + - ContDiffBump.neg + - ContDiffBump.one_of_mem_closedBall + - ContDiff.contDiffBump + - ContDiffBump.zero_of_le_dist + - ContDiffWithinAt.contDiffBump + - ContDiffBump.pos_of_mem_ball + - path: Mathlib/Deprecated/Subgroup.lean + theorems: + - IsSubgroup.mem_norm_comm + - Group.mem_closure_union_iff + - IsSubgroup.center_normal + - IsGroupHom.image_subgroup + - IsSubgroup.mul_mem_cancel_left + - IsSubgroup.trivial_normal + - IsSubgroup.normalizer_isSubgroup + - Group.image_closure + - Group.exists_list_of_mem_closure + - Additive.isNormalAddSubgroup_iff + - IsSubgroup.of_div + - IsGroupHom.inv_ker_one' + - Additive.isAddSubgroup_iff + - IsGroupHom.inv_iff_ker' + - IsGroupHom.preimage_normal + - IsSubgroup.univ_subgroup + - IsSubgroup.trivial_eq_closure + - IsSubgroup.subset_normalizer + - IsGroupHom.one_ker_inv + - IsGroupHom.trivial_ker_iff_eq_one + - IsGroupHom.inv_ker_one + - Group.conjugatesOf_subset + - IsGroupHom.preimage + - Group.closure_subset + - Multiplicative.isSubgroup_iff + - IsGroupHom.trivial_ker_of_injective + - IsSubgroup.eq_trivial_iff + - IsGroupHom.one_ker_inv' + - IsGroupHom.inv_iff_ker + - IsSubgroup.inv_mem_iff + - IsSubgroup.div_mem + - Multiplicative.isNormalSubgroup_iff + - IsSubgroup.mul_mem_cancel_right + - IsAddSubgroup.of_sub + - isNormalSubgroup_of_commGroup + - Group.normalClosure.is_normal + - path: Mathlib/Algebra/Parity.lean + theorems: + - Even.sub_odd + - IsSquare.pow + - even_neg_two + - Odd.pow_nonpos + - even_two + - IsSquare.zpow + - Odd.mul + - Odd.add_even + - IsSquare.map + - Odd.sub_even + - Odd.pow + - Even.neg_one_pow + - Even.tsub + - Even.neg_pow + - odd_add_one_self + - Even.add_odd + - Odd.strictMono_pow + - Odd.neg_one_pow + - isSquare_iff_exists_sq + - even_iff_exists_two_mul + - Even.pow_of_ne_zero + - Odd.sub_odd + - isSquare_inv + - Even.pow_pos_iff + - IsSquare.div + - even_abs + - Even.odd_add + - Odd.map + - even_iff_two_dvd + - Odd.pow_neg + - odd_neg_one + - Odd.pos + - Odd.neg_pow + - Even.pow_abs + - odd_abs + - odd_add_self_one' + - Even.isSquare_pow + - one_add_self_self + - Even.pow_pos + - odd_iff_exists_bit1 + - range_two_mul_add_one + - Even.pow_nonneg + - IsSquare.mul + - Even.neg_one_zpow + - Even.isSquare_zpow + - odd_add_one_self' + - Odd.neg + - Even.neg_zpow + - Odd.add_odd + - path: Mathlib/Algebra/Order/Field/Basic.lean + theorems: + - sub_inv_antitoneOn_Icc_left + - sub_one_div_inv_le_two + - sub_inv_antitoneOn_Icc_right + - IsLUB.mul_left + - inv_le_inv_of_neg + - one_div_lt_neg_one + - lt_one_div + - add_halves + - div_le_div_iff + - one_div_le + - div_lt_iff' + - inv_le + - inv_lt_one + - div_lt_one + - abs_one_div + - div_lt_div_of_pos_right + - left_lt_add_div_two + - le_inv + - mul_le_mul_of_mul_div_le + - le_one_div_of_neg + - inv_antitoneOn_Icc_right + - div_lt_one_of_neg + - mul_sub_mul_div_mul_nonpos_iff + - one_le_div + - mul_inv_le_one_of_le + - mul_inv_lt_iff' + - div_le_self + - div_le_one_of_le + - exists_pos_lt_mul + - inv_pos_lt_iff_one_lt_mul + - inv_mul_le_iff' + - IsGLB.mul_right + - div_le_one + - div_le_iff_of_neg' + - one_div_lt_one_div_of_lt + - inv_mul_le_one_of_le + - le_one_div + - one_div_le_of_neg + - div_nonneg_iff + - one_div_le_neg_one + - one_div_le_one_div_of_neg_of_le + - div_pos_iff_of_pos_left + - add_self_div_two + - one_le_div_of_neg + - one_div_le_one_div_of_le + - add_sub_div_two_lt + - div_lt_self + - div_le_div_of_nonneg_left + - div_le_one_of_ge + - div_le_of_nonneg_of_le_mul + - one_div_le_one_div_of_neg + - inv_antitoneOn_Icc_left + - one_div_lt + - le_div_self + - div_le_div + - div_lt_div_of_neg_of_lt + - exists_pos_mul_lt + - one_div_pow_le_one_div_pow_of_le + - inv_pos_le_iff_one_le_mul + - inv_mul_le_iff + - div_lt_iff_of_neg' + - inv_le_of_neg + - div_self_le_one + - inv_mul_lt_iff' + - inv_antitoneOn_Ioi + - one_div_pow_lt_one_div_pow_of_lt + - one_lt_inv + - add_thirds + - one_lt_div + - one_div_lt_of_neg + - le_inv_of_neg + - inv_antitoneOn_Iio + - one_le_inv + - StrictMono.div_const + - le_div_iff' + - mul_self_inj_of_nonneg + - div_nonpos_iff + - half_lt_self_iff + - inv_lt_one_iff + - div_lt_div_of_pos_left + - lt_div_iff_of_neg' + - IsLUB.mul_right + - mul_inv_le_iff' + - lt_div_iff' + - le_of_forall_sub_le + - IsGLB.mul_left + - div_two_sub_self + - inv_le_one_iff + - div_le_iff + - div_le_div_of_nonpos_of_le + - div_le_div_of_nonneg_right + - div_le_iff' + - mul_sub_mul_div_mul_neg_iff + - mul_inv_le_iff + - one_lt_one_div + - inv_pow_le_inv_pow_of_le + - one_le_div_iff + - div_lt_div_iff + - one_lt_div_iff + - mul_le_of_nonneg_of_le_div + - lt_one_div_of_neg + - inv_pos_le_iff_one_le_mul' + - le_div_iff_of_neg + - inv_le_inv_of_le + - one_div_lt_one_div_of_neg_of_lt + - div_neg_iff + - div_pos_iff_of_pos_right + - inv_pow_lt_inv_pow_of_lt + - div_lt_one_iff + - add_div_two_lt_right + - sub_self_div_two + - div_le_one_of_neg + - inv_pos_lt_iff_one_lt_mul' + - half_le_self_iff + - mul_inv_lt_iff + - inv_mul_lt_iff + - div_le_one_iff + - one_lt_div_of_neg + - div_lt_div_left + - path: Mathlib/Order/SupIndep.lean + theorems: + - CompleteLattice.SetIndependent.disjoint_sSup + - CompleteLattice.independent_of_independent_coe_Iic_comp + - Finset.supIndep_pair + - CompleteLattice.independent_def'' + - Finset.supIndep_product_iff + - CompleteLattice.Independent.setIndependent_range + - Finset.SupIndep.le_sup_iff + - CompleteLattice.setIndependent_iff + - CompleteLattice.independent_pair + - Finset.SupIndep.sigma + - Finset.supIndep_singleton + - Finset.supIndep_attach + - Finset.SupIndep.sup + - CompleteLattice.independent_def' + - CompleteLattice.Independent.injOn + - CompleteLattice.Independent.injective + - Finset.SupIndep.biUnion + - CompleteLattice.setIndependent_pair + - CompleteLattice.Independent.comp' + - CompleteLattice.Independent.comp + - Finset.supIndep_map + - CompleteLattice.independent_iff_supIndep + - Finset.SupIndep.attach + - CompleteLattice.independent_ne_bot_iff_independent + - Finset.SupIndep.image + - Finset.supIndep_univ_fin_two + - Finset.SupIndep.product + - path: Mathlib/Data/IsROrC/Basic.lean + theorems: + - IsROrC.neg_iff_exists_ofReal + - IsROrC.mul_self_norm + - IsROrC.conj_mul + - IsROrC.one_im + - IsROrC.inv_I + - IsROrC.re_ofReal_mul + - IsROrC.inv_def + - IsROrC.re_eq_norm_of_mul_conj + - IsROrC.normSq_neg + - IsROrC.abs_im_le_norm + - IsROrC.inv_re + - IsROrC.im_eq_zero + - IsROrC.nonpos_iff + - IsROrC.isCauSeq_im + - IsROrC.normSq_to_real + - IsROrC.ofNat_mul_im + - IsROrC.norm_I_of_ne_zero + - IsROrC.smul_im + - IsROrC.bit1_im + - IsROrC.exists_norm_mul_eq_self + - IsROrC.norm_sq_re_add_conj + - IsROrC.intCast_im + - IsROrC.conj_ofReal + - IsROrC.normSq_sub + - IsROrC.pos_iff + - IsROrC.smul_re + - IsROrC.conj_neg_I + - IsROrC.norm_conj + - IsROrC.im_eq_zero_of_le + - IsROrC.bit1_re + - IsROrC.abs_re_le_norm + - IsROrC.lt_iff_re_im + - IsROrC.sub_conj + - IsROrC.im_eq_conj_sub + - IsROrC.nonneg_iff_exists_ofReal + - IsROrC.I_eq_zero_or_im_I_eq_one + - IsROrC.conj_eq_re_sub_im + - IsROrC.re_eq_self_of_le + - IsROrC.isCauSeq_re + - IsROrC.abs_im_div_norm_le_one + - IsROrC.div_re_ofReal + - IsROrC.ratCast_re + - IsROrC.conj_eq_iff_real + - IsROrC.neg_iff + - IsROrC.ofNat_mul_re + - IsROrC.real_smul_eq_coe_smul + - IsROrC.inv_eq_conj + - IsROrC.abs_re_div_norm_le_one + - IsROrC.add_conj + - IsROrC.I_mul_re + - IsROrC.norm_sq_re_conj_add + - IsROrC.intCast_re + - IsROrC.normSq_pos + - IsROrC.is_real_TFAE + - IsROrC.natCast_im + - IsROrC.normSq_conj + - IsROrC.div_I + - IsROrC.mul_conj + - IsROrC.im_ofReal_mul + - IsROrC.ratCast_im + - IsROrC.I_im' + - IsROrC.norm_natCast + - IsROrC.inv_im + - IsROrC.pos_iff_exists_ofReal + - IsROrC.sqrt_normSq_eq_norm + - IsROrC.normSq_add + - IsROrC.toOrderedSMul + - IsROrC.natCast_re + - IsROrC.div_im + - IsROrC.norm_nsmul + - IsROrC.nonpos_iff_exists_ofReal + - path: Mathlib/Analysis/Complex/Basic.lean + theorems: + - Complex.norm_rat + - Complex.dist_of_im_eq + - Complex.edist_of_im_eq + - Complex.dist_of_re_eq + - Complex.nndist_conj_self + - Complex.restrictScalars_one_smulRight + - Complex.tendsto_normSq_cocompact_atTop + - Complex.restrictScalars_one_smulRight' + - Complex.ringHom_eq_ofReal_of_continuous + - Complex.mem_slitPlane_of_norm_lt_one + - Complex.antilipschitz_equivRealProd + - Complex.equivRealProd_apply_le' + - Complex.orderClosedTopology + - Complex.edist_of_re_eq + - Complex.norm_int_of_nonneg + - Complex.nat_cast_mem_slitPlane + - IsROrC.hasSum_ofReal + - Complex.dist_conj_self + - Complex.nndist_self_conj + - Complex.dist_eq_re_im + - IsROrC.ofReal_tsum + - Complex.dist_conj_comm + - Complex.ofReal_mem_slitPlane + - IsROrC.summable_ofReal + - Complex.norm_exp_ofReal_mul_I + - Complex.nnnorm_eq_one_of_pow_eq_one + - Complex.lipschitz_equivRealProd + - Complex.ringHom_eq_id_or_conj_of_continuous + - IsROrC.hasSum_iff + - Complex.dist_self_conj + - Complex.nnnorm_nat + - Complex.neg_ofReal_mem_slitPlane + - Complex.continuous_normSq + - Complex.ball_one_subset_slitPlane + - Complex.equivRealProd_apply_le + - Complex.eq_coe_norm_of_nonneg + - path: Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean + theorems: + - iInter_halfspaces_eq + - separate_convex_open_set + - geometric_hahn_banach_open_open + - geometric_hahn_banach_closed_compact + - geometric_hahn_banach_point_open + - geometric_hahn_banach_point_point + - geometric_hahn_banach_compact_closed + - geometric_hahn_banach_open + - path: Mathlib/Data/Nat/Squarefree.lean + theorems: + - Nat.minSqFac_dvd + - Nat.sum_divisors_filter_squarefree + - Nat.sq_mul_squarefree_of_pos + - Nat.squarefree_mul + - Nat.Squarefree.ext_iff + - Nat.squarefree_iff_nodup_factors + - Nat.prod_primeFactors_of_squarefree + - Nat.prod_primeFactors_sdiff_of_squarefree + - Nat.minSqFacProp_div + - Nat.divisors_filter_squarefree + - Nat.sq_mul_squarefree + - Nat.squarefree_pow_iff + - Nat.coprime_div_gcd_of_squarefree + - Nat.squarefree_and_prime_pow_iff_prime + - Nat.squarefree_two + - Squarefree.natFactorization_le_one + - Nat.squarefree_of_factorization_le_one + - Nat.sq_mul_squarefree_of_pos' + - Nat.squarefree_iff_minSqFac + - Nat.minSqFac_le_of_dvd + - Nat.primeFactors_div_gcd + - Nat.minSqFacAux_has_prop + - Nat.minSqFac_prime + - Nat.minSqFac_has_prop + - Nat.primeFactors_prod + - path: Mathlib/Data/Set/Intervals/OrderIso.lean + theorems: + - OrderIso.preimage_Iic + - OrderIso.preimage_Ioc + - OrderIso.image_Ico + - OrderIso.preimage_Ioo + - OrderIso.preimage_Icc + - OrderIso.image_Icc + - OrderIso.image_Iio + - OrderIso.image_Ioc + - OrderIso.preimage_Ici + - OrderIso.preimage_Ico + - OrderIso.preimage_Ioi + - OrderIso.preimage_Iio + - OrderIso.image_Iic + - OrderIso.image_Ioo + - path: Mathlib/Topology/Compactness/SigmaCompact.lean + theorems: + - IsSigmaCompact.of_isClosed_subset + - Inducing.isSigmaCompact_iff + - SigmaCompactSpace_iff_exists_compact_covering + - isSigmaCompact_sUnion_of_isCompact + - isSigmaCompact_iUnion + - LocallyFinite.countable_univ + - isSigmaCompact_biUnion + - ClosedEmbedding.sigmaCompactSpace + - isSigmaCompact_sUnion + - iUnion_compactCovering + - countable_cover_nhdsWithin_of_sigma_compact + - countable_cover_nhds_of_sigma_compact + - IsSigmaCompact.image_of_continuousOn + - isSigmaCompact_iUnion_of_isCompact + - CompactExhaustion.exists_superset_of_isCompact + - isSigmaCompact_iff_isSigmaCompact_univ + - path: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean + theorems: + - aemeasurable_iSup + - ContinuousOn.measurable_piecewise + - Measurable.isLUB + - IsCompact.closure_subset_measurableSet + - Real.borel_eq_generateFrom_Ici_rat + - AEMeasurable.nnreal_tsum + - aemeasurable_coe_nnreal_real_iff + - Measurable.ennreal_tsum' + - measurable_of_isOpen + - measurable_ereal_toReal + - Measurable.iSup_Prop + - Measurable.iInf_Prop + - Measurable.min + - Dense.borel_eq_generateFrom_Ioc_mem_aux + - MeasureTheory.Measure.ext_of_Ico' + - Measurable.nnreal_tsum + - tendsto_measure_thickening_of_isClosed + - measurable_of_Iic + - pi_le_borel_pi + - measurable_of_Ici + - tendsto_measure_cthickening_of_isClosed + - measurable_of_Ioi + - Real.isPiSystem_Ici_rat + - AEMeasurable.ennreal_tsum + - measurable_of_Iio + - AEMeasurable.isLUB + - measurable_liminf' + - borel_eq_generateFrom_of_subbasis + - Real.isPiSystem_Iio_rat + - Dense.borel_eq_generateFrom_Ico_mem + - borel_eq_top_of_countable + - NNReal.measurable_of_tendsto' + - measurable_of_isClosed' + - borel_eq_generateFrom_Ioi + - measurable_of_isClosed + - measurableSet_of_mem_nhdsWithin_Ioi + - Real.borel_eq_generateFrom_Iic_rat + - Real.measure_ext_Ioo_rat + - ENNReal.aemeasurable_of_tendsto' + - Measurable.max + - Dense.borel_eq_generateFrom_Ioc_mem + - Real.borel_eq_generateFrom_Iio_rat + - IsFiniteMeasureOnCompacts.map + - measurable_coe_nnreal_real_iff + - opensMeasurableSpace_iff_forall_measurableSet + - borel_eq_generateFrom_Iic + - measurable_sSup + - aemeasurable_biSup + - measurableSet_bddAbove_range + - tendsto_measure_cthickening + - measurable_iSup + - Set.OrdConnected.measurableSet + - exists_spanning_measurableSet_le + - Real.isPiSystem_Iic_rat + - MeasureTheory.Measure.ext_of_Ico_finite + - borel_eq_generateFrom_Iio + - Measurable.isLUB_of_mem + - "IsG\u03B4.measurableSet" + - MeasureTheory.Measure.ext_of_Iic + - MeasureTheory.Measure.ext_of_Ioc' + - measurableSet_of_mem_nhdsWithin_Ioi_aux + - Real.isPiSystem_Ioi_rat + - EReal.measurable_of_measurable_real + - MeasureTheory.Measure.ext_of_Ioc_finite + - IsCompact.measure_closure + - Real.isPiSystem_Ioo_rat + - measure_eq_measure_preimage_add_measure_tsum_Ico_zpow + - generateFrom_Ico_mem_le_borel + - Dense.borel_eq_generateFrom_Ico_mem_aux + - measurable_biSup + - Inseparable.mem_measurableSet_iff + - ENNReal.measurable_of_tendsto' + - prod_le_borel_prod + - MeasurableEmbedding.borelSpace + - tendsto_measure_thickening + - path: Mathlib/MeasureTheory/Measure/Haar/Quotient.lean + theorems: + - MeasureTheory.IsFundamentalDomain.isMulLeftInvariant_map + - QuotientAddGroup.integral_mul_eq_integral_automorphize_mul + - MeasurePreservingQuotientGroup.mk' + - MeasureTheory.IsFundamentalDomain.smulInvariantMeasure_map + - MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map + - MeasureTheory.IsFundamentalDomain.map_restrict_quotient + - QuotientGroup.integral_mul_eq_integral_automorphize_mul + - essSup_comp_quotientGroup_mk + - QuotientGroup.integral_eq_integral_automorphize + - path: Mathlib/Analysis/NormedSpace/Exponential.lean + theorems: + - NormedSpace.norm_expSeries_summable_of_mem_ball' + - NormedSpace.expSeries_eq_expSeries + - NormedSpace.expSeries_hasSum_exp_of_mem_ball' + - NormedSpace.map_exp_of_mem_ball + - NormedSpace.exp_add_of_commute_of_mem_ball + - NormedSpace.expSeries_radius_eq_top + - NormedSpace.norm_expSeries_div_summable_of_mem_ball + - NormedSpace.exp_eq_exp + - NormedSpace.expSeries_apply_eq_div + - NormedSpace.exp_mem_unitary_of_mem_skewAdjoint + - NormedSpace.expSeries_radius_pos + - NormedSpace.analyticAt_exp_of_mem_ball + - NormedSpace.exp_sum_of_commute + - Function.update_exp + - NormedSpace.invOf_exp_of_mem_ball + - NormedSpace.expSeries_apply_zero + - NormedSpace.exp_unop + - NormedSpace.exp_zsmul + - NormedSpace.exp_continuous + - NormedSpace.expSeries_div_hasSum_exp_of_mem_ball + - Commute.exp_right + - NormedSpace.exp_zero + - NormedSpace.exp_op + - NormedSpace.exp_nsmul + - NormedSpace.exp_sum + - NormedSpace.star_exp + - NormedSpace.expSeries_apply_eq + - path: Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean + theorems: + - MvPolynomial.NewtonIdentities.sum_filter_pairs_eq_sum_powersetCard_sum + - MvPolynomial.NewtonIdentities.disjUnion_filter_pairs_eq_pairs + - MvPolynomial.psum_eq_mul_esymm_sub_sum + - MvPolynomial.NewtonIdentities.pairMap_ne_self + - MvPolynomial.NewtonIdentities.weight_add_weight_pairMap + - MvPolynomial.sum_antidiagonal_card_esymm_psum_eq_zero + - MvPolynomial.NewtonIdentities.pairMap_involutive + - MvPolynomial.NewtonIdentities.disjoint_filter_pairs_lt_filter_pairs_eq + - MvPolynomial.NewtonIdentities.esymm_summand_to_weight + - MvPolynomial.mul_esymm_eq_sum + - MvPolynomial.NewtonIdentities.sum_filter_pairs_eq_sum_powersetCard_mem_filter_antidiagonal_sum + - MvPolynomial.NewtonIdentities.sum_filter_pairs_eq_sum_filter_antidiagonal_powersetCard_sum + - MvPolynomial.NewtonIdentities.pairMap_of_snd_nmem_fst + - MvPolynomial.NewtonIdentities.pairMap_mem_pairs + - MvPolynomial.NewtonIdentities.esymm_mul_psum_to_weight + - MvPolynomial.NewtonIdentities.pairMap_of_snd_mem_fst + - MvPolynomial.NewtonIdentities.filter_pairs_lt + - MvPolynomial.NewtonIdentities.esymm_mul_psum_summand_to_weight + - MvPolynomial.NewtonIdentities.mem_pairs + - MvPolynomial.NewtonIdentities.esymm_to_weight + - path: Mathlib/Order/Hom/Basic.lean + theorems: + - map_inv_lt_iff + - OrderIso.symm_symm + - OrderIso.complementedLattice_iff + - OrderHom.comp_id + - OrderHom.fst_prod_snd + - OrderIso.map_bot' + - OrderIso.ofRelIsoLT_toRelIsoLT + - lt_map_inv_iff + - OrderIso.complementedLattice + - Disjoint.map_orderIso + - le_map_inv_iff + - Codisjoint.map_orderIso + - OrderIso.toRelIsoLT_ofRelIsoLT + - OrderIso.trans_refl + - OrderIso.map_inf + - OrderHom.id_comp + - OrderIso.refl_trans + - map_inv_le_iff + - path: Mathlib/Data/BitVec/Lemmas.lean + theorems: + - BitVec.ofNat_toNat' + - BitVec.ofFin_and + - BitVec.ofFin_ofNat + - BitVec.ofFin_or + - BitVec.toFin_natCast + - BitVec.addLsb_eq_twice_add_one + - BitVec.ofFin_neg + - BitVec.toNat_extractLsb' + - BitVec.ofFin_natCast + - BitVec.decide_addLsb_mod_two + - BitVec.toFin_one + - BitVec.ofFin_one + - BitVec.toFin_ofFin + - BitVec.ofNat_toNat_of_eq + - BitVec.toFin_le_toFin_of_le + - BitVec.addLsb_div_two + - BitVec.ofFin_xor + - BitVec.toNat_ofNat_of_lt + - BitVec.toFin_val + - BitVec.toFin_neg + - path: Mathlib/Algebra/Group/Pi/Lemmas.lean + theorems: + - Pi.mulSingle_commute + - Pi.update_eq_div_mul_mulSingle + - Pi.mulSingle_apply_commute + - Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle + - path: Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean + theorems: + - UniqueMDiffWithinAt.preimage_partialHomeomorph + - UniqueMDiffWithinAt.image_denseRange + - UniqueMDiffOn.uniqueDiffOn_inter_preimage + - UniqueMDiffOn.uniqueDiffOn_target_inter + - UniqueMDiffWithinAt.smooth_bundle_preimage + - path: Mathlib/Data/Seq/Computation.lean + theorems: + - Computation.results_bind + - Computation.destruct_eq_think + - Computation.liftRel_pure_left + - Computation.mem_map + - Computation.length_think + - Computation.liftRel_def + - Computation.liftRel_pure_right + - Computation.bind_assoc + - Computation.destruct_eq_pure + - Computation.bind_pure' + - Computation.liftRel_think_right + - Computation.rel_of_liftRel + - Computation.orElse_pure + - Computation.corec_eq + - Computation.liftRel_map + - Computation.results_think_iff + - Computation.LiftRelAux.ret_left + - Computation.destruct_map + - Computation.map_id + - Computation.orElse_empty + - Computation.ret_orElse + - Computation.not_mem_empty + - Computation.LiftRelAux.swap + - Computation.empty_orElse + - Computation.terminates_congr + - Computation.bind_pure + - Computation.mem_unique + - Computation.eq_empty_of_not_terminates + - Computation.orElse_think + - Computation.map_think + - Computation.of_think_mem + - Computation.of_results_think + - Computation.LiftRelAux.ret_right + - Computation.of_results_bind + - Computation.map_congr + - Computation.terminates_def + - Computation.results_thinkN_pure + - Computation.lift_eq_iff_equiv + - Computation.LiftRelRec.lem + - Computation.equiv_of_mem + - Computation.liftRel_of_mem + - Computation.le_stable + - Computation.think_bind + - Computation.tail_think + - Computation.exists_of_mem_map + - Computation.map_comp + - Computation.liftRel_pure + - Computation.mem_of_promises + - Computation.ret_bind + - Computation.results_think + - Computation.eq_thinkN + - Computation.bind_promises + - Computation.mem_of_get_eq + - Computation.Results.len_unique + - Computation.eq_of_bisim + - path: Mathlib/Analysis/Convex/Strict.lean + theorems: + - StrictConvex.add_left + - StrictConvex.preimage_add_left + - StrictConvex.preimage_add_right + - StrictConvex.smul_mem_of_zero_mem + - StrictConvex.add_right + - StrictConvex.linear_image + - StrictConvex.mem_smul_of_zero_mem + - StrictConvex.inter + - StrictConvex.add + - StrictConvex.smul + - StrictConvex.affine_image + - strictConvex_iff_div + - StrictConvex.add_smul_sub_mem + - StrictConvex.linear_preimage + - StrictConvex.affine_preimage + - strictConvex_univ + - StrictConvex.preimage_smul + - StrictConvex.eq_of_openSegment_subset_frontier + - Directed.strictConvex_iUnion + - StrictConvex.add_smul_mem + - Set.OrdConnected.strictConvex + - path: Mathlib/Algebra/Order/Module/Defs.lean + theorems: + - smul_nonneg_iff_pos_imp_nonneg + - smul_lt_of_lt_one_left + - inv_smul_le_iff_of_neg + - smul_add_smul_lt_smul_add_smul + - smul_nonneg + - smul_nonneg_iff_neg_imp_nonpos + - pos_and_pos_or_neg_and_neg_of_smul_pos + - smul_le_smul_iff_of_neg_left + - smul_neg_iff_of_pos_left + - SMulPosMono.lift + - smul_eq_smul_iff_eq_and_eq_of_pos + - PosSMulReflectLT.lift + - smul_neg_of_pos_of_neg + - smul_nonneg' + - smul_pos' + - le_smul_of_one_le_left + - neg_of_smul_neg_left + - smul_nonpos_of_nonpos_of_nonneg + - smul_le_iff_le_one_left + - smul_add_smul_le_smul_add_smul' + - smul_add_smul_le_smul_add_smul + - smul_le_of_le_one_left + - smul_neg_iff_of_neg_left + - lt_smul_iff_one_lt_left + - lt_smul_of_one_lt_left + - PosSMulMono.lift + - inv_smul_lt_iff_of_pos + - smul_pos + - le_smul_iff_one_le_left + - SMulPosReflectLE.lift + - pos_of_smul_pos_right + - smul_inv_le_iff_of_neg + - pos_of_smul_pos_left + - smul_nonpos_of_nonneg_of_nonpos + - SMulPosMono.of_pos + - inv_smul_lt_iff_of_neg + - smul_lt_smul_iff_of_neg_left + - smul_lt_iff_lt_one_left + - SMulPosReflectLT.of_pos + - smul_inv_lt_iff_of_neg + - lt_of_smul_lt_smul_of_nonpos + - smul_nonpos_iff_pos_imp_nonpos + - nonneg_and_nonneg_or_nonpos_and_nonpos_of_smul_nonneg + - lt_inv_smul_iff_of_pos + - smul_pos_iff_of_pos_right + - smul_add_smul_lt_smul_add_smul' + - smul_nonpos_iff + - PosSMulMono.of_pos + - SMulPosStrictMono.lift + - smul_neg_of_neg_of_pos + - PosSMulStrictMono.lift + - smul_eq_smul_iff_eq_and_eq_of_pos' + - PosSMulReflectLT.of_pos + - smul_lt_smul_of_neg_left + - SMulPosReflectLT.lift + - smul_nonpos_iff_neg_imp_nonneg + - le_of_smul_le_smul_of_neg + - smul_pos_iff_of_pos_left + - inv_smul_le_iff_of_pos + - smul_le_smul_of_nonpos_left + - le_inv_smul_iff_of_pos + - path: Mathlib/Algebra/Ring/CompTypeclasses.lean + theorems: + - "RingHomInvPair.comp_apply_eq\u2082" + - RingHomSurjective.comp + - RingHomInvPair.comp_apply_eq + - path: Mathlib/Algebra/Homology/HomologicalComplex.lean + theorems: + - HomologicalComplex.Hom.comm + - HomologicalComplex.eqToHom_f + - HomologicalComplex.xPrevIso_comp_dTo + - HomologicalComplex.XIsoOfEq_inv_comp_XIsoOfEq_hom + - HomologicalComplex.image_to_eq_image + - HomologicalComplex.d_comp_d + - CochainComplex.mkHom_f_succ_succ + - ChainComplex.mk_d_1_0 + - HomologicalComplex.XIsoOfEq_inv_naturality + - CochainComplex.mk_d_1_0 + - HomologicalComplex.kernel_eq_kernel + - ChainComplex.mk_d_2_1 + - HomologicalComplex.eqToHom_comp_d + - HomologicalComplex.Hom.prev_eq + - ChainComplex.of_d_ne + - CochainComplex.mk'_d_1_0 + - HomologicalComplex.hom_ext + - HomologicalComplex.isZero_zero + - HomologicalComplex.dFrom_comp_xNextIso + - CochainComplex.of_d_ne + - HomologicalComplex.d_comp_XIsoOfEq_hom + - HomologicalComplex.ext + - HomologicalComplex.xPrevIsoSelf_comp_dTo + - ChainComplex.next_nat_zero + - ChainComplex.of_d + - HomologicalComplex.XIsoOfEq_inv_comp_d + - CochainComplex.of_d + - HomologicalComplex.XIsoOfEq_inv_comp_XIsoOfEq_inv + - CochainComplex.prev_nat_zero + - HomologicalComplex.Hom.next_eq + - HomologicalComplex.dFrom_comp_xNextIsoSelf + - HomologicalComplex.hom_f_injective + - ChainComplex.mk'_d_1_0 + - HomologicalComplex.XIsoOfEq_hom_comp_d + - HomologicalComplex.XIsoOfEq_hom_naturality + - HomologicalComplex.dTo_eq + - HomologicalComplex.XIsoOfEq_hom_comp_XIsoOfEq_hom + - CochainComplex.mk_d_2_0 + - HomologicalComplex.d_comp_eqToHom + - ChainComplex.mkHom_f_succ_succ + - HomologicalComplex.XIsoOfEq_hom_comp_XIsoOfEq_inv + - HomologicalComplex.kernel_from_eq_kernel + - HomologicalComplex.d_comp_XIsoOfEq_inv + - HomologicalComplex.dFrom_eq + - HomologicalComplex.image_eq_image + - HomologicalComplex.Hom.isoOfComponents_app + - path: Mathlib/Combinatorics/SimpleGraph/Subgraph.lean + theorems: + - SimpleGraph.Subgraph.sInf_adj_of_nonempty + - SimpleGraph.Subgraph.verts_spanningCoe_injective + - SimpleGraph.Subgraph.coeSubgraph_le + - SimpleGraph.Subgraph.iInf_adj + - SimpleGraph.Subgraph.le_induce_union_right + - SimpleGraph.Subgraph.le_induce_union_left + - SimpleGraph.Subgraph.edgeSet_iSup + - SimpleGraph.neighborSet_subgraphOfAdj + - SimpleGraph.Subgraph.spanningCoe_inj + - SimpleGraph.Subgraph.mem_verts_if_mem_edge + - SimpleGraph.neighborSet_fst_subgraphOfAdj + - SimpleGraph.Subgraph.comap_monotone + - SimpleGraph.Subgraph.degree_le + - SimpleGraph.Subgraph.IsSpanning.card_verts + - SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq + - SimpleGraph.Subgraph.coe_deleteEdges_eq + - SimpleGraph.Subgraph.neighborSet_sInf + - SimpleGraph.Subgraph.map_sup + - SimpleGraph.Subgraph.edgeSet_sSup + - SimpleGraph.Subgraph.deleteVerts_deleteVerts + - SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq + - SimpleGraph.Subgraph.edgeSet_sInf + - SimpleGraph.Subgraph.neighborSet_iSup + - SimpleGraph.Subgraph.neighborSet_iInf + - SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree + - SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj + - SimpleGraph.Subgraph.deleteVerts_inter_verts_set_right_eq + - SimpleGraph.Subgraph.le_induce_union + - SimpleGraph.Subgraph.deleteEdges_deleteEdges + - SimpleGraph.Subgraph.coeSubgraph_adj + - SimpleGraph.Subgraph.iInf_adj_of_nonempty + - SimpleGraph.Subgraph.restrict_coeSubgraph + - SimpleGraph.subgraphOfAdj_le_of_adj + - Disjoint.edgeSet + - SimpleGraph.Subgraph.deleteEdges_le + - SimpleGraph.Subgraph.edgeSet_bot + - SimpleGraph.Subgraph.inclusion.injective + - SimpleGraph.Subgraph.verts_iInf + - SimpleGraph.Subgraph.degree_spanningCoe + - SimpleGraph.Subgraph.subgraphOfAdj_eq_induce + - SimpleGraph.edgeSet_subgraphOfAdj + - SimpleGraph.map_singletonSubgraph + - SimpleGraph.neighborSet_snd_subgraphOfAdj + - SimpleGraph.Subgraph.singletonSubgraph_eq_induce + - SimpleGraph.induce_eq_coe_induce_top + - SimpleGraph.Subgraph.induce_self_verts + - SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj + - SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq + - SimpleGraph.Subgraph.induce_mono + - SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne + - SimpleGraph.Subgraph.neighborSet_sSup + - SimpleGraph.Subgraph.edgeSet_inf + - SimpleGraph.Subgraph.deleteEdges_coe_eq + - SimpleGraph.map_subgraphOfAdj + - SimpleGraph.Subgraph.edgeSet_sup + - SimpleGraph.Subgraph.coeSubgraph_restrict_eq + - SimpleGraph.Subgraph.map_monotone + - SimpleGraph.Subgraph.deleteVerts_empty + - SimpleGraph.Subgraph.map_le_iff_le_comap + - SimpleGraph.eq_singletonSubgraph_iff_verts_eq + - SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj + - SimpleGraph.Subgraph.iSup_adj + - SimpleGraph.Subgraph.deleteVerts_inter_verts_left_eq + - SimpleGraph.Subgraph.edgeSet_iInf + - SimpleGraph.singletonSubgraph_le_iff + - SimpleGraph.Subgraph.deleteVerts_adj + - SimpleGraph.Subgraph.deleteEdges_le_of_le + - SimpleGraph.Subgraph.coe_deleteEdges_le + - SimpleGraph.Subgraph.verts_iSup + - SimpleGraph.Subgraph.induce_empty + - SimpleGraph.Subgraph.deleteEdges_empty_eq + - SimpleGraph.subgraphOfAdj_symm + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean + theorems: + - MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite' + - MeasureTheory.integral_norm_le_of_forall_fin_meas_integral_eq + - MeasureTheory.Lp.ae_eq_zero_of_forall_set_integral_eq_zero' + - MeasureTheory.lintegral_nnnorm_le_of_forall_fin_meas_integral_eq + - MeasureTheory.lpMeas.ae_eq_zero_of_forall_set_integral_eq_zero + - MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq' + - path: Mathlib/Algebra/DualNumber.lean + theorems: + - DualNumber.lift_op_smul + - DualNumber.lift_apply_eps + - DualNumber.algHom_ext' + - DualNumber.algHom_ext + - DualNumber.lift_apply_inl + - DualNumber.lift_smul + - DualNumber.commute_eps_left + - path: Mathlib/Topology/ContinuousFunction/Algebra.lean + theorems: + - Subalgebra.SeparatesPoints.strongly + - ContinuousMap.prod_apply + - ContinuousMap.hasSum_apply + - Subalgebra.separatesPoints_monotone + - ContinuousMap.periodic_tsum_comp_add_zsmul + - path: Mathlib/NumberTheory/ArithmeticFunction.lean + theorems: + - ArithmeticFunction.IsMultiplicative.prodPrimeFactors_add_of_squarefree + - ArithmeticFunction.moebius_ne_zero_iff_eq_or + - ArithmeticFunction.ppow_zero + - ArithmeticFunction.cardFactors_multiset_prod + - ArithmeticFunction.coe_coe + - ArithmeticFunction.pow_zero_eq_zeta + - ArithmeticFunction.mul_smul' + - ArithmeticFunction.intCoe_mul + - ArithmeticFunction.IsMultiplicative.lcm_apply_mul_gcd_apply + - ArithmeticFunction.mul_apply_one + - ArithmeticFunction.ppow_succ + - ArithmeticFunction.coe_zeta_smul_apply + - ArithmeticFunction.pdiv_zeta + - ArithmeticFunction.IsMultiplicative.int_cast + - ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_of_nonzero + - ArithmeticFunction.sigma_one_apply + - ArithmeticFunction.coe_mul_zeta_apply + - ArithmeticFunction.sum_eq_iff_sum_mul_moebius_eq + - ArithmeticFunction.IsMultiplicative.ppow + - ArithmeticFunction.moebius_mul_coe_zeta + - ArithmeticFunction.IsMultiplicative.prodPrimeFactors + - ArithmeticFunction.natCoe_one + - ArithmeticFunction.zeta_pmul + - ArithmeticFunction.isMultiplicative_zeta + - ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on' + - ArithmeticFunction.IsMultiplicative.iff_ne_zero + - ArithmeticFunction.pow_apply + - ArithmeticFunction.cardDistinctFactors_eq_cardFactors_iff_squarefree + - ArithmeticFunction.cardFactors_mul + - ArithmeticFunction.one_smul' + - ArithmeticFunction.ppow_succ' + - ArithmeticFunction.cardDistinctFactors_zero + - ArithmeticFunction.pmul_zeta + - ArithmeticFunction.zeta_mul_pow_eq_sigma + - ArithmeticFunction.pmul_comm + - ArithmeticFunction.pmul_assoc + - ArithmeticFunction.IsMultiplicative.pmul + - ArithmeticFunction.coe_moebius_mul_coe_zeta + - ArithmeticFunction.moebius_apply_prime + - ArithmeticFunction.IsMultiplicative.eq_iff_eq_on_prime_powers + - ArithmeticFunction.isMultiplicative_sigma + - ArithmeticFunction.coe_zeta_mul_moebius + - ArithmeticFunction.cardFactors_eq_one_iff_prime + - ArithmeticFunction.moebius_apply_isPrimePow_not_prime + - ArithmeticFunction.coe_zeta_mul_coe_moebius + - ArithmeticFunction.IsMultiplicative.mul + - ArithmeticFunction.IsMultiplicative.prodPrimeFactors_one_add_of_squarefree + - ArithmeticFunction.prod_eq_iff_prod_pow_moebius_eq_on_of_nonzero + - ArithmeticFunction.isMultiplicative_moebius + - ArithmeticFunction.cardDistinctFactors_apply_prime_pow + - ArithmeticFunction.sigma_zero_apply + - ArithmeticFunction.isMultiplicative_one + - ArithmeticFunction.natCoe_mul + - ArithmeticFunction.IsMultiplicative.pdiv + - ArithmeticFunction.moebius_ne_zero_iff_squarefree + - ArithmeticFunction.cardFactors_apply_prime_pow + - ArithmeticFunction.moebius_apply_prime_pow + - ArithmeticFunction.IsMultiplicative.map_prod + - ArithmeticFunction.IsMultiplicative.nat_cast + - ArithmeticFunction.cardDistinctFactors_one + - ArithmeticFunction.sum_eq_iff_sum_mul_moebius_eq_on + - ArithmeticFunction.moebius_apply_one + - ArithmeticFunction.sigma_zero_apply_prime_pow + - ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq + - ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on + - ArithmeticFunction.IsMultiplicative.prodPrimeFactors_one_sub_of_squarefree + - ArithmeticFunction.cardDistinctFactors_apply_prime + - ArithmeticFunction.intCoe_one + - ArithmeticFunction.ppow_apply + - path: Mathlib/CategoryTheory/Quotient.lean + theorems: + - CategoryTheory.Quotient.natTrans_ext + - CategoryTheory.Quotient.compClosure_iff_self + - CategoryTheory.Quotient.functor_map_eq_iff + - CategoryTheory.Quotient.sound + - CategoryTheory.Quotient.natTransLift_id + - CategoryTheory.Quotient.comp_right + - CategoryTheory.Quotient.lift_unique' + - CategoryTheory.Quotient.comp_left + - CategoryTheory.Quotient.CompClosure.of + - CategoryTheory.Quotient.lift_unique + - CategoryTheory.Quotient.induction + - CategoryTheory.Quotient.lift_spec + - CategoryTheory.Quotient.comp_natTransLift + - CategoryTheory.Quotient.lift_map_functor_map + - path: Mathlib/CategoryTheory/Subobject/FactorThru.lean + theorems: + - CategoryTheory.Subobject.factorThru_comp_arrow + - CategoryTheory.MonoOver.factors_congr + - CategoryTheory.Subobject.factorThru_zero + - CategoryTheory.Subobject.factors_add + - CategoryTheory.Subobject.factors_right_of_factors_add + - CategoryTheory.Subobject.factorThru_eq_zero + - CategoryTheory.Subobject.factorThru_add + - CategoryTheory.Subobject.factors_left_of_factors_add + - CategoryTheory.Subobject.factors_of_le + - CategoryTheory.Subobject.mk_factors_self + - CategoryTheory.Subobject.factorThru_right + - CategoryTheory.Subobject.factors_self + - CategoryTheory.Subobject.factorThru_add_sub_factorThru_right + - CategoryTheory.Subobject.factorThru_self + - CategoryTheory.Subobject.factorThru_add_sub_factorThru_left + - CategoryTheory.Subobject.factorThru_ofLE + - CategoryTheory.Subobject.factors_of_factors_right + - CategoryTheory.Subobject.factorThru_mk_self + - CategoryTheory.Subobject.factors_zero + - path: .lake/packages/std/Std/Data/String/Lemmas.lean + theorems: + - Substring.Valid.data_takeWhile + - String.all_iff + - Substring.ValidFor.take + - String.Iterator.ValidFor.hasPrev + - String.Iterator.ValidFor.curr + - Substring.Valid.toString_extract + - String.valid_next + - String.get_cons_addChar + - Substring.ValidFor.next + - Substring.ValidFor.dropWhile + - Substring.ValidFor.foldl + - String.any_eq + - String.firstDiffPos_eq + - Substring.Valid.nextn_stop + - String.Iterator.Valid.next + - String.map_eq + - String.Pos.zero_addString_eq + - Substring.ValidFor.prevn + - String.validFor_toSubstring + - String.Iterator.ValidFor.prev + - String.foldr_eq + - Substring.ValidFor.valid + - String.foldrAux_of_valid + - String.Iterator.ValidFor.atEnd + - String.any_iff + - String.front_eq + - String.utf8GetAux?_of_valid + - Substring.Valid.nextn + - String.back_eq + - String.Iterator.ValidFor.valid + - Substring.Valid.isEmpty + - String.Iterator.ValidFor.next + - String.mapAux_of_valid + - String.prev_of_valid + - String.data_join + - Substring.Valid.next + - String.Iterator.hasNext_cons_addChar + - Substring.Valid.all + - String.modify_of_valid + - String.Iterator.ValidFor.remainingToString + - String.offsetOfPos_of_valid + - String.join_eq + - String.Pos.valid_endPos + - String.splitAux_of_valid + - Substring.ValidFor.get + - String.Iterator.Valid.validFor + - String.split_of_valid + - String.takeWhileAux_of_valid + - Substring.prevn_zero + - String.drop_empty + - Substring.Valid.get + - "String.extract.go\u2081_add_right_cancel" + - String.Iterator.ValidFor.prevn + - Substring.ValidFor.takeWhile + - String.extract_zero_endPos + - Substring.ValidFor.drop + - String.utf8Len_eq_zero + - String.Iterator.ValidFor.extract + - String.utf8GetAux_add_right_cancel + - String.Iterator.ValidFor.hasNext + - "String.extract.go\u2081_cons_addChar" + - Substring.ValidFor.any + - String.data_drop + - Substring.Valid.contains + - String.foldlAux_of_valid + - String.offsetOfPosAux_of_valid + - Substring.Valid.prevn + - Substring.ValidFor.bsize + - String.extract_cons_addChar + - String.findAux_of_valid + - String.Iterator.ValidFor.out' + - Substring.Valid.extract + - Substring.ValidFor.nextn_stop + - Substring.ValidFor.prev + - Substring.Valid.valid + - String.Pos.zero_addString_byteIdx + - String.all_eq + - String.revFindAux_of_valid + - String.Iterator.ValidFor.toEnd + - String.data_dropWhile + - "String.extract.go\u2082_append_left" + - String.Iterator.ValidFor.setCurr' + - Substring.ValidFor.contains + - Substring.ValidFor.atEnd + - Substring.Valid.any + - String.utf8PrevAux_of_valid + - "String.extract.go\u2081_zero_utf8Len" + - String.find_of_valid + - String.Pos.addChar_right_comm + - Substring.Valid.foldl + - String.data_takeWhile + - String.Iterator.Valid.remainingBytes_le + - String.revFind_of_valid + - String.Iterator.Valid.setCurr + - Substring.ValidFor.nextn + - Substring.ValidFor.foldr + - String.utf8SetAux_of_valid + - Substring.Valid.prev + - String.utf8GetAux_of_valid + - String.Iterator.ValidFor.pos_eq_endPos + - Substring.ValidFor.next_stop + - "String.extract.go\u2081_append_right" + - String.foldl_eq + - String.utf8Len_reverseAux + - String.Pos.zero_addChar_byteIdx + - Substring.ValidFor.toIterator + - String.firstDiffPos_loop_eq + - String.utf8Len_append + - Substring.Valid.validFor + - String.atEnd_of_valid + - Substring.Valid.data_take + - Substring.ValidFor.isEmpty + - String.Iterator.ValidFor.remainingBytes + - String.Iterator.ValidFor.toEnd' + - String.next_of_valid' + - String.extract_of_valid + - String.Iterator.ValidFor.mk' + - String.Pos.Valid.le_endPos + - Substring.Valid.data_dropWhile + - String.contains_iff + - "String.extract.go\u2082_add_right_cancel" + - Substring.Valid.data_drop + - Substring.Valid.atEnd + - Substring.ValidFor.extract + - Substring.Valid.bsize + - String.Pos.zero_addChar_eq + - String.prev_of_valid' + - Substring.Valid.next_stop + - String.Iterator.ValidFor.prev_nil + - String.length_push + - Substring.Valid.foldr + - String.Iterator.Valid.remainingToString + - Substring.prev_zero + - String.anyAux_of_valid + - String.Iterator.ValidFor.pos_eq_zero + - path: Mathlib/Data/Real/Sqrt.lean + theorems: + - Real.sqrt_eq_iff_sq_eq + - Real.lt_sq_of_sqrt_lt + - NNReal.sqrt_one + - Real.mul_self_sqrt + - Real.sqrt_div_self' + - NNReal.sqrt_mul_self + - NNReal.sqrt_pos + - Real.sqrt_zero + - Real.sqrt_lt_sqrt_iff_of_pos + - Real.sum_mul_le_sqrt_mul_sqrt + - Real.sqrt_eq_zero_of_nonpos + - Real.sqrt_eq_zero + - Real.sqrt_eq_iff_mul_self_eq + - Real.sqrt_ne_zero + - Real.sqrt_div + - Real.sq_le + - Real.sqrt_eq_cases + - Real.sqrt_lt + - Real.coe_sqrt + - NNReal.sqrt_zero + - Real.sqrt_div' + - Real.sqrt_inv + - Real.sqrt_mul' + - Real.nat_sqrt_le_real_sqrt + - Real.sqrt_le_sqrt_iff' + - Real.sqrt_mul + - Real.sq_lt + - Real.sqrt_le_iff + - Real.sqrt_le_one + - Real.lt_sqrt + - Real.sqrt_le_sqrt_iff + - Real.sqrt_eq_zero' + - NNReal.sqrt_eq_zero + - Real.sqrt_lt' + - Real.sqrtAux_nonneg + - Real.real_sqrt_le_nat_sqrt_succ + - NNReal.one_le_sqrt + - NNReal.mul_self_sqrt + - Real.sqrt_one + - NNReal.sqrt_eq_one + - Real.sqrt_ne_zero' + - Real.sqrt_div_self + - NNReal.sqrt_le_one + - Real.one_le_sqrt + - Real.sqrt_le_left + - Real.div_sqrt + - Real.sqrt_mul_self_eq_abs + - Real.sqrt_sq + - NNReal.sum_sqrt_mul_sqrt_le + - Real.abs_le_sqrt + - NNReal.sqrt_mul + - Real.sqrt_le_sqrt + - Real.sqrt_inj + - Real.sqrt_sq_eq_abs + - Real.sum_sqrt_mul_sqrt_le + - Real.sqrt_eq_iff_mul_self_eq_of_pos + - Real.sqrt_pos + - Real.sq_sqrt + - Real.sqrt_eq_one + - path: Mathlib/Topology/MetricSpace/Lipschitz.lean + theorems: + - LipschitzWith.max_const + - LipschitzWith.of_le_add_mul + - LipschitzWith.min_const + - LocallyLipschitz.const_max + - Real.lipschitzWith_toNNReal + - LipschitzOnWith.of_le_add + - LipschitzWith.dist_le_mul_of_le + - LipschitzWith.const_max + - LipschitzWith.of_dist_le' + - LipschitzWith.mk_one + - continuousAt_of_locally_lipschitz + - LipschitzOnWith.of_le_add_mul + - LipschitzWith.const_min + - LipschitzOnWith.extend_real + - lipschitzOnWith_iff_dist_le_mul + - LipschitzOnWith.extend_pi + - LocallyLipschitz.const_min + - LipschitzOnWith.of_dist_le' + - LipschitzWith.dist_iterate_succ_le_geometric + - LipschitzOnWith.mk_one + - LipschitzWith.of_le_add + - path: Mathlib/RingTheory/PolynomialAlgebra.lean + theorems: + - PolyEquivTensor.toFunLinear_mul_tmul_mul_aux_2 + - PolyEquivTensor.toFunBilinear_apply_eq_sum + - PolyEquivTensor.toFunLinear_one_tmul_one + - matPolyEquiv_symm_C + - PolyEquivTensor.toFunLinear_mul_tmul_mul + - matPolyEquiv_symm_X + - matPolyEquiv_coeff_apply + - matPolyEquiv_symm_apply_coeff + - PolyEquivTensor.right_inv + - matPolyEquiv_smul_one + - matPolyEquiv_coeff_apply_aux_1 + - PolyEquivTensor.invFun_add + - matPolyEquiv_diagonal_X + - matPolyEquiv_coeff_apply_aux_2 + - matPolyEquiv_map_smul + - PolyEquivTensor.left_inv + - PolyEquivTensor.toFunLinear_mul_tmul_mul_aux_1 + - support_subset_support_matPolyEquiv + - path: Mathlib/RingTheory/Ideal/Over.lean + theorems: + - Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrime + - Ideal.exists_ideal_over_prime_of_isIntegral_of_isDomain + - Ideal.comap_lt_comap_of_integral_mem_sdiff + - Ideal.exists_ideal_over_maximal_of_isIntegral + - Ideal.IsIntegralClosure.eq_bot_of_comap_eq_bot + - Ideal.exists_ideal_comap_le_prime + - Ideal.coeff_zero_mem_comap_of_root_mem_of_eval_mem + - Ideal.exists_ideal_over_prime_of_isIntegral + - Ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff + - Ideal.injective_quotient_le_comap_map + - Ideal.exists_nonzero_mem_of_ne_bot + - Ideal.exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem + - Ideal.map_eq_top_iff_of_ker_le + - Ideal.quotient_mk_maps_eq + - Ideal.isMaximal_comap_of_isIntegral_of_isMaximal + - Ideal.map_eq_top_iff + - Ideal.comap_eq_of_scalar_tower_quotient + - path: Mathlib/Topology/ContinuousFunction/Basic.lean + theorems: + - Homeomorph.toContinuousMap_comp_symm + - ContinuousMap.coe_injective + - ContinuousMap.liftCover_coe + - ContinuousMap.liftCover_coe' + - ContinuousMap.liftCover_restrict + - Homeomorph.symm_comp_toContinuousMap + - ContinuousMap.cancel_left + - QuotientMap.lift_comp + - path: Mathlib/Topology/OmegaCompletePartialOrder.lean + theorems: + - "Scott.is\u03C9Sup_iff_isLUB" + - notBelow_isOpen + - scottContinuous_of_continuous + - Scott.isOpen_sUnion + - "is\u03C9Sup_\u03C9Sup" + - continuous_of_scottContinuous + - path: Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean + theorems: + - FormalMultilinearSeries.order_eq_zero_iff + - FormalMultilinearSeries.norm_apply_eq_norm_coef + - FormalMultilinearSeries.order_eq_find + - FormalMultilinearSeries.coeff_fslope + - FormalMultilinearSeries.ne_zero_of_order_ne_zero + - FormalMultilinearSeries.apply_eq_pow_smul_coeff + - FormalMultilinearSeries.coeff_eq_zero + - FormalMultilinearSeries.congr + - FormalMultilinearSeries.order_zero + - FormalMultilinearSeries.apply_eq_prod_smul_coeff + - FormalMultilinearSeries.order_eq_zero_iff' + - FormalMultilinearSeries.coeff_iterate_fslope + - FormalMultilinearSeries.removeZero_of_pos + - path: Mathlib/Algebra/Homology/TotalComplex.lean + theorems: + - "HomologicalComplex\u2082.total.map_id" + - "HomologicalComplex\u2082.D\u2082_D\u2081" + - "HomologicalComplex\u2082.total.map_comp" + - "HomologicalComplex\u2082.total.mapAux.d\u2082_mapMap" + - "HomologicalComplex\u2082.D\u2082_shape" + - "HomologicalComplex\u2082.\u03B9_D\u2081" + - "HomologicalComplex\u2082.D\u2081_D\u2082" + - "HomologicalComplex\u2082.totalAux.d\u2082_eq'" + - "HomologicalComplex\u2082.total.hom_ext" + - "HomologicalComplex\u2082.totalAux.\u03B9MapObj_D\u2081" + - "HomologicalComplex\u2082.totalAux.d\u2081_eq" + - "HomologicalComplex\u2082.D\u2082_D\u2082" + - "HomologicalComplex\u2082.totalAux.d\u2082_eq" + - "HomologicalComplex\u2082.total.mapAux.mapMap_D\u2081" + - "HomologicalComplex\u2082.d\u2082_eq_zero" + - "HomologicalComplex\u2082.totalAux.d\u2081_eq'" + - "HomologicalComplex\u2082.\u03B9Total_map" + - "HomologicalComplex\u2082.\u03B9TotalOrZero_map" + - "HomologicalComplex\u2082.total.mapAux.d\u2081_mapMap" + - "HomologicalComplex\u2082.D\u2081_D\u2081" + - "HomologicalComplex\u2082.d\u2081_eq_zero'" + - "HomologicalComplex\u2082.d\u2081_eq_zero" + - "HomologicalComplex\u2082.d\u2082_eq_zero'" + - "HomologicalComplex\u2082.XXIsoOfEq_inv_\u03B9Total" + - "HomologicalComplex\u2082.total.mapAux.mapMap_D\u2082" + - "HomologicalComplex\u2082.XXIsoOfEq_hom_\u03B9Total" + - "HomologicalComplex\u2082.totalAux.\u03B9MapObj_D\u2082" + - "HomologicalComplex\u2082.\u03B9_D\u2082" + - "HomologicalComplex\u2082.D\u2081_shape" + - "HomologicalComplex\u2082.\u03B9_totalDesc" + - path: Mathlib/RingTheory/Polynomial/Basic.lean + theorems: + - Polynomial.coeff_mem_frange + - Polynomial.prime_C_iff + - Polynomial.degreeLE_eq_span_X_pow + - Polynomial.degreeLT_succ_eq_degreeLE + - Polynomial.frange_ofSubring + - Polynomial.toSubring_one + - Polynomial.ker_mapRingHom + - Polynomial.support_toSubring + - Polynomial.linearIndependent_powers_iff_aeval + - Polynomial.toSubring_zero + - "Polynomial.eval\u2082_restriction" + - Polynomial.frange_one + - MvPolynomial.aeval_natDegree_le + - "MvPolynomial.map_mvPolynomial_eq_eval\u2082" + - Polynomial.degree_restriction + - Ideal.mem_map_C_iff + - Ideal.isPrime_map_C_iff_isPrime + - MvPolynomial.noZeroDivisors_of_finite + - Polynomial.sup_aeval_range_eq_top_of_coprime + - Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime + - Polynomial.exists_monic_irreducible_factor + - Polynomial.isNoetherianRing + - Polynomial.map_toSubring + - Polynomial.eval_eq_sum_degreeLTEquiv + - Polynomial.not_finite + - Polynomial.coeff_prod_mem_ideal_pow_tsub + - Polynomial.mem_frange_iff + - Polynomial.mem_degreeLT + - Polynomial.exists_degree_le_of_mem_span_of_finite + - Ideal.eq_zero_of_constant_mem_of_maximal + - Polynomial.support_restriction + - Polynomial.natDegree_toSubring + - MvPolynomial.prime_C_iff_of_fintype + - MvPolynomial.isNoetherianRing_fin_0 + - Polynomial.sup_ker_aeval_le_ker_aeval_mul + - Polynomial.span_le_degreeLE_of_finite + - Polynomial.map_restriction + - MvPolynomial.prime_C_iff + - Ideal.mem_leadingCoeffNth + - MvPolynomial.uniqueFactorizationMonoid_of_fintype + - Polynomial.degreeLT_eq_span_X_pow + - Ideal.leadingCoeffNth_mono + - MvPolynomial.ker_map + - Polynomial.coeff_restriction + - Polynomial.monic_toSubring + - MvPolynomial.mem_ideal_of_coeff_mem_ideal + - Polynomial.Monic.geom_sum + - Polynomial.exists_degree_le_of_mem_span + - Polynomial.monic_restriction + - Polynomial.coeff_toSubring + - Polynomial.disjoint_ker_aeval_of_coprime + - Ideal.mem_leadingCoeff + - Polynomial.restriction_zero + - Ideal.mem_leadingCoeffNth_zero + - Polynomial.restriction_one + - Polynomial.degreeLTEquiv_eq_zero_iff_eq_zero + - Ideal.polynomial_not_isField + - Polynomial.geom_sum_X_comp_X_add_one_eq_sum + - Polynomial.degree_toSubring + - Polynomial.coeff_ofSubring + - Polynomial.monic_geom_sum_X + - Polynomial.natDegree_restriction + - Polynomial.mem_degreeLE + - Polynomial.span_of_finite_le_degreeLT + - MvPolynomial.mem_map_C_iff + - path: Mathlib/Analysis/InnerProductSpace/LinearPMap.lean + theorems: + - LinearPMap.adjoint_apply_of_dense + - LinearPMap.IsFormalAdjoint.symm + - IsSelfAdjoint.dense_domain + - ContinuousLinearMap.toPMap_adjoint_eq_adjoint_toPMap_of_dense + - LinearPMap.IsFormalAdjoint.le_adjoint + - LinearPMap.mem_adjoint_domain_of_exists + - LinearPMap.adjointAux_inner + - path: Mathlib/Data/Finsupp/Basic.lean + theorems: + - Finsupp.mapDomain_injective + - Finsupp.subtypeDomain_eq_zero_iff + - Finsupp.mem_graph_iff + - Finsupp.some_single_some + - Finsupp.mapRange.zeroHom_comp + - Finsupp.prod_option_index + - Finsupp.sum_comapDomain + - Finsupp.single_smul + - Finsupp.filter_smul + - Finsupp.mapDomain_support + - Finsupp.extendDomain_subtypeDomain + - Finsupp.mem_splitSupport_iff_nonzero + - Finsupp.mk_mem_graph_iff + - Finsupp.mem_support_multiset_sum + - Finsupp.support_curry + - Finsupp.filter_eq_sum + - Finsupp.support_extendDomain_subset + - Finsupp.support_smul_eq + - Finsupp.subtypeDomain_eq_zero_iff' + - Finsupp.equivMapDomain_zero + - Finsupp.mapDomain.addMonoidHom_comp_mapRange + - Finsupp.comapSMul_apply + - Finsupp.filter_eq_indicator + - Finsupp.equivMapDomain_single + - Finsupp.mapDomain_comp + - Finsupp.filter_zero + - Finsupp.filter_pos_add_filter_neg + - Finsupp.mapDomain_comapDomain + - Finsupp.prod_subtypeDomain_index + - Finsupp.mapDomain_apply' + - Finsupp.mapDomain_equiv_apply + - Finsupp.prod_filter_mul_prod_filter_not + - Finsupp.equivMapDomain_eq_mapDomain + - Finsupp.filter_eq_zero_iff + - Finsupp.mapRange.equiv_trans + - Finsupp.comapDomain_add + - Finsupp.prod_equivMapDomain + - Finsupp.some_zero + - Finsupp.mapDomain_notin_range + - Finsupp.filter_eq_self_iff + - Finsupp.filter_curry + - Finsupp.graph_zero + - Finsupp.equivMapDomain_trans' + - Finsupp.equivMapDomain_refl + - Finsupp.mapDomain_injOn + - Finsupp.frange_single + - Finsupp.eq_zero_of_comapDomain_eq_zero + - Finsupp.mapDomain_congr + - Finsupp.support_subtypeDomain + - Finsupp.prod_filter_index + - Finsupp.comapDomain_single + - Finsupp.equivMapDomain_refl' + - Finsupp.some_add + - Finsupp.extendDomain_eq_embDomain_subtype + - Finsupp.sum_update_add + - Finsupp.mapDomain_apply + - Finsupp.mapRange_smul + - Finsupp.mapRange.addMonoidHom_comp + - Finsupp.mapRange.addEquiv_trans + - Finsupp.support_smul + - Finsupp.split_apply + - Finsupp.sigma_sum + - Finsupp.sigma_support + - Finsupp.embDomain_comapDomain + - Finsupp.extendDomain_single + - Finsupp.curry_apply + - Finsupp.smul_single_one + - Finsupp.sum_curry_index + - Finsupp.mapDomain_support_of_injOn + - Finsupp.mem_frange + - Finsupp.graph_injective + - Finsupp.prod_mapDomain_index_inj + - Finsupp.comapDomain_smul + - Finsupp.some_single_none + - path: Mathlib/Probability/Kernel/CondCdf.lean + theorems: + - ProbabilityTheory.integrable_condCDF + - ProbabilityTheory.inf_gt_preCDF + - MeasureTheory.Measure.IicSnd_le_fst + - ProbabilityTheory.monotone_condCDFRat + - ProbabilityTheory.lintegral_condCDF + - ProbabilityTheory.set_lintegral_condCDF + - ProbabilityTheory.tendsto_preCDF_atTop_one + - ProbabilityTheory.monotone_condCDF' + - ProbabilityTheory.set_integral_condCDF + - ProbabilityTheory.hasCondCDF_ae + - ProbabilityTheory.measurable_measure_condCDF + - Real.iUnion_Iic_rat + - ProbabilityTheory.condCDFRat_ae_eq + - ProbabilityTheory.set_lintegral_iInf_gt_preCDF + - ProbabilityTheory.condCDF'_eq_condCDFRat + - ProbabilityTheory.condCDF_ae_eq + - ProbabilityTheory.ofReal_condCDF_ae_eq + - ProbabilityTheory.tendsto_condCDF_atBot + - ProbabilityTheory.continuousWithinAt_condCDF'_Ici + - ProbabilityTheory.tendsto_lintegral_preCDF_atBot + - ProbabilityTheory.integral_condCDF + - Real.iInter_Iic_rat + - ProbabilityTheory.tendsto_lintegral_preCDF_atTop + - ProbabilityTheory.condCDFRat_of_not_mem + - MeasureTheory.Measure.IicSnd_apply + - ProbabilityTheory.ofReal_condCDFRat_ae_eq + - ProbabilityTheory.mem_condCDFSet_ae + - ProbabilityTheory.tendsto_preCDF_atBot_zero + - ProbabilityTheory.condCDFRat_of_mem + - ProbabilityTheory.set_lintegral_condCDF_rat + - ProbabilityTheory.inf_gt_condCDFRat + - ProbabilityTheory.hasCondCDF_of_mem_condCDFSet + - ProbabilityTheory.measure_condCDF_Iic + - ProbabilityTheory.preCDF_le_one + - ProbabilityTheory.condCDF'_nonneg + - ProbabilityTheory.measurable_condCDF + - ProbabilityTheory.tendsto_condCDFRat_atTop + - ProbabilityTheory.measurable_condCDFRat + - ProbabilityTheory.condCDFRat_le_one + - MeasureTheory.Measure.tendsto_IicSnd_atTop + - MeasureTheory.Measure.iInf_IicSnd_gt + - ProbabilityTheory.set_lintegral_preCDF_fst + - MeasureTheory.Measure.tendsto_IicSnd_atBot + - ProbabilityTheory.bddBelow_range_condCDFRat_gt + - ProbabilityTheory.condCDFRat_nonneg + - ProbabilityTheory.tendsto_condCDF_atTop + - ProbabilityTheory.condCDF_le_one + - ProbabilityTheory.measure_condCDF_univ + - MeasureTheory.Measure.IicSnd_mono + - ProbabilityTheory.condCDF'_def' + - ProbabilityTheory.tendsto_condCDFRat_atBot + - ProbabilityTheory.monotone_preCDF + - path: Mathlib/Algebra/Group/Commute/Hom.lean + theorems: + - SemiconjBy.map + - SemiconjBy.of_map + - Commute.of_map + - path: Mathlib/Algebra/Category/ModuleCat/Images.lean + theorems: + - ModuleCat.image.lift_fac + - ModuleCat.imageIsoRange_hom_subtype + - path: Mathlib/Order/Atoms.lean + theorems: + - OrderIso.isCoatomic_iff + - Prop.isCoatom_iff + - BooleanAlgebra.le_iff_atom_le_imp + - Pi.isAtom_iff + - eq_iff_atom_le_iff + - sSup_atoms_le_eq + - sSup_atoms_eq_top + - isCoatomistic_dual_iff_isAtomistic + - OrderEmbedding.isAtom_of_map_bot_of_image + - le_iff_atom_le_imp + - Set.isCoatom_iff + - IsAtom.le_iff + - atom_le_iSup + - Pi.isAtom_single + - GaloisInsertion.isCoatom_iff + - OrderIso.isAtomic_iff + - isAtomistic_dual_iff_isCoatomistic + - covBy_iff_coatom_Iic + - isCoatomic_of_isAtomic_of_complementedLattice_of_isModular + - GaloisInsertion.isAtom_iff' + - isCoatomic_dual_iff_isAtomic + - Pi.isAtom_iff_eq_single + - Pi.eq_bot_iff + - isSimpleOrder_iff_isSimpleOrder_orderDual + - isAtom_iff_le_of_ge + - bot_covBy_iff + - Prop.isAtom_iff + - IsSimpleOrder.bot_ne_top + - BooleanAlgebra.eq_iff_atom_le_iff + - Set.Iic.isCoatom_iff + - OrderIso.isSimpleOrder_iff + - isAtomic_dual_iff_isCoatomic + - GaloisInsertion.isAtom_iff + - covBy_iff_atom_Ici + - Set.isAtom_iff + - Set.Ici.isAtom_iff + - path: Mathlib/Order/LiminfLimsup.lean + theorems: + - Filter.HasBasis.liminf_eq_ite + - Filter.isBounded_principal + - Filter.liminf_nat_add + - Filter.blimsup_sup_not + - Filter.isCobounded_bot + - GaloisConnection.l_limsup_le + - Filter.mem_limsup_iff_frequently_mem + - Filter.liminf_le_of_frequently_le' + - Filter.liminf_bot + - Filter.limsSup_bot + - Filter.HasBasis.liminf_eq_sSup_univ_of_empty + - Filter.exists_forall_mem_of_hasBasis_mem_blimsup' + - Filter.exists_forall_mem_of_hasBasis_mem_blimsup + - Filter.blimsup_eq_limsup + - Filter.IsBoundedUnder.mono_le + - Filter.limsup_bot + - OrderIso.isBoundedUnder_le_comp + - OrderIso.limsup_apply + - Monotone.isBoundedUnder_le_comp_iff + - Filter.cofinite.limsup_set_eq + - Filter.limsInf_bot + - Filter.sdiff_limsup + - Filter.iInf_le_liminf + - Filter.cofinite.liminf_set_eq + - Filter.limsSup_principal + - Filter.CompleteLatticeHom.apply_limsup_iterate + - Filter.blimsup_eq_iInf_biSup + - Filter.liminf_sdiff + - Filter.limsup_const_bot + - Filter.limsup_compl + - Filter.sup_liminf + - Filter.blimsup_congr' + - Filter.limsup_le_iSup + - Filter.IsBoundedUnder.bddAbove_range_of_cofinite + - Filter.sdiff_liminf + - Filter.cofinite.blimsup_set_eq + - Filter.OrderIso.apply_blimsup + - Filter.isCobounded_top + - Filter.limsup_piecewise + - Filter.HasBasis.blimsup_eq_iInf_iSup + - Filter.SupHom.apply_blimsup_le + - Filter.bliminf_antitone + - Filter.blimsup_congr + - Filter.limsup_eq_iInf_iSup_of_nat' + - Filter.blimsup_true + - Filter.IsBoundedUnder.bddAbove_range + - Filter.limsInf_top + - Filter.HasBasis.liminf_eq_ciSup_ciInf + - Filter.frequently_lt_of_lt_limsSup + - Filter.mem_liminf_iff_eventually_mem + - Filter.limsup_sup_filter + - Filter.HasBasis.limsup_eq_iInf_iSup + - Filter.blimsup_sup_le_or + - Filter.limsup_eq_sInf_sSup + - Filter.cofinite.bliminf_set_eq + - Filter.eventually_lt_of_lt_liminf + - Filter.bliminf_true + - Filter.limsup_const + - Filter.limsup_sdiff + - Filter.limsup_eq_iInf_iSup + - Filter.limsup_eq_iInf_iSup_of_nat + - Filter.IsBoundedUnder.sup + - Filter.liminf_compl + - Filter.blimsup_and_le_inf + - Filter.not_isBoundedUnder_of_tendsto_atTop + - Filter.blimsup_false + - Filter.blimsup_or_eq_sup + - Filter.le_limsup_of_frequently_le + - Filter.isBounded_top + - Filter.limsSup_top + - Filter.HasBasis.liminf_eq_sSup_iUnion_iInter + - Filter.sup_limsup + - Filter.blimsup_eq_iInf_biSup_of_nat + - Filter.isBounded_bot + - path: Mathlib/MeasureTheory/Measure/Restrict.lean + theorems: + - MeasureTheory.Measure.restrict_map + - "MeasureTheory.Measure.restrict_union\u2080" + - ae_eq_restrict_iff_indicator_ae_eq + - MeasureTheory.Measure.ext_of_generateFrom_of_iUnion + - MeasureTheory.Measure.restrict_compl_add_restrict + - MeasureTheory.Measure.restrict_eq_zero + - MeasureTheory.mem_map_restrict_ae_iff + - MeasureTheory.Measure.restrict_union_congr + - MeasurableSet.map_coe_volume + - MeasureTheory.ae_restrict_biUnion_eq + - MeasureTheory.Measure.Subtype.volume_univ + - "MeasureTheory.Measure.restrict_restrict\u2080'" + - MeasureTheory.Measure.restrict_comm + - MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl + - MeasureTheory.Measure.ext_iff_of_sUnion_eq_univ + - volume_preimage_coe + - "MeasureTheory.Measure.restrict_apply\u2080" + - MeasurableEmbedding.restrict_comap + - MeasureTheory.Measure.restrict_iUnion_apply_ae + - "MeasureTheory.Measure.restrict_inter_add_diff\u2080" + - "MeasureTheory.Measure.restrict_restrict\u2080" + - MeasureTheory.div_ae_eq_one + - MeasureTheory.Measure.restrict_union_add_inter' + - MeasureTheory.Measure.ext_iff_of_iUnion_eq_univ + - MeasureTheory.Measure.restrict_sum_of_countable + - MeasurableEmbedding.ae_map_iff + - MeasureTheory.Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict + - map_comap_subtype_coe + - MeasureTheory.Measure.restrict_iUnion_congr + - MeasureTheory.self_mem_ae_restrict + - MeasureTheory.ae_restrict_biUnion_finset_iff + - MeasurableEmbedding.comap_preimage + - MeasureTheory.Measure.restrict_union_le + - MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae + - indicator_ae_eq_restrict_compl + - MeasureTheory.ae_eq_restrict_biUnion_iff + - MeasureTheory.Measure.ext_of_generateFrom_of_cover_subset + - Filter.EventuallyEq.restrict + - MeasureTheory.ae_restrict_of_ae_eq_of_ae_restrict + - indicator_ae_eq_of_ae_eq_set + - MeasureTheory.ae_restrict_union_eq + - "MeasureTheory.ae_restrict_iff\u2080" + - MeasureTheory.Measure.restrict_congr_meas + - MeasureTheory.one_le_div_ae + - indicator_ae_eq_zero_of_restrict_ae_eq_zero + - indicator_ae_eq_of_restrict_compl_ae_eq_zero + - MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe + - indicator_ae_eq_restrict + - MeasurableEmbedding.restrict_map + - MeasureTheory.ae_imp_of_ae_restrict + - MeasureTheory.ae_restrict_iUnion_iff + - MeasureTheory.measure_setOf_frequently_eq_zero + - MeasureTheory.Measure.restrict_sum + - MeasureTheory.Measure.restrict_apply_eq_zero + - MeasureTheory.ae_smul_measure + - MeasureTheory.Measure.restrict_apply_eq_zero' + - MeasureTheory.ae_restrict_biUnion_iff + - MeasureTheory.Measure.restrict_iUnion_ae + - MeasureTheory.Measure.restrict_apply' + - MeasureTheory.Measure.restrict_sInf_eq_sInf_restrict + - piecewise_ae_eq_restrict + - MeasureTheory.ae_eq_restrict_iUnion_iff + - MeasurableEmbedding.map_comap + - MeasureTheory.Measure.restrict_univ + - MeasureTheory.Measure.ext_iff_of_biUnion_eq_univ + - MeasureTheory.Measure.restrict_restrict_of_subset + - "MeasureTheory.ae_restrict_iff'\u2080" + - MeasureTheory.ae_restrict_uIoc_iff + - MeasureTheory.ae_restrict_union_iff + - MeasureTheory.Measure.restrict_iUnion_apply_eq_iSup + - ae_restrict_iff_subtype + - MeasureTheory.Measure.ext_of_generateFrom_of_cover + - map_restrict_ae_le_map_indicator_ae + - "MeasureTheory.Measure.restrict_apply\u2080'" + - MeasurableEmbedding.comap_restrict + - MeasurableEmbedding.comap_apply + - MeasureTheory.ae_restrict_eq + - mem_map_indicator_ae_iff_of_zero_nmem + - MeasureTheory.Measure.restrict_biUnion_congr + - MeasureTheory.Measure.restrict_union' + - MeasureTheory.Measure.restrict_add_restrict_compl + - piecewise_ae_eq_of_ae_eq_set + - piecewise_ae_eq_restrict_compl + - MeasureTheory.Measure.AbsolutelyContinuous.restrict + - MeasureTheory.Measure.restrict_apply_univ + - MeasurableEmbedding.comap_map + - MeasureTheory.Measure.restrict_finset_biUnion_congr + - MeasureTheory.ae_restrict_uIoc_eq + - MeasureTheory.Measure.restrict_congr_mono + - "MeasureTheory.Measure.restrict_union_add_inter\u2080" + - path: Mathlib/Analysis/NormedSpace/lpSpace.lean + theorems: + - lp.isLUB_norm + - LipschitzOnWith.coordinate + - "Mem\u2113p.of_exponent_ge" + - lp.hasSum_norm + - lp.eq_zero_iff_coeFn_eq_zero + - lp.single_smul + - lp.norm_eq_zero_iff + - lp.norm_sum_single + - lp.norm_eq_tsum_rpow + - lp.norm_compl_sum_single + - lp.coeFn_sum + - "Mem\u2113p.finset_sum" + - lp.norm_apply_le_norm + - lp.norm_le_of_forall_le' + - "Mem\u2113p.const_smul" + - lp.hasSum_single + - "Mem\u2113p.add" + - lp.norm_zero + - lp.norm_const_smul + - lp.sum_rpow_le_of_tendsto + - lp.norm_le_of_forall_le + - lp.single_neg + - "mem\u2113p_gen_iff" + - "one_mem\u2113p_infty" + - lp.norm_single + - lp.single_apply_self + - "mem\u2113p_infty_iff" + - lp.single_apply_ne + - "Mem\u2113p.const_mul" + - lp.uniformContinuous_coe + - "mem\u2113p_gen'" + - lp.tsum_mul_le_mul_norm + - "Mem\u2113p.star_mem" + - lp.tendsto_lp_of_tendsto_pi + - lp.norm_neg + - "Mem\u2113p.neg" + - "lp.mem\u2113p_of_tendsto" + - lp.norm_const_smul_le + - "algebraMap_mem\u2113p_infty" + - "zero_mem\u2113p" + - lp.sum_rpow_le_norm_rpow + - LipschitzWith.coordinate + - lp.norm_eq_ciSup + - lp.norm_rpow_eq_tsum + - lp.norm_sub_norm_compl_sub_single + - "mem\u2113p_gen" + - lp.norm_le_of_tsum_le + - "Mem\u2113p.sub" + - lp.norm_apply_le_of_tendsto + - "mem\u2113p_zero_iff" + - lp.norm_nonneg' + - path: Mathlib/Analysis/Analytic/Composition.lean + theorems: + - FormalMultilinearSeries.id_apply_ne_one + - FormalMultilinearSeries.comp_coeff_one + - FormalMultilinearSeries.compAlongComposition_nnnorm + - FormalMultilinearSeries.id_comp + - FormalMultilinearSeries.comp_assoc + - Composition.sigma_composition_eq_iff + - FormalMultilinearSeries.mem_compPartialSumTarget_iff + - FormalMultilinearSeries.mem_compPartialSumSource_iff + - FormalMultilinearSeries.comp_summable_nnreal + - FormalMultilinearSeries.applyComposition_ones + - FormalMultilinearSeries.removeZero_comp_of_pos + - FormalMultilinearSeries.comp_coeff_zero'' + - FormalMultilinearSeries.removeZero_applyComposition + - FormalMultilinearSeries.compAlongComposition_bound + - FormalMultilinearSeries.compChangeOfVariables_length + - Composition.sizeUpTo_sizeUpTo_add + - FormalMultilinearSeries.le_comp_radius_of_summable + - FormalMultilinearSeries.comp_partialSum + - FormalMultilinearSeries.compPartialSumTarget_tendsto_atTop + - FormalMultilinearSeries.applyComposition_single + - Composition.blocksFun_sigmaCompositionAux + - FormalMultilinearSeries.compChangeOfVariables_blocksFun + - FormalMultilinearSeries.comp_coeff_zero + - FormalMultilinearSeries.id_apply_one' + - FormalMultilinearSeries.compPartialSumTargetSet_image_compPartialSumSource + - FormalMultilinearSeries.applyComposition_update + - Composition.sigma_pi_composition_eq_iff + - HasFPowerSeriesAt.comp + - FormalMultilinearSeries.comp_id + - FormalMultilinearSeries.compContinuousLinearMap_applyComposition + - FormalMultilinearSeries.compChangeOfVariables_sum + - Composition.length_sigmaCompositionAux + - Composition.length_gather + - path: Mathlib/GroupTheory/Perm/Cycle/Concrete.lean + theorems: + - Equiv.Perm.toList_formPerm_singleton + - Equiv.Perm.toList_formPerm_nil + - Cycle.support_formPerm + - Equiv.Perm.toList_eq_nil_iff + - Equiv.Perm.toList_formPerm_nontrivial + - Equiv.Perm.two_le_length_toList_iff_mem_support + - Equiv.Perm.toCycle_eq_toList + - List.formPerm_apply_mem_eq_next + - Equiv.Perm.pow_apply_mem_toList_iff_mem_support + - Equiv.Perm.nthLe_toList + - Equiv.Perm.toList_formPerm_isRotated_self + - Equiv.Perm.toList_one + - Equiv.Perm.nontrivial_toCycle + - Equiv.Perm.formPerm_toList + - Equiv.Perm.IsCycle.existsUnique_cycle_nontrivial_subtype + - Equiv.Perm.nodup_toCycle + - Equiv.Perm.IsCycle.existsUnique_cycle_subtype + - Cycle.formPerm_reverse + - Equiv.Perm.SameCycle.toList_isRotated + - Equiv.Perm.toList_nthLe_zero + - Equiv.Perm.nodup_toList + - Equiv.Perm.length_toList + - List.cycleType_formPerm + - Equiv.Perm.next_toList_eq_apply + - Cycle.isCycle_formPerm + - Cycle.formPerm_subsingleton + - Equiv.Perm.toList_pow_apply_eq_rotate + - Cycle.formPerm_eq_formPerm_iff + - List.cycleOf_formPerm + - Equiv.Perm.toList_ne_singleton + - Equiv.Perm.mem_toList_iff + - Equiv.Perm.IsCycle.existsUnique_cycle + - Cycle.formPerm_eq_self_of_not_mem + - List.formPerm_disjoint_iff + - Cycle.formPerm_apply_mem_eq_next + - List.isCycle_formPerm + - path: Mathlib/RingTheory/Etale.lean + theorems: + - Algebra.FormallySmooth.exists_lift + - Algebra.FormallySmooth.of_split + - Algebra.FormallySmooth.liftOfSurjective_apply + - Algebra.FormallySmooth.iff_split_surjection + - Algebra.FormallyUnramified.lift_unique_of_ringHom + - Algebra.FormallySmooth.of_isLocalization + - Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential + - Algebra.FormallyUnramified.of_isLocalization + - Algebra.FormallyUnramified.of_comp + - Algebra.FormallyUnramified.of_equiv + - Algebra.FormallySmooth.comp + - Algebra.FormallySmooth.localization_base + - Algebra.FormallySmooth.of_equiv + - Algebra.FormallyEtale.localization_map + - Algebra.FormallySmooth.localization_map + - Algebra.FormallyUnramified.comp + - Algebra.FormallyUnramified.lift_unique + - path: Mathlib/Analysis/Convex/Strong.lean + theorems: + - aux_sub + - UniformConcaveOn.add + - UniformConcaveOn.concaveOn + - StrongConvexOn.strictConvexOn + - UniformConvexOn.add + - UniformConcaveOn.sub + - UniformConvexOn.sub + - UniformConvexOn.convexOn + - strongConvexOn_zero + - strongConcaveOn_zero + - aux_add + - strongConvexOn_iff_convex + - UniformConvexOn.mono + - UniformConvexOn.strictConvexOn + - StrongConcaveOn.strictConcaveOn + - UniformConvexOn.neg + - StrongConcaveOn.mono + - UniformConcaveOn.mono + - StrongConvexOn.mono + - uniformConvexOn_zero + - UniformConcaveOn.neg + - uniformConcaveOn_zero + - strongConcaveOn_iff_convex + - UniformConcaveOn.strictConcaveOn + - path: Mathlib/CategoryTheory/Iso.lean + theorems: + - CategoryTheory.eq_of_inv_eq_inv + - CategoryTheory.Iso.hom_comp_eq_id + - CategoryTheory.isIso_of_comp_hom_eq_id + - CategoryTheory.IsIso.of_isIso_comp_left + - CategoryTheory.Iso.hom_eq_inv + - CategoryTheory.Iso.cancel_iso_hom_left + - CategoryTheory.Iso.cancel_iso_inv_right_assoc + - CategoryTheory.Functor.map_inv_hom + - CategoryTheory.Iso.inv_eq_inv + - CategoryTheory.Iso.trans_assoc + - CategoryTheory.IsIso.inv_eq_of_inv_hom_id + - CategoryTheory.isIso_of_hom_comp_eq_id + - CategoryTheory.Iso.cancel_iso_hom_right + - CategoryTheory.Functor.map_inv + - CategoryTheory.Iso.cancel_iso_inv_left + - CategoryTheory.Iso.refl_trans + - CategoryTheory.Iso.cancel_iso_inv_right + - CategoryTheory.IsIso.of_isIso_comp_right + - CategoryTheory.Iso.symm_self_id_assoc + - CategoryTheory.Iso.comp_hom_eq_id + - CategoryTheory.IsIso.inv_id + - CategoryTheory.Functor.mapIso_trans + - CategoryTheory.Iso.cancel_iso_hom_right_assoc + - CategoryTheory.IsIso.of_isIso_fac_right + - CategoryTheory.IsIso.inv_eq_of_hom_inv_id + - CategoryTheory.Iso.inv_comp_eq + - CategoryTheory.Iso.trans_refl + - CategoryTheory.Iso.symm_symm_eq + - CategoryTheory.Functor.map_hom_inv + - CategoryTheory.IsIso.inv_comp + - CategoryTheory.IsIso.hom_inv_id_assoc + - CategoryTheory.IsIso.of_isIso_fac_left + - CategoryTheory.IsIso.Iso.inv_inv + - CategoryTheory.Iso.comp_inv_eq + - CategoryTheory.Iso.self_symm_id_assoc + - CategoryTheory.Iso.ext + - CategoryTheory.IsIso.inv_inv + - CategoryTheory.IsIso.Iso.inv_hom + - CategoryTheory.IsIso.inv_hom_id_assoc + - path: Mathlib/Data/Finset/Lattice.lean + theorems: + - Finset.max'_insert + - Finset.toDual_max' + - Finset.min_of_mem + - Finset.sup_lt_iff + - Finset.le_sup'_iff + - Finset.sup_comm + - Finset.iSup_insert_update + - Finset.sup'_product_left + - Finset.sup'_biUnion + - Finset.iSup_union + - Finset.sup'_inf_sup' + - Finset.map_ofDual_max + - Finset.sup_product_left + - Finset.isGLB_mem + - Finset.map_ofDual_min + - Finset.inf_sdiff_right + - Finset.disjoint_sup_right + - Finset.exists_max_image + - Finset.iInf_finset_image + - Finset.sup_eq_bot_iff + - Finset.mem_sup + - Finset.max'_singleton + - Finset.exists_mem_eq_sup' + - Finset.sup_sup + - Finset.sup_le_of_le_directed + - iSup_eq_iSup_finset' + - map_finset_inf + - Finset.exists_next_right + - Finset.min'_insert + - Finset.prodMk_sup'_sup' + - Finset.sup'_congr + - Finset.map_toDual_min + - Finset.sup'_comm + - map_finset_sup + - Finset.mem_of_max + - Finset.sup_inf_distrib_left + - Finset.sup_biUnion + - Finset.comp_sup'_eq_sup'_comp + - Finset.sup_sdiff_right + - Finset.lt_sup_iff + - map_finset_inf' + - Finset.inf_congr + - Finset.sup'_lt_iff + - Finset.max_of_mem + - Finset.min_erase_ne_self + - Finset.induction_on_max + - Multiset.mem_sup + - Finset.iSup_insert + - Finset.subset_set_biUnion_of_mem + - Finset.max'_mem + - Finset.isGLB_iff_isLeast + - Finset.coe_sup' + - Finset.sup'_const + - Finset.iSup_finset_image + - Finset.map_toDual_max + - Finset.disjoint_sup_left + - Finset.le_sup_iff + - Finset.sup_inf_distrib_right + - Finset.sup'_inf_distrib_left + - Finset.comp_sup_eq_sup_comp + - Finset.sup_coe + - Finset.max'_image + - Finset.inf_sup + - Finset.min'_mem + - Finset.inf_union + - Finset.sup_product_right + - Finset.sup'_union + - Finset.iSup_option_toFinset + - Finset.min'_image + - Multiset.count_finset_sup + - Finset.sup'_image + - map_finset_sup' + - Finset.iSup_singleton + - Finset.sup_congr + - Finset.sup_erase_bot + - Finset.ofDual_max' + - Finset.min'_singleton + - Finset.max_singleton + - Finset.min_eq_top + - Finset.sup'_cons + - List.foldr_sup_eq_sup_toFinset + - Finset.sup_eq_biUnion + - Finset.iSup_biUnion + - Finset.sup'_le_iff + - Finset.sup'_product_right + - Finset.iInf_singleton + - Finset.sup'_inf_distrib_right + - Finset.sup_id_eq_sSup + - Finset.min_singleton + - Finset.max_erase_ne_self + - Finset.sup_prodMap + - Finset.sup_sdiff_left + - Finset.sup'_map + - Finset.toDual_min' + - Finset.sup'_induction + - Finset.ofDual_min' + - Finset.sup_inf_sup + - List.foldr_inf_eq_inf_toFinset + - Finset.lt_sup'_iff + - Finset.min'_lt_max'_of_card + - Finset.sup'_insert + - Finset.sup_singleton'' + - Finset.sup_union + - Finset.sup_induction + - Finset.max_eq_bot + - Finset.inf_sdiff_left + - Finset.induction_on_max_value + - Finset.sup_eq_sSup_image + - Finset.sup_bot + - iSup_eq_iSup_finset + - Finset.coe_sup_of_nonempty + - path: Mathlib/Data/Nat/Parity.lean + theorems: + - Nat.even_mul_succ_self + - Function.Involutive.iterate_even + - Nat.odd_sub' + - Nat.div_two_mul_two_add_one_of_odd + - Nat.bit1_div_bit0 + - Nat.succ_mod_two_eq_one_iff + - Nat.bit1_div_two + - Odd.not_two_dvd_nat + - Nat.bit0_div_two + - Function.Involutive.iterate_bit0 + - Nat.even_iff_not_odd + - Nat.bit0_mod_bit0 + - Nat.odd_sub + - Nat.succ_mod_two_add_mod_two + - pow_eq_one_iff_cases + - Nat.odd_add + - Nat.two_mul_div_two_add_one_of_odd + - Nat.odd_iff_not_even + - Function.Involutive.iterate_two_mul + - Nat.even_xor_odd' + - Nat.bit1_mod_bit0 + - Nat.bit0_div_bit0 + - Function.Involutive.iterate_eq_id + - Function.Involutive.iterate_bit1 + - Nat.even_or_odd' + - Nat.one_add_div_two_mul_two_of_odd + - Nat.mod_two_add_add_odd_mod_two + - Nat.even_mul + - Nat.succ_mod_two_eq_zero_iff + - Nat.even_add' + - Nat.not_odd_iff + - Nat.odd_mul + - pow_eq_one_iff_of_ne_zero + - Nat.even_div + - Nat.not_even_bit1 + - Nat.even_add_one + - Nat.odd_add' + - Nat.mod_two_ne_one + - Nat.isCompl_even_odd + - Nat.two_not_dvd_two_mul_add_one + - pow_eq_pow_iff_of_ne_zero + - Nat.even_xor_odd + - Nat.not_even_iff + - pow_eq_neg_pow_iff + - Nat.Odd.sub_odd + - Function.Involutive.iterate_eq_self + - Nat.even_sub + - Nat.even_pow + - pow_eq_pow_iff_cases + - Nat.mod_two_ne_zero + - Nat.even_iff + - Nat.odd_iff + - Nat.even_sub' + - Function.Involutive.iterate_odd + - Nat.ne_of_odd_add + - path: Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean + theorems: + - "ZMod.\u03C7\u2088'_int_eq_\u03C7\u2084_mul_\u03C7\u2088" + - ZMod.neg_one_pow_div_two_of_three_mod_four + - "ZMod.\u03C7\u2088_nat_mod_eight" + - "ZMod.isQuadratic_\u03C7\u2088" + - "ZMod.isQuadratic_\u03C7\u2088'" + - "ZMod.\u03C7\u2084_eq_neg_one_pow" + - "ZMod.\u03C7\u2084_nat_mod_four" + - "ZMod.\u03C7\u2088_int_eq_if_mod_eight" + - "ZMod.\u03C7\u2084_int_one_mod_four" + - "ZMod.\u03C7\u2084_nat_three_mod_four" + - "ZMod.\u03C7\u2084_int_mod_four" + - ZMod.neg_one_pow_div_two_of_one_mod_four + - "ZMod.\u03C7\u2088'_eq_\u03C7\u2084_mul_\u03C7\u2088" + - "ZMod.\u03C7\u2084_int_eq_if_mod_four" + - "ZMod.\u03C7\u2088'_int_eq_if_mod_eight" + - "ZMod.\u03C7\u2084_int_three_mod_four" + - "ZMod.isQuadratic_\u03C7\u2084" + - "ZMod.\u03C7\u2088_int_mod_eight" + - "ZMod.\u03C7\u2084_nat_one_mod_four" + - path: Mathlib/CategoryTheory/Limits/Preserves/Limits.lean + theorems: + - CategoryTheory.preserves_lift_mapCone + - CategoryTheory.preserves_desc_mapCocone + - CategoryTheory.preservesColimitsIso_inv_comp_desc + - CategoryTheory.lift_comp_preservesLimitsIso_hom + - path: Mathlib/Topology/FiberBundle/Trivialization.lean + theorems: + - Pretrivialization.symm_trans_symm + - Trivialization.continuousAt_of_comp_left + - Trivialization.toPretrivialization_injective + - Pretrivialization.symm_trans_source_eq + - Pretrivialization.mk_symm + - Pretrivialization.ext' + - Pretrivialization.preimage_symm_proj_baseSet + - Trivialization.frontier_preimage + - Trivialization.continuousAt_of_comp_right + - Trivialization.nhds_eq_inf_comap + - Pretrivialization.mem_source + - Trivialization.isImage_preimage_prod + - Trivialization.tendsto_nhds_iff + - Trivialization.coordChange_same_apply + - Trivialization.coordChange_apply_snd + - Pretrivialization.apply_mk_symm + - Pretrivialization.proj_symm_apply + - Pretrivialization.symm_trans_target_eq + - Trivialization.ext' + - Pretrivialization.trans_source + - Trivialization.continuousOn_symm + - Trivialization.coordChange_coordChange + - Pretrivialization.toPartialEquiv_injective + - Trivialization.mk_coordChange + - Pretrivialization.symm_proj_apply + - Pretrivialization.symm_apply_mk_proj + - Pretrivialization.ext + - Pretrivialization.mem_target + - Trivialization.map_proj_nhds + - Trivialization.continuous_coordChange + - Trivialization.mem_source + - Pretrivialization.target_inter_preimage_symm_source_eq + - Pretrivialization.preimage_symm_proj_inter + - Trivialization.preimageSingletonHomeomorph_symm_apply + - path: Mathlib/Init/Order/LinearOrder.lean + theorems: + - max_eq_left + - max_self + - le_max_left + - min_self + - le_min + - min_le_left + - min_eq_left + - max_def + - min_le_right + - le_max_right + - max_le + - max_assoc + - lt_min + - max_lt + - min_assoc + - min_def + - path: Mathlib/Topology/MetricSpace/Holder.lean + theorems: + - holderOnWith_one + - HolderOnWith.uniformContinuousOn + - HolderWith.dist_le_of_le + - HolderOnWith.comp + - HolderOnWith.edist_le_of_le + - holderOnWith_univ + - holderOnWith_singleton + - HolderWith.nndist_le_of_le + - path: Mathlib/RingTheory/Adjoin/Field.lean + theorems: + - Polynomial.lift_of_splits + - Subalgebra.adjoin_rank_le + - IsIntegral.mem_range_algHom_of_minpoly_splits + - IsIntegral.minpoly_splits_tower_top + - path: Mathlib/Data/Polynomial/FieldDivision.lean + theorems: + - Polynomial.map_div + - Polynomial.X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic + - Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors + - Polynomial.gcd_map + - Polynomial.rootSet_X_pow + - Polynomial.quotient_mul_add_remainder_eq_aux + - Polynomial.isUnit_iff_degree_eq_zero + - Polynomial.rootMultiplicity_sub_one_le_derivative_rootMultiplicity + - Polynomial.isCoprime_map + - Polynomial.degree_div_le + - Polynomial.rootSet_monomial + - Polynomial.C_mul_dvd + - Polynomial.isUnit_map + - Polynomial.degree_add_div + - Polynomial.leadingCoeff_normalize + - Polynomial.not_irreducible_C + - Polynomial.derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors + - Polynomial.map_dvd_map' + - Polynomial.coeff_inv_units + - Polynomial.div_C_mul + - Polynomial.divByMonic_eq_div + - Polynomial.coe_normUnit + - Polynomial.root_left_of_root_gcd + - Polynomial.coe_normUnit_of_ne_zero + - Polynomial.exists_root_of_degree_eq_one + - Polynomial.monic_map_iff + - Polynomial.normalize_monic + - Polynomial.map_eq_zero + - Polynomial.one_lt_rootMultiplicity_iff_isRoot + - Polynomial.X_sub_C_mul_divByMonic_eq_sub_modByMonic + - "Polynomial.eval\u2082_gcd_eq_zero" + - Polynomial.one_lt_rootMultiplicity_iff_isRoot_gcd + - Polynomial.irreducible_iff_degree_lt + - Polynomial.Monic.normalize_eq_self + - Polynomial.mem_roots_map + - Polynomial.roots_normalize + - Polynomial.leadingCoeff_div + - Polynomial.leadingCoeff_map + - Polynomial.map_mod + - Polynomial.modByMonic_eq_mod + - Polynomial.degree_div_lt + - Polynomial.degree_normalize + - Polynomial.isCoprime_of_is_root_of_eval_derivative_ne_zero + - Polynomial.prime_of_degree_eq_one + - Polynomial.dvd_C_mul + - Polynomial.remainder_lt_aux + - Polynomial.eval_iterate_derivative_rootMultiplicity + - Polynomial.divByMonic_add_X_sub_C_mul_derivate_divByMonic_eq_derivative + - Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors' + - Polynomial.derivative_rootMultiplicity_of_root + - Polynomial.div_eq_zero_iff + - Irreducible.natDegree_pos + - Polynomial.irreducible_iff_lt_natDegree_lt + - Polynomial.root_right_of_root_gcd + - Polynomial.rootSet_C_mul_X_pow + - Polynomial.degree_pos_of_ne_zero_of_nonunit + - path: Mathlib/Data/TypeVec.lean + theorems: + - TypeVec.dropFun_prod + - TypeVec.fst_diag + - TypeVec.prod_map_id + - TypeVec.prod_snd_mk + - TypeVec.toSubtype_of_subtype + - TypeVec.append_prod_appendFun + - TypeVec.snd_diag + - TypeVec.subtypeVal_toSubtype + - TypeVec.repeatEq_iff_eq + - TypeVec.diag_sub_val + - TypeVec.subtypeVal_diagSub + - TypeVec.repeat_eq_nil + - TypeVec.repeat_eq_append1 + - TypeVec.eq_of_drop_last_eq + - TypeVec.lastFun_toSubtype + - TypeVec.const_append1 + - TypeVec.const_nil + - TypeVec.toSubtype'_of_subtype' + - TypeVec.prod_id + - TypeVec.subtypeVal_nil + - TypeVec.const_iff_true + - TypeVec.id_eq_nilFun + - TypeVec.lastFun_prod + - TypeVec.toSubtype_of_subtype_assoc + - TypeVec.subtypeVal_toSubtype' + - TypeVec.fst_prod_mk + - TypeVec.lastFun_of_subtype + - TypeVec.prod_fst_mk + - TypeVec.append1_drop_last + - TypeVec.dropFun_diag + - TypeVec.snd_prod_mk + - TypeVec.splitFun_inj + - TypeVec.Arrow.ext + - TypeVec.eq_nilFun + - TypeVec.dropFun_toSubtype + - TypeVec.nilFun_comp + - TypeVec.dropFun_of_subtype + - path: Mathlib/Analysis/ODE/PicardLindelof.lean + theorems: + - PicardLindelof.proj_of_mem + - PicardLindelof.FunSpace.vComp_apply_coe + - PicardLindelof.FunSpace.dist_next_apply_le_of_le + - exists_forall_hasDerivAt_Ioo_eq_of_contDiffAt + - PicardLindelof.FunSpace.dist_iterate_next_apply_le + - exists_isPicardLindelof_const_of_contDiffAt + - "PicardLindelof.dist_t\u2080_le" + - PicardLindelof.FunSpace.range_toContinuousMap + - IsPicardLindelof.exists_forall_hasDerivWithinAt_Icc_eq + - PicardLindelof.FunSpace.dist_iterate_next_le + - PicardLindelof.FunSpace.continuous_vComp + - PicardLindelof.FunSpace.mem_closedBall + - PicardLindelof.FunSpace.hasDerivWithinAt_next + - PicardLindelof.exists_contracting_iterate + - PicardLindelof.exists_solution + - path: Mathlib/Data/Real/Hyperreal.lean + theorems: + - Hyperreal.infinite_abs_iff + - Hyperreal.infiniteNeg_mul_of_infinitePos_not_infinitesimal_neg + - Hyperreal.Infinitesimal.add + - Hyperreal.infiniteNeg_add_not_infinitePos + - Hyperreal.infinitePos_add_not_infiniteNeg + - Hyperreal.st_neg + - Hyperreal.IsSt.isSt_st + - Hyperreal.isSt_iff_abs_sub_lt_delta + - Hyperreal.infinitesimal_iff_infinite_inv + - Hyperreal.st_inv + - Hyperreal.infinitePos_iff_infinitesimal_inv_pos + - Hyperreal.infinitePos_of_tendsto_top + - Hyperreal.Infinitesimal.neg + - Hyperreal.IsSt.lt + - Hyperreal.st_eq_sSup + - Hyperreal.Infinitesimal.mul + - Hyperreal.infiniteNeg_mul_of_infiniteNeg_not_infinitesimal_pos + - Hyperreal.neg_lt_of_tendsto_zero_of_pos + - Hyperreal.IsSt.unique + - Hyperreal.isSt_symm_real + - Hyperreal.infiniteNeg_of_tendsto_bot + - Hyperreal.IsSt.st_eq + - Hyperreal.infinite_of_infinitesimal_inv + - Hyperreal.IsSt.map + - Hyperreal.gt_of_tendsto_zero_of_neg + - Hyperreal.IsSt.inv + - Hyperreal.infinitesimal_def + - Hyperreal.isSt_trans_real + - Hyperreal.infiniteNeg_iff_infinitesimal_inv_neg + - Hyperreal.infinitePos_mul_of_infinitePos_not_infinitesimal_pos + - Hyperreal.infinite_mul_of_not_infinitesimal_infinite + - Hyperreal.infinitePos_mul_of_infiniteNeg_not_infinitesimal_neg + - Hyperreal.not_infinite_real + - "Hyperreal.IsSt.map\u2082" + - Hyperreal.isSt_iff_tendsto + - Hyperreal.infiniteNeg_mul_of_not_infinitesimal_pos_infiniteNeg + - Hyperreal.IsSt.infinitesimal_sub + - Hyperreal.infinitesimal_neg_iff_infiniteNeg_inv + - path: Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean + theorems: + - CategoryTheory.ShortComplex.ShortExact.unop + - CategoryTheory.ShortComplex.ShortExact.map_of_exact + - CategoryTheory.ShortComplex.shortExact_of_iso + - CategoryTheory.ShortComplex.ShortExact.isIso_f_iff + - "CategoryTheory.ShortComplex.isIso\u2082_of_shortExact_of_isIso\u2081\u2083" + - CategoryTheory.ShortComplex.ShortExact.op + - CategoryTheory.ShortComplex.ShortExact.isIso_g_iff + - CategoryTheory.ShortComplex.shortExact_iff_of_iso + - path: Mathlib/Topology/LocalAtTarget.lean + theorems: + - openEmbedding_iff_openEmbedding_of_iSup_eq_top + - inducing_iff_inducing_of_iSup_eq_top + - embedding_iff_embedding_of_iSup_eq_top + - closedEmbedding_iff_closedEmbedding_of_iSup_eq_top + - isOpen_iff_coe_preimage_of_iSup_eq_top + - Set.restrictPreimage_inducing + - Set.restrictPreimage_isClosedMap + - isClosedMap_iff_isClosedMap_of_iSup_eq_top + - isOpen_iff_inter_of_iSup_eq_top + - isClosed_iff_coe_preimage_of_iSup_eq_top + - path: Mathlib/LinearAlgebra/FiniteDimensional.lean + theorems: + - Submodule.finrank_add_le_finrank_add_finrank + - FiniteDimensional.exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card + - LinearMap.ker_eq_bot_iff_range_eq_top + - Module.End.ker_pow_eq_ker_pow_finrank_of_le + - Submodule.finrank_sup_add_finrank_inf_eq + - finrank_span_singleton + - LinearMap.injective_iff_surjective_of_finrank_eq_finrank + - finrank_le_one_iff + - LinearMap.isUnit_iff_range_eq_top + - Submodule.finrank_lt + - LinearMap.mul_eq_one_of_mul_eq_one + - Submodule.finrank_le_one_iff_isPrincipal + - cardinal_lt_aleph0_of_finiteDimensional + - FiniteDimensional.eq_of_le_of_finrank_le + - LinearMap.injOn_iff_surjOn + - cardinal_mk_eq_cardinal_mk_field_pow_rank + - FiniteDimensional.basisSingleton_apply + - Module.finrank_le_one_iff_top_isPrincipal + - LinearMap.ker_comp_eq_of_commute_of_disjoint_ker + - finrank_eq_one_iff_of_nonzero' + - Module.End.ker_pow_constant + - Module.End.ker_pow_le_ker_pow_finrank + - coe_finsetBasisOfLinearIndependentOfCardEqFinrank + - LinearMap.isUnit_iff_ker_eq_bot + - Subalgebra.bot_eq_top_iff_finrank_eq_one + - Subalgebra.isSimpleOrder_of_finrank + - FiniteDimensional.range_basisSingleton + - Submodule.eq_top_of_finrank_eq + - Submodule.finrank_add_eq_of_isCompl + - FiniteDimensional.exists_mul_eq_one + - Module.End.exists_ker_pow_eq_ker_pow_succ + - LinearMap.ker_noncommProd_eq_of_supIndep_ker + - LinearMap.ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank + - finrank_eq_one_iff' + - LinearMap.comap_eq_sup_ker_of_disjoint + - LinearIndependent.span_eq_top_of_card_eq_finrank' + - coe_setBasisOfLinearIndependentOfCardEqFinrank + - Subalgebra.bot_eq_top_iff_rank_eq_one + - Submodule.eq_top_of_disjoint + - Submodule.finrank_quotient_add_finrank + - Subalgebra.eq_bot_of_finrank_one + - finrank_eq_one_iff_of_nonzero + - LinearMap.surjective_of_injective + - exists_smul_eq_of_finrank_eq_one + - FiniteDimensional.finite_of_finite + - Subalgebra.eq_bot_of_rank_le_one + - is_simple_module_of_finrank_eq_one + - surjective_of_nonzero_of_finrank_eq_one + - finrank_eq_one_iff + - path: Mathlib/Analysis/NormedSpace/Ray.lean + theorems: + - norm_injOn_ray_right + - SameRay.norm_sub + - sameRay_iff_inv_norm_smul_eq + - sameRay_iff_of_norm_eq + - sameRay_iff_inv_norm_smul_eq_of_ne + - SameRay.norm_smul_eq + - norm_injOn_ray_left + - SameRay.norm_add + - path: Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean + theorems: + - "groupCohomology.groupCohomology\u03C0_comp_isoH2_hom" + - groupCohomology.twoCocycles_map_one_fst + - groupCohomology.mem_range_of_mem_twoCoboundaries + - "groupCohomology.groupCohomology\u03C0_comp_isoH1_hom" + - groupCohomology.isTwoCoboundary_of_twoCoboundaries + - groupCohomology.oneCocycles_map_inv + - groupCohomology.oneCocycles_map_one + - groupCohomology.map_one_fst_of_isMulTwoCocycle + - groupCohomology.map_one_of_isMulOneCocycle + - groupCohomology.map_inv_of_isMulOneCocycle + - groupCohomology.twoCocycles_map_one_snd + - groupCohomology.isOneCoboundary_of_oneCoboundaries + - groupCohomology.mem_oneCocycles_def + - groupCohomology.smul_map_inv_div_map_inv_of_isMulTwoCocycle + - groupCohomology.oneCoboundaries_eq_bot_of_isTrivial + - groupCohomology.dZero_comp_H0_subtype + - groupCohomology.map_inv_of_isOneCocycle + - groupCohomology.dZero_eq_zero + - groupCohomology.smul_map_inv_sub_map_inv_of_isTwoCocycle + - groupCohomology.isMulTwoCoboundary_of_twoCoboundaries + - groupCohomology.mem_oneCocycles_iff + - groupCohomology.map_one_of_isOneCocycle + - "groupCohomology.twoCocycles_\u03C1_map_inv_sub_map_inv" + - groupCohomology.shortComplexH0_exact + - groupCohomology.dZero_ker_eq_invariants + - groupCohomology.mem_twoCocycles_iff + - groupCohomology.mem_twoCocycles_def + - groupCohomology.isMulOneCoboundary_of_oneCoboundaries + - "groupCohomology.H1LequivOfIsTrivial_comp_H1_\u03C0" + - groupCohomology.toCocycles_comp_isoOneCocycles_hom + - groupCohomology.dOne_comp_eq + - groupCohomology.dZero_comp_eq + - groupCohomology.mem_range_of_mem_oneCoboundaries + - groupCohomology.map_one_fst_of_isTwoCocycle + - groupCohomology.isoOneCocycles_hom_comp_subtype + - groupCohomology.dTwo_comp_eq + - groupCohomology.map_one_snd_of_isMulTwoCocycle + - groupCohomology.dOne_comp_dZero + - groupCohomology.mem_oneCocycles_of_addMonoidHom + - groupCohomology.oneCocycles_map_mul_of_isTrivial + - groupCohomology.toCocycles_comp_isoTwoCocycles_hom + - "groupCohomology.groupCohomology\u03C0_comp_isoH0_hom" + - groupCohomology.dTwo_comp_dOne + - groupCohomology.isoTwoCocycles_hom_comp_subtype + - groupCohomology.isoZeroCocycles_hom_comp_subtype + - path: Mathlib/Data/Finset/Sort.lean + theorems: + - Finset.orderEmbOfFin_singleton + - Finset.sort_perm_toList + - Finset.sorted_zero_eq_min' + - Fin.sort_univ + - Finset.orderEmbOfFin_apply + - Finset.orderEmbOfFin_unique + - Finset.range_orderEmbOfFin + - Finset.sorted_zero_eq_min'_aux + - Finset.sort_nodup + - Finset.orderEmbOfFin_zero + - Finset.sorted_last_eq_max' + - Finset.min'_eq_sorted_zero + - Finset.orderEmbOfFin_last + - Finset.orderEmbOfCardLe_mem + - Finset.max'_eq_sorted_last + - Finset.sorted_last_eq_max'_aux + - Finset.orderEmbOfFin_eq_orderEmbOfFin_iff + - path: Mathlib/MeasureTheory/Constructions/Projective.lean + theorems: + - MeasureTheory.IsProjectiveLimit.measure_univ_unique + - MeasureTheory.IsProjectiveLimit.isFiniteMeasure + - MeasureTheory.IsProjectiveMeasureFamily.measure_univ_eq_of_subset + - MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder + - MeasureTheory.IsProjectiveLimit.measure_univ_eq + - MeasureTheory.IsProjectiveLimit.unique + - MeasureTheory.IsProjectiveLimit.isProbabilityMeasure + - MeasureTheory.IsProjectiveMeasureFamily.measure_univ_eq + - MeasureTheory.IsProjectiveLimit.measure_cylinder + - MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder_of_subset + - path: Mathlib/Order/Filter/Bases.lean + theorems: + - Filter.HasBasis.prod_self + - Filter.HasBasis.to_hasBasis' + - Filter.ofSets_filter_eq_generate + - Filter.hasBasis_biInf_of_directed' + - Filter.map_sigma_mk_comap + - FilterBasis.generate + - Filter.IsBasis.filter_eq_generate + - Filter.HasBasis.eq_bot_iff + - Filter.HasBasis.congr + - Filter.HasBasis.principal_inf + - Filter.HasBasis.inf_principal + - Filter.generate_neBot_iff + - Filter.exists_antitone_seq + - Filter.isCountablyGenerated_of_seq + - Filter.HasBasis.to_image_id' + - Filter.HasBasis.disjoint_iff + - Filter.HasBasis.sup' + - Filter.HasBasis.le_iff + - Filter.HasBasis.eq_biInf + - Filter.disjoint_principal_right + - Filter.HasBasis.isBasis + - Filter.disjoint_pure_pure + - Filter.HasBasis.hasBasis_self_subset + - Filter.HasBasis.eq_iInf + - Filter.eventually_prod_self_iff + - Filter.hasBasis_iInf_principal + - Filter.HasAntitoneBasis.mem_iff + - Filter.antitone_seq_of_seq + - Filter.HasBasis.eq_generate + - Filter.compl_diagonal_mem_prod + - Filter.hasBasis_iInf + - Filter.HasBasis.restrict + - Set.PairwiseDisjoint.exists_mem_filter_basis + - Filter.HasBasis.exists_antitone_subbasis + - Filter.isCountablyGenerated_iff_exists_antitone_basis + - Filter.HasBasis.eq_of_same_basis + - Filter.hasBasis_principal + - Filter.HasBasis.tendsto_right_iff + - Filter.HasBasis.map + - Filter.hasBasis_iInf_of_directed' + - Filter.hasBasis_iSup + - FilterBasis.eq_iInf_principal + - Filter.HasBasis.le_basis_iff + - Filter.HasBasis.prod_same_index + - Filter.HasBasis.ext + - Filter.HasBasis.tendsto_iff + - Filter.hasBasis_self + - Filter.HasBasis.tendsto_left_iff + - Filter.hasBasis_pure + - Filter.HasBasis.filter_eq + - Filter.countable_biInf_eq_iInf_seq' + - Filter.isCountablyGenerated_pure + - Filter.HasBasis.sup_principal + - Filter.isCountablyGenerated_seq + - Filter.inf_neBot_iff_frequently_left + - Filter.disjoint_principal_left + - Filter.inf_neBot_iff_frequently_right + - Filter.HasBasis.comap + - Filter.IsBasis.hasBasis + - Filter.hasBasis_iInf_of_directed + - Filter.HasBasis.disjoint_iff_left + - Pairwise.exists_mem_filter_basis_of_disjoint + - Filter.hasBasis_generate + - Filter.HasBasis.sup_pure + - Filter.mem_iff_inf_principal_compl + - Filter.disjoint_principal_principal + - Filter.HasBasis.frequently_iff + - Filter.HasBasis.eventually_iff + - Filter.hasBasis_iInf_principal_finite + - Filter.hasBasis_iInf' + - Filter.HasBasis.inf_basis_neBot_iff + - Filter.hasBasis_biInf_principal + - Filter.HasBasis.inf' + - Filter.HasBasis.forall_mem_mem + - Filter.hasBasis_biInf_of_directed + - path: Mathlib/MeasureTheory/Function/AEEqFun.lean + theorems: + - MeasureTheory.AEEqFun.mk_coeFn + - MeasureTheory.AEEqFun.le_inf + - MeasureTheory.AEEqFun.coeFn_compQuasiMeasurePreserving + - MeasureTheory.AEEqFun.coeFn_comp + - "MeasureTheory.AEEqFun.comp\u2082Measurable_eq_mk" + - MeasureTheory.AEEqFun.comp_eq_mk + - MeasureTheory.AEEqFun.compQuasiMeasurePreserving_eq_mk + - MeasureTheory.AEEqFun.coeFn_compMeasurable + - MeasureTheory.AEEqFun.compMeasurable_toGerm + - MeasureTheory.AEEqFun.coeFn_abs + - "MeasureTheory.AEEqFun.comp\u2082_eq_mk" + - MeasureTheory.AEEqFun.liftRel_iff_coeFn + - MeasureTheory.AEEqFun.compMeasurable_eq_mk + - MeasureTheory.AEEqFun.sup_le + - MeasureTheory.AEEqFun.lintegral_add + - MeasureTheory.AEEqFun.le_sup_left + - MeasureTheory.AEEqFun.toGerm_eq + - MeasureTheory.AEEqFun.le_sup_right + - MeasureTheory.AEEqFun.coeFn_mk + - "MeasureTheory.AEEqFun.comp\u2082_toGerm" + - MeasureTheory.AEEqFun.pair_eq_mk + - MeasureTheory.AEEqFun.inf_le_left + - MeasureTheory.AEEqFun.lintegral_coeFn + - MeasureTheory.AEEqFun.comp_toGerm + - "MeasureTheory.AEEqFun.comp\u2082Measurable_toGerm" + - "MeasureTheory.AEEqFun.coeFn_comp\u2082Measurable" + - MeasureTheory.AEEqFun.ext_iff + - MeasureTheory.AEEqFun.inf_le_right + - MeasureTheory.AEEqFun.toGerm_injective + - MeasureTheory.AEEqFun.coeFn_pair + - MeasureTheory.AEEqFun.ext + - path: Mathlib/Order/CountableDenseLinearOrder.lean + theorems: + - Order.embedding_from_countable_to_dense + - Order.PartialIso.exists_across + - Order.iso_of_countable_dense + - path: Mathlib/Algebra/Group/Units.lean + theorems: + - isUnit_iff_exists_inv' + - IsUnit.isUnit_iff_mulLeft_bijective + - IsUnit.pow + - mul_eq_one + - IsUnit.div_self + - IsUnit.isUnit_iff_mulRight_bijective + - IsUnit.div_mul_left + - Units.inv_eq_of_mul_eq_one_right + - IsUnit.one_div_mul_cancel + - Units.isUnit_mul_units + - Units.val_div_eq_div_val + - IsUnit.inv_mul_cancel + - IsUnit.div_eq_iff + - IsUnit.exists_right_inv + - Units.val_eq_one + - Units.inv_mul_of_eq + - divp_mul_divp + - Units.mul_inv_cancel_left + - IsUnit.mul_div_mul_left + - IsUnit.div_mul_cancel_left + - Units.eq_mul_inv_iff_mul_eq + - Units.ext + - mul_divp_cancel + - eq_one_of_mul_left + - IsUnit.mul_div_cancel_left + - IsUnit.eq_div_iff + - IsUnit.mul_mul_div + - divp_mul_cancel + - isUnit_of_mul_eq_one_right + - Units.inv_eq_of_mul_eq_one_left + - Units.mul_right_inj + - Units.mul_inv_cancel_right + - Units.inv_unique + - IsUnit.mul_val_inv + - divp_mul_eq_mul_divp + - isUnit_of_mul_isUnit_left + - divp_divp_eq_divp_mul + - Units.mul_inv_eq_iff_eq_mul + - isUnit_iff_eq_one + - IsUnit.div_div_cancel_left + - Units.eq_inv_mul_iff_mul_eq + - IsUnit.mul_div_cancel + - IsUnit.mul_div_cancel_right + - Units.inv_mul_cancel_left + - inv_eq_one_divp' + - Units.mul_eq_one_iff_eq_inv + - isUnit_iff_exists + - Units.mul_eq_one_iff_inv_eq + - eq_one_of_mul_right + - IsUnit.div_mul_cancel + - IsUnit.mul + - IsUnit.div_div_cancel + - IsUnit.div_eq_div_iff + - IsUnit.exists_left_inv + - divp_eq_iff_mul_eq + - Units.inv_mul_cancel_right + - isUnit_of_mul_isUnit_right + - divp_eq_div + - Units.mul_left_inj + - val_div_eq_divp + - IsUnit.mul_inv_cancel + - IsUnit.div + - inv_eq_one_divp + - IsUnit.mul_div_mul_right + - Units.mul_inv_of_eq + - IsUnit.inv + - IsUnit.div_mul_right + - Units.commute_coe_inv + - Units.inv_mul_eq_iff_eq_mul + - IsUnit.mul_one_div_cancel + - divp_eq_divp_iff + - IsUnit.div_mul_cancel_right + - divp_eq_one_iff_eq + - IsUnit.mul_eq_mul_of_div_eq_div + - path: Mathlib/Algebra/Order/Group/Abs.lean + theorems: + - abs_sub + - abs_sub_abs_le_abs_sub + - abs_le_max_abs_abs + - abs_cases + - abs_sub_nonpos + - inv_mabs_le + - inv_mabs_le_inv + - one_le_mul_mabs + - mabs_pow + - sup_sq_eq_mul_mul_mabs_div + - abs_sub_le_iff + - abs_eq_self + - abs_abs_sub_abs_le_abs_sub + - mabs_lt + - mabs_le_mabs_of_le_one + - mabs_inf_div_inf_le_mabs + - inf_sq_eq_mul_div_mabs_div + - abs_le + - mabs_div_sup_mul_mabs_div_inf + - mabs_le_mabs_of_one_le + - mabs_mul_le + - mabs_mabs_div_mabs_le + - abs_eq + - abs_sub_lt_iff + - mabs_mul_eq_mul_mabs_iff + - mabs_sup_div_sup_le_mabs + - abs_sub_lt_of_nonneg_of_lt + - eq_or_eq_inv_of_mabs_eq + - sup_div_inf_eq_mabs_div + - mabs_inv + - max_zero_add_max_neg_zero_eq_abs_self + - abs_sub_le + - one_le_mabs + - le_abs' + - one_lt_mabs + - abs_sub_le_of_nonneg_of_le + - abs_eq_neg_self + - max_div_min_eq_mabs + - mabs_eq_mabs + - abs_add' + - max_div_min_eq_mabs' + - mabs_mul_eq_mul_mabs_le + - path: Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean + theorems: + - MeasureTheory.ae_eq_empty + - MeasurableSpace.ae_induction_on_inter + - MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_right + - MeasureTheory.ae_le_set + - MeasureTheory.exists_measurable_superset_forall_eq + - "MeasureTheory.exists_measurable_superset\u2082" + - MeasureTheory.measure_toMeasurable + - MeasureTheory.ae_eq_set_compl_compl + - MeasureTheory.union_ae_eq_right_of_ae_eq_empty + - MeasureTheory.union_ae_eq_left_of_ae_eq_empty + - MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left + - MeasureTheory.measure_eq_extend + - MeasureTheory.Measure.ext_iff + - MeasureTheory.exists_measurable_superset + - MeasureTheory.measure_eq_iInf + - MeasureTheory.measure_null_iff_singleton + - MeasureTheory.measure_union_eq_top_iff + - MeasureTheory.measure_union_lt_top_iff + - MeasureTheory.union_ae_eq_right + - MeasureTheory.measure_iUnion_fintype_le + - MeasureTheory.diff_ae_eq_self + - MeasureTheory.measure_biUnion_le + - MeasureTheory.compl_mem_ae_iff + - MeasureTheory.all_ae_of + - MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_left + - MeasureTheory.measure_biUnion_lt_top + - MeasureTheory.Measure.toOuterMeasure_injective + - MeasureTheory.measure_mono_ae + - MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right + - MeasureTheory.subset_toMeasurable + - MeasureTheory.Measure.ext_iff' + - Set.mulIndicator_ae_eq_one + - MeasureTheory.ae_iff_of_countable + - MeasureTheory.ae_eq_set_compl + - MeasureTheory.inter_ae_eq_left_of_ae_eq_univ + - MeasureTheory.measure_eq_iInf' + - MeasureTheory.measurableSet_toMeasurable + - MeasureTheory.measure_biUnion_finset_le + - MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null + - MeasureTheory.inter_ae_eq_right_of_ae_eq_univ + - MeasureTheory.measure_symmDiff_eq_zero_iff + - MeasureTheory.ae_eq_set + - MeasureTheory.Measure.ext + - path: Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean + theorems: + - AkraBazziRecurrence.GrowsPolynomially.add_isLittleO + - AkraBazziRecurrence.GrowsPolynomially.neg + - AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_le + - AkraBazziRecurrence.GrowsPolynomially.rpow + - AkraBazziRecurrence.GrowsPolynomially.add + - AkraBazziRecurrence.GrowsPolynomially.zpow + - AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_ge + - AkraBazziRecurrence.GrowsPolynomially.inv + - AkraBazziRecurrence.growsPolynomially_const + - AkraBazziRecurrence.GrowsPolynomially.of_isTheta + - AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_le_nat + - AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_ge_nat + - AkraBazziRecurrence.GrowsPolynomially.pow + - AkraBazziRecurrence.GrowsPolynomially.of_isEquivalent + - AkraBazziRecurrence.GrowsPolynomially.congr_of_eventuallyEq + - AkraBazziRecurrence.GrowsPolynomially.mul + - AkraBazziRecurrence.GrowsPolynomially.abs + - AkraBazziRecurrence.GrowsPolynomially.div + - AkraBazziRecurrence.GrowsPolynomially.eventually_zero_of_frequently_zero + - AkraBazziRecurrence.growsPolynomially_id + - AkraBazziRecurrence.growsPolynomially_log + - AkraBazziRecurrence.GrowsPolynomially.norm + - AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos + - path: Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean + theorems: + - IsBoundedLinearMap.snd + - ContinuousLinearEquiv.nhds + - IsBoundedBilinearMap.isBigO + - IsBoundedLinearMap.neg + - isBoundedLinearMap_continuousMultilinearMap_comp_linear + - IsBoundedLinearMap.id + - "ContinuousLinearMap.map_add\u2082" + - IsBoundedLinearMap.add + - IsBoundedLinearMap.zero + - "ContinuousLinearMap.map_smul\u2082" + - IsBoundedBilinearMap.continuous + - isBoundedBilinearMap_mul + - "ContinuousLinearMap.map_neg\u2082" + - IsBoundedLinearMap.fst + - IsBoundedLinearMap.sub + - "ContinuousLinearMap.map_sub\u2082" + - IsBoundedLinearMap.tendsto + - ContinuousLinearEquiv.isOpen + - "ContinuousLinearMap.map_smul\u209B\u2097\u2082" + - ContinuousLinearMap.isBoundedBilinearMap + - "ContinuousLinearMap.map_zero\u2082" + - isBoundedLinearMap_prod_multilinear + - path: Mathlib/ModelTheory/Encoding.lean + theorems: + - FirstOrder.Language.BoundedFormula.listDecode_encode_list + - FirstOrder.Language.Term.listDecode_encode_list + - FirstOrder.Language.Term.card_sigma + - FirstOrder.Language.BoundedFormula.card_le + - path: Mathlib/Data/ENNReal/Real.lean + theorems: + - ENNReal.lt_ofReal_iff_toReal_lt + - "ENNReal.trichotomy\u2082" + - ENNReal.toReal_iInf + - ENNReal.toNNReal_pos_iff + - ENNReal.smul_toNNReal + - ENNReal.nat_cast_le_ofReal + - ENNReal.toReal_iSup + - ENNReal.toReal_sInf + - ENNReal.ofReal_eq_ofReal_iff + - ENNReal.toNNReal_iInf + - ENNReal.ofReal_pos + - ENNReal.toReal_smul + - ENNReal.iInf_mul + - ENNReal.toReal_add + - ENNReal.iInf_add_iInf + - ENNReal.dichotomy + - ENNReal.le_toReal_sub + - ENNReal.iInf_sum + - ENNReal.ofReal_lt_ofReal_iff_of_nonneg + - ENNReal.toNNReal_sSup + - ENNReal.toReal_le_add' + - ENNReal.ofReal_nsmul + - ENNReal.toNNReal_iSup + - ENNReal.trichotomy + - ENNReal.ofReal_lt_ofReal_iff + - ENNReal.iSup_zero_eq_zero + - ENNReal.ofReal_prod_of_nonneg + - ENNReal.toNNReal_sInf + - ENNReal.ofReal_le_ofReal + - ENNReal.toReal_max + - ENNReal.ofReal_lt_nat_cast + - ENNReal.toReal_le_toReal + - ENNReal.mul_iInf_of_ne + - ENNReal.toNNReal_strict_mono + - ENNReal.toNNReal_lt_toNNReal + - ENNReal.ofReal_div_of_pos + - ENNReal.ofReal_lt_iff_lt_toReal + - ENNReal.ofReal_lt_coe_iff + - ENNReal.sInf_add + - ENNReal.ofReal_pow + - ENNReal.ofReal_mul + - ENNReal.ofReal_le_ofReal_iff + - ENNReal.add_iInf + - ENNReal.toReal_div + - ENNReal.toReal_sub_of_le + - ENNReal.one_le_ofReal + - ENNReal.toReal_add_le + - ENNReal.ofReal_inv_of_pos + - ENNReal.toNNReal_inv + - ENNReal.sub_iInf + - ENNReal.le_ofReal_iff_toReal_le + - ENNReal.ofReal_mul' + - ENNReal.toNNReal_mul_top + - ENNReal.toReal_nsmul + - ENNReal.toReal_mul_top + - ENNReal.ofReal_add + - ENNReal.toReal_mono' + - ENNReal.toReal_inv + - ENNReal.ofReal_eq_zero + - ENNReal.toReal_eq_toReal + - ENNReal.toNNReal_top_mul + - ENNReal.toNNReal_div + - ENNReal.toReal_ofReal_mul + - ENNReal.toReal_lt_toReal + - ENNReal.ofReal_le_iff_le_toReal + - ENNReal.toReal_top_mul + - ENNReal.toNNReal_le_toNNReal + - ENNReal.toReal_sSup + - ENNReal.toReal_min + - path: Mathlib/Topology/UniformSpace/Separation.lean + theorems: + - SeparationQuotient.uniformContinuous_lift' + - isClosed_range_of_spaced_out + - isClosed_of_spaced_out + - t0Space_iff_uniformity + - t0Space_iff_uniformity' + - SeparationQuotient.map_unique + - Inseparable.nhds_le_uniformity + - inseparable_iff_clusterPt_uniformity + - SeparationQuotient.lift'_mk + - SeparationQuotient.map_comp + - eq_of_forall_symmetric + - SeparationQuotient.map_mk + - t0Space_iff_ker_uniformity + - Filter.Tendsto.inseparable_iff_uniformity + - SeparationQuotient.comap_map_mk_uniformity + - "SeparationQuotient.uniformContinuous_dom\u2082" + - path: Mathlib/Algebra/Order/ToIntervalMod.lean + theorems: + - toIcoMod_mem_Ico' + - toIcoMod_add_right_eq_add + - toIocDiv_sub_zsmul' + - toIocDiv_neg + - toIcoMod_add_toIocMod_zero + - toIocMod_zsmul_add' + - toIxxMod_total' + - self_sub_toIocMod + - toIcoDiv_sub + - toIcoMod_neg' + - toIocDiv_apply_left + - iUnion_Icc_zsmul + - toIcoDiv_neg' + - toIocDiv_add_zsmul + - toIocMod_add_right' + - toIcoDiv_add_right' + - toIcoMod_sub + - toIcoMod_inj + - toIocDiv_add_zsmul' + - AddCommGroup.modEq_iff_toIocMod_eq_right + - toIocMod_add_left + - toIcoMod_zero_sub_comm + - iUnion_Icc_add_zsmul + - toIcoDiv_sub' + - AddCommGroup.tfae_modEq + - Ico_eq_locus_Ioc_eq_iUnion_Ioo + - toIcoDiv_zsmul_add + - toIocMod_add_left' + - toIocDiv_eq_neg_floor + - toIocMod_apply_left + - toIocMod_add_toIcoMod_zero + - toIocMod_zero_sub_comm + - toIcoMod_sub_self + - toIcoMod_add_left' + - toIocMod_add_toIocDiv_zsmul + - toIocMod_neg' + - toIocDiv_add_left + - toIcoDiv_sub_zsmul' + - iUnion_Icc_int_cast + - iUnion_Ioc_add_int_cast + - AddCommGroup.modEq_iff_toIcoDiv_eq_toIocDiv_add_one + - toIocDiv_add_left' + - toIcoMod_neg + - toIocDiv_neg' + - toIocMod_sub_eq_sub + - iUnion_Ioc_add_zsmul + - toIocMod_eq_toIocMod + - toIocMod_sub_self + - toIcoMod_eq_add_fract_mul + - toIcoMod_add_right + - toIcoDiv_apply_right + - toIocDiv_zsmul_sub_toIocMod + - toIocMod_sub_zsmul' + - toIocMod_add_right_eq_add + - self_sub_toIcoMod + - toIcoMod_eq_sub + - toIocDiv_sub + - toIcoMod_add_zsmul' + - toIocDiv_add_right + - toIxxMod_cyclic_left + - toIcoDiv_eq_floor + - toIocMod_eq_sub_fract_mul + - toIocMod_sub + - toIxxMod_iff + - iUnion_Ico_add_zsmul + - toIocDiv_zsmul_add + - toIcoMod_sub' + - iUnion_Ico_zsmul + - toIocMod_add_zsmul' + - toIocDiv_eq_sub + - toIocMod_zsmul_add + - toIcoMod_zsmul_add + - toIcoDiv_add_zsmul' + - toIcoDiv_neg + - AddCommGroup.modEq_iff_toIcoMod_eq_left + - toIcoMod_add_right' + - toIocMod_eq_sub + - toIocDiv_sub_zsmul + - toIcoDiv_add_left + - toIocMod_sub' + - toIocDiv_sub_eq_toIocDiv_add' + - toIcoDiv_sub_zsmul + - toIcoDiv_sub_eq_toIcoDiv_add' + - toIocDiv_add_right' + - iUnion_Icc_add_int_cast + - toIcoMod_apply_right + - toIcoMod_eq_fract_mul + - AddCommGroup.not_modEq_iff_toIcoDiv_eq_toIocDiv + - toIocDiv_sub_eq_toIocDiv_add + - toIcoDiv_add_left' + - toIcoMod_add_toIcoDiv_zsmul + - toIcoDiv_apply_left + - toIcoMod_sub_eq_sub + - toIocMod_add_zsmul + - toIcoDiv_add_zsmul + - toIcoMod_zero_one + - toIcoDiv_zsmul_sub_toIcoMod + - toIxxMod_trans + - toIcoDiv_zero_one + - toIocMod_apply_right + - toIcoMod_sub_zsmul' + - iUnion_Ioc_zsmul + - toIocDiv_zsmul_sub_self + - toIcoDiv_add_right + - toIcoMod_eq_toIcoMod + - toIcoDiv_zsmul_sub_self + - toIocMod_le_toIcoMod_add + - toIxxMod_antisymm + - toIcoMod_add_left + - toIocMod_add_right + - iUnion_Ico_int_cast + - iUnion_Ico_add_int_cast + - toIocDiv_wcovBy_toIcoDiv + - toIocMod_neg + - QuotientAddGroup.btw_coe_iff + - toIocDiv_sub' + - toIcoDiv_sub_eq_toIcoDiv_add + - toIcoDiv_eq_sub + - toIocMod_sub_zsmul + - toIocMod_eq_self + - toIcoMod_eq_self + - toIocDiv_apply_right + - path: Mathlib/NumberTheory/DiophantineApproximation.lean + theorems: + - Real.exists_int_int_abs_mul_sub_le + - Rat.den_le_and_le_num_le_of_sub_lt_one_div_den_sq + - "Real.aux\u2081" + - Real.convergent_of_zero + - Real.exists_rat_abs_sub_le_and_den_le + - Rat.finite_rat_abs_sub_lt_one_div_den_sq + - "Real.aux\u2082" + - "Real.aux\u2083" + - Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational + - Real.convergent_of_int + - Real.exists_rat_eq_convergent' + - Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational + - "Real.aux\u2080" + - Real.continued_fraction_convergent_eq_convergent + - Real.exists_continued_fraction_convergent_eq_rat + - Real.exists_rat_abs_sub_lt_and_lt_of_irrational + - Real.exists_nat_abs_mul_sub_round_le + - Real.invariant + - path: Mathlib/Topology/Algebra/InfiniteSum/Basic.lean + theorems: + - HasSum.add + - tsum_bool + - tsum_fintype + - Summable.map_iff_of_leftInverse + - tsum_extend_zero + - Equiv.tsum_eq + - hasSum_sum + - tsum_setElem_eq_tsum_setElem_diff + - tsum_zero + - hasSum_zero + - HasSum.update' + - HasSum.add_disjoint + - tsum_dite_right + - tsum_eq_tsum_of_ne_zero_bij + - Finset.tsum_subtype' + - summable_of_finite_support + - hasSum_subtype_iff_indicator + - tsum_eq_finsum + - Function.Surjective.tsum_eq_tsum_of_hasSum_iff_hasSum + - tsum_subtype + - tsum_eq_single + - HasSum.add_isCompl + - hasSum_single + - hasSum_sum_disjoint + - tsum_eq_sum' + - tsum_empty + - hasSum_ite_eq + - tsum_dite_left + - tsum_range + - Equiv.tsum_eq_tsum_of_support + - Equiv.hasSum_iff + - hasSum_empty + - tsum_eq_add_tsum_ite' + - tsum_congr_set_coe + - Finset.tsum_subtype + - tsum_ite_eq + - Function.Injective.tsum_eq + - Equiv.hasSum_iff_of_support + - tsum_singleton + - sum_eq_tsum_indicator + - eq_add_of_hasSum_ite + - path: Mathlib/Algebra/DirectSum/Module.lean + theorems: + - DirectSum.isInternal_ne_bot_iff + - DirectSum.linearEquivFunOnFintype_lof + - DirectSum.IsInternal.collectedBasis_mem + - DirectSum.IsInternal.collectedBasis_repr_of_mem_ne + - DirectSum.linearEquivFunOnFintype_symm_coe + - DirectSum.linearEquivFunOnFintype_symm_single + - DirectSum.IsInternal.collectedBasis_coe + - DirectSum.isInternal_submodule_iff_isCompl + - DirectSum.IsInternal.isCompl + - DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne + - DirectSum.ext_iff + - DirectSum.IsInternal.submodule_iSup_eq_top + - DirectSum.isInternal_biSup_submodule_of_independent + - DirectSum.IsInternal.collectedBasis_repr_of_mem + - DirectSum.IsInternal.ofBijective_coeLinearMap_same + - path: Mathlib/MeasureTheory/Measure/Haar/Unique.lean + theorems: + - MeasureTheory.Measure.measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport + - MeasureTheory.Measure.absolutelyContinuous_isHaarMeasure + - MeasureTheory.Measure.isMulLeftInvariant_eq_smul_of_regular + - MeasureTheory.Measure.smul_measure_isMulInvariant_le_of_isCompact_closure + - MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop + - MeasureTheory.Measure.haarScalarFactor_eq_mul + - MeasureTheory.Measure.exists_integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport + - MeasureTheory.Measure.integral_isMulLeftInvariant_isMulRightInvariant_combo + - IsCompact.measure_eq_biInf_integral_hasCompactSupport + - MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isEverywherePos + - MeasureTheory.Measure.haarScalarFactor_pos_of_isHaarMeasure + - MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measurableSet + - MeasureTheory.Measure.integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport + - MeasureTheory.Measure.haarScalarFactor_self + - MeasureTheory.Measure.measurePreserving_zpow + - MeasureTheory.Measure.isMulInvariant_eq_smul_of_compactSpace + - MeasureTheory.continuous_integral_apply_inv_mul + - MeasureTheory.Measure.measure_isMulLeftInvariant_eq_smul_of_ne_top + - MeasureTheory.Measure.isMulLeftInvariant_eq_smul_of_innerRegular + - path: Mathlib/RingTheory/IntegralRestrict.lean + theorems: + - Algebra.algebraMap_intNorm_of_isGalois + - Algebra.intNorm_eq_zero + - Algebra.intNorm_ne_zero + - Algebra.intNorm_eq_norm + - Algebra.intNorm_eq_of_isLocalization + - Algebra.algebraMap_intNorm_fractionRing + - prod_galRestrict_eq_norm + - Algebra.intTrace_eq_of_isLocalization + - Algebra.map_intNormAux + - Algebra.algebraMap_intTrace_fractionRing + - algebraMap_galRestrictHom_apply + - Algebra.intTrace_eq_trace + - galRestrictHom_symm_algebraMap_apply + - Algebra.algebraMap_intNorm + - Algebra.intNorm_zero + - Algebra.algebraMap_intTrace + - path: Mathlib/Topology/Maps.lean + theorems: + - Inducing.inducing_iff + - IsOpenMap.id + - IsClosedMap.lift'_closure_map_eq + - Inducing.continuousAt_iff' + - IsOpenMap.isOpen_range + - Inducing.nhdsSet_eq_comap + - Inducing.closure_eq_preimage_closure_image + - OpenEmbedding.isOpenMap_iff + - ClosedEmbedding.closed_iff_preimage_closed + - IsOpenMap.of_sections + - ClosedEmbedding.closed_iff_image_closed + - QuotientMap.continuous_iff + - Inducing.continuous_iff + - IsClosedMap.comp + - Inducing.dense_iff + - IsOpenMap.preimage_frontier_subset_frontier_preimage + - IsOpenMap.preimage_frontier_eq_frontier_preimage + - IsClosedMap.of_inverse + - IsClosedMap.id + - Inducing.isClosedMap + - Inducing.mapClusterPt_iff + - Inducing.isClosed_iff + - Inducing.isOpen_iff + - QuotientMap.comp + - quotientMap_iff_closed + - IsClosedMap.mapClusterPt_iff_lift'_closure + - IsOpenMap.preimage_closure_subset_closure_preimage + - Inducing.tendsto_nhds_iff + - OpenEmbedding.tendsto_nhds_iff' + - Inducing.comp + - isOpenMap_iff_interior + - QuotientMap.of_quotientMap_compose + - Inducing.isClosed_iff' + - IsClosedMap.of_nonempty + - openEmbedding_of_continuous_injective_open + - closedEmbedding_of_continuous_injective_closed + - isClosedMap_iff_clusterPt + - IsOpenMap.comp + - isClosedMap_iff_closure_image + - OpenEmbedding.open_iff_preimage_open + - inducing_of_inducing_compose + - OpenEmbedding.of_comp_iff + - path: Mathlib/Data/List/Sublists.lean + theorems: + - List.sublists'Aux_eq_map + - List.length_sublists' + - List.sublistsLenAux_zero + - List.sublists'Aux_eq_array_foldl + - List.sublists_cons_perm_append + - List.length_of_sublistsLen + - List.sublistsLen_length + - List.sublistsLen_sublist_of_sublist + - List.sublists'_cons + - List.sublists_perm_sublists' + - List.sublistsLen_succ_cons + - List.sublistsLen_sublist_sublists' + - List.sublists_map + - List.sublistsLen_one + - List.sublistsLenAux_append + - List.sublists_eq_sublists' + - List.nodup_sublistsLen + - List.sublists_concat + - List.revzip_sublists' + - List.sublists_cons + - List.sublistsAux_eq_bind + - List.map_ret_sublist_sublists + - List.Pairwise.sublists' + - List.nodup_sublists + - List.sublistsAux_eq_array_foldl + - List.sublists'_reverse + - List.range_bind_sublistsLen_perm + - List.length_sublists + - List.sublists_eq_sublistsFast + - List.mem_sublistsLen_self + - List.revzip_sublists + - List.sublists'_eq_sublists'Aux + - List.mem_sublists' + - List.nodup_sublists' + - List.length_sublistsLen + - List.sublists'_map + - List.pairwise_sublists + - List.sublistsLenAux_eq + - List.sublists_append + - List.mem_sublists + - path: Mathlib/Topology/Algebra/Module/Cardinality.lean + theorems: + - cardinal_eq_of_isOpen + - continuum_le_cardinal_of_isOpen + - Set.Countable.dense_compl + - continuum_le_cardinal_of_module + - cardinal_eq_of_mem_nhds + - cardinal_eq_of_mem_nhds_zero + - continuum_le_cardinal_of_nontriviallyNormedField + - path: Mathlib/AlgebraicTopology/DoldKan/Faces.lean + theorems: + - AlgebraicTopology.DoldKan.HigherFacesVanish.of_succ + - AlgebraicTopology.DoldKan.HigherFacesVanish.induction + - "AlgebraicTopology.DoldKan.HigherFacesVanish.comp_H\u03C3_eq" + - "AlgebraicTopology.DoldKan.HigherFacesVanish.comp_\u03B4_eq_zero" + - "AlgebraicTopology.DoldKan.HigherFacesVanish.comp_H\u03C3_eq_zero" + - AlgebraicTopology.DoldKan.HigherFacesVanish.of_comp + - path: Mathlib/Algebra/Homology/Augment.lean + theorems: + - ChainComplex.augment_d_succ_succ + - ChainComplex.chainComplex_d_succ_succ_zero + - CochainComplex.cochainComplex_d_succ_succ_zero + - path: Mathlib/Combinatorics/Additive/Behrend.lean + theorems: + - Behrend.map_injOn + - Behrend.exp_neg_two_mul_le + - Behrend.sum_eq + - Behrend.addSalemSpencer_image_sphere + - Behrend.map_eq_iff + - Behrend.lower_bound_le_one' + - Behrend.exists_large_sphere + - Behrend.mem_box + - Behrend.card_box + - Behrend.three_le_nValue + - Behrend.roth_lower_bound + - Behrend.dValue_pos + - Behrend.map_mod + - Behrend.ceil_lt_mul + - Behrend.two_div_one_sub_two_div_e_le_eight + - Behrend.le_sqrt_log + - Behrend.addSalemSpencer_sphere + - Behrend.sphere_zero_right + - Behrend.exists_large_sphere_aux + - Behrend.sphere_zero_subset + - Behrend.bound_aux + - Behrend.le_N + - Behrend.sphere_subset_preimage_metric_sphere + - Behrend.exp_four_lt + - Behrend.lower_bound_le_one + - Behrend.roth_lower_bound_explicit + - Behrend.four_zero_nine_six_lt_exp_sixteen + - Behrend.card_sphere_le_rothNumberNat + - Behrend.log_two_mul_two_le_sqrt_log_eight + - Behrend.bound + - Behrend.norm_of_mem_sphere + - Behrend.box_zero + - Behrend.map_monotone + - Behrend.sum_sq_le_of_mem_box + - Behrend.map_zero + - Behrend.div_lt_floor + - path: Mathlib/Topology/MetricSpace/ThickenedIndicator.lean + theorems: + - mulIndicator_thickening_eventually_eq_mulIndicator_closure + - tendsto_mulIndicator_thickening_mulIndicator_closure + - thickenedIndicatorAux_one_of_mem_closure + - thickenedIndicatorAux_le_one + - thickenedIndicator_le_one + - tendsto_mulIndicator_cthickening_mulIndicator_closure + - thickenedIndicator_zero + - indicator_le_thickenedIndicatorAux + - thickenedIndicatorAux_tendsto_indicator_closure + - thickenedIndicator_one_of_mem_closure + - thickenedIndicatorAux_one + - indicator_le_thickenedIndicator + - thickenedIndicator_tendsto_indicator_closure + - one_le_thickenedIndicator_apply' + - thickenedIndicator_mono + - mulIndicator_cthickening_eventually_eq_mulIndicator_closure + - continuous_thickenedIndicatorAux + - thickenedIndicatorAux_closure_eq + - thickenedIndicatorAux_zero + - path: Mathlib/Data/List/EditDistance/Bounds.lean + theorems: + - le_levenshtein_cons + - suffixLevenshtein_minimum_le_levenshtein_append + - le_suffixLevenshtein_cons_minimum + - suffixLevenshtein_minimum_le_levenshtein_cons + - le_suffixLevenshtein_append_minimum + - le_levenshtein_append + - path: Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean + theorems: + - mfderivWithin_id + - HasMFDerivWithinAt.mul + - mfderiv_prod_right + - hasMFDerivAt_snd + - mdifferentiableAt_neg + - mfderiv_prod_eq_add + - hasMFDerivAt_neg + - tangentMapWithin_prod_fst + - mfderiv_add + - tangentMap_id + - tangentMap_prod_snd + - hasMFDerivAt_const + - mfderivWithin_fst + - tangentMapWithin_prod_snd + - mfderiv_neg + - mfderiv_prod_left + - hasMFDerivAt_id + - tangentMapWithin_id + - HasMFDerivWithinAt.mul' + - tangentMap_prod_fst + - hasMFDerivAt_fst + - MDifferentiableAt.mfderiv_prod + - mfderivWithin_snd + - path: Mathlib/MeasureTheory/Group/Measure.lean + theorems: + - MulEquiv.isHaarMeasure_map + - MeasureTheory.map_div_right_eq_self + - MeasureTheory.eventually_div_right_iff + - MeasureTheory.forall_measure_preimage_mul_iff + - MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_compact + - MeasureTheory.Measure.measurePreserving_div_left + - MeasureTheory.Measure.measure_inv + - MeasureTheory.measure_mul_closure_one + - MeasureTheory.eventually_nhds_one_measure_smul_diff_lt + - MeasureTheory.measure_preimage_mul + - MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant + - MeasurableSet.mul_closure_one_eq + - MeasureTheory.measurePreserving_div_right + - MeasureTheory.Measure.haar_singleton + - MeasureTheory.measure_preimage_mul_right + - MeasureTheory.isMulLeftInvariant_map + - MeasureTheory.eventually_mul_right_iff + - MeasureTheory.forall_measure_preimage_mul_right_iff + - MeasureTheory.measure_univ_of_isMulLeftInvariant + - MeasureTheory.eventually_mul_left_iff + - MeasureTheory.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group + - MeasureTheory.null_iff_of_isMulLeftInvariant + - MeasureTheory.measure_ne_zero_iff_nonempty_of_isMulLeftInvariant + - path: Mathlib/Data/Vector/MapLemmas.lean + theorems: + - "Vector.mapAccumr\u2082_bisim_tail" + - "Vector.mapAccumr\u2082_mapAccumr\u2082_left_right" + - "Vector.map_map\u2082" + - "Vector.mapAccumr\u2082_mapAccumr\u2082_right_right" + - "Vector.mapAccumr\u2082_mapAccumr\u2082_left_left" + - Vector.mapAccumr_redundant_pair + - "Vector.map\u2082_map_right" + - "Vector.map\u2082_eq_mapAccumr\u2082" + - "Vector.map\u2082_flip" + - "Vector.mapAccumr\u2082_eq_map\u2082" + - "Vector.mapAccumr\u2082_mapAccumr_left" + - "Vector.mapAccumr\u2082_bisim" + - "Vector.mapAccumr\u2082_redundant_pair" + - "Vector.map\u2082_comm" + - Vector.map_mapAccumr + - "Vector.mapAccumr\u2082_unused_input_left" + - "Vector.mapAccumr\u2082_mapAccumr\u2082_right_left" + - Vector.mapAccumr_eq_map + - Vector.mapAccumr_bisim_tail + - "Vector.mapAccumr\u2082_comm" + - "Vector.mapAccumr\u2082_mapAccumr_right" + - Vector.mapAccumr_eq_map_of_constant_state + - Vector.mapAccumr_map + - Vector.mapAccumr_bisim + - "Vector.mapAccumr\u2082_flip" + - "Vector.map\u2082_map_left" + - "Vector.mapAccumr_mapAccumr\u2082" + - "Vector.mapAccumr\u2082_unused_input_right" + - Vector.map_map + - Vector.map_eq_mapAccumr + - Vector.mapAccumr_mapAccumr + - "Vector.mapAccumr\u2082_eq_map\u2082_of_constant_state" + - path: Mathlib/RingTheory/AlgebraTower.lean + theorems: + - Basis.smul_repr + - Basis.smul_apply + - linearIndependent_smul + - path: Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean + theorems: + - PrimeSpectrum.image_comap_zeroLocus_eq_zeroLocus_comap + - PrimeSpectrum.mem_vanishingIdeal + - PrimeSpectrum.zeroLocus_span + - PrimeSpectrum.isRadical_vanishingIdeal + - PrimeSpectrum.basicOpen_eq_zeroLocus_compl + - PrimeSpectrum.gc_set + - PrimeSpectrum.isOpen_iff + - PrimeSpectrum.comap_id + - PrimeSpectrum.preimage_comap_zeroLocus_aux + - PrimeSpectrum.zeroLocus_empty_of_one_mem + - PrimeSpectrum.zeroLocus_empty_iff_eq_top + - PrimeSpectrum.isIrreducible_iff_vanishingIdeal_isPrime + - PrimeSpectrum.vanishingIdeal_singleton + - PrimeSpectrum.basicOpen_le_basicOpen_iff + - PrimeSpectrum.vanishingIdeal_eq_top_iff + - LocalRing.PrimeSpectrum.comap_residue + - PrimeSpectrum.basicOpen_mul + - PrimeSpectrum.localization_comap_range + - PrimeSpectrum.localization_comap_inducing + - PrimeSpectrum.vanishingIdeal_isClosed_isIrreducible + - PrimeSpectrum.le_iff_mem_closure + - PrimeSpectrum.sup_vanishingIdeal_le + - PrimeSpectrum.zeroLocus_ideal_mem_irreducibleComponents + - PrimeSpectrum.zeroLocus_minimalPrimes + - PrimeSpectrum.isClosed_singleton_iff_isMaximal + - LocalRing.closedPoint_mem_iff + - LocalRing.isLocalRingHom_iff_comap_closedPoint + - PrimeSpectrum.closure_singleton + - PrimeSpectrum.localization_away_comap_range + - PrimeSpectrum.basicOpen_eq_bot_iff + - PrimeSpectrum.t1Space_iff_isField + - PrimeSpectrum.isTopologicalBasis_basic_opens + - PrimeSpectrum.range_comap_of_surjective + - PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure + - PrimeSpectrum.isClosed_iff_zeroLocus + - PrimeSpectrum.isClosed_zeroLocus + - PrimeSpectrum.isBasis_basic_opens + - PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal + - PrimeSpectrum.coe_vanishingIdeal + - PrimeSpectrum.vanishingIdeal_irreducibleComponents + - PrimeSpectrum.vanishingIdeal_univ + - PrimeSpectrum.zeroLocus_subset_zeroLocus_iff + - PrimeSpectrum.basicOpen_pow + - PrimeSpectrum.basicOpen_mul_le_right + - PrimeSpectrum.union_zeroLocus + - PrimeSpectrum.isIrreducible_zeroLocus_iff_of_radical + - PrimeSpectrum.localization_away_openEmbedding + - PrimeSpectrum.vanishingIdeal_anti_mono_iff + - PrimeSpectrum.basicOpen_one + - PrimeSpectrum.vanishingIdeal_mem_minimalPrimes + - PrimeSpectrum.basicOpen_zero + - PrimeSpectrum.zeroLocus_subset_zeroLocus_singleton_iff + - PrimeSpectrum.zeroLocus_bUnion + - PrimeSpectrum.zeroLocus_singleton_pow + - PrimeSpectrum.isClosed_range_comap_of_surjective + - PrimeSpectrum.isCompact_basicOpen + - PrimeSpectrum.mem_compl_zeroLocus_iff_not_mem + - PrimeSpectrum.localization_comap_injective + - PrimeSpectrum.zeroLocus_singleton_mul + - PrimeSpectrum.primeSpectrumProd_symm_inl_asIdeal + - PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical + - PrimeSpectrum.comap_inducing_of_surjective + - path: Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean + theorems: + - EuclideanGeometry.preimage_inversion_sphere_dist_center + - EuclideanGeometry.image_inversion_perpBisector + - EuclideanGeometry.preimage_inversion_perpBisector + - EuclideanGeometry.image_inversion_sphere_dist_center + - EuclideanGeometry.inversion_mem_perpBisector_inversion_iff + - EuclideanGeometry.inversion_mem_perpBisector_inversion_iff' + - path: Mathlib/AlgebraicGeometry/StructureSheaf.lean + theorems: + - AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkSpecializes + - AlgebraicGeometry.StructureSheaf.toBasicOpen_mk' + - AlgebraicGeometry.StructureSheaf.germ_toOpen + - AlgebraicGeometry.StructureSheaf.comap_comp + - AlgebraicGeometry.StructureSheaf.isUnit_toStalk + - AlgebraicGeometry.StructureSheaf.const_mul_rev + - AlgebraicGeometry.StructureSheaf.const_mul_cancel' + - AlgebraicGeometry.StructureSheaf.const_mul + - AlgebraicGeometry.StructureSheaf.comap_id_eq_map + - AlgebraicGeometry.StructureSheaf.comap_id' + - AlgebraicGeometry.StructureSheaf.locally_const_basicOpen + - AlgebraicGeometry.StructureSheaf.const_ext + - AlgebraicGeometry.StructureSheaf.comap_id + - AlgebraicGeometry.StructureSheaf.toStalk_comp_stalkToFiberRingHom + - AlgebraicGeometry.StructureSheaf.localizationToStalk_mk' + - AlgebraicGeometry.StructureSheaf.isUnit_to_basicOpen_self + - AlgebraicGeometry.StructureSheaf.toStalk_stalkSpecializes + - AlgebraicGeometry.StructureSheaf.toBasicOpen_surjective + - AlgebraicGeometry.StructureSheaf.localization_toBasicOpen + - AlgebraicGeometry.StructureSheaf.stalkToFiberRingHom_germ + - AlgebraicGeometry.StructureSheaf.toBasicOpen_injective + - AlgebraicGeometry.StructureSheaf.to_global_factors + - AlgebraicGeometry.StructureSheaf.const_mul_cancel + - AlgebraicGeometry.StructureSheaf.const_congr + - AlgebraicGeometry.StructureSheaf.comapFunIsLocallyFraction + - AlgebraicGeometry.StructureSheaf.IsFraction.eq_mk' + - AlgebraicGeometry.StructureSheaf.toOpen_germ + - AlgebraicGeometry.StructureSheaf.stalkSpecializes_stalk_to_fiber + - AlgebraicGeometry.StructureSheaf.normalize_finite_fraction_representation + - AlgebraicGeometry.StructureSheaf.const_zero + - path: Mathlib/Algebra/Order/Group/Defs.lean + theorems: + - Right.inv_lt_one_iff + - cmp_div_one' + - le_inv_iff_mul_le_one_right + - le_inv_mul_iff_mul_le + - mul_inv_lt_one_iff + - OrderedCommGroup.to_contravariantClass_right_le + - div_le_iff_le_mul' + - inv_lt_iff_one_lt_mul + - mul_inv_lt_mul_inv_iff' + - lt_inv_mul_iff_mul_lt + - div_le_iff_le_mul + - lt_div_iff_mul_lt' + - le_div_iff_mul_le + - Right.one_le_inv_iff + - OrderedCommGroup.to_contravariantClass_left_le + - le_div_iff_mul_le' + - mul_inv_le_mul_inv_iff' + - le_div_self_iff + - div_lt_div_iff_right + - div_le_div_iff_left + - inv_mul_lt_one_iff_lt + - lt_or_lt_of_div_lt_div + - mul_inv_le_one_iff_le + - le_inv_iff_mul_le_one_left + - lt_mul_inv_iff_mul_lt + - mul_inv_le_iff_le_mul' + - div_le_div_flip + - inv_lt_iff_one_lt_mul' + - inv_mul_lt_iff_lt_mul' + - mul_inv_lt_inv_mul_iff + - mul_inv_le_inv_mul_iff + - inv_le_inv_iff + - inv_lt' + - lt_inv_self_iff + - inv_mul_le_iff_le_mul + - inv_le_iff_one_le_mul' + - Right.one_lt_inv_iff + - Left.inv_lt_one_iff + - inv_lt_div_iff_lt_mul + - exists_one_lt' + - lt_mul_inv_iff_lt + - div_lt_iff_lt_mul + - lt_inv_iff_mul_lt_one + - div_lt_one' + - div_le_div'' + - div_le_div_iff' + - lt_inv_iff_mul_lt_one' + - le_mul_inv_iff_le + - le_mul_inv_iff_mul_le + - le_of_forall_one_lt_lt_mul + - lt_inv' + - inv_lt_inv_iff + - Left.one_le_inv_iff + - inv_le_iff_one_le_mul + - one_lt_div' + - div_lt_self_iff + - Right.inv_le_one_iff + - lt_inv_mul_iff_lt + - inv_le_self_iff + - inv_mul_le_one_iff + - div_le_one' + - inv_mul_lt_iff_lt_mul + - le_inv_self_iff + - le_inv_mul_iff_le + - inv_lt_self_iff + - inv_le_div_iff_le_mul' + - mul_inv_lt_iff_le_mul' + - one_le_div' + - mul_inv_le_one_iff + - div_le_self_iff + - Left.inv_le_one_iff + - inv_mul_le_iff_le_mul' + - div_lt_div_iff_left + - mul_inv_le_iff_le_mul + - Left.one_lt_inv_iff + - lt_div_iff_mul_lt + - div_lt_iff_lt_mul' + - div_lt_div'' + - div_le_inv_mul_iff + - div_lt_div_iff' + - mul_inv_lt_iff_lt_mul + - path: Mathlib/Analysis/Complex/Isometry.lean + theorems: + - toMatrix_rotation + - det_rotation + - rotation_ne_conjLIE + - rotation_trans + - linear_isometry_complex + - LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re + - LinearIsometry.re_apply_eq_re_of_add_conj_eq + - linearEquiv_det_rotation + - LinearIsometry.im_apply_eq_im + - rotationOf_rotation + - LinearIsometry.re_apply_eq_re + - linear_isometry_complex_aux + - path: Mathlib/GroupTheory/GroupAction/Defs.lean + theorems: + - smul_div_assoc + - IsCentralScalar.unop_smul_eq_smul + - smul_pow + - smul_smul_smul_comm + - Function.Surjective.smulCommClass + - smul_ite_zero + - smul_mul_smul + - smul_one_smul + - smul_sub + - SMul.comp.isScalarTower + - MulAction.isPretransitive_compHom + - mul_smul_one + - SMulCommClass.of_mul_smul_one + - smul_one_mul + - Function.Injective.smulCommClass + - IsScalarTower.of_smul_one_mul + - smul_neg + - path: Mathlib/SetTheory/Cardinal/Basic.lean + theorems: + - Cardinal.prod_eq_of_fintype + - Cardinal.exists_not_mem_of_length_lt + - Cardinal.mk_preimage_of_subset_range_lift + - Cardinal.mk_insert + - Cardinal.lift_umax_eq + - Cardinal.lift_sSup + - Cardinal.mk_biUnion_le + - Cardinal.lift_two_power + - Cardinal.lift_succ + - Cardinal.one_le_iff_pos + - Cardinal.mk_finsupp_of_fintype + - Cardinal.aleph0_le_mul_iff' + - Cardinal.lt_one_iff_zero + - Cardinal.two_le_iff' + - Cardinal.mk_lt_aleph0_iff + - Cardinal.lt_wf + - Cardinal.exists_eq_of_iSup_eq_of_not_isLimit + - Cardinal.bddAbove_image + - Cardinal.nat_eq_lift_iff + - Cardinal.le_sum + - Cardinal.mk_iUnion_le_lift + - Cardinal.prod_le_prod + - Cardinal.range_natCast + - Cardinal.IsLimit.aleph0_le + - Cardinal.natCast_add_one_le_iff + - Cardinal.lift_sum + - Cardinal.bddAbove_range + - Cardinal.power_le_max_power_one + - Cardinal.mk_le_mk_mul_of_mk_preimage_le + - Cardinal.lift_prod + - Cardinal.aleph0_le_mul_iff + - Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit + - Cardinal.mk_vector + - Cardinal.one_lt_aleph0 + - Cardinal.lift_iSup + - Cardinal.power_bit1 + - Cardinal.power_lt_aleph0 + - Cardinal.nat_lt_aleph0 + - Cardinal.mk_sUnion_le + - Cardinal.mk_option + - Cardinal.lift_lt_nat_iff + - Cardinal.isSuccLimit_aleph0 + - Cardinal.lt_aleph0 + - Cardinal.mk_eq_two_iff' + - Cardinal.mk_preimage_of_injective_of_subset_range + - Cardinal.prod_ne_zero + - Cardinal.prod_eq_zero + - Cardinal.mk_preimage_of_injective + - Cardinal.one_le_lift_iff + - Cardinal.one_le_iff_ne_zero + - Cardinal.card_le_of + - Cardinal.lift_le_nat_iff + - Cardinal.lift_natCast + - Cardinal.nat_mul_aleph0 + - Cardinal.mk_preimage_of_subset_range + - Cardinal.lift_mk_shrink'' + - Cardinal.zero_powerlt + - Cardinal.mk_preimage_of_injective_lift + - Cardinal.succ_eq_of_lt_aleph0 + - Cardinal.natCast_lt + - Cardinal.lift_mk_le_lift_mk_of_injective + - Cardinal.infinite_iff + - Cardinal.add_le_add' + - Cardinal.mk_emptyCollection_iff + - Cardinal.powerlt_le + - Cardinal.lift_sInf + - Cardinal.lift_down + - Cardinal.bddAbove_range_comp + - Cardinal.zero_lt_lift_iff + - Cardinal.natCast_pow + - Cardinal.mul_lt_aleph0_iff_of_ne_zero + - Cardinal.sum_lt_prod + - Cardinal.power_le_power_left + - Cardinal.le_mk_iff_exists_subset + - Cardinal.mk_subset_ge_of_subset_image + - Cardinal.mk_le_of_module + - Cardinal.zero_le + - Cardinal.nsmul_lt_aleph0_iff + - Cardinal.add_one_le_succ + - Cardinal.le_powerlt + - Cardinal.aleph0_le_lift + - Cardinal.nat_succ + - Cardinal.nat_lt_lift_iff + - Cardinal.mk_eq_nat_iff_finset + - Cardinal.mk_biUnion_le_lift + - Cardinal.lift_le + - Cardinal.sum_const' + - Cardinal.cantor + - Cardinal.succ_zero + - Cardinal.zero_eq_lift_iff + - Cardinal.lift_iSup_le_lift_iSup + - Cardinal.mk_set_eq_nat_iff_finset + - Cardinal.lift_lt_aleph0 + - Cardinal.add_lt_aleph0 + - Cardinal.mk_coe_finset + - Cardinal.mk_union_le_aleph0 + - Cardinal.mk_fintype + - Cardinal.lift_iSup_le + - Cardinal.aleph0_lt_lift + - Cardinal.lift_bit1 + - Cardinal.lift_two + - Cardinal.sum_nat_eq_add_sum_succ + - Cardinal.bddAbove_iff_small + - Cardinal.sum_le_sum + - Cardinal.mk_le_aleph0_iff + - Cardinal.exists_finset_le_card + - Cardinal.lt_aleph0_iff_finite + - Cardinal.nat_add_aleph0 + - Cardinal.mk_le_iff_forall_finset_subset_card_le + - Cardinal.sInf_eq_zero_iff + - Cardinal.mk_set + - Cardinal.aleph0_mul_nat + - Cardinal.sum_le_iSup_lift + - Cardinal.mk_list_eq_sum_pow + - Cardinal.mk_finsupp_lift_of_fintype + - Cardinal.mk_fin + - Cardinal.cast_succ + - Cardinal.out_embedding + - Cardinal.mk_eq_nat_iff + - Cardinal.mk_eq_two_iff + - Cardinal.mk_Prop + - Cardinal.mk_finset_of_fintype + - Cardinal.three_le + - Cardinal.one_eq_lift_iff + - Cardinal.one_lt_lift_iff + - Cardinal.two_le_iff_one_lt + - Cardinal.succ_natCast + - Cardinal.zero_power_le + - Cardinal.exists_nat_eq_of_le_nat + - Cardinal.iInf_eq_zero_iff + - Cardinal.aleph0_le_add_iff + - Cardinal.lift_iInf + - Cardinal.mk_eq_nat_iff_fintype + - Cardinal.powerlt_zero + - Cardinal.denumerable_iff + - Cardinal.mk_preimage_down + - Cardinal.sum_add_distrib + - Cardinal.mul_lt_aleph0_iff + - Cardinal.aleph0_le + - Cardinal.lift_iSup_le_iff + - Cardinal.mul_lt_aleph0 + - Cardinal.nat_le_lift_iff + - Cardinal.sum_le_iSup + - Cardinal.two_le_iff + - Cardinal.lift_le_one_iff + - Cardinal.natCast_le + - Cardinal.mk_bool + - Cardinal.mk_diff_add_mk + - Cardinal.cantor' + - path: Mathlib/RingTheory/Adjoin/Basic.lean + theorems: + - Algebra.restrictScalars_adjoin_of_algEquiv + - Algebra.adjoin_inl_union_inr_eq_prod + - Algebra.adjoin_union_coe_submodule + - AlgHom.map_adjoin_singleton + - Algebra.adjoin_adjoin_of_tower + - Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin + - Algebra.adjoin_toSubmodule_le + - Submonoid.adjoin_eq_span_of_eq_span + - Algebra.adjoin_image + - Algebra.adjoin_union_eq_adjoin_adjoin + - Algebra.adjoin_attach_biUnion + - Algebra.restrictScalars_adjoin + - Algebra.adjoin_eq_span + - Subalgebra.adjoin_eq_span_basis + - Algebra.adjoin_induction' + - "Algebra.adjoin_induction\u2082" + - Algebra.adjoin_empty + - Algebra.mem_adjoin_of_map_mul + - path: Mathlib/CategoryTheory/EqToHom.lean + theorems: + - CategoryTheory.Functor.postcomp_map_heq + - CategoryTheory.eqToIso_trans + - CategoryTheory.congrArg_cast_hom_right + - CategoryTheory.Functor.congr_map + - CategoryTheory.dcongr_arg + - CategoryTheory.Functor.congr_obj + - CategoryTheory.Functor.ext_of_iso + - CategoryTheory.NatTrans.congr + - CategoryTheory.eqToHom_op + - CategoryTheory.eqToHom_map_comp + - CategoryTheory.Functor.ext + - CategoryTheory.Functor.map_comp_heq' + - CategoryTheory.eqToHom_comp_iff + - CategoryTheory.Functor.congr_hom + - CategoryTheory.eqToHom_trans + - CategoryTheory.congrArg_mpr_hom_right + - CategoryTheory.eq_conj_eqToHom + - CategoryTheory.eqToHom_app + - CategoryTheory.eqToHom_iso_inv_naturality + - CategoryTheory.eqToHom_map + - CategoryTheory.inv_eqToHom + - CategoryTheory.eqToIso_map + - CategoryTheory.congrArg_mpr_hom_left + - CategoryTheory.Functor.hcongr_hom + - CategoryTheory.congrArg_cast_hom_left + - CategoryTheory.Functor.map_comp_heq + - CategoryTheory.Functor.postcomp_map_heq' + - CategoryTheory.eqToHom_iso_hom_naturality + - CategoryTheory.comp_eqToHom_iff + - CategoryTheory.eqToHom_unop + - CategoryTheory.eqToHom_naturality + - CategoryTheory.Functor.conj_eqToHom_iff_heq + - CategoryTheory.Functor.congr_inv_of_congr_hom + - path: Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean + theorems: + - "CategoryTheory.Limits.MultispanIndex.\u03B9_sndSigmaMap" + - "CategoryTheory.Limits.MulticospanIndex.sndPiMap_\u03C0" + - CategoryTheory.Limits.Multicoequalizer.hom_ext + - "CategoryTheory.Limits.Multifork.app_right_eq_\u03B9_comp_snd" + - "CategoryTheory.Limits.MultispanIndex.\u03B9_fstSigmaMap" + - "CategoryTheory.Limits.MulticospanIndex.fstPiMap_\u03C0" + - "CategoryTheory.Limits.Multiequalizer.\u03B9Pi_\u03C0" + - "CategoryTheory.Limits.Multifork.app_right_eq_\u03B9_comp_fst" + - CategoryTheory.Limits.Multicofork.sigma_condition + - CategoryTheory.Limits.Multicofork.fst_app_right + - "CategoryTheory.Limits.Multicoequalizer.\u03B9_sigma\u03C0" + - CategoryTheory.Limits.Multicofork.condition + - CategoryTheory.Limits.Multifork.condition + - CategoryTheory.Limits.Multicofork.snd_app_right + - CategoryTheory.Limits.Multifork.pi_condition + - path: Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean + theorems: + - HasDerivAt.cpow + - Real.contDiffAt_rpow_of_ne + - HasStrictDerivAt.cpow + - Real.hasDerivAt_rpow_const + - HasDerivWithinAt.rpow + - Complex.hasStrictFDerivAt_cpow + - HasDerivAt.rpow + - Real.hasStrictFDerivAt_rpow_of_neg + - isTheta_deriv_rpow_const_atTop + - aux + - HasStrictDerivAt.rpow + - hasDerivAt_ofReal_cpow + - tendsto_one_plus_div_rpow_exp + - Real.hasStrictDerivAt_const_rpow_of_neg + - Real.hasStrictFDerivAt_rpow_of_pos + - HasFDerivAt.cpow + - HasDerivWithinAt.cpow + - Real.contDiff_rpow_const_of_le + - Complex.hasStrictDerivAt_const_cpow + - Complex.hasStrictDerivAt_cpow_const + - HasFDerivWithinAt.cpow + - HasDerivWithinAt.rpow_const + - Real.hasStrictDerivAt_rpow_const + - tendsto_one_plus_div_pow_exp + - Real.hasStrictDerivAt_rpow_const_of_ne + - isBigO_deriv_rpow_const_atTop + - ContDiffAt.rpow_const_of_le + - HasDerivAt.rpow_const + - HasStrictFDerivAt.cpow + - Real.hasStrictDerivAt_const_rpow + - path: Mathlib/Topology/Separation.lean + theorems: + - t2_iff_ultrafilter + - R1Space.inf + - exists_isOpen_singleton_of_isOpen_finite + - ker_nhds + - t0Space_iff_not_inseparable + - Pi.isCompact_iff + - nhdsWithin_insert_of_ne + - Dense.diff_finset + - infinite_of_mem_nhds + - disjoint_nhds_nhds + - exists_isOpen_superset_and_isCompact_closure + - loc_compact_Haus_tot_disc_of_zero_dim + - minimal_nonempty_open_subsingleton + - isCompact_closure_singleton + - TopologicalSpace.IsTopologicalBasis.inseparable_iff + - continuousOn_update_iff + - Embedding.t5Space + - nhds_le_nhdsSet_iff + - t2Space_iff_nhds + - SeparationQuotient.t2Space_iff + - IsCompact.binary_compact_cover + - IsCompact.closure + - isOpen_iff_ultrafilter' + - connectedComponent_eq_iInter_isClopen + - SeparatedNhds.union_left + - separatedNhds_iff_disjoint + - TopologicalSpace.IsTopologicalBasis.nhds_basis_closure + - isIrreducible_iff_singleton + - IsCompact.closure_subset_of_isOpen + - t2_separation_compact_nhds + - Inducing.r0Space + - exists_open_between_and_isCompact_closure + - singleton_mem_nhdsWithin_of_mem_discrete + - IsClosed.exists_closed_singleton + - IsCompact.isClosed + - regularSpace_TFAE + - IsCompact.finite_compact_cover + - SeparatedNhds.of_isCompact_isCompact + - compact_t2_tot_disc_iff_tot_sep + - PreconnectedSpace.trivial_of_discrete + - disjoint_nested_nhds + - ClosedEmbedding.normalSpace + - t0Space_iff_or_not_mem_closure + - isTopologicalBasis_isClopen + - continuousAt_of_tendsto_nhds + - R1Space.t2Space_iff_t0Space + - nhds_inter_eq_singleton_of_mem_discrete + - loc_compact_t2_tot_disc_iff_tot_sep + - nhds_basis_clopen + - t2_iff_nhds + - RegularSpace.inf + - RegularSpace.t3Space_iff_t0Space + - nhdsSet_inj_iff + - Set.InjOn.exists_mem_nhdsSet + - IsCompact.closure_eq_biUnion_inseparable + - Ne.nhdsWithin_diff_singleton + - insert_mem_nhdsWithin_of_subset_insert + - TopologicalSpace.IsTopologicalBasis.exists_mem_of_ne + - exists_mem_nhds_isClosed_subset + - discrete_of_t1_of_finite + - nhdsSet_le_iff + - Inducing.regularSpace + - t2Space_iff_disjoint_nhds + - SeparatedNhds.of_isCompact_isClosed + - disjoint_nhdsWithin_of_mem_discrete + - disjoint_nhdsSet_nhds + - disjoint_nhds_nhds_iff_not_inseparable + - Set.Subsingleton.closure + - Dense.diff_finite + - IsCompact.mem_closure_iff_exists_inseparable + - t1Space_TFAE + - Set.Finite.isClosed + - exists_isOpen_mem_isCompact_closure + - Pi.isCompact_closure_iff + - TopologicalSpace.IsTopologicalBasis.exists_closure_subset + - "eqOn_closure\u2082'" + - minimal_nonempty_closed_subsingleton + - isOpen_singleton_of_finite_mem_nhds + - R1Space.sInf + - biInter_basis_nhds + - regularSpace_sInf + - t2_iff_isClosed_diagonal + - disjoint_lift'_closure_nhds + - normal_exists_closure_subset + - exists_open_nhds_disjoint_closure + - IsCompact.closure_eq_biUnion_closure_singleton + - exists_mem_nhds_isCompact_mapsTo_of_isCompact_mem_nhds + - continuousAt_update_of_ne + - isPreirreducible_iff_subsingleton + - isOpen_inter_eq_singleton_of_mem_discrete + - tendsto_const_nhds_iff + - totallySeparatedSpace_of_t1_of_basis_clopen + - isOpen_setOf_eventually_nhdsWithin + - t0Space_iff_exists_isOpen_xor'_mem + - Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete + - isClosedMap_const + - T0Space.of_cover + - isClosed_setOf_inseparable + - SeparatedNhds.of_isCompact_isCompact_isClosed + - nhdsWithin_compl_singleton_le + - IsPreconnected.infinite_of_nontrivial + - path: Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean + theorems: + - CategoryTheory.IsPullback.of_map + - CategoryTheory.IsPullback.of_isBilimit + - CategoryTheory.IsPullback.of_map_of_faithful + - CategoryTheory.IsPullback.isoPullback_inv_snd + - CategoryTheory.IsPushout.of_hasBinaryCoproduct + - CategoryTheory.IsPullback.of_isLimit + - CategoryTheory.IsPushout.of_map + - CategoryTheory.Functor.map_isPullback + - CategoryTheory.IsPullback.isoPullback_hom_snd + - CategoryTheory.IsPullback.isoPullback_hom_fst + - CategoryTheory.IsPushout.of_isBilimit + - CategoryTheory.IsPullback.inl_snd' + - "CategoryTheory.BicartesianSq.of_has_biproduct\u2081" + - CategoryTheory.IsPushout.of_horiz_isIso + - CategoryTheory.IsPushout.zero_left + - "CategoryTheory.BicartesianSq.of_has_biproduct\u2082" + - CategoryTheory.IsPushout.of_is_coproduct + - CategoryTheory.IsPullback.of_horiz_isIso + - CategoryTheory.IsPullback.of_hasBinaryProduct + - CategoryTheory.IsPushout.inr_isoPushout_inv + - CategoryTheory.IsPullback.of_is_product + - CategoryTheory.IsPushout.of_map_of_faithful + - CategoryTheory.IsPushout.of_isColimit + - CategoryTheory.IsPullback.zero_right + - CategoryTheory.IsPushout.inr_fst' + - CategoryTheory.IsPushout.zero_right + - CategoryTheory.IsPushout.inl_isoPushout_inv + - CategoryTheory.IsPushout.inl_snd' + - CategoryTheory.IsPullback.inr_fst' + - CategoryTheory.IsPushout.inl_isoPushout_hom + - CategoryTheory.IsPullback.zero_left + - CategoryTheory.IsPushout.of_is_bilimit' + - CategoryTheory.IsPullback.of_is_bilimit' + - CategoryTheory.Functor.map_isPushout + - CategoryTheory.IsPullback.isoPullback_inv_fst + - CategoryTheory.IsPushout.inr_isoPushout_hom + - path: Mathlib/Analysis/Convex/Combination.lean + theorems: + - Finset.centerMass_of_sum_add_sum_eq_zero + - Finset.centerMass_subset + - Finset.centerMass_pair + - convexHull_eq_union_convexHull_finite_subsets + - Finset.centerMass_filter_ne_zero + - Finset.centerMass_smul + - Finset.convexHull_eq + - Finset.centerMass_ite_eq + - convexHull_sub + - Finset.centerMass_eq_of_sum_1 + - Finset.centerMass_segment + - Set.Finite.convexHull_eq + - AffineIndependent.convexHull_inter + - affineCombination_eq_centerMass + - Finset.centerMass_smul_left + - Convex.centerMass_mem + - Finset.centerMass_neg_left + - Convex.finsum_mem + - affineCombination_mem_convexHull + - convexHull_range_eq_exists_affineCombination + - Finset.mem_convexHull + - convexHull_eq + - Finset.centerMass_singleton + - convexHull_basis_eq_stdSimplex + - Finset.centerMass_mem_convexHull_of_nonpos + - Finset.centerMass_insert + - Finset.centerMass_empty + - AffineBasis.convexHull_eq_nonneg_coord + - Finset.centroid_mem_convexHull + - mk_mem_convexHull_prod + - convexHull_add + - Finset.inf_le_centerMass + - Finset.centerMass_segment' + - convex_iff_sum_mem + - Finset.mem_convexHull' + - Finset.centerMass_le_sup + - Convex.sum_mem + - path: Mathlib/Algebra/Star/Order.lean + theorems: + - star_le_one_iff + - star_pos_iff + - StarOrderedRing.nonneg_iff + - conjugate_le_conjugate + - star_le_star_iff + - IsSelfAdjoint.mono + - star_lt_one_iff + - star_nonneg_iff + - star_le_iff + - star_neg_iff + - star_lt_iff + - NonUnitalRingHom.map_le_map_of_map_star + - one_lt_star_iff + - conjugate_nonneg + - mul_star_self_nonneg + - conjugate_le_conjugate' + - conjugate_nonneg' + - one_le_star_iff + - star_nonpos_iff + - star_lt_star_iff + - path: Mathlib/Topology/Order/OrderClosed.lean + theorems: + - disjoint_nhds_atBot + - Filter.tendsto_nhds_max_right + - Dense.exists_ge' + - continuousWithinAt_Ico_iff_Ici + - continuous_if_le + - bddBelow_closure + - bddAbove_closure + - upperBounds_closure + - Filter.tendsto_nhds_max_left + - frontier_le_subset_eq + - isOpen_Ioi + - Ioo_mem_nhdsWithin_Ioi + - continuousWithinAt_Ioc_iff_Ioi + - continuousWithinAt_Icc_iff_Iic + - isOpen_lt + - disjoint_nhds_atBot_iff + - Ioo_mem_nhdsWithin_Iio' + - lowerBounds_closure + - continuousWithinAt_Ioc_iff_Iic + - continuousWithinAt_Ioo_iff_Iio + - Filter.Tendsto.max_left + - Filter.Tendsto.max_right + - Dense.Ioi_eq_biUnion + - frontier_lt_subset_eq + - continuousWithinAt_Ioo_iff_Ioi + - continuousWithinAt_Icc_iff_Ici + - continuousWithinAt_Ico_iff_Iio + - Continuous.min + - Ioo_subset_closure_interior + - path: Mathlib/CategoryTheory/Sites/Sieves.lean + theorems: + - CategoryTheory.Presieve.ofArrows_bind + - CategoryTheory.Sieve.sSup_apply + - CategoryTheory.Sieve.functor_galoisConnection + - CategoryTheory.Sieve.ofArrows_mk + - CategoryTheory.Sieve.generate_of_contains_isSplitEpi + - CategoryTheory.Sieve.pullback_id + - CategoryTheory.Presieve.singleton_eq_iff_domain + - CategoryTheory.Sieve.pullback_comp + - CategoryTheory.Sieve.pushforward_le_bind_of_mem + - CategoryTheory.Presieve.ofArrows_surj + - CategoryTheory.Sieve.ofArrows_le_ofObjects + - CategoryTheory.Presieve.functorPushforward_comp + - CategoryTheory.Sieve.functorPullback_comp + - CategoryTheory.Sieve.pullback_eq_top_iff_mem + - CategoryTheory.Sieve.sieveOfSubfunctor_functorInclusion + - CategoryTheory.Presieve.ofArrows_pullback + - CategoryTheory.Presieve.ofArrows_pUnit + - CategoryTheory.Sieve.sets_iff_generate + - CategoryTheory.Sieve.pullback_ofObjects_eq_top + - CategoryTheory.Presieve.image_mem_functorPushforward + - CategoryTheory.Sieve.arrows_ext + - CategoryTheory.Sieve.image_mem_functorPushforward + - CategoryTheory.Sieve.functorPushforward_extend_eq + - CategoryTheory.Sieve.pushforward_comp + - CategoryTheory.Sieve.mem_ofObjects_iff + - CategoryTheory.Sieve.pullbackArrows_comm + - CategoryTheory.Sieve.functorPushforward_id + - CategoryTheory.Sieve.pullback_inter + - CategoryTheory.Presieve.pullback_singleton + - CategoryTheory.Sieve.id_mem_iff_eq_top + - CategoryTheory.Sieve.functorPushforward_comp + - CategoryTheory.Sieve.functorPullback_id + - CategoryTheory.Sieve.ofArrows_eq_ofObjects + - CategoryTheory.Sieve.le_pullback_bind + - CategoryTheory.Sieve.mem_ofArrows_iff + - CategoryTheory.Sieve.functorPushforward_top + - path: Mathlib/Data/Nat/Order/Basic.lean + theorems: + - Nat.zero_eq_mul + - Nat.two_le_iff + - Nat.le_add_one_iff + - Nat.bit1_le_bit0_iff + - Nat.add_eq_three_iff + - Nat.find_add + - Nat.add_eq_min_iff + - Nat.le_and_le_add_one_iff + - Nat.div_mul_div_le_div + - Nat.add_pos_iff_pos_or_pos + - Nat.diag_induction + - Nat.half_le_of_sub_le_half + - Nat.add_eq_max_iff + - Nat.add_mod_eq_ite + - Nat.add_succ_lt_add + - Nat.sub_mod_eq_zero_of_mod_eq + - Nat.pred_le_iff + - Nat.one_mod_of_ne_one + - Nat.le_or_le_of_add_eq_add_pred + - Nat.le_mul_self + - Nat.div_le_of_le_mul' + - Nat.eq_one_of_mul_eq_one_left + - Nat.le_add_pred_of_pos + - Nat.not_dvd_of_between_consec_multiples + - Nat.eq_zero_of_le_div + - Nat.bit0_le_bit1_iff + - Nat.findGreatest_pos + - Nat.two_mul_odd_div_two + - Nat.not_dvd_of_pos_of_lt + - Nat.findGreatest_eq_iff + - Nat.le_half_of_half_lt_sub + - Nat.findGreatest_mono_left + - Nat.findGreatest_mono_right + - Nat.div_le_self' + - Nat.div_div_self + - Nat.add_eq_one_iff + - Nat.find_pos + - Nat.mul_div_mul_comm_of_dvd_dvd + - Nat.div_eq_sub_mod_div + - Nat.add_eq_two_iff + - Nat.findGreatest_eq_zero_iff + - path: Mathlib/Algebra/Star/StarAlgHom.lean + theorems: + - StarAlgHom.snd_prod + - StarAlgHom.fst_prod + - NonUnitalStarAlgHom.snd_prod + - NonUnitalStarAlgHom.fst_prod + - StarAlgEquiv.symm_symm + - StarAlgHom.mk_coe + - path: Mathlib/Algebra/Module/Submodule/Map.lean + theorems: + - Submodule.comap_zero + - Submodule.map_smul + - Submodule.map_comp + - LinearMap.iInf_invariant + - Submodule.map_smul' + - Submodule.inf_comap_le_comap_add + - LinearEquiv.map_eq_comap + - Submodule.map_comap_subtype + - Submodule.map_add_le + - Submodule.comap_smul + - Submodule.disjoint_iff_comap_eq_bot + - LinearMap.comap_codRestrict + - Submodule.comap_smul' + - Submodule.mem_map_equiv + - Submodule.map_id + - Submodule.map_inf_eq_map_inf_comap + - Submodule.map_equivMapOfInjective_symm_apply + - Submodule.comap_le_comap_smul + - Submodule.le_comap_pow_of_le_comap + - Submodule.map_symm_eq_iff + - Submodule.comap_neg + - Submodule.map_zero + - Submodule.map_equiv_eq_comap_symm + - path: Mathlib/Algebra/Lie/Subalgebra.lean + theorems: + - LieSubalgebra.submodule_span_le_lieSpan + - LieSubalgebra.sInf_glb + - LieSubalgebra.eq_bot_iff + - LieSubalgebra.coe_lieSpan_submodule_eq_iff + - Submodule.exists_lieSubalgebra_coe_eq_iff + - LieSubalgebra.lieSpan_le + - LieSubalgebra.inclusion_injective + - LieSubalgebra.subset_lieSpan + - LieSubalgebra.sInf_coe + - LieSubalgebra.incl_range + - LieSubalgebra.coe_to_submodule_mk + - LieSubalgebra.mem_lieSpan + - LieSubalgebra.mem_ofLe + - LieSubalgebra.subsingleton_bot + - LieSubalgebra.ofLe_eq_comap_incl + - LieHom.surjective_rangeRestrict + - LieSubalgebra.lieSpan_mono + - LieSubalgebra.mem_inf + - LieHom.range_eq_map + - LieSubalgebra.to_submodule_injective + - path: Mathlib/FieldTheory/PerfectClosure.lean + theorems: + - PerfectClosure.mul_aux_left + - PerfectClosure.mk_eq_iff + - PerfectClosure.frobenius_mk + - PerfectClosure.add_aux_right + - PerfectClosure.int_cast + - PerfectClosure.mk_zero + - PerfectClosure.nat_cast_eq_iff + - PerfectClosure.iterate_frobenius_mk + - PerfectClosure.add_aux_left + - PerfectClosure.nat_cast + - PerfectClosure.eq_iff + - PerfectClosure.mk_pow + - PerfectClosure.R.sound + - PerfectClosure.mul_aux_right + - path: Mathlib/RingTheory/Polynomial/Selmer.lean + theorems: + - Polynomial.X_pow_sub_X_sub_one_irreducible + - Polynomial.X_pow_sub_X_sub_one_irreducible_aux + - Polynomial.X_pow_sub_X_sub_one_irreducible_rat + - path: Mathlib/Analysis/InnerProductSpace/TwoDim.lean + theorems: + - Orientation.areaForm_neg_orientation + - Orientation.kahler_comp_rightAngleRotation + - Orientation.areaForm_map + - Orientation.kahler_ne_zero_iff + - Orientation.inner_rightAngleRotation_swap + - Orientation.areaForm_comp_linearIsometryEquiv + - Orientation.kahler_rightAngleRotation_right + - Orientation.kahler_apply_self + - Orientation.inner_mul_inner_add_areaForm_mul_areaForm' + - Orientation.normSq_kahler + - Orientation.linearIsometryEquiv_comp_rightAngleRotation + - Complex.kahler + - Orientation.kahler_swap + - Orientation.areaForm_le + - Orientation.rightAngleRotation_symm + - Orientation.kahler_neg_orientation + - Orientation.kahler_map_complex + - Orientation.kahler_comp_linearIsometryEquiv + - Orientation.rightAngleRotation_map + - Orientation.rightAngleRotation_map_complex + - Orientation.nonneg_inner_and_areaForm_eq_zero_iff_sameRay + - Orientation.kahler_eq_zero_iff + - Orientation.eq_zero_or_eq_zero_of_kahler_eq_zero + - Orientation.kahler_map + - Orientation.abs_areaForm_le + - Orientation.areaForm_to_volumeForm + - Complex.rightAngleRotation + - Orientation.rightAngleRotation_rightAngleRotation + - "Orientation.inner_rightAngleRotationAux\u2081_right" + - Orientation.inner_sq_add_areaForm_sq + - Orientation.areaForm_apply_self + - Orientation.inner_rightAngleRotation_self + - Orientation.kahler_comp_rightAngleRotation' + - "Orientation.inner_rightAngleRotationAux\u2081_left" + - Orientation.areaForm_rightAngleRotation_left + - Orientation.areaForm_map_complex + - Complex.areaForm + - Orientation.inner_rightAngleRotation_right + - Orientation.areaForm_swap + - Orientation.abs_areaForm_of_orthogonal + - "Orientation.rightAngleRotationAux\u2081_rightAngleRotationAux\u2081" + - Orientation.rightAngleRotation_trans_rightAngleRotation + - Orientation.rightAngleRotation_neg_orientation + - Orientation.kahler_rightAngleRotation_left + - Orientation.inner_rightAngleRotation_swap' + - Orientation.kahler_ne_zero + - Orientation.inner_rightAngleRotation_left + - Orientation.abs_kahler + - Orientation.areaForm_rightAngleRotation_right + - Orientation.areaForm_comp_rightAngleRotation + - path: Mathlib/Data/Fin/Tuple/Basic.lean + theorems: + - Fin.cons_self_tail + - Fin.cons_injective_of_injective + - Fin.snoc_comp_nat_add + - Fin.init_snoc + - Fin.insertNth_last' + - Fin.cons_eq_append + - Fin.insertNth_apply_above + - Fin.append_right_eq_snoc + - Fin.insertNth_comp_rev + - Fin.snoc_last + - Fin.nat_find_mem_find + - Fin.repeat_add + - Fin.init_update_last + - Fin.find_min + - Fin.insertNth_zero_right + - Fin.cons_rev + - Fin.insertNth_mem_Icc + - Fin.insertNth_sub_same + - Fin.snoc_castSucc + - Fin.append_cast_right + - Fin.mem_find_iff + - Fin.cons_le_cons + - Fin.snoc_comp_castSucc + - Fin.tail_cons + - Fin.insertNth_apply_below + - Fin.comp_cons + - Fin.comp_tail + - Fin.insertNth_rev + - Fin.consCases_cons + - Fin.append_left_snoc + - Fin.insertNth_binop + - Fin.insertNth_comp_succAbove + - Fin.pi_lex_lt_cons_cons + - Fin.find_eq_none_iff + - Fin.cons_comp_rev + - Fin.init_update_castSucc + - Fin.cons_injective_iff + - Fin.comp_init + - Fin.repeat_one + - Fin.cons_zero + - Fin.range_cons + - Fin.cons_succ + - Fin.sigma_eq_of_eq_comp_cast + - Fin.cons_one + - Fin.append_cons + - Fin.tail_init_eq_init_tail + - Fin.snocCases_snoc + - Fin.contractNth_apply_of_ne + - Fin.preimage_insertNth_Icc_of_not_mem + - Fin.update_snoc_last + - Fin.append_right_nil + - Fin.append_left_nil + - Fin.append_right_cons + - Fin.tail_update_zero + - Fin.append_cast_left + - Fin.cons_injective2 + - Fin.insertNth_eq_iff + - Fin.append_snoc + - Fin.isSome_find_iff + - Fin.le_cons + - Fin.insertNth_le_iff + - Fin.tail_update_succ + - Fin.find_spec + - Fin.preimage_insertNth_Icc_of_mem + - Fin.repeat_succ + - Fin.update_cons_zero + - Fin.comp_snoc + - Fin.le_insertNth_iff + - Fin.insertNth_zero + - Fin.snoc_update + - Fin.append_left_eq_cons + - Fin.cons_update + - Fin.contractNth_apply_of_eq + - Fin.snoc_rev + - Fin.insertNth_apply_same + - Fin.append_assoc + - Fin.insertNth_zero' + - Fin.cons_snoc_eq_snoc_cons + - Fin.insertNth_last + - Fin.append_rev + - Fin.snoc_comp_cast_add + - Fin.contractNth_apply_of_gt + - path: Mathlib/Data/List/OfFn.lean + theorems: + - List.ofFn_mul + - List.length_ofFn + - List.nthLe_ofFn + - List.ofFn_succ' + - List.ofFn_add + - List.pairwise_ofFn + - List.ofFn_succ + - List.ofFn_congr + - List.mem_ofFn + - List.forall_mem_ofFn_iff + - List.ofFn_injective + - List.ofFn_fin_append + - List.ofFn_const + - List.get?_ofFn + - List.ofFn_eq_nil_iff + - List.get_ofFn + - List.ofFn_fin_repeat + - List.ofFnRec_ofFn + - List.ofFn_mul' + - List.map_ofFn + - List.ofFn_get + - path: Mathlib/MeasureTheory/Integral/Gamma.lean + theorems: + - integral_exp_neg_mul_rpow + - Complex.integral_rpow_mul_exp_neg_rpow + - integral_exp_neg_rpow + - Complex.integral_rpow_mul_exp_neg_mul_rpow + - Complex.integral_exp_neg_rpow + - Complex.integral_exp_neg_mul_rpow + - integral_rpow_mul_exp_neg_mul_rpow + - integral_rpow_mul_exp_neg_rpow + - path: Mathlib/RingTheory/Jacobson.lean + theorems: + - Ideal.isMaximal_iff_isMaximal_disjoint + - Ideal.Polynomial.isMaximal_comap_C_of_isJacobson' + - Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson + - Ideal.disjoint_powers_iff_not_mem + - Ideal.Polynomial.isIntegral_isLocalization_polynomial_quotient + - Ideal.Polynomial.isJacobson_polynomial_iff_isJacobson + - Ideal.isJacobson_of_surjective + - Ideal.isMaximal_of_isMaximal_disjoint + - Ideal.Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson' + - Ideal.Polynomial.isMaximal_comap_C_of_isJacobson + - Ideal.Polynomial.isMaximal_comap_C_of_isMaximal + - Ideal.isJacobson_iff_prime_eq + - Ideal.Polynomial.mem_closure_X_union_C + - Ideal.MvPolynomial.aux_IH + - Ideal.MvPolynomial.comp_C_integral_of_surjective_of_jacobson + - Ideal.Polynomial.comp_C_integral_of_surjective_of_jacobson + - Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson' + - Ideal.Polynomial.isJacobson_polynomial_of_domain + - Ideal.isJacobson_of_isIntegral + - Ideal.isJacobson_localization + - Ideal.Polynomial.Subring.mem_closure_image_of + - Ideal.Polynomial.quotient_mk_comp_C_isIntegral_of_jacobson + - path: Mathlib/Control/Functor/Multivariate.lean + theorems: + - MvFunctor.LiftR_def + - MvFunctor.LiftR_RelLast_iff + - MvFunctor.LiftP_PredLast_iff + - MvFunctor.LiftP_def + - MvFunctor.exists_iff_exists_of_mono + - path: Mathlib/FieldTheory/Cardinality.lean + theorems: + - Field.nonempty_iff + - Infinite.nonempty_field + - Fintype.nonempty_field_iff + - Fintype.isPrimePow_card_of_field + - path: .lake/packages/std/Std/Data/Sum/Lemmas.lean + theorems: + - Sum.isLeft_map + - Sum.isRight_swap + - Sum.isLeft_eq_false + - Sum.liftRel_swap_iff + - Sum.isLeft_iff + - Sum.LiftRel.lex + - Sum.lex_acc_inl + - Sum.getLeft?_eq_some_iff + - Sum.bnot_isLeft + - Sum.lex_acc_inr + - Sum.getLeft?_eq_none_iff + - Sum.isRight_iff + - Sum.getRight_eq_iff + - Sum.isLeft_swap + - Sum.elim_const_const + - Sum.LiftRel.swap + - Sum.LiftRel.mono + - Sum.getRight?_eq_some_iff + - Sum.getLeft_eq_iff + - Sum.not_isRight + - Sum.elim_map + - Sum.Lex.mono + - Sum.getLeft?_swap + - Sum.not_isLeft + - Sum.getLeft?_map + - Sum.bnot_isRight + - Sum.getRight?_swap + - Sum.getRight?_eq_none_iff + - Sum.forall_sum + - Sum.isRight_eq_false + - Sum.swap_swap + - Sum.isRight_map + - Sum.getRight?_map + - path: Mathlib/Data/Setoid/Basic.lean + theorems: + - Setoid.eq_top_iff + - Setoid.eqvGen_of_setoid + - Setoid.sSup_def + - Setoid.sSup_eq_eqvGen + - Setoid.sInf_def + - Quotient.subsingleton_iff + - Setoid.eqvGen_le + - Setoid.comap_eq + - Setoid.mapOfSurjective_eq_map + - Setoid.lift_unique + - Setoid.sup_def + - Quot.subsingleton_iff + - Setoid.sup_eq_eqvGen + - path: Mathlib/Data/List/Lattice.lean + theorems: + - List.inter_eq_nil_iff_disjoint + - List.mem_of_mem_inter_right + - List.cons_bagInter_of_pos + - List.sublist_suffix_of_union + - List.forall_mem_union + - List.bagInter_nil_iff_inter_nil + - List.cons_bagInter_of_neg + - List.bagInter_sublist_left + - List.mem_inter_of_mem_of_mem + - List.bagInter_nil + - List.count_bagInter + - List.inter_cons_of_not_mem + - List.inter_cons_of_mem + - List.mem_bagInter + - List.inter_reverse + - path: Mathlib/Combinatorics/SetFamily/Shadow.lean + theorems: + - Set.Sized.shadow_iterate + - Finset.shadow_iterate_empty + - Finset.mem_shadow_iterate_iff_exists_sdiff + - Set.Sized.shadow + - Finset.mem_shadow_iterate_iff_exists_card + - Finset.mem_shadow_iterate_iff_exists_mem_card_add + - Finset.mem_upShadow_iff + - Finset.mem_upShadow_iff_exists_mem_card_add_one + - Finset.shadow_compls + - Finset.mem_shadow_iff_exists_sdiff + - Finset.mem_shadow_iff_exists_mem_card_add_one + - Finset.mem_upShadow_iterate_iff_exists_sdiff + - Finset.mem_upShadow_iterate_iff_exists_card + - Finset.mem_shadow_iff + - Finset.upShadow_compls + - Finset.mem_upShadow_iff_exists_mem_card_add + - Finset.mem_shadow_iff_insert_mem + - Finset.sized_shadow_iff + - Finset.mem_upShadow_iterate_iff_exists_mem_card_add + - Finset.mem_upShadow_iff_exists_sdiff + - Finset.mem_upShadow_iff_erase_mem + - path: Mathlib/Data/Complex/Basic.lean + theorems: + - Complex.ofReal_add + - Complex.div_ofReal_re + - Complex.re_ofReal_mul + - Complex.conj_bit1 + - Complex.mul_I_im + - Complex.ofReal_sub + - Complex.I_sq + - Complex.normSq_ofReal + - Complex.inv_I + - Complex.conj_I + - Complex.div_nat_cast_im + - Complex.conj_eq_iff_re + - Complex.div_rat_cast_re + - Complex.sub_conj + - Complex.normSq_eq_conj_mul_self + - Complex.re_eq_add_conj + - Complex.div_I + - Complex.I_zpow_bit0 + - Complex.mul_conj + - Complex.mk_eq_add_mul_I + - Complex.normSq_add + - Complex.I_pow_bit1 + - Complex.I_mul_re + - Complex.ofReal_zsmul + - Complex.inv_re + - Complex.conj_ofReal + - Complex.ofReal_inv + - Complex.div_ofReal_im + - Complex.im_eq_sub_conj + - Complex.ext_iff + - Complex.I_zpow_bit1 + - Complex.I_mul_im + - Complex.div_ofReal + - Complex.div_int_cast_im + - Complex.re_add_im + - Complex.mul_I_re + - Complex.im_mul_ofReal + - Complex.add_conj + - Complex.ofReal_bit0 + - Complex.div_im + - Complex.I_pow_bit0 + - Complex.mul_inv_cancel + - Complex.equivRealProd_symm_apply + - Complex.ofReal_inj + - Complex.ofReal_pow + - Complex.div_nat_cast_re + - Complex.normSq_I + - Complex.div_re + - Complex.I_pow_four + - Complex.normSq_add_mul_I + - Complex.conj_eq_iff_real + - Complex.im_ofReal_mul + - Complex.conj_bit0 + - Complex.ofReal_nsmul + - Complex.ofReal_neg + - Complex.inv_zero + - Complex.normSq_neg + - Complex.div_rat_cast_im + - Complex.inv_im + - Complex.div_int_cast_re + - Complex.ofReal_bit1 + - Complex.I_mul_I + - Complex.I_mul + - Complex.ofReal_mul + - Complex.normSq_conj + - Complex.conj_neg_I + - Complex.re_mul_ofReal + - path: Mathlib/Order/SuccPred/LinearLocallyFinite.lean + theorems: + - LinearLocallyFiniteOrder.succFn_le_of_lt + - toZ_neg + - iterate_succ_toZ + - toZ_iterate_succ_of_not_isMax + - toZ_mono + - LinearLocallyFiniteOrder.isMax_of_succFn_le + - toZ_nonneg + - LinearLocallyFiniteOrder.isGLB_Ioc_of_isGLB_Ioi + - toZ_of_eq + - le_of_toZ_le + - toZ_iterate_pred_ge + - toZ_iterate_succ_le + - LinearLocallyFiniteOrder.le_succFn + - toZ_iterate_pred_of_not_isMin + - iterate_pred_toZ + - LinearLocallyFiniteOrder.le_of_lt_succFn + - path: Mathlib/Analysis/SpecificLimits/FloorPow.lean + theorems: + - tendsto_div_of_monotone_of_exists_subseq_tendsto_div + - mul_pow_le_nat_floor_pow + - tendsto_div_of_monotone_of_tendsto_div_floor_pow + - sum_div_nat_floor_pow_sq_le_div_sq + - sum_div_pow_sq_le_div_sq + - path: Mathlib/FieldTheory/SeparableClosure.lean + theorems: + - IntermediateField.finInsepDegree_bot + - IsSeparable.sepDegree_eq + - IntermediateField.sepDegree_bot + - Field.insepDegree_self + - IsSeparable.insepDegree_eq + - separableClosure.map_eq_of_algEquiv + - separableClosure.map_eq_of_separableClosure_eq_bot + - separableClosure.comap_eq_of_algHom + - Field.finInsepDegree_self + - le_separableClosure_iff + - le_separableClosure' + - Field.sepDegree_self + - map_mem_separableClosure_iff + - Field.finInsepDegree_eq_of_equiv + - IntermediateField.finInsepDegree_bot' + - IntermediateField.insepDegree_bot + - IntermediateField.finInsepDegree_top + - IsSepClosed.separableClosure_eq_bot_iff + - path: Mathlib/Analysis/Convex/Between.lean + theorems: + - affineSegment_comm + - Collinear.wbtw_or_wbtw_or_wbtw + - wbtw_self_iff + - mem_vadd_const_affineSegment + - Wbtw.swap_left_iff + - mem_vsub_const_affineSegment + - wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg + - affineSegment_eq_segment + - sbtw_const_vsub_iff + - wbtw_or_wbtw_smul_vadd_of_nonpos + - Function.Injective.wbtw_map_iff + - Wbtw.swap_right_iff + - Wbtw.map + - AffineEquiv.sbtw_map_iff + - sbtw_vadd_const_iff + - sbtw_zero_one_iff + - Wbtw.collinear + - Wbtw.mem_affineSpan + - wbtw_swap_right_iff + - AffineEquiv.wbtw_map_iff + - mem_const_vsub_affineSegment + - wbtw_zero_one_iff + - sbtw_comm + - affineSegment_same + - wbtw_iff_left_eq_or_right_mem_image_Ici + - wbtw_swap_left_iff + - sbtw_const_vadd_iff + - Wbtw.trans_sbtw_right + - Wbtw.trans_sbtw_left + - wbtw_iff_sameRay_vsub + - sbtw_one_zero_iff + - sbtw_midpoint_of_ne + - wbtw_smul_vadd_smul_vadd_of_nonpos_of_le + - mem_const_vadd_affineSegment + - affineSegment_image + - sbtw_lineMap_iff + - wbtw_one_zero_iff + - Wbtw.sameRay_vsub_right + - mem_segment_iff_wbtw + - wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos + - wbtw_midpoint + - Sbtw.mem_image_Ioo + - Wbtw.trans_right_left + - sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair + - sbtw_vsub_const_iff + - Wbtw.rotate_iff + - Wbtw.trans_right + - wbtw_comm + - Wbtw.trans_left_ne + - wbtw_rotate_iff + - Wbtw.right_mem_affineSpan_of_left_ne + - Wbtw.trans_right_ne + - wbtw_smul_vadd_smul_vadd_of_nonneg_of_le + - Wbtw.trans_left_right + - sbtw_pointReflection_of_ne + - Function.Injective.sbtw_map_iff + - Wbtw.sameRay_vsub + - sbtw_iff_right_ne_and_left_mem_image_Ioi + - wbtw_or_wbtw_smul_vadd_of_nonneg + - wbtw_lineMap_iff + - Wbtw.trans_left + - Wbtw.sameRay_vsub_left + - sbtw_iff_mem_image_Ioo_and_ne + - Sbtw.affineCombination_of_mem_affineSpan_pair + - sbtw_iff_left_ne_and_right_mem_image_Ioi + - wbtw_pointReflection + - path: Mathlib/RingTheory/AdjoinRoot.lean + theorems: + - AdjoinRoot.minpoly_powerBasis_gen_of_monic + - Irreducible.exists_dvd_monic_irreducible_of_isIntegral + - AdjoinRoot.Minpoly.toAdjoin.surjective + - AdjoinRoot.of.injective_of_degree_ne_zero + - AdjoinRoot.nontrivial + - AdjoinRoot.quotEquivQuotMap_apply_mk + - AdjoinRoot.minpoly_root + - AdjoinRoot.quotMapOfEquivQuotMapCMapSpanMk_symm_mk + - AdjoinRoot.aeval_eq + - AdjoinRoot.Polynomial.quotQuotEquivComm_symm_mk_mk + - AdjoinRoot.lift_of + - AdjoinRoot.Polynomial.quotQuotEquivComm_mk + - AdjoinRoot.smul_of + - AdjoinRoot.mk_leftInverse + - AdjoinRoot.root_isInv + - AdjoinRoot.Minpoly.toAdjoin_apply' + - AdjoinRoot.adjoinRoot_eq_top + - AdjoinRoot.mk_eq_zero + - AdjoinRoot.algHom_subsingleton + - AdjoinRoot.mk_self + - AdjoinRoot.isRoot_root + - AdjoinRoot.aeval_algHom_eq_zero + - AdjoinRoot.Minpoly.toAdjoin.apply_X + - "AdjoinRoot.eval\u2082_root" + - AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk + - AdjoinRoot.lift_root + - PowerBasis.quotientEquivQuotientMinpolyMap_apply_mk + - PowerBasis.quotientEquivQuotientMinpolyMap_symm_apply_mk + - AdjoinRoot.liftHom_eq_algHom + - AdjoinRoot.minpoly_powerBasis_gen + - AdjoinRoot.quotEquivQuotMap_symm_apply_mk + - path: Mathlib/RingTheory/Ideal/Operations.lean + theorems: + - Ideal.comap_eq_top_iff + - Submodule.mem_ideal_smul_span_iff_exists_sum + - RingHom.ker_coe_equiv + - Ideal.isUnit_iff + - RingHom.ker_eq_bot_iff_eq_zero + - Ideal.IsRadical.inf + - Ideal.comap_of_equiv + - Submodule.mem_smul_top_iff + - Submodule.smul_comap_le_comap_smul + - RingHom.comap_ker + - Ideal.prod_sup_eq_top + - Associates.mk_ne_zero' + - Ideal.radical_mul + - LinearMap.annihilator_le_of_injective + - Submodule.smul_induction_on + - Submodule.mem_colon_singleton + - Submodule.mul_annihilator + - Pi.ker_ringHom + - Ideal.comap_isPrime + - Ideal.pow_sup_eq_top + - Submodule.mem_smul_span_singleton + - Ideal.constr_basisSpanSingleton + - RingHom.mem_ker + - Submodule.colon_bot + - Ideal.multiset_prod_le_inf + - Ideal.subset_union_prime' + - Submodule.span_smul_eq + - Ideal.map_isPrime_of_equiv + - Ideal.isCoprime_tfae + - Ideal.sup_pow_eq_top + - Ideal.top_pow + - Ideal.IsPrime.mul_le + - RingHom.eq_liftOfRightInverse + - Ideal.one_eq_top + - Ideal.map_mul + - Ideal.annihilator_quotient + - Ideal.finsuppTotal_apply + - Ideal.le_span_singleton_mul_iff + - Submodule.mem_annihilator_span + - Ideal.pow_right_mono + - Ideal.isCoprime_iff_add + - Ideal.mul_sup_eq_of_coprime_left + - Ideal.isCoprime_iff_exists + - Ideal.span_singleton_mul_span_singleton + - Ideal.map.isMaximal + - Ideal.range_finsuppTotal + - LinearMap.annihilator_le_of_surjective + - Ideal.mul_bot + - Ideal.radical_pow + - Submodule.mem_of_span_top_of_smul_mem + - Ideal.span_singleton_mul_le_iff + - Ideal.comap_bot_le_of_injective + - Ideal.mul_sup_eq_of_coprime_right + - Ideal.prod_mem_prod + - RingHom.ker_isMaximal_of_surjective + - Ideal.comap_radical + - Ideal.mem_span_singleton_mul + - Ideal.span_singleton_mul_left_inj + - Ideal.map_radical_of_surjective + - Ideal.multiset_prod_span_singleton + - Ideal.le_comap_pow + - Ideal.span_singleton_mul_right_inj + - Ideal.mul_eq_bot + - Submodule.mem_annihilator + - Ideal.subset_union_prime + - Ideal.span_mul_span' + - Ideal.pow_le_pow_right + - Ideal.isCoprime_iff_sup_eq + - Ideal.isPrime_radical + - Ideal.comap_sInf' + - Ideal.map_isPrime_of_surjective + - Ideal.bot_mul + - Ideal.mem_colon_singleton + - Ideal.map_le_comap_of_inv_on + - Ideal.add_eq_one_iff + - RingHom.injective_iff_ker_eq_bot + - Ideal.finsuppTotal_apply_eq_of_fintype + - Ideal.map_eq_iff_sup_ker_eq_of_surjective + - Ideal.span_pair_mul_span_pair + - Ideal.span_singleton_mul_right_mono + - Ideal.eq_span_singleton_mul + - Ideal.finset_inf_span_singleton + - Ideal.isCoprime_biInf + - RingHom.not_one_mem_ker + - Submodule.mem_ideal_smul_span_iff_exists_sum' + - Submodule.map_le_smul_top + - Ideal.span_singleton_mul_le_span_singleton_mul + - Ideal.isCoprime_iff_codisjoint + - Ideal.map_of_equiv + - Ideal.map_span + - Submodule.smul_induction_on' + - Submodule.annihilator_quotient + - Ideal.span_singleton_mul_eq_span_singleton_mul + - Ideal.comap_map_of_surjective + - Ideal.map_eq_bot_iff_of_injective + - Ideal.isPrimary_inf + - Submodule.mem_of_span_eq_top_of_smul_pow_mem + - Submodule.ideal_span_singleton_smul + - Submodule.colon_top + - Ideal.sup_prod_eq_top + - Ideal.map_sInf + - Ideal.sup_mul_eq_of_coprime_right + - RingHom.ker_equiv + - Ideal.map_eq_bot_iff_le_ker + - Ideal.basisSpanSingleton_apply + - Ideal.sup_eq_top_iff_isCoprime + - RingHom.ker_isPrime + - Ideal.span_singleton_pow + - Ideal.iInf_sup_eq_top + - Ideal.radical_eq_sInf + - Ideal.isCoprime_span_singleton_iff + - Ideal.iInf_span_singleton_natCast + - Ideal.smul_top_eq_map + - Ideal.IsRadical.comap + - Submodule.mem_annihilator_span_singleton + - Ideal.IsPrime.multiset_prod_map_le + - Ideal.span_singleton_mul_left_mono + - Ideal.comap_ne_top + - RingHom.sub_mem_ker_iff + - Ideal.comap_isMaximal_of_surjective + - Ideal.prod_eq_bot + - Ideal.IsPrime.prod_mem_iff_exists_mem + - Ideal.map_eq_top_or_isMaximal_of_surjective + - path: Mathlib/SetTheory/Game/Basic.lean + theorems: + - SetTheory.PGame.invVal_isEmpty + - SetTheory.PGame.leftMoves_mul_cases + - SetTheory.PGame.rightMoves_mul_cases + - SetTheory.Game.bddAbove_range_of_small + - SetTheory.Game.add_lf_add_left + - SetTheory.PGame.quot_eq_of_mk'_quot_eq + - SetTheory.Game.bddBelow_range_of_small + - SetTheory.PGame.quot_mul_assoc + - SetTheory.PGame.mul_moveLeft_inl + - SetTheory.PGame.mul_moveRight_inl + - SetTheory.PGame.mul_moveRight_inr + - SetTheory.PGame.mul_moveLeft_inr + - SetTheory.Game.bddAbove_of_small + - SetTheory.PGame.inv_eq_of_equiv_zero + - SetTheory.Game.bddBelow_of_small + - SetTheory.Game.not_le + - SetTheory.PGame.quot_left_distrib_sub + - SetTheory.Game.add_lf_add_right + - SetTheory.PGame.quot_right_distrib + - SetTheory.PGame.quot_right_distrib_sub + - SetTheory.Game.not_lf + - SetTheory.PGame.inv_eq_of_pos + - SetTheory.PGame.zero_lf_inv' + - SetTheory.PGame.inv_eq_of_lf_zero + - SetTheory.PGame.quot_left_distrib + - path: Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean + theorems: + - "CategoryTheory.ShortComplex.SnakeInput.exact_C\u2082_down" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2082'_exact" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2081_f_\u03C6\u2081" + - "CategoryTheory.ShortComplex.SnakeInput.w\u2080\u2082_\u03C4\u2082" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2080'_exact" + - "CategoryTheory.ShortComplex.SnakeInput.w\u2080\u2082_\u03C4\u2081" + - "CategoryTheory.ShortComplex.SnakeInput.exact_C\u2083_up" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2080_g_\u03B4" + - "CategoryTheory.ShortComplex.SnakeInput.\u03B4_L\u2083_f" + - "CategoryTheory.ShortComplex.SnakeInput.w\u2081\u2083_\u03C4\u2081" + - "CategoryTheory.ShortComplex.SnakeInput.op_\u03B4" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2081'_exact" + - "CategoryTheory.ShortComplex.SnakeInput.exact_C\u2081_up" + - "CategoryTheory.ShortComplex.SnakeInput.exact_C\u2081_down" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2080_exact" + - "CategoryTheory.ShortComplex.SnakeInput.naturality_\u03C6\u2081" + - "CategoryTheory.ShortComplex.SnakeInput.naturality_\u03C6\u2082" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2080X\u2082ToP_comp_pullback_snd" + - "CategoryTheory.ShortComplex.SnakeInput.exact_C\u2082_up" + - "CategoryTheory.ShortComplex.SnakeInput.snd_\u03B4_inr" + - "CategoryTheory.ShortComplex.SnakeInput.L\u2080X\u2082ToP_comp_\u03C6\u2081" + - "CategoryTheory.ShortComplex.SnakeInput.w\u2081\u2083_\u03C4\u2082" + - "CategoryTheory.ShortComplex.SnakeInput.\u03B4_eq" + - "CategoryTheory.ShortComplex.SnakeInput.exact_C\u2083_down" + - "CategoryTheory.ShortComplex.SnakeInput.w\u2081\u2083_\u03C4\u2083" + - "CategoryTheory.ShortComplex.SnakeInput.lift_\u03C6\u2082" + - "CategoryTheory.ShortComplex.SnakeInput.w\u2080\u2082_\u03C4\u2083" + - path: Mathlib/RingTheory/MvPowerSeries/Basic.lean + theorems: + - MvPolynomial.coe_eq_zero_iff + - MvPowerSeries.mul_zero + - MvPowerSeries.coeff_monomial_same + - MvPowerSeries.mul_add + - MvPowerSeries.coeff_zero_mul_X + - MvPowerSeries.X_inj + - MvPowerSeries.one_mul + - MvPowerSeries.map_monomial + - MvPolynomial.coe_monomial + - MvPowerSeries.coeff_add_monomial_mul + - MvPowerSeries.coeff_prod + - MvPowerSeries.coeff_mul_monomial + - MvPowerSeries.coeff_mul_C + - MvPowerSeries.coeff_monomial + - MvPolynomial.coe_injective + - MvPolynomial.coe_bit1 + - MvPowerSeries.map_X + - MvPowerSeries.add_mul + - MvPolynomial.coe_eq_one_iff + - MvPowerSeries.monomial_def + - MvPolynomial.coe_mul + - MvPowerSeries.coeff_monomial_mul + - MvPowerSeries.coeff_add_mul_monomial + - MvPowerSeries.coeff_monomial_ne + - MvPowerSeries.coeff_zero_X + - MvPowerSeries.X_pow_dvd_iff + - MvPowerSeries.smul_eq_C_mul + - MvPowerSeries.coeff_index_single_X + - MvPowerSeries.mul_assoc + - MvPowerSeries.coeff_mul + - MvPowerSeries.zero_mul + - MvPowerSeries.coeff_C_mul + - MvPowerSeries.coeff_X_pow + - MvPowerSeries.X_dvd_iff + - MvPowerSeries.mul_one + - MvPowerSeries.coeff_zero_X_mul + - MvPowerSeries.X_pow_eq + - path: Mathlib/MeasureTheory/Integral/FundThmCalculus.lean + theorems: + - intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae + - intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le + - intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_left + - intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae + - intervalIntegral.integral_le_sub_of_hasDeriv_right_of_le + - intervalIntegral.integral_eq_sub_of_hasDeriv_right + - intervalIntegral.integral_deriv_comp_smul_deriv' + - intervalIntegral.integral_deriv_eq_sub' + - intervalIntegral.intervalIntegrable_deriv_of_nonneg + - intervalIntegral.integral_mul_deriv_eq_deriv_mul + - intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae + - intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae' + - intervalIntegral.integrableOn_deriv_right_of_nonneg + - intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' + - intervalIntegral.integral_unitInterval_deriv_eq_sub + - intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right + - intervalIntegral.integral_hasStrictFDerivAt_of_tendsto_ae + - intervalIntegral.integral_sub_linear_isLittleO_of_tendsto_ae + - intervalIntegral.integral_hasStrictDerivAt_left + - intervalIntegral.integral_comp_smul_deriv''' + - intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left + - intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left + - intervalIntegral.integral_comp_mul_deriv''' + - intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_left + - intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le + - intervalIntegral.integral_comp_mul_deriv'' + - intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico + - intervalIntegral.integral_comp_mul_deriv' + - intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right + - intervalIntegral.integral_deriv_comp_mul_deriv + - intervalIntegral.integral_deriv_comp_mul_deriv' + - intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' + - intervalIntegral.integral_eq_sub_of_hasDerivAt_of_tendsto + - path: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean + theorems: + - MeasureTheory.Measure.rnDeriv_add_of_mutuallySingular + - MeasureTheory.Measure.rnDeriv_zero + - MeasureTheory.Measure.rnDeriv_add' + - MeasureTheory.Measure.rnDeriv_smul_right' + - MeasureTheory.Measure.LebesgueDecomposition.zero_mem_measurableLE + - MeasureTheory.Measure.rnDeriv_add + - MeasureTheory.Measure.rnDeriv_add_singularPart + - MeasureTheory.Measure.rnDeriv_restrict_self + - MeasureTheory.Measure.singularPart_of_not_haveLebesgueDecomposition + - MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top + - MeasureTheory.Measure.rnDeriv_lt_top + - MeasureTheory.Measure.rnDeriv_self + - MeasureTheory.Measure.withDensity_rnDeriv_le + - MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top + - MeasureTheory.Measure.singularPart_eq_self + - MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top + - MeasureTheory.Measure.LebesgueDecomposition.iSup_succ_eq_sup + - MeasureTheory.Measure.withDensity_rnDeriv_eq_zero + - MeasureTheory.Measure.eq_withDensity_rnDeriv + - MeasureTheory.Measure.LebesgueDecomposition.sup_mem_measurableLE + - MeasureTheory.Measure.singularPart_eq_zero_of_ac + - MeasureTheory.Measure.singularPart_eq_zero + - MeasureTheory.Measure.MutuallySingular.rnDeriv_ae_eq_zero + - MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top' + - MeasureTheory.Measure.rnDeriv_of_not_haveLebesgueDecomposition + - MeasureTheory.Measure.haveLebesgueDecomposition_spec + - MeasureTheory.Measure.lintegral_rnDeriv_lt_top + - MeasureTheory.Measure.rnDeriv_smul_right + - MeasureTheory.Measure.eq_singularPart + - MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE + - MeasureTheory.Measure.singularPart_add + - MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure + - MeasureTheory.Measure.singularPart_smul + - MeasureTheory.Measure.rnDeriv_singularPart + - MeasureTheory.Measure.singularPart_le + - MeasureTheory.Measure.absolutelyContinuous_withDensity_rnDeriv + - MeasureTheory.Measure.rnDeriv_smul_left + - MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE' + - "MeasureTheory.Measure.rnDeriv_withDensity\u2080" + - MeasureTheory.Measure.singularPart_smul_right + - MeasureTheory.Measure.rnDeriv_ne_top + - MeasureTheory.Measure.rnDeriv_smul_left_of_ne_top' + - MeasureTheory.Measure.measurable_rnDeriv + - "MeasureTheory.Measure.eq_withDensity_rnDeriv\u2080" + - MeasureTheory.Measure.rnDeriv_eq_zero + - MeasureTheory.Measure.exists_positive_of_not_mutuallySingular + - path: Mathlib/CategoryTheory/Monoidal/Mon_.lean + theorems: + - Mon_.assoc_flip + - Mon_.one_leftUnitor + - Mon_.one_associator + - Mon_.one_rightUnitor + - Mon_.mul_associator + - Mon_.Mon_tensor_mul_assoc + - Mon_.Mon_tensor_mul_one + - Mon_.whiskerRight_hom + - Mon_.mul_rightUnitor + - Mon_.mul_one_hom + - Mon_.mul_leftUnitor + - Mon_.whiskerLeft_hom + - Mon_.one_mul_hom + - Mon_.Mon_tensor_one_mul + - path: Mathlib/NumberTheory/SumPrimeReciprocals.lean + theorems: + - Nat.Primes.not_summable_one_div + - Nat.Primes.summable_rpow + - not_summable_one_div_on_primes + - Nat.roughNumbersUpTo_card_le' + - path: Mathlib/NumberTheory/Harmonic/Int.lean + theorems: + - harmonic_not_int + - padicNorm_two_harmonic + - padicValRat_two_harmonic + - path: Mathlib/GroupTheory/GroupAction/Basic.lean + theorems: + - MulAction.mem_orbit_smul + - MulAction.le_stabilizer_smul_right + - smul_cancel_of_non_zero_divisor + - MulAction.stabilizer_smul_eq_right + - MulAction.orbit_smul + - MulAction.mem_fixedPoints_iff_card_orbit_eq_one + - MulAction.orbitRel.Quotient.orbit_eq_orbit_out + - AddAction.stabilizer_vadd_eq_stabilizer_map_conj + - MulAction.quotient_preimage_image_eq_union_mul + - Equiv.swap_mem_stabilizer + - MulAction.smul_mem_orbit_smul + - MulAction.stabilizer_smul_eq_left + - MulAction.stabilizer_smul_eq_stabilizer_map_conj + - MulAction.le_stabilizer_smul_left + - MulAction.mem_orbit_self + - MulAction.orbitRel.Quotient.mem_orbit + - MulAction.pretransitive_iff_unique_quotient_of_nonempty + - MulAction.pretransitive_iff_subsingleton_quotient + - path: Mathlib/Analysis/Distribution/SchwartzSpace.lean + theorems: + - SchwartzMap.iteratedPDeriv_succ_right + - SchwartzMap.seminormAux_zero + - SchwartzMap.isBigO_cocompact_zpow_neg_nat + - SchwartzMap.decay_neg_aux + - SchwartzMap.integrable + - Function.HasTemperateGrowth.of_fderiv + - SchwartzMap.integralCLM_dirac_eq_delta + - SchwartzMap.coeHom_injective + - SchwartzMap.isBigO_cocompact_rpow + - SchwartzMap.seminorm_le_bound' + - SchwartzMap.pderivCLM_eq_lineDeriv + - SchwartzMap.isBigO_cocompact_zpow + - SchwartzMap.decay_smul_aux + - SchwartzMap.seminormAux_smul_le + - SchwartzMap.le_seminorm' + - SchwartzMap.decay_add_le_aux + - Function.HasTemperateGrowth.const + - schwartz_withSeminorms + - ContinuousLinearMap.hasTemperateGrowth + - SchwartzMap.norm_pow_mul_le_seminorm + - SchwartzMap.iteratedPDeriv_eq_iteratedFDeriv + - SchwartzMap.decay + - Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux + - SchwartzMap.norm_le_seminorm + - SchwartzMap.integrable_pow_mul + - SchwartzMap.norm_iteratedFDeriv_le_seminorm + - Function.HasTemperateGrowth.zero + - SchwartzMap.one_add_le_sup_seminorm_apply + - SchwartzMap.integralCLM_apply + - path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean + theorems: + - ProjectiveSpectrum.le_iff_mem_closure + - ProjectiveSpectrum.basicOpen_mul_le_right + - ProjectiveSpectrum.isClosed_iff_zeroLocus + - ProjectiveSpectrum.isOpen_iff + - ProjectiveSpectrum.isTopologicalBasis_basic_opens + - ProjectiveSpectrum.zeroLocus_bUnion + - ProjectiveSpectrum.zeroLocus_singleton_pow + - ProjectiveSpectrum.mem_vanishingIdeal + - ProjectiveSpectrum.coe_vanishingIdeal + - ProjectiveSpectrum.vanishingIdeal_union + - ProjectiveSpectrum.basicOpen_eq_zeroLocus_compl + - ProjectiveSpectrum.gc_set + - ProjectiveSpectrum.sup_vanishingIdeal_le + - ProjectiveSpectrum.vanishingIdeal_closure + - ProjectiveSpectrum.basicOpen_zero + - ProjectiveSpectrum.zeroLocus_singleton_mul + - ProjectiveSpectrum.zeroLocus_span + - ProjectiveSpectrum.vanishingIdeal_univ + - ProjectiveSpectrum.isClosed_zeroLocus + - ProjectiveSpectrum.union_zeroLocus + - ProjectiveSpectrum.basicOpen_eq_union_of_projection + - ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure + - ProjectiveSpectrum.vanishingIdeal_singleton + - ProjectiveSpectrum.basicOpen_mul + - ProjectiveSpectrum.basicOpen_one + - ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem + - ProjectiveSpectrum.basicOpen_pow + - ProjectiveSpectrum.gc_homogeneousIdeal + - path: Mathlib/Algebra/Algebra/Basic.lean + theorems: + - Algebra.right_comm + - NoZeroSMulDivisors.trans + - Algebra.mul_sub_algebraMap_pow_commutes + - Algebra.mul_smul_comm + - Algebra.algebra_ext + - LinearMap.map_mul_algebraMap + - NoZeroSMulDivisors.algebraMap_injective + - smul_algebraMap + - LinearMap.map_algebraMap_mul + - Module.End_algebraMap_isUnit_inv_apply_eq_iff' + - algebra_compatible_smul + - Module.End_algebraMap_isUnit_inv_apply_eq_iff + - Algebra.left_comm + - Algebra.mul_sub_algebraMap_commutes + - path: Mathlib/LinearAlgebra/Alternating/Basic.lean + theorems: + - AlternatingMap.compLinearMap_zero + - AlternatingMap.map_perm + - AlternatingMap.map_swap_add + - AlternatingMap.map_add_swap + - MultilinearMap.alternization_map_eq_zero_of_eq_aux + - AlternatingMap.curryLeft_compLinearMap + - AlternatingMap.coe_multilinearMap_mk + - AlternatingMap.add_compLinearMap + - MultilinearMap.alternatization_apply + - LinearMap.compMultilinearMap_alternatization + - AlternatingMap.map_update_self + - AlternatingMap.map_eq_zero_of_not_injective + - AlternatingMap.coe_alternatization + - AlternatingMap.map_linearDependent + - AlternatingMap.map_update_update + - AlternatingMap.compLinearMap_injective + - Basis.ext_alternating + - AlternatingMap.zero_compLinearMap + - AlternatingMap.map_congr_perm + - path: Mathlib/Tactic/Ring/Basic.lean + theorems: + - Mathlib.Tactic.Ring.natCast_nat + - Mathlib.Tactic.Ring.div_congr + - Mathlib.Tactic.Ring.smul_eq_cast + - Mathlib.Tactic.Ring.mul_zero + - Mathlib.Tactic.Ring.mul_add + - Mathlib.Tactic.Ring.of_lift + - Mathlib.Tactic.Ring.neg_zero + - Mathlib.Tactic.Ring.cast_neg + - Mathlib.Tactic.Ring.neg_one_mul + - Mathlib.Tactic.Ring.zero_pow + - Mathlib.Tactic.Ring.pow_bit0 + - Mathlib.Tactic.Ring.smul_nat + - Mathlib.Tactic.Ring.pow_prod_atom + - Mathlib.Tactic.Ring.add_pf_add_overlap_zero + - Mathlib.Tactic.Ring.of_eq + - Mathlib.Tactic.Ring.add_pf_zero_add + - Mathlib.Tactic.Ring.mul_pow + - Mathlib.Tactic.Ring.mul_congr + - Mathlib.Tactic.Ring.one_pow + - Mathlib.Tactic.Ring.pow_bit1 + - Mathlib.Tactic.Ring.one_mul + - Mathlib.Tactic.Ring.pow_atom + - Mathlib.Tactic.Ring.inv_congr + - Mathlib.Tactic.Ring.add_congr + - Mathlib.Tactic.Ring.coeff_mul + - Mathlib.Tactic.Ring.mul_pp_pf_overlap + - Mathlib.Tactic.Ring.single_pow + - Mathlib.Tactic.Ring.pow_congr + - Mathlib.Tactic.Ring.zero_mul + - Mathlib.Tactic.Ring.pow_nat + - Mathlib.Tactic.Ring.atom_pf + - Mathlib.Tactic.Ring.cast_pos + - Mathlib.Tactic.Ring.pow_add + - Mathlib.Tactic.Ring.add_pf_add_overlap + - Mathlib.Tactic.Ring.pow_one + - Mathlib.Tactic.Ring.sub_pf + - Mathlib.Tactic.Ring.inv_mul + - Mathlib.Tactic.Ring.inv_add + - Mathlib.Tactic.Ring.sub_congr + - Mathlib.Tactic.Ring.add_overlap_pf_zero + - Mathlib.Tactic.Ring.neg_congr + - Mathlib.Tactic.Ring.toProd_pf + - Mathlib.Tactic.Ring.atom_pf' + - Mathlib.Tactic.Ring.coeff_one + - Mathlib.Tactic.Ring.add_pf_add_lt + - Mathlib.Tactic.Ring.inv_single + - Mathlib.Tactic.Ring.add_mul + - Mathlib.Tactic.Ring.pow_zero + - Mathlib.Tactic.Ring.add_overlap_pf + - Mathlib.Tactic.Ring.neg_mul + - Mathlib.Tactic.Ring.add_pf_add_zero + - Mathlib.Tactic.Ring.mul_pf_right + - Mathlib.Tactic.Ring.add_pf_add_gt + - Mathlib.Tactic.Ring.cast_zero + - Mathlib.Tactic.Ring.nsmul_congr + - Mathlib.Tactic.Ring.natCast_mul + - Mathlib.Tactic.Ring.natCast_add + - Mathlib.Tactic.Ring.mul_pf_left + - Mathlib.Tactic.Ring.cast_rat + - Mathlib.Tactic.Ring.pow_one_cast + - Mathlib.Tactic.Ring.neg_add + - path: Mathlib/Data/Set/MulAntidiagonal.lean + theorems: + - Set.swap_mem_mulAntidiagonal + - Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd + - Set.MulAntidiagonal.finite_of_isPWO + - Set.swap_mem_mulAntidiagonal_aux + - path: Mathlib/Algebra/GroupPower/Order.lean + theorems: + - pow_right_strictMono + - strictMono_pow_bit1 + - pow_bit1_neg + - pow_right_strictAnti + - sq_eq_sq + - MonoidHom.map_neg + - pow_bit1_nonpos_iff + - pow_lt_one + - pow_eq_one_iff_of_nonneg + - pow_right_mono + - pow_right_injective + - le_self_pow + - lt_of_mul_self_lt_mul_self + - Even.add_pow_le + - MonoidHom.map_neg_one + - zpow_left_injective + - zpow_right_strictMono + - zpow_mono_right + - one_lt_pow_iff_of_nonneg + - pow_add_pow_le + - pow_lt_pow_left + - pow_bit0_pos_iff + - zpow_strictMono_left + - lt_self_pow + - one_lt_zpow' + - one_le_pow_of_one_le + - zpow_mono_left + - one_le_pow_iff_of_nonneg + - pow_lt_self_of_lt_one + - pow_le_one_iff_of_nonneg + - pow_add_pow_le' + - add_sq_le + - add_pow_le + - MonoidHom.map_sub_swap + - one_lt_pow + - pow_bit0_pos_of_neg + - pow_bit0_nonneg + - pow_le_pow_left + - path: Mathlib/Topology/Algebra/Module/Basic.lean + theorems: + - ContinuousLinearMap.smulRight_one_eq_iff + - ContinuousLinearMap.projKerOfRightInverse_comp_inv + - ContinuousLinearMap.toSpanSingleton_add + - ContinuousLinearEquiv.image_symm_eq_preimage + - DenseRange.topologicalClosure_map_submodule + - ContinuousLinearEquiv.symm_comp_self + - ContinuousLinearMap.comp_neg + - ContinuousLinearEquiv.symm_symm + - ContinuousLinearMap.pi_eq_zero + - ContinuousLinearEquiv.trans_toLinearEquiv + - "ContinuousLinearMap.comp_smul\u209B\u2097" + - Submodule.ClosedComplemented.isClosed + - Submodule.closedComplemented_bot + - ContinuousLinearMap.coe_eq_id + - ContinuousLinearMap.comp_smul + - ContinuousLinearMap.smulRight_one_pow + - ContinuousLinearMap.coe_sum' + - Submodule.closure_subset_topologicalClosure_span + - ContinuousLinearMap.inverse_equiv + - ContinuousLinearMap.smulRight_comp + - ContinuousSMul.of_nhds_zero + - ContinuousLinearMap.smulRight_one_one + - ContinuousLinearMap.prod_ext_iff + - ContinuousLinearMap.ring_inverse_equiv + - ContinuousLinearEquiv.self_comp_symm + - ContinuousLinearMap.ring_inverse_eq_map_inverse + - ContinuousLinearMap.coprod_inl_inr + - ContinuousLinearMap.comp_sub + - ContinuousLinearMap.neg_comp + - Submodule.eq_top_of_nonempty_interior' + - LinearMap.continuous_on_pi + - Module.punctured_nhds_neBot + - ContinuousLinearMap.comp_add + - TopologicalSpace.IsSeparable.span + - ContinuousLinearMap.projKerOfRightInverse_apply_idem + - ContinuousLinearMap.map_sub + - Submodule.closedComplemented_top + - ContinuousLinearMap.isOpenMap_of_ne_zero + - ContinuousLinearMap.sub_comp + - "ContinuousLinearEquiv.ext\u2081" + - Submodule.dense_iff_topologicalClosure_eq_top + - ContinuousLinearEquiv.toLinearEquiv_injective + - ContinuousLinearMap.sum_apply + - ContinuousLinearMap.zero_comp + - ContinuousLinearMap.map_smul + - ContinuousLinearMap.toSpanSingleton_smul' + - ContinuousLinearEquiv.symm_toLinearEquiv + - ContinuousLinearMap.exists_ne_zero + - ContinuousLinearMap.coe_injective + - Submodule.isClosed_or_dense_of_isCoatom + - ContinuousLinearMap.to_ring_inverse + - ContinuousLinearMap.add_comp + - ContinuousLinearMap.comp_zero + - path: Mathlib/GroupTheory/Perm/Sign.lean + theorems: + - Equiv.Perm.sign_surjective + - Equiv.Perm.signAux_inv + - Equiv.Perm.swap_induction_on + - Equiv.Perm.signAux3_symm_trans_trans + - Equiv.Perm.sign_bij + - Equiv.Perm.signAux_eq_signAux2 + - Equiv.Perm.sign_prodCongrLeft + - Equiv.Perm.sign_prodExtendRight + - Equiv.Perm.sign_extendDomain + - Equiv.Perm.signAux_swap_zero_one + - Equiv.Perm.signAux_swap_zero_one' + - Equiv.Perm.sign_subtypePerm + - Equiv.Perm.signBijAux_surj + - Equiv.Perm.sign_swap' + - Equiv.Perm.sign_trans + - Equiv.Perm.sign_sumCongr + - Equiv.Perm.signBijAux_injOn + - Equiv.Perm.sign_permCongr + - Equiv.Perm.signAux_swap + - Equiv.Perm.isConj_swap + - Equiv.Perm.eq_sign_of_surjective_hom + - Equiv.Perm.signAux3_mul_and_swap + - Equiv.Perm.sign_prodCongrRight + - Equiv.Perm.signBijAux_mem + - Equiv.Perm.signAux_one + - Equiv.Perm.sign_eq_sign_of_equiv + - Equiv.Perm.closure_isSwap + - Equiv.Perm.prod_prodExtendRight + - Equiv.Perm.mem_finPairsLT + - Equiv.Perm.sign_subtypeCongr + - Equiv.Perm.sign_inv + - Equiv.Perm.signAux_mul + - Equiv.Perm.sign_prod_list_swap + - path: Mathlib/Combinatorics/Partition.lean + theorems: + - Nat.Partition.indiscrete_parts + - Nat.Partition.ofComposition_surj + - path: Mathlib/Data/Matroid/Dual.lean + theorems: + - Matroid.Indep.ssubset_ground + - Matroid.setOf_dual_base_eq + - Matroid.dual_indep_iff_exists + - Matroid.eq_dual_comm + - Matroid.base_iff_dual_base_compl + - Matroid.dual_coindep_iff + - Matroid.coindep_iff_exists' + - Matroid.Base.inter_basis_iff_compl_inter_basis_dual + - Matroid.Base.compl_inter_basis_of_inter_basis + - Matroid.coindep_iff_subset_compl_base + - Matroid.coindep_iff_exists + - Matroid.dual_indep_iff_exists' + - Matroid.dual_base_iff + - Matroid.Base.ssubset_ground + - Matroid.ground_not_base + - Matroid.Base.compl_base_dual + - Matroid.dual_dep_iff_forall + - Matroid.dual_base_iff' + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean + theorems: + - Matrix.Represents.smul + - Matrix.isRepresentation.toEnd_surjective + - Matrix.Represents.algebraMap + - Matrix.isRepresentation.toEnd_exists_mem_ideal + - Matrix.Represents.zero + - PiToModule.fromMatrix_apply_single_one + - PiToModule.fromEnd_injective + - PiToModule.fromEnd_apply_single_one + - Matrix.Represents.add + - LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul + - Matrix.represents_iff' + - LinearMap.exists_monic_and_aeval_eq_zero + - Matrix.Represents.mul + - path: Mathlib/CategoryTheory/Comma/Presheaf.lean + theorems: + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2082_yonedaEquivFst" + - CategoryTheory.OverPresheafAux.counitForward_counitBackward + - "CategoryTheory.OverPresheafAux.unitForward_naturality\u2081" + - CategoryTheory.OverPresheafAux.OverArrows.map_val + - CategoryTheory.OverPresheafAux.counitForward_val_fst + - "CategoryTheory.OverPresheafAux.counitForward_naturality\u2081" + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2081_comp" + - CategoryTheory.OverPresheafAux.MakesOverArrow.of_yoneda_arrow + - CategoryTheory.OverPresheafAux.counitBackward_counitForward + - CategoryTheory.OverPresheafAux.YonedaCollection.ext + - CategoryTheory.OverPresheafAux.OverArrows.yonedaCollectionPresheafToA_val_fst + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2081_yonedaEquivFst" + - CategoryTheory.OverPresheafAux.unitForward_unitBackward + - "CategoryTheory.OverPresheafAux.unitForward_naturality\u2082" + - CategoryTheory.OverPresheafAux.app_unitForward + - CategoryTheory.OverPresheafAux.map_mkPrecomp_eqToHom + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2082_snd" + - "CategoryTheory.OverPresheafAux.counitForward_naturality\u2082" + - "CategoryTheory.OverPresheafAux.MakesOverArrow.map\u2082" + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2082_id" + - CategoryTheory.OverPresheafAux.YonedaCollection.mk_snd + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2081_map\u2082" + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2082_comp" + - CategoryTheory.OverPresheafAux.unitBackward_unitForward + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2082_fst" + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2081_id" + - "CategoryTheory.OverPresheafAux.YonedaCollection.map\u2081_snd" + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean + theorems: + - ContinuousLinearEquiv.coord_norm + - LinearIsometry.norm_toContinuousLinearMap + - ContinuousLinearMap.homothety_norm + - ContinuousLinearMap.norm_id + - ContinuousLinearMap.norm_smulRight_apply + - LinearMap.bound_of_ball_bound + - ContinuousLinearEquiv.one_le_norm_mul_norm_symm + - ContinuousLinearEquiv.subsingleton_or_norm_symm_pos + - ContinuousLinearMap.opNorm_zero_iff + - LinearMap.antilipschitz_of_comap_nhds_le + - ContinuousLinearMap.opNorm_comp_linearIsometryEquiv + - NormedSpace.equicontinuous_TFAE + - path: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean + theorems: + - CategoryTheory.Limits.map_lift_pullbackComparison + - CategoryTheory.Limits.spanExt_app_right + - CategoryTheory.Limits.inr_inl_pushoutLeftPushoutInrIso_hom + - CategoryTheory.Limits.cospanExt_inv_app_right + - CategoryTheory.Limits.pullbackAssoc_inv_fst_snd + - CategoryTheory.Limits.pullbackAssoc_inv_snd + - CategoryTheory.Limits.cospanExt_inv_app_one + - CategoryTheory.Limits.pushout.mapLift_comp + - CategoryTheory.Limits.PushoutCocone.coequalizer_ext + - CategoryTheory.Limits.spanExt_hom_app_left + - "CategoryTheory.Limits.pullbackConeOfRightIso_\u03C0_app_none" + - CategoryTheory.Limits.PullbackCone.mono_snd_of_is_pullback_of_mono + - CategoryTheory.Limits.pullbackSymmetry_hom_comp_fst + - CategoryTheory.Limits.inr_pushoutAssoc_hom + - CategoryTheory.Limits.inl_inr_pushoutAssoc_inv + - CategoryTheory.Limits.spanExt_app_one + - CategoryTheory.Limits.cospanExt_hom_app_right + - CategoryTheory.Limits.pullbackSymmetry_hom_comp_snd + - CategoryTheory.Limits.cospanExt_hom_app_one + - CategoryTheory.Limits.spanExt_hom_app_right + - CategoryTheory.Limits.pushout.congrHom_inv + - CategoryTheory.Limits.inl_comp_pushoutSymmetry_inv + - CategoryTheory.Limits.spanExt_inv_app_left + - CategoryTheory.Limits.pullback.mapDesc_comp + - "CategoryTheory.Limits.pushoutCoconeOfRightIso_\u03B9_app_none" + - CategoryTheory.Limits.inr_inr_pushoutAssoc_inv + - CategoryTheory.Limits.pullbackSymmetry_inv_comp_snd + - CategoryTheory.Limits.inl_inl_pushoutAssoc_hom + - CategoryTheory.Limits.cospanExt_app_right + - CategoryTheory.Limits.pullbackAssoc_hom_snd_snd + - CategoryTheory.Limits.cospanExt_app_one + - CategoryTheory.Limits.PushoutCocone.epi_of_isColimitMkIdId + - CategoryTheory.Limits.spanExt_inv_app_right + - CategoryTheory.Limits.PushoutCocone.epi_inr_of_is_pushout_of_epi + - CategoryTheory.Limits.pullbackAssoc_hom_fst + - CategoryTheory.Limits.inr_inl_pushoutAssoc_hom + - "CategoryTheory.Limits.pullbackConeOfLeftIso_\u03C0_app_none" + - CategoryTheory.Limits.inr_comp_pushoutSymmetry_inv + - CategoryTheory.Limits.PullbackCone.equalizer_ext + - CategoryTheory.Limits.PullbackCone.condition_one + - "CategoryTheory.Limits.pushoutCoconeOfLeftIso_\u03B9_app_none" + - CategoryTheory.Limits.PushoutCocone.epi_inl_of_is_pushout_of_epi + - CategoryTheory.Limits.PushoutCocone.condition_zero + - CategoryTheory.Limits.cospanExt_app_left + - CategoryTheory.Limits.pullback.congrHom_inv + - CategoryTheory.Limits.cospanExt_hom_app_left + - CategoryTheory.Limits.inr_pushoutLeftPushoutInrIso_inv + - CategoryTheory.Limits.pullbackAssoc_inv_fst_fst + - CategoryTheory.Limits.pullbackAssoc_hom_snd_fst + - CategoryTheory.Limits.pullbackRightPullbackFstIso_inv_snd_fst + - CategoryTheory.Limits.PullbackCone.mono_fst_of_is_pullback_of_mono + - CategoryTheory.Limits.pushoutComparison_map_desc + - CategoryTheory.Limits.PullbackCone.mono_of_isLimitMkIdId + - CategoryTheory.Limits.pullbackSymmetry_hom_of_mono_eq + - CategoryTheory.Limits.pullbackSymmetry_inv_comp_fst + - CategoryTheory.Limits.cospanExt_inv_app_left + - CategoryTheory.Limits.spanExt_hom_app_zero + - CategoryTheory.Limits.inl_pushoutAssoc_inv + - CategoryTheory.Limits.spanExt_inv_app_zero + - CategoryTheory.Limits.inl_inl_pushoutLeftPushoutInrIso_hom + - CategoryTheory.Limits.spanExt_app_left + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean + theorems: + - MeasureTheory.mem_lpMeasSubgroup_toLp_of_trim + - MeasureTheory.AEStronglyMeasurable'.const_smul + - MeasureTheory.isometry_lpMeasSubgroupToLpTrim + - MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable' + - MeasureTheory.lpMeasSubgroupToLpTrim_add + - MeasureTheory.lpMeasToLpTrimLie_symm_indicator + - MeasureTheory.AEStronglyMeasurable'.add + - MeasureTheory.lpMeasSubgroupToLpTrim_neg + - "MeasureTheory.mem\u2112p_trim_of_mem_lpMeasSubgroup" + - MeasureTheory.AEStronglyMeasurable'.sub + - MeasureTheory.AEStronglyMeasurable'.aeStronglyMeasurable'_of_measurableSpace_le_on + - MeasureTheory.isComplete_aeStronglyMeasurable' + - MeasureTheory.lpMeasToLpTrim_smul + - MeasureTheory.Lp.induction_stronglyMeasurable + - MeasureTheory.AEStronglyMeasurable'.mono + - MeasureTheory.aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim + - MeasureTheory.lpMeasSubgroupToLpTrim_sub + - MeasureTheory.Lp.induction_stronglyMeasurable_aux + - "MeasureTheory.Mem\u2112p.induction_stronglyMeasurable" + - MeasureTheory.lpMeasSubgroupToLpTrim_right_inv + - MeasureTheory.AEStronglyMeasurable'.const_inner + - MeasureTheory.lpMeasSubgroupToLpTrim_norm_map + - MeasureTheory.mem_lpMeas_iff_aeStronglyMeasurable' + - MeasureTheory.AEStronglyMeasurable'.congr + - MeasureTheory.lpMeasToLpTrimLie_symm_toLp + - MeasureTheory.AEStronglyMeasurable'.neg + - MeasureTheory.lpMeasSubgroupToLpTrim_left_inv + - path: Mathlib/RingTheory/HahnSeries/Multiplication.lean + theorems: + - HahnSeries.order_pow + - HahnModule.smul_coeff_left + - HahnSeries.support_mul_subset_add_support + - HahnSeries.mul_single_coeff_add + - HahnSeries.embDomainRingHom_C + - HahnSeries.single_zero_mul_coeff + - HahnSeries.mul_single_zero_coeff + - HahnSeries.mul_coeff_order_add_order + - HahnModule.smul_coeff_right + - HahnSeries.C_ne_zero + - HahnSeries.single_zero_mul_eq_smul + - HahnSeries.order_mul + - HahnSeries.single_mul_single + - HahnSeries.order_one + - HahnSeries.order_C + - HahnSeries.mul_assoc' + - HahnSeries.C_injective + - HahnSeries.embDomain_mul + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean + theorems: + - MeasureTheory.ae_eq_condexp_of_forall_set_integral_eq + - MeasureTheory.condexp_bot_ae_eq + - MeasureTheory.condexp_finset_sum + - MeasureTheory.condexp_neg + - MeasureTheory.condexp_bot + - MeasureTheory.condexp_of_sigmaFinite + - MeasureTheory.condexp_nonneg + - MeasureTheory.condexp_of_aestronglyMeasurable' + - MeasureTheory.condexp_of_not_sigmaFinite + - MeasureTheory.set_integral_condexp + - MeasureTheory.condexp_bot' + - MeasureTheory.condexp_nonpos + - MeasureTheory.integral_condexp + - MeasureTheory.condexp_add + - MeasureTheory.condexp_undef + - MeasureTheory.condexp_of_not_le + - MeasureTheory.stronglyMeasurable_condexp + - MeasureTheory.condexp_sub + - MeasureTheory.condexp_of_stronglyMeasurable + - MeasureTheory.condexp_smul + - MeasureTheory.tendsto_condexp_unique + - MeasureTheory.integrable_condexp + - MeasureTheory.condexp_ae_eq_condexpL1CLM + - MeasureTheory.condexp_mono + - MeasureTheory.condexp_ae_eq_condexpL1 + - MeasureTheory.condexp_zero + - MeasureTheory.condexp_congr_ae + - path: Mathlib/Data/Seq/WSeq.lean + theorems: + - Stream'.WSeq.mem_rec_on + - Stream'.WSeq.mem_cons_iff + - Stream'.WSeq.LiftRel.symm + - Stream'.WSeq.destruct_flatten + - Stream'.WSeq.liftRel_destruct_iff + - Stream'.WSeq.dropn_nil + - Stream'.WSeq.flatten_pure + - Stream'.WSeq.toList'_map + - Stream'.WSeq.tail_nil + - Stream'.WSeq.tail_think + - Stream'.WSeq.exists_of_mem_map + - Stream'.WSeq.dropn_think + - Stream'.WSeq.liftRel_dropn_destruct + - Stream'.WSeq.destruct_cons + - Stream'.WSeq.destruct_join + - Stream'.WSeq.liftRel_map + - Stream'.WSeq.LiftRelO.swap + - Stream'.WSeq.length_eq_map + - Stream'.WSeq.toList_ofList + - Stream'.WSeq.think_equiv + - Stream'.WSeq.flatten_equiv + - Stream'.WSeq.head_ofSeq + - Stream'.WSeq.cons_congr + - Stream'.WSeq.head_think + - Stream'.WSeq.mem_think + - Stream'.WSeq.Equiv.ext + - Stream'.WSeq.head_cons + - Stream'.WSeq.exists_of_liftRel_right + - Stream'.WSeq.liftRel_join.lem + - Stream'.WSeq.toSeq_ofSeq + - Stream'.WSeq.destruct_think + - Stream'.WSeq.map_join + - Stream'.WSeq.bind_assoc + - Stream'.WSeq.exists_of_mem_join + - Stream'.WSeq.liftRel_destruct + - Stream'.WSeq.liftRel_flatten + - Stream'.WSeq.ofList_cons + - Stream'.WSeq.toList_cons + - Stream'.WSeq.destruct_dropn + - Stream'.WSeq.head_nil + - Stream'.WSeq.head_congr + - Stream'.WSeq.liftRel_cons + - Stream'.WSeq.destruct_ofSeq + - Stream'.WSeq.liftRel_think_right + - Stream'.WSeq.get?_terminates_le + - Stream'.WSeq.map_comp + - Stream'.WSeq.bind_ret + - Stream'.WSeq.join_map_ret + - Stream'.WSeq.tail_cons + - Stream'.WSeq.destruct_tail + - Stream'.WSeq.destruct_map + - Stream'.WSeq.join_think + - Stream'.WSeq.exists_of_liftRel_left + - Stream'.WSeq.toList'_think + - Stream'.WSeq.toList'_cons + - Stream'.WSeq.liftRel_nil + - Stream'.WSeq.ret_bind + - Stream'.WSeq.join_append + - Stream'.WSeq.dropn_tail + - Stream'.WSeq.join_ret + - Stream'.WSeq.head_terminates_of_head_tail_terminates + - Stream'.WSeq.exists_get?_of_mem + - Stream'.WSeq.tail_congr + - Stream'.WSeq.exists_of_mem_bind + - Stream'.WSeq.dropn_ofSeq + - Stream'.WSeq.join_join + - Stream'.WSeq.LiftRel.swap_lem + - Stream'.WSeq.bind_congr + - Stream'.WSeq.get?_mem + - Stream'.WSeq.think_congr + - Stream'.WSeq.flatten_think + - Stream'.WSeq.join_cons + - Stream'.WSeq.exists_dropn_of_mem + - Stream'.WSeq.dropn_cons + - Stream'.WSeq.head_some_of_head_tail_some + - Stream'.WSeq.destruct_some_of_destruct_tail_some + - Stream'.WSeq.eq_or_mem_iff_mem + - Stream'.WSeq.map_id + - Stream'.WSeq.productive_congr + - Stream'.WSeq.liftRel_think_left + - Stream'.WSeq.dropn_congr + - Stream'.WSeq.LiftRel.trans + - Stream'.WSeq.head_some_of_get?_some + - Stream'.WSeq.destruct_append + - Stream'.WSeq.drop.aux_none + - Stream'.WSeq.map_ret + - Stream'.WSeq.mem_of_mem_tail + - Stream'.WSeq.tail_ofSeq + - Stream'.WSeq.get?_ofSeq + - Stream'.WSeq.liftRel_join + - Stream'.WSeq.liftRel_append + - path: Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean + theorems: + - CategoryTheory.Subgroupoid.sInf_isNormal + - CategoryTheory.Subgroupoid.le_iff + - CategoryTheory.Subgroupoid.full_univ + - CategoryTheory.Subgroupoid.inclusion_inj_on_objects + - CategoryTheory.Subgroupoid.obj_surjective_of_im_eq_top + - CategoryTheory.Subgroupoid.isTotallyDisconnected_iff + - CategoryTheory.Subgroupoid.subset_generated + - CategoryTheory.Subgroupoid.disconnect_isTotallyDisconnected + - CategoryTheory.Subgroupoid.mem_im_iff + - CategoryTheory.Subgroupoid.Map.arrows_iff + - CategoryTheory.Subgroupoid.isNormal_comap + - CategoryTheory.Subgroupoid.coe_inv_coe' + - CategoryTheory.Subgroupoid.IsWide.eqToHom_mem + - CategoryTheory.Subgroupoid.mem_discrete_iff + - CategoryTheory.Subgroupoid.full_empty + - CategoryTheory.Subgroupoid.mul_mem_cancel_right + - CategoryTheory.Subgroupoid.map_objs_eq + - CategoryTheory.Subgroupoid.hom.faithful + - CategoryTheory.Subgroupoid.id_mem_of_nonempty_isotropy + - CategoryTheory.Subgroupoid.mul_mem_cancel_left + - CategoryTheory.Subgroupoid.IsNormal.conj' + - CategoryTheory.Subgroupoid.galoisConnection_map_comap + - CategoryTheory.Subgroupoid.inclusion_faithful + - CategoryTheory.Subgroupoid.IsNormal.generatedNormal_le + - CategoryTheory.Subgroupoid.mem_im_objs_iff + - CategoryTheory.Subgroupoid.mem_full_objs_iff + - CategoryTheory.Subgroupoid.disconnect_le + - CategoryTheory.Subgroupoid.full_mono + - CategoryTheory.Subgroupoid.IsNormal.vertexSubgroup + - CategoryTheory.Subgroupoid.mem_map_objs_iff + - CategoryTheory.Subgroupoid.mem_top_objs + - CategoryTheory.Subgroupoid.hom.inj_on_objects + - CategoryTheory.Subgroupoid.isNormal_map + - CategoryTheory.Subgroupoid.isWide_iff_objs_eq_univ + - CategoryTheory.Subgroupoid.IsNormal.conjugation_bij + - CategoryTheory.Subgroupoid.inv_mem_iff + - CategoryTheory.Subgroupoid.discrete_isNormal + - path: Mathlib/Algebra/Field/Power.lean + theorems: + - Odd.neg_one_zpow + - zpow_bit1_neg + - Odd.neg_zpow + - path: Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean + theorems: + - Bornology.IsVonNBounded.image_multilinear + - Bornology.IsVonNBounded.image_multilinear' + - path: Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean + theorems: + - Complex.mem_slitPlane_iff_arg + - Complex.arg_cos_add_sin_mul_I_sub + - Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero + - Complex.tan_arg + - Complex.arg_mem_Ioc + - Complex.arg_I + - Complex.continuousAt_arg + - Complex.arg_mul_cos_add_sin_mul_I_sub + - Complex.arg_neg_eq_arg_add_pi_iff + - Complex.neg_pi_div_two_le_arg_iff + - Complex.tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero + - Complex.arg_eq_pi_iff + - Complex.arg_cos_add_sin_mul_I + - Complex.arg_neg_eq_arg_add_pi_of_im_neg + - Complex.arg_zero + - Complex.arg_ofReal_of_nonneg + - Complex.continuousAt_arg_coe_angle + - Complex.arg_neg_I + - Complex.cos_arg + - Complex.arg_of_im_neg + - Complex.arg_coe_angle_eq_iff + - Complex.arg_lt_pi_div_two_iff + - Complex.arg_cos_add_sin_mul_I_eq_toIocMod + - Complex.arg_mul_coe_angle + - Complex.arg_of_re_neg_of_im_neg + - Complex.arg_inv_coe_angle + - Complex.ext_abs_arg + - Complex.arg_conj + - Complex.arg_exp_mul_I + - Complex.arg_eq_zero_iff + - Complex.abs_mul_exp_arg_mul_I + - Complex.abs_mul_sin_arg + - Complex.arg_eq_zero_iff_zero_le + - Complex.arg_neg_coe_angle + - Complex.abs_eq_one_iff' + - Complex.arg_mul_cos_add_sin_mul_I_coe_angle + - Complex.arg_cos_add_sin_mul_I_coe_angle + - Complex.range_exp_mul_I + - Complex.image_exp_Ioc_eq_sphere + - Complex.arg_mul_cos_add_sin_mul_I + - Complex.abs_mul_cos_arg + - Complex.arg_real_mul + - Complex.arg_neg_eq_arg_sub_pi_iff + - Complex.arg_conj_coe_angle + - Complex.sin_arg + - Complex.arg_one + - Complex.abs_eq_one_iff + - Complex.continuousWithinAt_arg_of_re_neg_of_im_zero + - Complex.arg_inv + - Complex.arg_nonneg_iff + - Complex.arg_mul_eq_add_arg_iff + - Complex.neg_pi_div_two_lt_arg_iff + - Complex.abs_arg_le_pi_div_two_iff + - Complex.arg_eq_arg_iff + - Complex.arg_eq_pi_div_two_iff + - Complex.arg_eq_nhds_of_re_neg_of_im_neg + - Complex.arg_of_im_nonneg_of_ne_zero + - Complex.arg_le_pi_div_two_iff + - Complex.abs_arg_inv + - Complex.arg_neg_one + - Complex.arg_div_coe_angle + - Complex.arg_coe_angle_toReal_eq_arg + - Complex.arg_of_re_neg_of_im_nonneg + - Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod + - Complex.arg_lt_pi_iff + - Complex.arg_eq_neg_pi_div_two_iff + - Complex.abs_arg_lt_pi_div_two_iff + - Complex.arg_eq_nhds_of_re_neg_of_im_pos + - Complex.arg_neg_eq_arg_sub_pi_of_im_pos + - Complex.arg_coe_angle_eq_iff_eq_toReal + - path: Mathlib/Analysis/Convex/KreinMilman.lean + theorems: + - surjOn_extremePoints_image + - IsCompact.extremePoints_nonempty + - closure_convexHull_extremePoints + - path: Mathlib/Data/Int/Units.lean + theorems: + - Int.ofNat_isUnit + - Int.eq_one_or_neg_one_of_mul_eq_one' + - Int.units_ne_iff_eq_neg + - Int.isUnit_add_isUnit_eq_isUnit_add_isUnit + - Int.mul_eq_neg_one_iff_eq_one_or_neg_one + - Int.isUnit_iff_natAbs_eq + - Int.units_eq_one_or + - Int.mul_eq_one_iff_eq_one_or_neg_one + - Int.units_natAbs + - Int.isUnit_ne_iff_eq_neg + - Int.eq_one_or_neg_one_of_mul_eq_neg_one' + - path: Mathlib/SetTheory/Ordinal/Principal.lean + theorems: + - Ordinal.principal_iff_principal_swap + - Ordinal.principal_mul_isLimit + - Ordinal.op_eq_self_of_principal + - Ordinal.add_omega_opow + - Ordinal.mul_principal_add_is_principal_add + - Ordinal.Principal.iterate_lt + - Ordinal.mul_omega_opow_opow + - Ordinal.principal_add_of_principal_mul + - Ordinal.principal_add_of_principal_mul_opow + - Ordinal.principal_one_iff + - Ordinal.principal_mul_iff_le_two_or_omega_opow_opow + - Ordinal.principal_mul_of_le_two + - Ordinal.principal_add_of_le_one + - Ordinal.add_absorp + - Ordinal.principal_add_iff_zero_or_omega_opow + - Ordinal.principal_mul_two + - Ordinal.principal_opow_omega + - Ordinal.add_omega + - Ordinal.principal_mul_iff_mul_left_eq + - Ordinal.principal_add_iff_add_left_eq_self + - Ordinal.mul_omega_dvd + - Ordinal.principal_add_iff_add_lt_ne_self + - Ordinal.principal_mul_one + - Ordinal.mul_lt_omega_opow + - Ordinal.principal_add_isLimit + - Ordinal.opow_principal_add_of_principal_add + - Ordinal.principal_mul_omega + - Ordinal.mul_eq_opow_log_succ + - Ordinal.exists_lt_add_of_not_principal_add + - "Ordinal.principal_nfp_blsub\u2082" + - path: Mathlib/Topology/Inseparable.lean + theorems: + - specializes_TFAE + - specializes_pi + - SeparationQuotient.isOpenMap_mk + - not_specializes_iff_exists_open + - not_specializes_iff_exists_closed + - Inducing.specializes_iff + - SeparationQuotient.preimage_image_mk_open + - inseparable_iff_forall_open + - SeparationQuotient.tendsto_lift_nhds_mk + - specializes_prod + - SeparationQuotient.map_mk_nhdsSet + - SeparationQuotient.isClosedMap_mk + - IsOpen.continuous_piecewise_of_specializes + - ker_nhds_eq_specializes + - inseparable_pi + - SeparationQuotient.preimage_image_mk_closed + - SeparationQuotient.map_prod_map_mk_nhds + - "SeparationQuotient.tendsto_lift\u2082_nhds" + - inseparable_iff_mem_closure + - SeparationQuotient.tendsto_lift_nhdsWithin_mk + - Specializes.not_disjoint + - SeparationQuotient.continuousOn_lift + - Inducing.inseparable_iff + - SeparationQuotient.map_mk_nhdsWithin_preimage + - IsClosed.continuous_piecewise_of_specializes + - "SeparationQuotient.continuous_lift\u2082" + - "SeparationQuotient.tendsto_lift\u2082_nhdsWithin" + - continuous_congr_of_inseparable + - "SeparationQuotient.continuousOn_lift\u2082" + - inseparable_iff_forall_closed + - SeparationQuotient.continuous_lift + - not_inseparable_iff_exists_open + - inseparable_iff_closure_eq + - SeparationQuotient.comap_mk_nhdsSet + - SeparationQuotient.map_mk_nhds + - inseparable_prod + - path: Mathlib/Analysis/Normed/Group/Pointwise.lean + theorems: + - inv_cthickening + - infEdist_inv_inv + - ball_mul + - closedBall_div_singleton + - ediam_mul_le + - div_ball + - closedBall_one_div_singleton + - ball_div_one + - Bornology.IsBounded.inv + - ball_div_singleton + - singleton_mul_closedBall_one + - mul_ball_one + - singleton_mul_closedBall + - closedBall_mul_singleton + - IsCompact.mul_closedBall_one + - ball_div + - Bornology.IsBounded.mul + - IsCompact.mul_closedBall + - smul_closedBall_one + - smul_ball_one + - IsCompact.closedBall_one_mul + - closedBall_one_mul_singleton + - IsCompact.closedBall_div + - mul_ball + - singleton_div_ball + - singleton_mul_ball + - singleton_mul_ball_one + - IsCompact.closedBall_one_div + - singleton_div_closedBall_one + - IsCompact.div_closedBall_one + - ball_one_mul_singleton + - ball_mul_one + - ball_mul_singleton + - ball_one_div_singleton + - inv_thickening + - infEdist_inv + - IsCompact.div_closedBall + - singleton_div_ball_one + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean + theorems: + - EuclideanGeometry.dist_div_tan_oangle_right_of_oangle_eq_pi_div_two + - Orientation.norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two + - EuclideanGeometry.sin_oangle_right_mul_dist_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two + - EuclideanGeometry.oangle_right_eq_arcsin_of_oangle_eq_pi_div_two + - EuclideanGeometry.sin_oangle_right_of_oangle_eq_pi_div_two + - EuclideanGeometry.oangle_left_eq_arcsin_of_oangle_eq_pi_div_two + - Orientation.oangle_sub_right_eq_arccos_of_oangle_eq_pi_div_two + - EuclideanGeometry.oangle_left_eq_arctan_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_sub_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.oangle_right_eq_arccos_of_oangle_eq_pi_div_two + - EuclideanGeometry.cos_oangle_right_mul_dist_of_oangle_eq_pi_div_two + - Orientation.norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two + - EuclideanGeometry.sin_oangle_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.oangle_right_eq_arctan_of_oangle_eq_pi_div_two + - Orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two + - Orientation.oangle_add_left_eq_arccos_of_oangle_eq_pi_div_two + - Orientation.oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_add_left_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_sub_right_of_oangle_eq_pi_div_two + - Orientation.oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_add_left_smul_rotation_pi_div_two + - Orientation.tan_oangle_add_left_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_sub_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.dist_div_sin_oangle_left_of_oangle_eq_pi_div_two + - Orientation.norm_div_cos_oangle_add_right_of_oangle_eq_pi_div_two + - Orientation.oangle_sub_right_eq_arcsin_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two + - EuclideanGeometry.tan_oangle_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.tan_oangle_right_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two + - Orientation.oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two + - Orientation.norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.dist_div_tan_oangle_left_of_oangle_eq_pi_div_two + - Orientation.oangle_sub_right_smul_rotation_pi_div_two + - Orientation.norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_sub_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.cos_oangle_right_of_oangle_eq_pi_div_two + - Orientation.norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two + - Orientation.norm_div_cos_oangle_add_left_of_oangle_eq_pi_div_two + - Orientation.oangle_sub_left_smul_rotation_pi_div_two + - EuclideanGeometry.sin_oangle_left_mul_dist_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two + - Orientation.norm_div_cos_oangle_sub_left_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two + - EuclideanGeometry.tan_oangle_left_mul_dist_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two + - EuclideanGeometry.dist_div_cos_oangle_left_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_sub_right_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two + - Orientation.norm_div_sin_oangle_add_right_of_oangle_eq_pi_div_two + - Orientation.norm_div_sin_oangle_add_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.cos_oangle_left_of_oangle_eq_pi_div_two + - Orientation.sin_oangle_add_right_of_oangle_eq_pi_div_two + - Orientation.oangle_add_right_smul_rotation_pi_div_two + - Orientation.oangle_sub_left_eq_arcsin_of_oangle_eq_pi_div_two + - EuclideanGeometry.oangle_left_eq_arccos_of_oangle_eq_pi_div_two + - Orientation.oangle_sub_left_eq_arccos_of_oangle_eq_pi_div_two + - EuclideanGeometry.dist_div_cos_oangle_right_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two + - Orientation.norm_div_tan_oangle_sub_left_of_oangle_eq_pi_div_two + - Orientation.norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two + - Orientation.oangle_add_left_smul_rotation_pi_div_two + - Orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two + - Orientation.cos_oangle_add_left_of_oangle_eq_pi_div_two + - EuclideanGeometry.tan_oangle_right_mul_dist_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_add_right_of_oangle_eq_pi_div_two + - Orientation.oangle_add_left_eq_arctan_of_oangle_eq_pi_div_two + - EuclideanGeometry.cos_oangle_left_mul_dist_of_oangle_eq_pi_div_two + - Orientation.tan_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two + - path: Mathlib/MeasureTheory/MeasurableSpace/Card.lean + theorems: + - MeasurableSpace.compl_mem_generateMeasurableRec + - MeasurableSpace.generateMeasurable_eq_rec + - MeasurableSpace.empty_mem_generateMeasurableRec + - MeasurableSpace.generateMeasurableRec_subset + - MeasurableSpace.iUnion_mem_generateMeasurableRec + - MeasurableSpace.self_subset_generateMeasurableRec + - MeasurableSpace.cardinal_generateMeasurable_le + - MeasurableSpace.cardinal_generateMeasurableRec_le + - MeasurableSpace.cardinal_generateMeasurable_le_continuum + - path: Mathlib/Analysis/Fourier/AddCircle.lean + theorems: + - fourier_one + - fourier_coe_apply' + - fourier_zero + - has_antideriv_at_fourier_neg + - fourier_norm + - fourier_eval_zero + - fourierCoeff.const_smul + - AddCircle.toCircle_add + - fourierCoeffOn.const_smul + - span_fourierLp_closure_eq_top + - hasDerivAt_fourier_neg + - fourierBasis_repr + - AddCircle.scaled_exp_map_periodic + - fourierSubalgebra_separatesPoints + - fourierCoeff_liftIoc_eq + - fourierCoeff_eq_intervalIntegral + - hasSum_fourier_series_L2 + - fourier_add' + - tsum_sq_fourierCoeff + - hasDerivAt_fourier + - span_fourier_closure_eq_top + - fourier_zero' + - fourier_add_half_inv_index + - has_pointwise_sum_fourier_series_of_summable + - fourier_add + - fourierCoeffOn_eq_integral + - fourierCoeff_liftIco_eq + - AddCircle.injective_toCircle + - fourier_neg' + - fourier_neg + - hasSum_fourier_series_of_summable + - fourierSubalgebra_coe + - fourierCoeffOn_of_hasDerivAt + - orthonormal_fourier + - fourier_coe_apply + - fourierCoeff_toLp + - path: Mathlib/Geometry/Manifold/MFDeriv/Basic.lean + theorems: + - mfderiv_comp + - uniqueMDiffWithinAt_iff + - MDifferentiableAt.hasMFDerivAt + - HasMFDerivWithinAt.hasMFDerivAt + - mfderivWithin_of_mem_nhds + - HasMFDerivAt.comp + - HasMFDerivWithinAt.mfderivWithin + - hasMFDerivWithinAt_univ + - MDifferentiableWithinAt.mfderivWithin + - HasMFDerivWithinAt.comp + - hasMFDerivWithinAt_inter' + - MDifferentiableAt.mfderiv + - mdifferentiableWithinAt_iff + - tangentMapWithin_subset + - tangentMap_comp_at + - mdifferentiableOn_univ + - UniqueMDiffWithinAt.mono_nhds + - mfderivWithin_eq_mfderiv + - tangentMapWithin_univ + - MDifferentiable.mfderivWithin + - tangentMapWithin_comp_at + - HasMFDerivAt.comp_hasMFDerivWithinAt + - tangentMapWithin_eq_tangentMap + - mfderivWithin_univ + - Filter.EventuallyEq.mfderivWithin_eq + - tangentMap_comp + - MDifferentiableWithinAt.mdifferentiableAt + - hasMFDerivAt_unique + - mfderivWithin_zero_of_not_mdifferentiableWithinAt + - mfderiv_congr + - Filter.EventuallyEq.mdifferentiableWithinAt_iff + - Filter.EventuallyEq.mfderiv_eq + - mfderiv_zero_of_not_mdifferentiableAt + - uniqueMDiffWithinAt_univ + - HasMFDerivWithinAt.union + - UniqueMDiffWithinAt.prod + - HasMFDerivWithinAt.congr_of_eventuallyEq + - HasMFDerivAt.congr_of_eventuallyEq + - mfderivWithin_inter + - UniqueMDiffWithinAt.eq + - mdifferentiableWithinAt_inter' + - MDifferentiableWithinAt.comp + - mdifferentiableWithinAt_inter + - hasMFDerivWithinAt_inter + - MDifferentiableWithinAt.hasMFDerivWithinAt + - mfderivWithin_comp + - mdifferentiableWithinAt_univ + - mdifferentiableOn_of_locally_mdifferentiableOn + - tangentMapWithin_congr + - ContMDiffWithinAt.mdifferentiableWithinAt + - mdifferentiableAt_iff_of_mem_source + - writtenInExtChartAt_comp + - path: Mathlib/CategoryTheory/Sites/Subsheaf.lean + theorems: + - CategoryTheory.GrothendieckTopology.Subpresheaf.family_of_elements_compatible + - CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify_isSheaf + - CategoryTheory.GrothendieckTopology.imagePresheaf_id + - CategoryTheory.GrothendieckTopology.Subpresheaf.le_sheafify + - "CategoryTheory.GrothendieckTopology.Subpresheaf.homOfLe_\u03B9" + - CategoryTheory.GrothendieckTopology.Subpresheaf.eq_top_iff_isIso + - CategoryTheory.GrothendieckTopology.Subpresheaf.to_sheafify_lift_unique + - "CategoryTheory.GrothendieckTopology.Subpresheaf.lift_\u03B9" + - CategoryTheory.GrothendieckTopology.Subpresheaf.isSheaf_iff + - "CategoryTheory.GrothendieckTopology.toImageSheaf_\u03B9" + - CategoryTheory.GrothendieckTopology.Subpresheaf.eq_sheafify + - CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify_le + - CategoryTheory.GrothendieckTopology.Subpresheaf.to_sheafifyLift + - path: Mathlib/GroupTheory/Perm/Fin.lean + theorems: + - Fin.cycleRange_symm_zero + - Fin.cycleRange_of_gt + - Fin.cycleRange_succAbove + - Equiv.Perm.decomposeFin_symm_of_refl + - Fin.cycleRange_of_le + - isCycle_finRotate_of_le + - Fin.cycleRange_of_eq + - support_finRotate_of_le + - Equiv.Perm.decomposeFin_symm_apply_one + - Fin.coe_cycleRange_of_lt + - Fin.cycleRange_zero' + - Fin.isThreeCycle_cycleRange_two + - Fin.isCycle_cycleRange + - Fin.sign_cycleRange + - cycleType_finRotate_of_le + - cycleType_finRotate + - Fin.cycleType_cycleRange + - Fin.coe_cycleRange_of_le + - Fin.cycleRange_zero + - isCycle_finRotate + - Fin.cycleRange_of_lt + - finRotate_succ_eq_decomposeFin + - Fin.succAbove_cycleRange + - sign_finRotate + - support_finRotate + - Equiv.Perm.decomposeFin_symm_apply_succ + - Equiv.Perm.decomposeFin_symm_apply_zero + - Equiv.Perm.decomposeFin.symm_sign + - Fin.cycleRange_last + - path: Mathlib/SetTheory/Ordinal/Arithmetic.lean + theorems: + - Ordinal.lift_is_succ + - Ordinal.div_one + - Ordinal.pred_eq_iff_not_succ + - Ordinal.add_le_add_iff_right + - Ordinal.isLimit_iff_omega_dvd + - Ordinal.sup_eq_sup + - Ordinal.lsub_le_iff + - Ordinal.lt_mul_of_limit + - Ordinal.mod_eq_of_lt + - Ordinal.lt_blsub_iff + - Ordinal.limitRecOn_limit + - Ordinal.bsup_le_iff + - Ordinal.sup_eq_lsub_iff_lt_sup + - Ordinal.sup_eq_lsub_iff_succ + - Ordinal.bsup_eq_sup' + - Ordinal.mul_div_le + - Ordinal.nat_cast_inj + - Ordinal.sSup_ord + - Ordinal.lt_mul_div_add + - Ordinal.lsub_eq_zero_iff + - Ordinal.mul_mod + - Ordinal.lsub_empty + - Ordinal.enumOrd_le_of_subset + - Ordinal.le_sup_shrink_equiv + - Ordinal.div_pos + - Ordinal.add_sub_add_cancel + - Ordinal.mod_zero + - Ordinal.blsub_congr + - Ordinal.mod_lt + - Ordinal.mk_initialSeg + - Ordinal.bddAbove_range + - Ordinal.omega_le + - Ordinal.pred_le_self + - Ordinal.mod_self + - Ordinal.mex_monotone + - Ordinal.sup_le_sup + - Ordinal.le_bmex_of_forall + - Ordinal.lt_div + - Ordinal.IsNormal.bsup_eq + - Ordinal.one_add_omega + - Ordinal.IsNormal.le_set' + - Ordinal.bsup_comp + - Ordinal.bsup_le_of_brange_subset + - Ordinal.add_lt_add_iff_left' + - Ordinal.mex_le_of_ne + - Ordinal.eq_enumOrd + - Ordinal.blsub_le_of_brange_subset + - Ordinal.smul_eq_mul + - Ordinal.enumOrd_surjective + - Ordinal.limitRecOn_succ + - Ordinal.add_eq_zero_iff + - Ordinal.bsup_eq_bsup + - Ordinal.familyOfBFamily'_enum + - Ordinal.IsNormal.inj + - Ordinal.mod_mod_of_dvd + - Ordinal.sup_not_succ_of_ne_sup + - "Ordinal.lt_blsub\u2082" + - Ordinal.bmex_not_mem_brange + - Ordinal.blsub_zero + - Ordinal.lsub_le_of_range_subset + - Ordinal.add_right_cancel + - Ordinal.lsub_eq_lsub + - Ordinal.bsup_congr + - Ordinal.mul_le_of_limit_aux + - Ordinal.sub_self + - Ordinal.IsNormal.le_set + - Ordinal.dvd_add_iff + - Ordinal.le_mex_of_forall + - Ordinal.dvd_of_mod_eq_zero + - Ordinal.blsub_le_iff + - Ordinal.isLimit_add_iff + - Ordinal.one_lt_omega + - Ordinal.bsup_eq_blsub_or_succ_bsup_eq_blsub + - Ordinal.mul_isNormal + - Acc.rank_eq + - Ordinal.ne_mex + - Ordinal.mul_div_cancel + - Ordinal.add_sub_cancel_of_le + - Ordinal.le_mul_right + - Ordinal.pred_lt_iff_is_succ + - Ordinal.add_mul_limit_aux + - Ordinal.enumOrd_univ + - Ordinal.div_mul_cancel + - Cardinal.ord_aleph0 + - Ordinal.IsNormal.bsup + - Ordinal.sup_sum + - Ordinal.sub_zero + - Ordinal.pred_eq_iff_not_succ' + - Ordinal.sup_eq_zero_iff + - Ordinal.familyOfBFamily_enum + - Ordinal.one_add_nat_cast + - Ordinal.lt_mul_succ_div + - Ordinal.mex_lt_ord_succ_mk + - Ordinal.le_mul_left + - Ordinal.succ_pred_iff_is_succ + - Ordinal.nat_cast_mod + - Ordinal.enumOrd_mem_aux + - Ordinal.lt_omega + - Ordinal.brange_bfamilyOfFamily' + - Ordinal.bsup_eq_blsub_of_lt_succ_limit + - Ordinal.mul_sub + - Ordinal.sSup_eq_bsup + - Ordinal.exists_of_lt_bmex + - Ordinal.sup_le_iff + - Ordinal.blsub_eq_zero_iff + - Ordinal.zero_mod + - Ordinal.range_familyOfBFamily' + - Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub + - Ordinal.range_enumOrd + - Ordinal.mul_eq_zero' + - Ordinal.sub_isLimit + - Ordinal.isNormal_iff_lt_succ_and_blsub_eq + - Ordinal.nat_cast_le + - Ordinal.bmex_monotone + - Ordinal.enumOrd_def + - Ordinal.IsNormal.apply_omega + - Ordinal.lt_limit + - Ordinal.nat_cast_div + - Ordinal.limitRecOn_zero + - Ordinal.enumOrd_zero + - Ordinal.mul_pos + - Ordinal.mul_isLimit_left + - Ordinal.isNormal_iff_lt_succ_and_bsup_eq + - Ordinal.lt_lsub_iff + - Ordinal.exists_of_lt_mex + - Ordinal.div_self + - Ordinal.dvd_antisymm + - Ordinal.ne_bmex + - Ordinal.enumOrd_range + - Ordinal.lift_isLimit + - Ordinal.bsup_succ_le_blsub + - Ordinal.le_of_dvd + - Ordinal.lt_bsup + - Ordinal.mul_add_mod_self + - Ordinal.enumOrd_succ_le + - Ordinal.lift_pred + - Ordinal.lift_succ + - Ordinal.brange_const + - Ordinal.type_subrel_lt + - Ordinal.le_div + - Ordinal.IsNormal.limit_lt + - Ordinal.sub_eq_zero_iff_le + - Ordinal.out_no_max_of_succ_lt + - Ordinal.sub_sub + - Ordinal.add_left_cancel + - Ordinal.bfamilyOfFamily'_typein + - WellFounded.rank_eq + - Ordinal.has_succ_of_type_succ_lt + - Ordinal.mul_le_of_limit + - Ordinal.mul_ne_zero + - Ordinal.lift_nat_cast + - Ordinal.bsup_one + - Ordinal.sup_typein_limit + - Ordinal.blsub_comp + - Cardinal.add_one_of_aleph0_le + - Ordinal.le_sub_of_le + - Ordinal.lt_sup + - Ordinal.sup_mul_nat + - Ordinal.iSup_ord + - Ordinal.lt_add_of_limit + - Ordinal.add_mul_succ + - Ordinal.add_le_of_forall_add_lt + - Ordinal.mod_eq_zero_of_dvd + - Ordinal.bounded_singleton + - Ordinal.IsLimit.one_lt + - Ordinal.bsup_eq_blsub_iff_lt_bsup + - Ordinal.add_le_of_limit + - Ordinal.omega_isLimit + - Ordinal.nat_cast_sub + - Ordinal.nat_cast_mul + - Ordinal.blsub_type + - Ordinal.div_eq_zero_of_lt + - Ordinal.mul_add_div + - Ordinal.sup_succ_le_lsub + - Ordinal.IsNormal.sup + - Ordinal.div_le + - Ordinal.lsub_typein + - Ordinal.bsup_eq_zero_iff + - Ordinal.one_add_of_omega_le + - Ordinal.lt_pred + - Ordinal.pred_succ + - Ordinal.zero_sub + - Ordinal.blsub_eq_blsub + - Ordinal.mod_one + - Ordinal.div_le_of_le_mul + - Ordinal.bmex_lt_ord_succ_card + - path: Mathlib/ModelTheory/Satisfiability.lean + theorems: + - FirstOrder.Language.Theory.IsMaximal.mem_of_models + - FirstOrder.Language.Theory.SemanticallyEquivalent.symm + - FirstOrder.Language.Theory.SemanticallyEquivalent.all + - FirstOrder.Language.Theory.exists_large_model_of_infinite_model + - FirstOrder.Language.Theory.exists_model_card_eq + - FirstOrder.Language.Theory.IsComplete.models_not_iff + - FirstOrder.Language.BoundedFormula.induction_on_all_ex + - FirstOrder.Language.Theory.SemanticallyEquivalent.ex + - FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable + - FirstOrder.Language.BoundedFormula.sup_semanticallyEquivalent_not_inf_not + - FirstOrder.Language.Theory.models_of_models_theory + - FirstOrder.Language.Theory.models_iff_not_satisfiable + - FirstOrder.Language.exists_elementaryEmbedding_card_eq + - FirstOrder.Language.Theory.isSatisfiable_union_distinctConstantsTheory_of_card_le + - FirstOrder.Language.BoundedFormula.imp_semanticallyEquivalent_not_sup + - FirstOrder.Language.BoundedFormula.semanticallyEquivalent_toPrenex + - FirstOrder.Language.exists_elementaryEmbedding_card_eq_of_ge + - FirstOrder.Language.Theory.ModelsBoundedFormula.realize_sentence + - Cardinal.empty_infinite_Theory_isComplete + - FirstOrder.Language.Theory.models_iff_finset_models + - Cardinal.empty_theory_categorical + - FirstOrder.Language.completeTheory.mem_or_not_mem + - FirstOrder.Language.Theory.SemanticallyEquivalent.refl + - FirstOrder.Language.Theory.IsComplete.realize_sentence_iff + - Cardinal.Categorical.isComplete + - FirstOrder.Language.Theory.isSatisfiable_directed_union_iff + - FirstOrder.Language.BoundedFormula.all_semanticallyEquivalent_not_ex_not + - FirstOrder.Language.Theory.SemanticallyEquivalent.imp + - FirstOrder.Language.exists_elementarilyEquivalent_card_eq + - FirstOrder.Language.exists_elementaryEmbedding_card_eq_of_le + - FirstOrder.Language.BoundedFormula.inf_semanticallyEquivalent_not_sup_not + - FirstOrder.Language.Theory.isSatisfiable_onTheory_iff + - FirstOrder.Language.Theory.isSatisfiable_union_distinctConstantsTheory_of_infinite + - FirstOrder.Language.BoundedFormula.semanticallyEquivalent_all_liftAt + - FirstOrder.Language.BoundedFormula.ex_semanticallyEquivalent_not_all_not + - FirstOrder.Language.BoundedFormula.semanticallyEquivalent_not_not + - FirstOrder.Language.Theory.isSatisfiable_iUnion_iff_isSatisfiable_iUnion_finset + - FirstOrder.Language.Theory.SemanticallyEquivalent.not + - FirstOrder.Language.Theory.SemanticallyEquivalent.trans + - path: Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean + theorems: + - rank_span_set + - FiniteDimensional.finrank_self + - Basis.card_le_card_of_linearIndependent_aux + - Basis.mk_eq_rank'' + - Basis.card_le_card_of_linearIndependent + - linearIndependent_le_span_aux' + - basis_le_span' + - finrank_eq_nat_card_basis + - Basis.mk_eq_rank + - Ideal.rank_eq + - mk_eq_mk_of_basis + - FiniteDimensional.finrank_eq_card_finset_basis + - rank_span + - linearIndependent_le_span + - maximal_linearIndependent_eq_infinite_basis + - finrank_eq_rank + - rank_self + - linearIndependent_le_basis + - Basis.le_span + - rank_lt_aleph0 + - linearIndependent_le_span_finset + - linearIndependent_le_infinite_basis + - Basis.le_span'' + - rank_eq_card_basis + - LinearIndependent.finite_of_le_span_finite + - Module.mk_finrank_eq_card_basis + - linearIndependent_le_span' + - path: Mathlib/Analysis/Convex/Star.lean + theorems: + - StarConvex.affine_preimage + - StarConvex.union + - StarConvex.linear_image + - StarConvex.mem + - StarConvex.preimage_add_right + - starConvex_iff_segment_subset + - StarConvex.affinity + - StarConvex.add_left + - StarConvex.mem_smul + - starConvex_iUnion + - starConvex_iff_ordConnected + - starConvex_iff_forall_pos + - starConvex_iff_pointwise_add_subset + - starConvex_iff_div + - StarConvex.add_smul_mem + - starConvex_singleton + - StarConvex.add + - StarConvex.add_right + - starConvex_compl_Iic + - Set.OrdConnected.starConvex + - starConvex_zero_iff + - starConvex_iff_forall_ne_pos + - StarConvex.smul_mem + - StarConvex.linear_preimage + - StarConvex.affine_image + - StarConvex.add_smul_sub_mem + - starConvex_sUnion + - StarConvex.neg + - path: Mathlib/LinearAlgebra/Matrix/BilinearForm.lean + theorems: + - BilinForm.nondegenerate_toBilin'_iff_det_ne_zero + - BilinForm.toMatrix_basisFun + - Matrix.Nondegenerate.toBilin' + - BilinForm.nondegenerate_iff_det_ne_zero + - Matrix.toBilin_basisFun + - Matrix.toBilin'_comp + - mem_skewAdjointMatricesSubmodule' + - mem_selfAdjointMatricesSubmodule' + - mem_pairSelfAdjointMatricesSubmodule' + - toBilin'Aux_toMatrixAux + - Matrix.nondegenerate_toBilin_iff + - path: Mathlib/Analysis/NormedSpace/Units.lean + theorems: + - NormedRing.inverse_add + - Units.isOpen + - NormedRing.inverse_one_sub + - NormedRing.inverse_add_norm + - NormedRing.inverse_one_sub_nth_order' + - NormedRing.inverse_continuousAt + - NormedRing.inverse_one_sub_norm + - NormedRing.inverse_add_nth_order + - Ideal.eq_top_of_norm_lt_one + - NormedRing.inverse_one_sub_nth_order + - NormedRing.inverse_add_norm_diff_first_order + - Ideal.closure_ne_top + - NormedRing.inverse_add_norm_diff_second_order + - nonunits.subset_compl_ball + - NormedRing.inverse_add_norm_diff_nth_order + - path: Mathlib/Data/Nat/Lattice.lean + theorems: + - Nat.iSup_le_succ' + - Nat.iSup_lt_succ' + - Nat.not_mem_of_lt_sInf + - Nat.iSup_le_succ + - Nat.sInf_mem + - Nat.iInf_const_zero + - Nat.sInf_empty + - Nat.iSup_lt_succ + - Nat.sInf_le + - Nat.sInf_add' + - Nat.sInf_add + - Nat.sInf_eq_zero + - Nat.sInf_upward_closed_eq_succ_iff + - Nat.nonempty_of_pos_sInf + - Nat.iInf_of_empty + - path: Mathlib/Probability/Martingale/Upcrossing.lean + theorems: + - MeasureTheory.upperCrossingTime_mono + - MeasureTheory.Adapted.upcrossingStrat_adapted + - MeasureTheory.Adapted.measurable_upcrossingsBefore + - MeasureTheory.upperCrossingTime_le_lowerCrossingTime + - MeasureTheory.Submartingale.sum_mul_upcrossingStrat_le + - MeasureTheory.Submartingale.sum_sub_upcrossingStrat_mul + - MeasureTheory.upcrossingsBefore_zero' + - MeasureTheory.upcrossingsBefore_zero + - MeasureTheory.upperCrossingTime_bound_eq + - MeasureTheory.upcrossingsBefore_lt_of_exists_upcrossing + - MeasureTheory.Adapted.isStoppingTime_crossing + - MeasureTheory.upcrossingsBefore_le + - MeasureTheory.stoppedValue_upperCrossingTime + - MeasureTheory.lowerCrossingTime_lt_upperCrossingTime + - MeasureTheory.upperCrossingTime_succ + - MeasureTheory.sub_eq_zero_of_upcrossingsBefore_lt + - MeasureTheory.Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part + - MeasureTheory.lowerCrossingTime_mono + - MeasureTheory.upcrossingStrat_le_one + - MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ + - MeasureTheory.mul_integral_upcrossingsBefore_le_integral_pos_part_aux + - MeasureTheory.Adapted.integrable_upcrossingsBefore + - MeasureTheory.integral_mul_upcrossingsBefore_le_integral + - MeasureTheory.crossing_pos_eq + - MeasureTheory.upperCrossingTime_lt_lowerCrossingTime + - MeasureTheory.upcrossingsBefore_eq_sum + - MeasureTheory.mul_upcrossingsBefore_le + - MeasureTheory.upcrossingsBefore_mono + - MeasureTheory.upcrossings_lt_top_iff + - MeasureTheory.upperCrossingTime_eq_of_upcrossingsBefore_lt + - MeasureTheory.upcrossingsBefore_pos_eq + - MeasureTheory.exists_upperCrossingTime_eq + - MeasureTheory.crossing_eq_crossing_of_lowerCrossingTime_lt + - MeasureTheory.Submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part + - MeasureTheory.upperCrossingTime_succ_eq + - MeasureTheory.upperCrossingTime_eq_upperCrossingTime_of_lt + - MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt + - MeasureTheory.lowerCrossingTime_le + - MeasureTheory.upperCrossingTime_lt_bddAbove + - MeasureTheory.stoppedValue_lowerCrossingTime + - path: Mathlib/Analysis/Seminorm.lean + theorems: + - Seminorm.rescale_to_shell_zpow + - Seminorm.ball_mem_nhds + - Seminorm.closedBall_eq_biInter_ball + - Seminorm.continuousAt_zero_of_forall' + - Seminorm.ball_zero' + - Seminorm.ball_sup + - Seminorm.bddAbove_iff + - Seminorm.uniformSpace_eq_of_hasBasis + - Seminorm.mem_ball_self + - Seminorm.smul_closedBall_subset + - Seminorm.closedBall_finset_sup_eq_iInter + - Seminorm.continuous_of_le + - Seminorm.ball_zero_absorbs_ball_zero + - Seminorm.closedBall_smul + - Seminorm.ball_comp + - Seminorm.convex_ball + - Seminorm.neg_ball + - Seminorm.finset_sup_smul + - Seminorm.exists_apply_eq_finset_sup + - Seminorm.finset_sup_apply_lt + - Seminorm.ball_finset_sup' + - ball_normSeminorm + - Seminorm.smul_ball_preimage + - Seminorm.ball_eq_emptyset + - Seminorm.iSup_apply + - balanced_ball_zero + - Seminorm.closedBall_smul_closedBall + - Seminorm.bound_of_shell_sup + - Seminorm.absorbent_closedBall + - Seminorm.ball_add_ball_subset + - Seminorm.continuous_of_continuousAt_zero + - Seminorm.balanced_closedBall_zero + - Seminorm.sSup_empty + - Seminorm.continuousAt_zero' + - Seminorm.convexOn + - Seminorm.closedBall_comp + - Seminorm.smul_sup + - absorbent_ball + - Seminorm.mem_closedBall_self + - Seminorm.smul_le_smul + - Seminorm.ball_smul + - Seminorm.closedBall_zero_eq_preimage_closedBall + - Seminorm.smul_closedBall_zero + - Seminorm.closedBall_sup + - Seminorm.closedBall_add_closedBall_subset + - Seminorm.closedBall_iSup + - Seminorm.mem_ball_zero + - Seminorm.ball_finset_sup_eq_iInter + - Seminorm.ball_smul_ball + - Seminorm.sub_mem_ball + - Seminorm.bound_of_shell + - Seminorm.bddAbove_range_iff + - Seminorm.neg_mem_ball_zero + - Seminorm.isLUB_sSup + - Seminorm.ball_norm_mul_subset + - Seminorm.finset_sup_apply_le + - Seminorm.ball_zero_eq_preimage_ball + - Seminorm.balanced_ball_zero + - Seminorm.closedBall_finset_sup' + - Seminorm.preimage_metric_closedBall + - Seminorm.closedBall_finset_sup + - Seminorm.sSup_apply + - Seminorm.mem_closedBall_zero + - Seminorm.uniformContinuous_of_continuousAt_zero + - Seminorm.absorbent_ball + - Seminorm.convex_closedBall + - Seminorm.uniformity_eq_of_hasBasis + - Seminorm.smul_ball_zero + - Seminorm.ball_finset_sup + - Seminorm.coe_iSup_eq + - Seminorm.ball_smul_closedBall + - Seminorm.zero_or_exists_apply_eq_finset_sup + - Seminorm.bddAbove_of_absorbent + - Seminorm.finset_sup_apply + - Seminorm.closedBall_smul_ball + - Seminorm.closedBall_eq_emptyset + - Seminorm.finset_sup_le_sum + - path: Mathlib/Data/Set/Intervals/ProjIcc.lean + theorems: + - Set.projIcc_of_right_le + - Set.strictMonoOn_projIci + - Set.IccExtend_range + - Set.projIic_of_mem + - Set.strictMonoOn_projIic + - Set.range_IicExtend + - Set.projIcc_eq_left + - Set.projIcc_of_mem + - Set.range_IciExtend + - Set.projIci_of_mem + - Set.projIci_coe + - Set.projIic_coe + - Set.projIcc_eq_right + - Set.projIci_eq_self + - Set.projIcc_val + - Set.IccExtend_eq_self + - Set.projIic_eq_self + - Set.strictMonoOn_projIcc + - Set.projIcc_of_le_left + - path: Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean + theorems: + - EuclideanGeometry.Sphere.secondInter_eq_self_iff + - EuclideanGeometry.Sphere.secondInter_dist + - EuclideanGeometry.Sphere.wbtw_secondInter + - EuclideanGeometry.Sphere.secondInter_smul + - EuclideanGeometry.Sphere.sbtw_secondInter + - EuclideanGeometry.Sphere.eq_or_eq_secondInter_of_mem_mk'_span_singleton_iff_mem + - EuclideanGeometry.Sphere.secondInter_mem + - EuclideanGeometry.Sphere.secondInter_secondInter + - EuclideanGeometry.Sphere.secondInter_zero + - EuclideanGeometry.Sphere.secondInter_collinear + - EuclideanGeometry.Sphere.secondInter_neg + - path: Mathlib/Topology/Order/MonotoneConvergence.lean + theorems: + - isLUB_of_tendsto_atTop + - tendsto_atBot_ciInf + - tendsto_atTop_isLUB + - tendsto_atTop_isGLB + - tendsto_atBot_isGLB + - tendsto_iff_tendsto_subseq_of_monotone + - tendsto_atTop_ciSup + - tendsto_atBot_ciSup + - path: Mathlib/Analysis/NormedSpace/Star/Spectrum.lean + theorems: + - IsSelfAdjoint.val_re_map_spectrum + - StarAlgEquiv.nnnorm_map + - IsStarNormal.spectralRadius_eq_nnnorm + - IsSelfAdjoint.mem_spectrum_eq_re + - IsSelfAdjoint.spectralRadius_eq_nnnorm + - StarAlgHom.nnnorm_apply_le + - unitary.spectrum_subset_circle + - path: Mathlib/Algebra/Order/Floor.lean + theorems: + - Int.ceil_intCast + - Int.ceil_sub_self_eq + - Int.preimage_Ico + - Int.ceil_eq_zero_iff + - Nat.add_one_le_ceil_iff + - Int.map_ceil + - Int.preimage_Ioi + - Nat.ceil_zero + - Int.map_round + - Int.preimage_fract + - Int.fract_add_fract_le + - Int.fract_int_add + - round_two_inv + - round_eq_zero_iff + - Int.ceil_neg + - round_sub_int + - Int.floor_int_add + - Int.add_one_le_ceil_iff + - round_eq + - Int.floor_natCast + - Int.ceil_add_ceil_le + - Int.floor_nonpos + - Nat.floor_le_ceil + - Nat.ceil_add_le + - Int.ceil_eq_add_one_sub_fract + - Nat.floor_mono + - Nat.floor_pos + - Nat.floor_zero + - Nat.preimage_Iic + - round_add_one + - Int.floor_fract + - Int.ceil_add_nat + - Int.fract_add_nat + - Nat.floor_eq_iff' + - Nat.preimage_Ioc + - Nat.pos_of_floor_pos + - Int.ceil_sub_int + - Nat.ceil_pos + - round_nat_add + - Int.floor_intCast + - Int.floor_eq_iff + - Nat.floor_div_eq_div + - Int.fract_sub_nat + - round_zero + - Nat.floor_eq_iff + - Nat.floor_lt_ceil_of_lt_of_pos + - Int.fract_div_mul_self_add_zsmul_eq + - Int.image_fract + - Int.floor_sub_nat + - Nat.le_floor_iff' + - Int.ceil_one + - Int.floor_one + - Nat.preimage_Iio + - Nat.floor_coe + - Int.preimage_Ici + - Nat.preimage_Ico + - Int.fract_sub_int + - abs_sub_round_div_natCast_eq + - abs_sub_round_eq_min + - Int.sub_floor_div_mul_lt + - Int.fract_neg + - Nat.ceil_eq_iff + - Int.ceil_lt_add_one + - Int.fract_div_intCast_eq_div_intCast_mod + - Nat.preimage_Ici + - Nat.ceil_intCast + - Int.lt_floor_add_one + - Nat.ceil_le_floor_add_one + - Nat.floor_of_nonpos + - Nat.floor_add_nat + - Int.floor_add_int + - Nat.preimage_Icc + - natCast_floor_eq_intCast_floor + - Int.fract_one + - Int.fract_eq_zero_or_add_one_sub_ceil + - Nat.map_floor + - Nat.ceil_natCast + - Int.preimage_Ioo + - subsingleton_floorSemiring + - Int.fract_mul_nat + - Int.floor_le_sub_one_iff + - Int.ceil_pos + - round_int_add + - round_intCast + - Int.ceil_sub_one + - Nat.ceil_add_one + - Int.floor_pos + - round_one + - Int.ceil_add_le + - Int.fract_zero + - Int.ofNat_floor_eq_floor + - Int.fract_int + - Int.floor_eq_zero_iff + - Nat.floor_one + - Nat.lt_floor_add_one + - Int.ceil_le_floor_add_one + - Int.fract_add + - Nat.map_ceil + - Int.abs_sub_lt_one_of_floor_eq_floor + - Int.ceil_zero + - Int.floor_neg + - Nat.preimage_Ioi + - Int.le_floor_add_floor + - Int.preimage_Ioc + - Int.one_le_ceil_iff + - Int.preimage_Iio + - Int.ofNat_ceil_eq_ceil + - Int.fract_eq_self + - Int.fract_eq_fract + - Int.fract_add_int + - round_add_int + - Nat.floor_add_one + - Int.ceil_natCast + - Int.preimage_Icc + - Int.fract_div_natCast_eq_div_natCast_mod + - natCast_ceil_eq_intCast_ceil + - Nat.floor_div_nat + - Int.fract_eq_iff + - Int.ceil_sub_nat + - Int.ceil_add_int + - Nat.preimage_Ioo + - Int.floor_zero + - Int.map_fract + - round_neg_two_inv + - Int.le_floor_add + - round_sub_one + - abs_sub_round + - Int.floor_nat_add + - Int.fract_intCast + - Nat.ceil_one + - Int.fract_neg_eq_zero + - Nat.floor_eq_zero + - Int.floor_nonneg + - subsingleton_floorRing + - Int.ceil_eq_iff + - Int.ceil_add_one + - Int.preimage_Iic + - Int.floor_add_nat + - Nat.ceil_add_nat + - Nat.ceil_eq_zero + - Int.floor_le_neg_one_iff + - Nat.floor_congr + - round_le + - Int.fract_add_le + - Int.fract_nat_add + - Int.map_floor + - Int.floor_add_one + - Int.floor_sub_int + - Nat.floor_lt_one + - Nat.floor_sub_nat + - round_natCast + - path: Mathlib/MeasureTheory/Measure/VectorMeasure.lean + theorems: + - MeasureTheory.VectorMeasure.trim_eq_self + - MeasureTheory.VectorMeasure.zero_trim + - MeasureTheory.VectorMeasure.of_diff + - MeasureTheory.VectorMeasure.ext_iff + - MeasureTheory.VectorMeasure.mapRange_add + - MeasureTheory.Measure.toENNRealVectorMeasure_zero + - MeasureTheory.VectorMeasure.of_union + - MeasureTheory.VectorMeasure.restrict_eq_self + - MeasureTheory.VectorMeasure.restrict_le_restrict_of_subset_le + - MeasureTheory.VectorMeasure.map_id + - MeasureTheory.SignedMeasure.toMeasureOfLEZero_apply + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.add + - MeasureTheory.VectorMeasure.of_add_of_diff + - MeasureTheory.VectorMeasure.restrict_apply + - MeasureTheory.Measure.toSignedMeasure_eq_toSignedMeasure_iff + - MeasureTheory.VectorMeasure.map_smul + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.neg_right + - MeasureTheory.VectorMeasure.restrict_trim + - MeasureTheory.VectorMeasure.restrict_univ + - MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply + - MeasureTheory.Measure.zero_le_toSignedMeasure + - MeasureTheory.VectorMeasure.of_nonpos_disjoint_union_eq_zero + - MeasureTheory.VectorMeasure.le_restrict_empty + - MeasureTheory.VectorMeasure.of_nonneg_disjoint_union_eq_zero + - MeasureTheory.VectorMeasure.mapRange_id + - MeasureTheory.VectorMeasure.ennrealToMeasure_apply + - MeasureTheory.SignedMeasure.toMeasureOfLEZero_toSignedMeasure + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.smul + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.mk + - MeasureTheory.VectorMeasure.mapRange_zero + - MeasureTheory.VectorMeasure.ext_iff' + - MeasureTheory.Measure.toSignedMeasure_toMeasureOfZeroLE + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.map + - MeasureTheory.VectorMeasure.coe_injective + - MeasureTheory.VectorMeasure.subset_le_of_restrict_le_restrict + - MeasureTheory.VectorMeasure.restrict_le_restrict_union + - MeasureTheory.VectorMeasure.zero_le_restrict_not_measurable + - MeasureTheory.VectorMeasure.restrict_empty + - MeasureTheory.VectorMeasure.le_iff' + - MeasureTheory.Measure.toSignedMeasure_add + - MeasureTheory.VectorMeasure.MutuallySingular.add_left + - MeasureTheory.VectorMeasure.neg_le_neg + - MeasureTheory.VectorMeasure.map_zero + - MeasureTheory.VectorMeasure.MutuallySingular.neg_left + - MeasureTheory.VectorMeasure.restrict_zero + - MeasureTheory.VectorMeasure.MutuallySingular.smul_right + - MeasureTheory.VectorMeasure.nonneg_of_zero_le_restrict + - MeasureTheory.Measure.toSignedMeasure_smul + - MeasureTheory.VectorMeasure.exists_pos_measure_of_not_restrict_le_zero + - MeasureTheory.VectorMeasure.restrict_le_zero_of_not_measurable + - MeasureTheory.Measure.toENNRealVectorMeasure_add + - MeasureTheory.VectorMeasure.restrict_le_restrict_iUnion + - MeasureTheory.VectorMeasure.restrict_smul + - MeasureTheory.VectorMeasure.le_restrict_univ_iff_le + - MeasureTheory.Measure.toSignedMeasure_zero + - MeasureTheory.VectorMeasure.of_diff_of_diff_eq_zero + - MeasureTheory.VectorMeasure.MutuallySingular.mk + - MeasureTheory.VectorMeasure.map_add + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.sub + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.ennrealToMeasure + - MeasureTheory.VectorMeasure.map_apply + - MeasureTheory.Measure.toSignedMeasure_sub_apply + - MeasureTheory.VectorMeasure.AbsolutelyContinuous.neg_left + - MeasureTheory.VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure + - MeasureTheory.VectorMeasure.restrict_le_restrict_countable_iUnion + - MeasureTheory.SignedMeasure.toMeasureOfZeroLE_toSignedMeasure + - path: Mathlib/Topology/Category/TopCat/Limits/Products.lean + theorems: + - TopCat.prodIsoProd_inv_snd + - "TopCat.sigmaIsoSigma_hom_\u03B9" + - TopCat.sigmaIsoSigma_inv_apply + - TopCat.prodIsoProd_hom_snd + - "TopCat.piIsoPi_inv_\u03C0" + - TopCat.prodIsoProd_inv_fst + - TopCat.piIsoPi_hom_apply + - TopCat.range_prod_map + - TopCat.binaryCofan_isColimit_iff + - TopCat.embedding_prod_map + - TopCat.prod_topology + - TopCat.inducing_prod_map + - TopCat.induced_of_isLimit + - TopCat.prodIsoProd_hom_fst + - path: Mathlib/MeasureTheory/Measure/Hausdorff.lean + theorems: + - MeasureTheory.Measure.mkMetric_mono + - MeasureTheory.OuterMeasure.mkMetric_top + - MeasureTheory.OuterMeasure.mkMetric_mono + - MeasureTheory.OuterMeasure.isometry_comap_mkMetric + - MeasureTheory.Measure.hausdorffMeasure_zero_singleton + - Isometry.hausdorffMeasure_preimage + - MeasureTheory.Measure.mkMetric_top + - MeasureTheory.OuterMeasure.mkMetric_smul + - MeasureTheory.OuterMeasure.trim_mkMetric + - Isometry.hausdorffMeasure_image + - LipschitzOnWith.hausdorffMeasure_image_le + - MeasureTheory.hausdorffMeasure_segment + - MeasureTheory.hausdorffMeasure_prod_real + - IsometryEquiv.hausdorffMeasure_preimage + - AntilipschitzWith.hausdorffMeasure_preimage_le + - MeasureTheory.hausdorffMeasure_homothety_preimage + - MeasureTheory.OuterMeasure.isometry_map_mkMetric + - MeasureTheory.hausdorffMeasure_affineSegment + - MeasureTheory.OuterMeasure.mkMetric_nnreal_smul + - MeasureTheory.OuterMeasure.isometryEquiv_map_mkMetric + - MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingleton + - MeasureTheory.hausdorffMeasure_homothety_image + - MeasureTheory.OuterMeasure.coe_mkMetric + - MeasureTheory.OuterMeasure.mkMetric'.eq_iSup_nat + - MeasureTheory.hausdorffMeasure_pi_real + - MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre + - MeasureTheory.OuterMeasure.mkMetric'.trim_pre + - MeasureTheory.hausdorffMeasure_lineMap_image + - MeasureTheory.OuterMeasure.mkMetric'_isMetric + - MeasureTheory.Measure.mkMetric_mono_smul + - "MeasureTheory.Measure.hausdorffMeasure_smul\u2080" + - MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre_nat + - MeasureTheory.Measure.mkMetric_le_liminf_tsum + - MeasureTheory.OuterMeasure.mkMetric_mono_smul + - MeasureTheory.Measure.mkMetric_apply + - MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory + - MeasureTheory.Measure.one_le_hausdorffMeasure_zero_of_nonempty + - MeasureTheory.OuterMeasure.mkMetric'.le_pre + - Isometry.map_hausdorffMeasure + - MeasureTheory.Measure.hausdorffMeasure_zero_or_top + - MeasureTheory.OuterMeasure.mkMetric'.mono_pre_nat + - HolderOnWith.hausdorffMeasure_image_le + - MeasureTheory.OuterMeasure.IsMetric.finset_iUnion_of_pairwise_separated + - MeasureTheory.hausdorffMeasure_smul_right_image + - MeasureTheory.Measure.hausdorffMeasure_mono + - MeasureTheory.Measure.le_mkMetric + - IsometryEquiv.map_hausdorffMeasure + - MeasureTheory.Measure.noAtoms_hausdorff + - MeasureTheory.hausdorffMeasure_real + - MeasureTheory.OuterMeasure.IsMetric.le_caratheodory + - path: Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean + theorems: + - "Units.continuousOn_zpow\u2080_spectrum" + - cfc_comp + - cfc_apply_of_not_predicate + - cfc_zpow + - cfc_comp_neg + - cfc_inv + - "Units.continuousOn_inv\u2080_spectrum" + - cfc_comp_zpow + - cfc_congr + - cfc_neg_id + - cfc_smul_id + - cfc_apply + - cfc_add + - cfc_apply_of_not_and + - eqOn_of_cfc_eq_cfc + - eq_algebraMap_of_spectrum_subset_singleton + - cfc_eval_C + - cfc_map_polynomial + - cfc_map_div + - cfc_pow + - cfc_pow_id + - cfc_zero + - cfc_map_spectrum + - cfc_neg + - eq_zero_of_spectrum_subset_zero + - cfc_apply_of_not_continuousOn + - cfc_polynomial + - cfc_star + - cfc_comp_inv + - cfcUnits_zpow + - cfc_comp_star + - cfcUnits_pow + - cfc_inv_id + - cfc_const + - cfcHom_comp + - cfc_mul + - cfc_comp_polynomial + - cfcHom_eq_of_continuous_of_map_id + - eq_one_of_spectrum_subset_one + - cfc_smul + - cfc_eval_X + - cfc_star_id + - isUnit_cfc_iff + - cfc_sub + - path: Mathlib/Order/Bounds/OrderIso.lean + theorems: + - OrderIso.isLUB_preimage + - OrderIso.isLUB_preimage' + - OrderIso.isLUB_image + - OrderIso.isLUB_image' + - path: Mathlib/Data/DFinsupp/Interval.lean + theorems: + - DFinsupp.support_rangeIcc_subset + - DFinsupp.card_Ioc + - DFinsupp.card_pi + - Finset.mem_dfinsupp_iff_of_support_subset + - DFinsupp.card_uIcc + - DFinsupp.card_Ico + - Finset.mem_dfinsupp_iff + - DFinsupp.card_Iic + - DFinsupp.card_Ioo + - DFinsupp.card_Iio + - path: Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean + theorems: + - "LinearMap.toMatrix\u2082_compl\u2082" + - "LinearMap.toMatrix\u2082'_compl\u2082" + - "LinearMap.toMatrix\u2082'_compl\u2081\u2082" + - "LinearMap.toMatrix\u2082_compl\u2081\u2082" + - "LinearMap.toMatrix\u2082_apply" + - "LinearMap.toMatrix\u2082_mul_basis_toMatrix" + - "LinearMap.toMatrix\u2082_mul" + - "LinearMap.toMatrix\u2082'_mul" + - LinearMap.separatingLeft_iff_det_ne_zero + - "Matrix.Nondegenerate.toLinearMap\u2082'" + - "LinearMap.toMatrix\u2082Aux_eq" + - "LinearMap.mul_toMatrix\u2082_mul" + - "Matrix.toLinearMap\u2082'_apply'" + - "Matrix.toLinearMap\u2082'_comp" + - mem_skewAdjointMatricesSubmodule + - "Matrix.toLinearMap\u2082'Aux_stdBasis" + - "LinearMap.mul_toMatrix\u2082" + - "Matrix.separatingLeft_toLinearMap\u2082_iff" + - "isAdjointPair_toLinearMap\u2082'" + - "Matrix.toLinearMap\u2082_compl\u2081\u2082" + - LinearMap.mul_toMatrix' + - mem_selfAdjointMatricesSubmodule + - "Matrix.toLinearMap\u2082_basisFun" + - "LinearMap.toLinearMap\u2082'Aux_toMatrix\u2082Aux" + - "LinearMap.mul_toMatrix\u2082'_mul" + - mem_pairSelfAdjointMatricesSubmodule + - "LinearMap.toMatrix\u2082'_comp" + - "LinearMap.toMatrix\u2082_basisFun" + - "LinearMap.toMatrix\u2082_comp" + - "Matrix.toMatrix\u2082Aux_toLinearMap\u2082'Aux" + - "LinearMap.separatingLeft_toLinearMap\u2082'_iff_det_ne_zero" + - Matrix.isAdjointPair_equiv + - "isAdjointPair_toLinearMap\u2082" + - path: Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean + theorems: + - gelfandTransform_bijective + - gelfandStarTransform_naturality + - spectrum.gelfandTransform_eq + - WeakDual.CharacterSpace.exists_apply_eq_zero + - Ideal.toCharacterSpace_apply_eq_zero_of_mem + - WeakDual.CharacterSpace.mem_spectrum_iff_exists + - gelfandTransform_isometry + - path: Mathlib/Algebra/RingQuot.lean + theorems: + - RingQuot.ringQuotToIdealQuotient_apply + - RingQuot.eqvGen_rel_eq + - RingQuot.Rel.star + - RingQuot.lift_unique + - RingQuot.Rel.add_right + - RingQuot.zero_quot + - RingQuot.neg_quot + - RingQuot.liftAlgHom_unique + - RingQuot.Rel.sub_right + - RingQuot.mkAlgHom_coe + - RingQuot.Rel.smul + - RingQuot.mkAlgHom_surjective + - RingQuot.smul_quot + - RingQuot.one_quot + - RingQuot.eq_liftAlgHom_comp_mkAlgHom + - RingQuot.add_quot + - RingQuot.Rel.neg + - RingQuot.mul_quot + - RingQuot.eq_lift_comp_mkRingHom + - RingQuot.liftAlgHom_mkAlgHom_apply + - RingQuot.sub_quot + - RingQuot.lift_mkRingHom_apply + - RingQuot.ringQuot_ext + - RingQuot.mkRingHom_surjective + - RingQuot.mkRingHom_rel + - RingQuot.Rel.sub_left + - RingQuot.pow_quot + - RingQuot.ringQuot_ext' + - RingQuot.mkAlgHom_rel + - path: Mathlib/MeasureTheory/Integral/SetIntegral.lean + theorems: + - continuousOn_integral_bilinear_of_locally_integrable_of_compact_support + - integral_withDensity_eq_integral_smul + - "set_integral_withDensity_eq_set_integral_smul\u2080'" + - MeasureTheory.Integrable.measure_le_integral + - MeasureTheory.integral_indicator + - Filter.Tendsto.integral_sub_linear_isLittleO_ae + - MeasureTheory.set_integral_eq_zero_of_ae_eq_zero + - MeasureTheory.set_integral_eq_integral_of_ae_compl_eq_zero + - MeasureTheory.Integrable.simpleFunc_mul' + - MeasureTheory.set_integral_ge_of_const_le + - MeasureTheory.integral_finset_biUnion + - MeasureTheory.tendsto_set_integral_of_monotone + - MeasureTheory.set_integral_mono_on + - continuousOn_integral_of_compact_support + - "set_integral_withDensity_eq_set_integral_smul\u2080" + - MeasureTheory.set_integral_trim + - MeasureTheory.set_integral_gt_gt + - fst_integral + - MeasureTheory.set_integral_mono_on_ae + - MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict + - MeasureTheory.integrable_of_summable_norm_restrict + - measure_le_lintegral_thickenedIndicatorAux + - continuous_parametric_integral_of_continuous + - MeasureTheory.set_integral_pos_iff_support_of_nonneg_ae + - snd_integral + - MeasureTheory.norm_set_integral_le_of_norm_le_const_ae' + - MeasureTheory.norm_Lp_toLp_restrict_le + - MeasureTheory.Lp_toLp_restrict_add + - MeasureTheory.integral_union_eq_left_of_ae + - measure_le_lintegral_thickenedIndicator + - integral_re_add_im + - "MeasureTheory.integral_add_compl\u2080" + - MeasureTheory.set_integral_neg_eq_set_integral_nonpos + - MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero_aux + - MeasureTheory.ofReal_set_integral_one_of_measure_ne_top + - ContinuousLinearMap.continuous_integral_comp_L1 + - ContinuousLinearEquiv.integral_comp_comm + - MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero + - MeasureTheory.set_integral_map + - MeasureTheory.set_integral_nonpos_le + - MeasureTheory.norm_set_integral_le_of_norm_le_const_ae + - "MeasureTheory.integral_inter_add_diff\u2080" + - MeasurableEmbedding.set_integral_map + - MeasureTheory.integral_indicator_const + - MeasureTheory.set_integral_nonpos_ae + - MeasureTheory.set_integral_le_nonneg + - MeasureTheory.hasSum_integral_iUnion_ae + - MeasureTheory.norm_set_integral_le_of_norm_le_const_ae'' + - MeasureTheory.integral_univ + - MeasureTheory.integral_union_eq_left_of_ae_aux + - set_integral_withDensity_eq_set_integral_smul + - MeasureTheory.integral_union_ae + - MeasureTheory.set_integral_nonneg_ae + - MeasureTheory.integral_Icc_eq_integral_Ioo + - MeasureTheory.Lp_toLp_restrict_smul + - MeasureTheory.integral_norm_eq_pos_sub_neg + - MeasureTheory.continuous_set_integral + - MeasureTheory.integral_fintype_iUnion + - MeasureTheory.set_integral_indicator + - MeasureTheory.integral_piecewise + - MeasureTheory.set_integral_const + - ContinuousLinearMap.integral_comp_comm + - MeasureTheory.Integrable.simpleFunc_mul + - integral_coe_re_add_coe_im + - MeasureTheory.set_integral_congr_set_ae + - MeasureTheory.integral_diff + - MeasureTheory.integral_empty + - MeasureTheory.set_integral_indicatorConstLp + - MeasureTheory.integral_indicatorConstLp + - ContinuousLinearMap.integral_comp_comm' + - MeasureTheory.integrableOn_iUnion_of_summable_integral_norm + - integral_smul_const + - MeasureTheory.integral_le_measure + - path: Mathlib/Algebra/Algebra/Quasispectrum.lean + theorems: + - PreQuasiregular.add_inv_add_mul_eq_zero + - Unitization.zero_mem_spectrum_inr + - isQuasiregular_iff_isUnit + - quasispectrum.zero_mem + - quasispectrum_eq_spectrum_union + - PreQuasiregular.inv_add_add_mul_eq_zero + - isQuasiregular_iff_isUnit' + - IsQuasiregular.map + - IsQuasiregular.isUnit_one_add + - Unitization.mem_spectrum_inr_of_not_isUnit + - Unitization.isQuasiregular_inr_iff + - Unitization.quasispectrum_eq_spectrum_inr + - isQuasiregular_iff + - quasispectrum_eq_spectrum_union_zero + - Unitization.quasispectrum_eq_spectrum_inr' + - quasispectrum.mem_of_not_quasiregular + - path: Mathlib/RingTheory/Ideal/QuotientOperations.lean + theorems: + - Ideal.mk_ker + - Ideal.snd_comp_quotientInfEquivQuotientProd + - RingHom.kerLift_injective + - "Ideal.quotient_map_comp_mk\u2090" + - Ideal.algebraMap_quotient_injective + - Ideal.quotientMap_surjective + - DoubleQuot.quotQuotEquivQuotOfLE_comp_quotQuotMk + - Ideal.KerLift.map_smul + - "DoubleQuot.quotQuotEquivComm_symm\u2090" + - DoubleQuot.quotQuotEquivQuotOfLE_symm_comp_mk + - RingHom.lift_injective_of_ker_le_ideal + - Ideal.quotientInfToPiQuotient_surj + - Ideal.kerLiftAlg_mk + - Ideal.quotientEquivAlgOfEq_symm + - Ideal.bot_quotient_isMaximal_iff + - DoubleQuot.quotQuotEquivComm_symm + - Ideal.quotientInfToPiQuotient_inj + - Ideal.injective_lift_iff + - Ideal.ker_quotient_lift + - Ideal.mem_quotient_iff_mem_sup + - DoubleQuot.ker_quotLeftToQuotSup + - Ideal.map_mk_eq_bot_of_le + - Ideal.quotientMap_injective' + - Ideal.quotientMap_comp_mk + - Ideal.ker_Pi_Quotient_mk + - Ideal.comp_quotientMap_eq_of_comp_eq + - DoubleQuot.ker_quotQuotMk + - Ideal.comap_map_mk + - path: Mathlib/InformationTheory/Hamming.lean + theorems: + - hammingDist_pos + - hammingDist_zero_left + - hammingNorm_comp + - hammingDist_triangle + - hammingDist_eq_hammingNorm + - hammingDist_eq_zero + - hamming_zero_eq_dist + - hammingDist_triangle_left + - hammingNorm_smul + - hammingNorm_comp_le_hammingNorm + - hammingNorm_smul_le_hammingNorm + - hammingDist_triangle_right + - hammingDist_comm + - hammingDist_self + - swap_hammingDist + - eq_of_hammingDist_eq_zero + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean + theorems: + - WeierstrassCurve.Projective.nonsingular_iff + - WeierstrassCurve.Projective.equiv_some_of_Z_ne_zero + - WeierstrassCurve.Projective.eval_polynomialX + - WeierstrassCurve.Projective.equiv_of_Z_eq_zero + - WeierstrassCurve.Projective.equation_iff + - WeierstrassCurve.Projective.polynomial_relation + - WeierstrassCurve.Projective.equation_zero + - WeierstrassCurve.Projective.smul_fin3 + - WeierstrassCurve.Projective.polynomialY_eq + - WeierstrassCurve.Projective.eval_polynomialZ + - WeierstrassCurve.Projective.nonsingular_zero + - WeierstrassCurve.Projective.eval_polynomial + - WeierstrassCurve.Projective.eval_polynomialY + - WeierstrassCurve.Projective.nonsingular_zero' + - WeierstrassCurve.Projective.equation_some + - WeierstrassCurve.Projective.polynomialZ_eq + - WeierstrassCurve.Projective.smul_fin3_ext + - WeierstrassCurve.Projective.nonsingular_of_equiv + - WeierstrassCurve.Projective.fin3_def + - WeierstrassCurve.Projective.nonsingular_smul_iff + - WeierstrassCurve.Projective.equation_smul_iff + - WeierstrassCurve.Projective.polynomialX_eq + - WeierstrassCurve.Projective.nonsingular_some + - path: Mathlib/NumberTheory/EulerProduct/Basic.lean + theorems: + - EulerProduct.eulerProduct + - EulerProduct.prod_primesBelow_geometric_eq_tsum_smoothNumbers + - EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_geometric + - EulerProduct.summable_and_hasSum_smoothNumbers_prod_primesBelow_tsum + - EulerProduct.eulerProduct_completely_multiplicative + - Summable.norm_lt_one + - EulerProduct.norm_tsum_smoothNumbers_sub_tsum_lt + - path: Mathlib/GroupTheory/SpecificGroups/KleinFour.lean + theorems: + - IsKleinFour.mul_self + - IsKleinFour.eq_finset_univ + - IsKleinFour.not_isCyclic + - IsKleinFour.eq_mul_of_ne_all + - IsKleinFour.nonempty_mulEquiv + - IsKleinFour.inv_eq_self + - path: Mathlib/Algebra/GeomSum.lean + theorems: + - one_geom_sum + - geom_sum_alternating_of_lt_neg_one + - "geom_sum\u2082_with_one" + - "op_geom_sum\u2082" + - geom_sum_eq + - geom_sum_ne_zero + - geom_sum_pos_iff + - geom_sum_Ico_le_of_lt_one + - Nat.geom_sum_le + - geom_sum_inv + - nat_sub_dvd_pow_sub_pow + - geom_sum_succ + - Nat.pred_mul_geom_sum_le + - "Commute.geom_sum\u2082_Ico_mul" + - geom_sum_one + - geom_sum_mul_neg + - mul_geom_sum + - "RingHom.map_geom_sum\u2082" + - "Commute.geom_sum\u2082" + - Nat.geomSum_eq + - geom_sum_Ico' + - "Commute.geom_sum\u2082_mul_add" + - Nat.geomSum_lt + - sub_one_dvd_pow_sub_one + - "Commute.mul_neg_geom_sum\u2082" + - geom_sum_pos + - op_geom_sum + - geom_sum_two + - Odd.geom_sum_pos + - Odd.add_dvd_pow_add_pow + - geom_sum_eq_zero_iff_neg_one + - Nat.geom_sum_Ico_le + - geom_sum_mul + - geom_sum_Ico + - "Commute.mul_geom_sum\u2082" + - geom_sum_Ico_mul_neg + - zero_geom_sum + - "Commute.mul_geom_sum\u2082_Ico" + - geom_sum_pos_and_lt_one + - RingHom.map_geom_sum + - "Commute.geom_sum\u2082_comm" + - geom_sum_mul_add + - neg_one_geom_sum + - one_sub_dvd_one_sub_pow + - geom_sum_Ico_mul + - "Commute.geom_sum\u2082_mul" + - geom_sum_alternating_of_le_neg_one + - mul_neg_geom_sum + - geom_sum_neg_iff + - "Commute.geom_sum\u2082_Ico" + - "geom_sum\u2082_self" + - "Commute.geom_sum\u2082_succ_eq" + - path: Mathlib/Logic/Equiv/Set.lean + theorems: + - Equiv.Set.sumCompl_symm_apply_of_not_mem + - Equiv.prod_assoc_preimage + - Equiv.Set.sumCompl_symm_apply_compl + - Equiv.coe_ofInjective_symm + - Equiv.prod_assoc_symm_image + - Equiv.prod_assoc_symm_preimage + - Equiv.Set.sumDiffSubset_symm_apply_of_not_mem + - Equiv.Set.sumCompl_symm_apply_of_mem + - Equiv.Set.sumCompl_symm_apply + - Equiv.prod_assoc_image + - Equiv.preimage_piEquivPiSubtypeProd_symm_pi + - Equiv.subset_symm_image + - dite_comp_equiv_update + - Equiv.ofLeftInverse'_eq_ofInjective + - Equiv.ofLeftInverse_eq_ofInjective + - Equiv.Set.image_symm_preimage + - Equiv.ofInjective_symm_apply + - Equiv.symm_image_subset + - Equiv.Set.sumDiffSubset_symm_apply_of_mem + - path: Mathlib/MeasureTheory/Measure/Typeclasses.lean + theorems: + - MeasureTheory.Measure.exists_isOpen_measure_lt_top + - MeasureTheory.not_isFiniteMeasure_iff + - "MeasureTheory.prob_compl_eq_one_sub\u2080" + - MeasureTheory.Measure.add_right_inj + - MeasureTheory.ae_of_forall_measure_lt_top_ae_restrict' + - Set.Countable.ae_not_mem + - MeasureTheory.ae_iff_measure_eq + - MeasureTheory.IsProbabilityMeasure.ne_zero + - MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff' + - MeasureTheory.Measure.isFiniteMeasure_map + - "MeasureTheory.prob_compl_eq_zero_iff\u2080" + - MeasureTheory.sigmaFinite_bot_iff + - Set.Infinite.meas_eq_top + - MeasureTheory.sum_restrict_disjointed_spanningSets + - MeasureTheory.exists_pos_measure_of_cover + - MeasureTheory.Measure.finiteAtBot + - MeasureTheory.ae_eq_univ_iff_measure_eq + - MeasureTheory.Measure.restrict_toMeasurable_of_cover + - MeasureTheory.exists_ne_forall_mem_nhds_pos_measure_preimage + - MeasureTheory.Measure.measure_toMeasurable_inter_of_cover + - "MeasureTheory.prob_compl_eq_one_iff\u2080" + - MeasureTheory.ite_ae_eq_of_measure_compl_zero + - Set.Countable.measure_zero + - MeasureTheory.measureUnivNNReal_pos + - MeasureTheory.iUnion_spanningSets + - MeasureTheory.Measure.sigmaFinite_of_countable + - isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace + - MeasureTheory.summable_measure_toReal + - MeasureTheory.Measure.smul_finite + - MeasureTheory.measureUnivNNReal_eq_zero + - MeasureTheory.ite_ae_eq_of_measure_zero + - MeasureTheory.Measure.add_left_inj + - MeasureTheory.Measure.iSup_restrict_spanningSets + - MeasureTheory.spanningSetsIndex_eq_iff + - MeasureTheory.Measure.restrict_singleton' + - MeasureTheory.Measure.measure_toMeasurable_inter_of_sum + - MeasureTheory.Measure.forall_measure_inter_spanningSets_eq_zero + - "MeasureTheory.Measure.countable_meas_level_set_pos\u2080" + - MeasureTheory.measure_compl_le_add_of_le_add + - IsCompact.measure_lt_top_of_nhdsWithin + - MeasureTheory.Measure.exists_measure_inter_spanningSets_pos + - MeasureTheory.Measure.isTopologicalBasis_isOpen_lt_top + - IsCompact.measure_zero_of_nhdsWithin + - MeasureTheory.SigmaFinite.of_map + - "MeasureTheory.Measure.countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top\u2080" + - MeasureTheory.isProbabilityMeasure_map + - MeasureTheory.isProbabilityMeasure_comap + - MeasureTheory.Measure.exists_subset_measure_lt_top + - MeasureTheory.exists_pos_preimage_ball + - MeasureTheory.ext_on_measurableSpace_of_generate_finite + - MeasureTheory.sfinite_sum_of_countable + - MeasureTheory.Measure.restrict_toMeasurable_of_sFinite + - MeasurableEquiv.sigmaFinite_map + - MeasureTheory.Measure.FiniteAtFilter.inf_ae_iff + - IsCompact.exists_open_superset_measure_lt_top' + - path: .lake/packages/std/Std/Data/List/Perm.lean + theorems: + - List.subperm_append_diff_self_of_count_le + - List.Subperm.filter + - List.Perm.erase + - List.Perm.foldl_eq' + - List.perm_ext_iff_of_nodup + - List.Perm.countP_eq + - List.subperm_ext_iff + - List.subperm_of_subset + - List.filter_append_perm + - List.subperm_cons + - List.Subperm.diff_right + - List.Perm.filterMap + - List.isPerm_iff + - List.perm_insert_swap + - List.perm_append_comm + - List.Subperm.cons_left + - List.Perm.sizeOf_eq_sizeOf + - List.Perm.union_left + - List.Perm.diff_right + - List.perm_replicate + - List.subperm_cons_diff + - List.perm_append_right_iff + - List.Subperm.exists_of_length_lt + - List.exists_perm_sublist + - List.Perm.countP_congr + - List.Perm.diff_left + - List.perm_inv_core + - List.perm_merge + - List.Perm.length_eq + - List.Perm.insert + - List.Perm.rec_heq + - List.Perm.union_right + - List.Perm.pairwise_iff + - List.singleton_perm_singleton + - List.perm_iff_count + - List.Perm.append_right + - List.erase_cons_subperm_cons_erase + - List.perm_append_singleton + - List.Perm.inter_left + - List.concat_perm + - List.cons_perm_iff_perm_erase + - List.Perm.filter + - List.singleton_subperm_iff + - List.Perm.mem_iff + - List.perm_insertNth + - List.cons_subperm_of_not_mem_of_mem + - List.Nodup.perm_iff_eq_of_sublist + - List.isSubperm_iff + - List.Perm.join + - List.Perm.eraseP + - path: Mathlib/Dynamics/Ergodic/Ergodic.lean + theorems: + - Ergodic.ae_empty_or_univ_of_ae_le_preimage' + - MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate + - "QuasiErgodic.ae_mem_or_ae_nmem\u2080" + - QuasiErgodic.ae_empty_or_univ' + - PreErgodic.measure_self_or_compl_eq_zero + - MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff + - PreErgodic.prob_eq_zero_or_one + - Ergodic.ae_empty_or_univ_of_image_ae_le' + - MeasureTheory.MeasurePreserving.ergodic_conjugate_iff + - Ergodic.ae_empty_or_univ_of_preimage_ae_le' + - path: Mathlib/Data/Num/Lemmas.lean + theorems: + - Num.ofNat'_one + - Num.dvd_iff_mod_eq_zero + - Num.castNum_testBit + - Num.castNum_and + - ZNum.le_to_int + - ZNum.cast_add + - PosNum.cmp_to_nat_lemma + - Num.cmp_to_nat + - PosNum.succ'_pred' + - ZNum.of_int_cast + - Num.cmp_eq + - Num.ofNat'_eq + - ZNum.cast_bit1 + - Num.ppred_to_nat + - ZNum.cast_le + - Num.mod_zero + - Num.add_zero + - ZNum.cast_mul + - Num.div_zero + - PosNum.divMod_to_nat + - PosNum.cast_sub' + - PosNum.pred_to_nat + - Num.cmp_swap + - ZNum.cast_bitm1 + - ZNum.zneg_zneg + - Num.castNum_shiftRight + - Num.cast_add + - PosNum.size_to_nat + - Num.cast_succ' + - Num.toZNum_inj + - Num.cast_lt + - Num.lt_to_nat + - ZNum.div_zero + - PosNum.cast_lt + - ZNum.neg_of_int + - ZNum.dvd_to_int + - ZNum.bit1_of_bit1 + - Num.mod_to_nat + - Num.zneg_toZNum + - PosNum.size_eq_natSize + - ZNum.mul_comm + - ZNum.add_one + - ZNum.gcd_to_nat + - ZNum.add_le_add_left + - PosNum.cmp_swap + - PosNum.of_to_nat' + - ZNum.zneg_bitm1 + - Num.div_to_nat + - ZNum.cast_inj + - Num.zero_add + - ZNum.cast_sub + - ZNum.zneg_pred + - Num.le_to_nat + - Num.natSize_to_nat + - PosNum.cmp_to_nat + - ZNum.of_to_int' + - ZNum.cmp_to_int + - Num.sub_to_nat + - Num.succ_ofInt' + - PosNum.add_succ + - Num.dvd_to_nat + - PosNum.cast_inj + - Num.cast_bit1 + - Num.gcd_to_nat + - Num.ofZNum_toNat + - ZNum.ofInt'_neg + - Num.castNum_eq_bitwise + - PosNum.bit0_of_bit0 + - PosNum.sub'_one + - Num.add_one + - ZNum.ofInt'_eq + - PosNum.to_nat_eq_succ_pred + - Num.of_nat_cast + - ZNum.zneg_bit1 + - Num.pred_succ + - ZNum.mod_to_int + - Num.castNum_or + - Num.castNum_xor + - ZNum.zero_add + - Num.ofNat'_succ + - PosNum.one_le_cast + - Num.mem_ofZNum' + - Num.add_ofNat' + - Num.castNum_ldiff + - PosNum.pred'_to_nat + - ZNum.of_to_int + - PosNum.cast_to_num + - PosNum.mul_to_nat + - Num.cast_ofZNum + - PosNum.cast_succ + - Num.ofNat'_zero + - Num.pred_to_nat + - Num.cast_bit0 + - Num.to_of_nat + - PosNum.cast_le + - PosNum.one_sub' + - PosNum.add_one + - Num.zneg_toZNumNeg + - PosNum.succ_to_nat + - ZNum.cast_bit0 + - Num.cast_le + - Num.castNum_shiftLeft + - ZNum.zneg_succ + - PosNum.lt_to_nat + - Num.cast_inj + - ZNum.to_of_int + - PosNum.cast_mul + - PosNum.pred'_succ' + - Num.le_iff_cmp + - Num.ofZNum'_toNat + - Num.ofInt'_toZNum + - Num.gcd_to_nat_aux + - PosNum.cast_to_int + - ZNum.div_to_int + - Num.add_succ + - ZNum.cast_to_int + - PosNum.cast_add + - ZNum.dvd_iff_mod_eq_zero + - PosNum.cmp_eq + - ZNum.lt_to_int + - ZNum.of_nat_cast + - ZNum.add_zero + - ZNum.of_nat_toZNumNeg + - ZNum.cast_succ + - PosNum.bit_to_nat + - PosNum.add_to_nat + - PosNum.le_iff_cmp + - PosNum.to_int_eq_succ_pred + - PosNum.one_add + - Num.cast_to_int + - PosNum.bit1_of_bit1 + - PosNum.natSize_to_nat + - ZNum.abs_to_nat + - PosNum.divMod_to_nat_aux + - PosNum.le_to_nat + - ZNum.cast_lt + - Num.bit_to_nat + - ZNum.mul_to_int + - PosNum.natSize_pos + - path: Mathlib/Data/Rat/Defs.lean + theorems: + - Rat.divInt_eq_div + - Rat.eq_iff_mul_eq_mul + - Rat.num_div_den + - Rat.add_zero + - Rat.add_left_neg + - Rat.add_divInt + - Rat.divInt_zero_one + - Rat.coe_int_num_of_den_eq_one + - Rat.mk_denom_ne_zero_of_ne_zero + - Rat.divInt_one_one + - Rat.zero_of_num_zero + - Rat.divInt_mul_divInt_cancel + - Rat.zero_mk + - Rat.mul_num_den + - Rat.divInt_div_divInt_cancel_left + - Rat.divInt_eq_zero + - Rat.eq_num_of_isInt + - Rat.divInt_neg_one_one + - Rat.add_comm + - Rat.mul_assoc + - Rat.divInt_div_divInt_cancel_right + - Rat.divInt_one + - Rat.add_assoc + - Rat.zero_iff_num_zero + - Rat.mul_inv_cancel + - Rat.zero_ne_one + - Rat.add_mul + - Rat.mk_num_ne_zero_of_ne_zero + - Rat.coe_int_div_eq_divInt + - Rat.mkRat_eq_div + - Rat.div_num_den + - Rat.lift_binop_eq + - Rat.mkRat_one + - Rat.num_eq_zero + - path: .lake/packages/std/Std/Data/Rat/Lemmas.lean + theorems: + - Rat.mul_one + - Rat.normalize_mul_left + - Rat.mkRat_eq_zero + - Rat.mul_zero + - Rat.mul_def + - Rat.normalize.reduced' + - Rat.normalize_mul_right + - Rat.normalize_eq_mkRat + - Rat.zero_divInt + - Rat.neg_divInt_neg + - Rat.normalize_zero + - Rat.normalize_eq_iff + - Rat.neg_normalize + - Rat.normalize_add_normalize + - Rat.sub_def' + - Rat.divInt_self + - Rat.normalize_eq + - Rat.zero_mul + - Rat.divInt_mul_left + - Rat.divInt_sub_divInt + - Rat.maybeNormalize_eq_normalize + - Rat.intCast_add + - Rat.zero_mkRat + - Rat.mul_comm + - Rat.divInt_eq_iff + - Rat.add_def + - Rat.normalize_num_den + - Rat.inv_zero + - Rat.divInt_ofNat + - Rat.mk_eq_divInt + - Rat.intCast_sub + - Rat.inv_divInt + - Rat.divInt_mul_divInt + - Rat.mkRat_self + - Rat.intCast_mul + - Rat.divInt_num_den + - Rat.mkRat_zero + - Rat.one_mul + - Rat.divInt_add_divInt + - Rat.divInt_neg' + - Rat.inv_def + - Rat.mk_eq_mkRat + - Rat.normalize_eq_zero + - Rat.ofScientific_false_def + - Rat.normalize_mul_normalize + - Rat.ofScientific_def + - Rat.mk_eq_normalize + - Rat.normalize_num_den' + - Rat.add_def' + - Rat.mkRat_mul_left + - Rat.divInt_mul_right + - Rat.mkRat_mul_right + - Rat.mkRat_add_mkRat + - Rat.mkRat_eq_iff + - Rat.intCast_inj + - Rat.sub_eq_add_neg + - Rat.ofScientific_true_def + - Rat.sub_def + - path: Mathlib/Analysis/Calculus/ContDiff/Basic.lean + theorems: + - LinearIsometryEquiv.norm_iteratedFDeriv_comp_right + - ContDiff.prod_map + - ContDiffWithinAt.iteratedFderivWithin_right + - ContDiffWithinAt.div_const + - ContDiff.div + - iteratedFDerivWithin_zero_fun + - ContDiffWithinAt.hasFDerivWithinAt_nhds + - contDiff_pi + - ContDiff.contDiff_fderiv_apply + - ContDiffOn.comp_same_univ + - PartialHomeomorph.contDiffAt_symm + - iteratedFDerivWithin_neg_apply + - contDiffAt_inv + - contDiffWithinAt_pi + - iteratedFDerivWithin_const_of_ne + - ContDiffAt.smul + - iteratedFDeriv_add_apply + - LinearIsometry.norm_iteratedFDerivWithin_comp_left + - ContDiffAt.const_smul + - ContDiffAt.neg + - iteratedFDerivWithin_clm_apply_const_apply + - hasFTaylorSeriesUpToOn_pi + - ContinuousLinearEquiv.comp_contDiff_iff + - contDiffOn_succ_iff_deriv_of_isOpen + - ContDiffOn.sub + - ContDiffAt.fderiv + - LinearIsometryEquiv.norm_iteratedFDeriv_comp_left + - iteratedFDeriv_succ_const + - contDiffAt_of_subsingleton + - ContDiff.inv + - ContDiffWithinAt.prod + - ContinuousLinearEquiv.contDiffAt_comp_iff + - contDiffOn_of_subsingleton + - LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right + - HasFTaylorSeriesUpToOn.add + - contDiff_of_subsingleton + - LinearIsometry.norm_iteratedFDeriv_comp_left + - ContDiffAt.prod_map' + - ContDiff.sum + - ContDiffAt.add + - hasFTaylorSeriesUpToOn_pi' + - ContDiffWithinAt.continuousLinearMap_comp + - contDiffOn_top_iff_deriv_of_isOpen + - ContDiffWithinAt.restrict_scalars + - ContDiffWithinAt.sub + - ContinuousLinearMap.iteratedFDeriv_comp_right + - iteratedFDeriv_sum + - ContDiffAt.sum + - ContDiffWithinAt.comp + - contDiff_succ_iff_deriv + - contDiff_zero_fun + - ContinuousLinearEquiv.comp_contDiffAt_iff + - ContDiffWithinAt.fderivWithin'' + - contDiffWithinAt_of_subsingleton + - ContinuousLinearMap.iteratedFDeriv_comp_left + - ContinuousLinearEquiv.contDiff_comp_iff + - contDiffAt_map_inverse + - contDiffWithinAt_prod + - ContDiffWithinAt.comp_continuousLinearMap + - ContDiff.div_const + - LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left + - ContDiffOn.continuousOn_fderivWithin_apply + - iteratedFDeriv_neg_apply + - ContDiffOn.derivWithin + - HasFTaylorSeriesUpToOn.restrictScalars + - ContinuousLinearEquiv.iteratedFDerivWithin_comp_left + - ContDiffWithinAt.div + - contDiff_const + - iteratedFDeriv_clm_apply_const_apply + - ContDiff.sub + - HasFTaylorSeriesUpToOn.prod + - IsBoundedBilinearMap.contDiff + - ContDiffOn.comp + - ContDiffAt.sub + - contDiff_top_iff_deriv + - HasFTaylorSeriesUpToOn.compContinuousLinearMap + - ContDiffWithinAt.fderivWithin_right + - iteratedFDeriv_const_smul_apply + - ContDiffAt.iteratedFDeriv_right + - iteratedFDeriv_zero_fun + - ContinuousLinearEquiv.contDiffWithinAt_comp_iff + - iteratedFDerivWithin_succ_const + - ContinuousLinearEquiv.comp_contDiffOn_iff + - ContDiffAt.prod_map + - contDiff_update + - IsBoundedLinearMap.contDiff + - ContDiffWithinAt.sum + - contDiffOn_succ_iff_derivWithin + - iteratedFDeriv_const_of_ne + - contDiffAt_ring_inverse + - iteratedFDerivWithin_sum_apply + - ContDiffWithinAt.fderivWithin + - path: Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean + theorems: + - midpoint_add_sub + - AffineEquiv.pointReflection_midpoint_left + - Equiv.pointReflection_midpoint_right + - AffineEquiv.pointReflection_midpoint_right + - midpoint_eq_midpoint_iff_vsub_eq_vsub + - midpoint_add_self + - midpoint_comm + - midpoint_neg_self + - midpoint_eq_smul_add + - midpoint_vsub_right + - right_eq_midpoint_iff + - midpoint_eq_left_iff + - Equiv.pointReflection_midpoint_left + - midpoint_eq_right_iff + - right_vsub_midpoint + - left_eq_midpoint_iff + - midpoint_self_neg + - vsub_midpoint + - midpoint_vsub + - midpoint_zero_add + - path: Mathlib/RingTheory/PowerSeries/Trunc.lean + theorems: + - PowerSeries.trunc_mul_trunc + - PowerSeries.coeff_trunc + - PowerSeries.trunc_C + - PowerSeries.degree_trunc_lt + - PowerSeries.natDegree_trunc_lt + - PowerSeries.trunc_trunc_pow + - PowerSeries.trunc_trunc_mul_trunc + - PowerSeries.trunc_trunc_of_le + - PowerSeries.trunc_trunc_mul + - "PowerSeries.eval\u2082_trunc_eq_sum_range" + - PowerSeries.trunc_add + - PowerSeries.trunc_X_of + - PowerSeries.trunc_succ + - PowerSeries.trunc_zero + - PowerSeries.trunc_one + - PowerSeries.trunc_X + - PowerSeries.coeff_coe_trunc_of_lt + - "PowerSeries.coeff_mul_eq_coeff_trunc_mul_trunc\u2082" + - path: Mathlib/NumberTheory/PellMatiyasevic.lean + theorems: + - Pell.d_pos + - Pell.eq_pell + - Pell.strictMono_x + - Pell.isPell_mul + - Pell.isPell_norm + - Pell.yn_add + - Pell.yn_one + - Pell.dz_val + - Pell.xn_one + - Pell.xn_modEq_x4n_add + - Pell.x_sub_y_dvd_pow + - Pell.dvd_of_ysq_dvd + - Pell.strictMono_y + - Pell.eq_pell_lem + - Pell.yn_modEq_two + - Pell.pell_eq + - Pell.eq_of_xn_modEq_lem3 + - Pell.modEq_of_xn_modEq + - Pell.xy_modEq_yn + - Pell.xn_modEq_x2n_sub_lem + - Pell.pellZd_succ_succ + - Pell.eq_pow_of_pell + - Pell.eq_pellZd + - Pell.yn_modEq_a_sub_one + - Pell.pellZd_sub + - Pell.eq_of_xn_modEq_lem2 + - Pell.yz_sub + - Pell.pellZd_succ + - Pell.xn_modEq_x2n_add + - Pell.n_lt_a_pow + - Pell.matiyasevic + - Pell.asq_pos + - Pell.xy_succ_succ + - Pell.x_sub_y_dvd_pow_lem + - Pell.xn_add + - Pell.eq_pow_of_pell_lem + - Pell.eq_of_xn_modEq_le + - Pell.xy_modEq_of_modEq + - Pell.pellZd_add + - Pell.y_dvd_iff + - Pell.y_mul_dvd + - Pell.isPell_nat + - Pell.yz_succ_succ + - Pell.isPell_star + - Pell.isPell_pellZd + - Pell.eq_of_xn_modEq_lem1 + - Pell.xz_succ_succ + - Pell.eq_of_xn_modEq' + - Pell.isPell_iff_mem_unitary + - Pell.xn_ge_a_pow + - Pell.xy_coprime + - Pell.xn_modEq_x4n_sub + - Pell.xn_modEq_x2n_add_lem + - Pell.ysq_dvd_yy + - Pell.xn_modEq_x2n_sub + - Pell.isPell_one + - Pell.xz_sub + - path: Mathlib/GroupTheory/Sylow.lean + theorems: + - Sylow.subtype_injective + - Subgroup.sylow_mem_fixedPoints_iff + - Sylow.prime_dvd_card_quotient_normalizer + - not_dvd_index_sylow + - Sylow.exists_subgroup_le_card_le + - card_sylow_modEq_one + - Sylow.smul_eq_of_normal + - Sylow.exists_subgroup_card_pow_prime_le + - Sylow.normal_of_all_max_subgroups_normal + - Sylow.normalizer_sup_eq_top + - Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer + - Sylow.normalizer_normalizer + - Sylow.normalizer_sup_eq_top' + - Sylow.card_quotient_normalizer_modEq_card_quotient + - Sylow.ext + - Sylow.card_normalizer_modEq_card + - Sylow.ne_bot_of_dvd_card + - Sylow.card_eq_multiplicity + - Sylow.smul_eq_iff_mem_normalizer + - Sylow.stabilizer_eq_normalizer + - Sylow.exists_subgroup_card_pow_prime + - not_dvd_index_sylow' + - Sylow.conj_eq_normalizer_conj_of_mem_centralizer + - Sylow.dvd_card_of_dvd_card + - Sylow.exists_subgroup_le_card_pow_prime_of_le_card + - Sylow.characteristic_of_normal + - QuotientGroup.card_preimage_mk + - Sylow.exists_subgroup_card_pow_succ + - IsPGroup.exists_le_sylow + - Sylow.exists_subgroup_card_pow_prime_of_le_card + - Sylow.normal_of_normalizer_normal + - path: Mathlib/Analysis/Convex/Normed.lean + theorems: + - convexOn_dist + - Convex.cthickening + - convexHull_ediam + - convexOn_norm + - isConnected_setOf_sameRay + - isConnected_setOf_sameRay_and_ne_zero + - convexHull_diam + - isBounded_convexHull + - convex_ball + - convexHull_exists_dist_ge2 + - convex_closedBall + - Wbtw.dist_add_dist + - path: Mathlib/Order/Filter/Archimedean.lean + theorems: + - Rat.comap_cast_atTop + - Filter.Tendsto.atBot_zsmul_neg_const + - Int.comap_cast_atBot + - tendsto_rat_cast_atBot_iff + - Filter.Eventually.int_cast_atTop + - Filter.Tendsto.atTop_nsmul_const + - Filter.Tendsto.atBot_mul_const' + - Filter.Tendsto.const_mul_atTop' + - Filter.Eventually.rat_cast_atBot + - Filter.Tendsto.atTop_nsmul_neg_const + - tendsto_rat_cast_atTop_iff + - Filter.Tendsto.atTop_zsmul_const + - tendsto_int_cast_atTop_iff + - Rat.comap_cast_atBot + - Filter.Tendsto.atTop_mul_neg_const' + - Filter.Eventually.rat_cast_atTop + - Filter.Tendsto.atTop_zsmul_neg_const + - Filter.Tendsto.atBot_mul_neg_const' + - tendsto_int_cast_atBot_iff + - path: Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean + theorems: + - liouville_liouvilleNumber + - LiouvilleNumber.aux_calc + - LiouvilleNumber.remainder_lt' + - LiouvilleNumber.remainder_pos + - LiouvilleNumber.remainder_summable + - LiouvilleNumber.partialSum_eq_rat + - path: Mathlib/GroupTheory/GroupAction/Hom.lean + theorems: + - DistribMulActionHom.toMulActionHom_injective + - MulActionHom.comp_id + - DistribMulActionHom.comp_id + - MulActionHom.id_comp + - DistribMulActionHom.toAddMonoidHom_injective + - MulSemiringActionHom.comp_id + - DistribMulActionHom.id_comp + - DistribMulActionHom.id_apply + - IsScalarTower.smulHomClass + - path: Mathlib/GroupTheory/MonoidLocalization.lean + theorems: + - Submonoid.LocalizationMap.mk'_eq_of_eq' + - Submonoid.LocalizationMap.mk'_self' + - Submonoid.LocalizationMap.mk'_spec' + - Submonoid.LocalizationMap.sec_spec' + - Submonoid.LocalizationMap.mk'_one + - Localization.r_iff_exists + - Submonoid.LocalizationMap.mul_inv_left + - Submonoid.LocalizationMap.ofMulEquivOfDom_id + - Localization.smul_mk + - Submonoid.LocalizationMap.mk'_eq_iff_eq_mul + - Submonoid.LocalizationMap.lift_surjective_iff + - Submonoid.LocalizationMap.mk'_cancel + - Submonoid.LocalizationMap.mk'_mul_cancel_right + - Submonoid.LocalizationMap.lift_mul_right + - Submonoid.LocalizationMap.mulEquivOfLocalizations_left_inv_apply + - Localization.r_of_eq + - Submonoid.LocalizationMap.mk'_mul_cancel_left + - Submonoid.LocalizationMap.mk'_spec + - Submonoid.LocalizationMap.map_injective_of_injective + - Localization.mk_eq_mk_iff' + - Localization.mk_zero + - Submonoid.LocalizationWithZeroMap.isLeftRegular_of_le_isCancelMulZero + - Submonoid.LocalizationMap.inv_unique + - Submonoid.LocalizationMap.mul_inv_right + - Submonoid.LocalizationMap.ofMulEquivOfLocalizations_comp + - Localization.mulEquivOfQuotient_monoidOf + - Submonoid.LocalizationMap.ext + - Submonoid.LocalizationMap.lift_comp + - Submonoid.LocalizationMap.lift_eq + - Submonoid.LocalizationMap.eq_of_eq + - Submonoid.LocalizationMap.mul_mk'_one_eq_mk' + - Submonoid.LocalizationMap.mul_mk'_eq_mk'_of_mul + - Localization.mulEquivOfQuotient_symm_mk + - Submonoid.LocalizationMap.lift_eq_iff + - Submonoid.LocalizationMap.mk'_eq_iff_eq + - Localization.liftOn_zero + - Localization.r_eq_r' + - Submonoid.LocalizationMap.lift_of_comp + - Submonoid.LocalizationMap.mk'_sec + - Submonoid.LocalizationMap.lift_mk'_spec + - Submonoid.LocalizationMap.map_map + - Submonoid.LocalizationMap.lift_comp_lift + - Submonoid.LocalizationMap.eq' + - Submonoid.LocalizationWithZeroMap.leftCancelMulZero_of_le_isLeftRegular + - Submonoid.LocalizationMap.inv_inj + - Submonoid.LocalizationMap.sec_zero_fst + - Localization.mulEquivOfQuotient_mk + - Submonoid.LocalizationMap.mulEquivOfLocalizations_right_inv_apply + - Submonoid.LocalizationMap.lift_spec_mul + - Submonoid.LocalizationMap.mul_inv + - Localization.mk_one + - "Localization.liftOn\u2082_mk'" + - Localization.Away.mk_eq_monoidOf_mk' + - "Submonoid.LocalizationMap.surj\u2082" + - Submonoid.LocalizationMap.eq_mk'_iff_mul_eq + - Submonoid.LocalizationMap.map_left_cancel + - Submonoid.LocalizationMap.map_mul_left + - Submonoid.LocalizationMap.subsingleton + - Localization.mk_eq_monoidOf_mk'_apply + - Submonoid.LocalizationMap.ofMulEquivOfLocalizations_id + - Submonoid.LocalizationMap.map_mk' + - Localization.liftOn_mk' + - Submonoid.LocalizationMap.mk'_eq_iff_eq' + - Submonoid.LocalizationMap.lift_injective_iff + - Submonoid.LocalizationMap.toMap_injective_iff + - Localization.mk_left_injective + - Submonoid.LocalizationMap.lift_unique + - Submonoid.LocalizationMap.lift_mk' + - Submonoid.LocalizationMap.lift_mul_left + - Submonoid.LocalizationMap.map_comp_map + - Localization.mk_mul + - Submonoid.LocalizationMap.eq_iff_exists + - Submonoid.LocalizationMap.mk'_eq_of_same + - Submonoid.LocalizationMap.mk'_mul_eq_mk'_of_mul + - Submonoid.LocalizationMap.map_right_cancel + - Localization.mk_pow + - Submonoid.LocalizationMap.mk'_mul + - Localization.mk_self + - Submonoid.LocalizationMap.epic_of_localizationMap + - path: Mathlib/Topology/MetricSpace/PiNat.lean + theorems: + - PiNat.firstDiff_le_longestPrefix + - PiNat.dist_triangle_nonarch + - PiCountable.dist_le_dist_pi_of_dist_lt + - PiNat.dist_nonneg + - PiNat.exists_retraction_subtype_of_isClosed + - PiNat.mem_cylinder_comm + - PiNat.update_mem_cylinder + - PiNat.res_eq_res + - PiNat.cylinder_eq_pi + - PiNat.mem_cylinder_iff_dist_le + - PiNat.res_injective + - exists_nat_nat_continuous_surjective_of_completeSpace + - PiNat.firstDiff_lt_shortestPrefixDiff + - PiNat.exists_lipschitz_retraction_of_isClosed + - PiNat.isOpen_cylinder + - PiNat.dist_eq_of_ne + - PiNat.shortestPrefixDiff_pos + - PiNat.dist_comm + - PiNat.firstDiff_comm + - PiCountable.min_dist_le_dist_pi + - PiNat.isOpen_iff_dist + - PiNat.apply_eq_of_lt_firstDiff + - PiNat.inter_cylinder_longestPrefix_nonempty + - PiNat.exists_retraction_of_isClosed + - PiNat.mem_cylinder_iff_eq + - PiNat.eq_of_dist_eq_zero + - PiNat.dist_self + - PiNat.min_firstDiff_le + - PiNat.completeSpace + - PiNat.apply_firstDiff_ne + - PiNat.apply_eq_of_dist_lt + - PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff + - PiCountable.dist_summable + - PiNat.self_mem_cylinder + - PiNat.disjoint_cylinder_of_longestPrefix_lt + - PiNat.exists_disjoint_cylinder + - PiNat.cylinder_zero + - PiNat.res_length + - PiNat.cylinder_eq_cylinder_of_le_firstDiff + - PiNat.isTopologicalBasis_cylinders + - PiNat.cylinder_eq_res + - PiNat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder + - PiNat.iUnion_cylinder_update + - path: Mathlib/Algebra/GroupWithZero/Units/Basic.lean + theorems: + - Units.mk0_mul + - Ring.inverse_mul_cancel + - div_helper + - Ring.inverse_mul_cancel_right + - div_mul_div_cancel + - "eq_mul_of_mul_inv_eq\u2080" + - div_mul_cancel_of_imp + - div_eq_div_iff_div_eq_div' + - Ring.inverse_mul_eq_iff_eq_mul + - Ring.eq_mul_inverse_iff_mul_eq + - Ring.inverse_mul_cancel_left + - Ring.inverse_unit + - Ring.mul_inverse_cancel + - div_div_div_cancel_left' + - Ring.mul_inverse_cancel_left + - mul_div_cancel_left_of_imp + - Units.mul_left_eq_zero + - mul_eq_mul_of_div_eq_div + - Ring.mul_inverse_cancel_right + - Ring.inverse_zero + - "eq_mul_of_inv_mul_eq\u2080" + - div_eq_zero_iff + - isUnit_zero_iff + - Ring.inverse_eq_inv + - "mul_eq_of_eq_mul_inv\u2080" + - Units.exists_iff_ne_zero + - Units.exists0' + - div_div_div_cancel_right + - Units.mul_right_eq_zero + - mul_div_cancel_of_imp' + - GroupWithZero.eq_zero_or_unit + - div_ne_zero + - "mul_eq_of_eq_inv_mul\u2080" + - isUnit_ring_inverse + - Units.mk0_inj + - path: Mathlib/LinearAlgebra/Matrix/ToLin.lean + theorems: + - Matrix.toLin_apply + - Matrix.vecMulLinear_transpose + - LinearMap.toMatrix_mul + - Matrix.vecMul_injective_iff + - range_vecMulLinear + - LinearMap.toMatrix'_comp + - Matrix.toLinAlgEquiv_apply + - Matrix.vecMul_stdBasis + - Matrix.ker_mulVecLin_eq_bot_iff + - LinearMap.toMatrix_reindexRange + - Matrix.toLin_toMatrix + - Matrix.mulVecLin_transpose + - LinearMap.toMatrixAlgEquiv'_apply + - LinearMap.toMatrix_id + - Matrix.toLin_mul_apply + - LinearMap.toMatrixAlgEquiv_toLinAlgEquiv + - toMatrix_distrib_mul_action_toLinearMap + - Matrix.mulVec_injective_iff + - Algebra.smul_leftMulMatrix_algebraMap_ne + - LinearMap.toMatrix'_id + - Algebra.smul_leftMulMatrix_algebraMap_eq + - Matrix.toLin_mul + - Algebra.smul_leftMulMatrix_algebraMap + - LinearMap.toMatrix_apply + - LinearMap.toMatrix_mulVec_repr + - Matrix.toLinAlgEquiv_one + - Algebra.smul_leftMulMatrix + - Matrix.toLinAlgEquiv_mul + - LinearMap.toMatrix'_algebraMap + - LinearMap.toMatrix_toLin + - Algebra.leftMulMatrix_eq_repr_mul + - Matrix.toLinAlgEquiv_toMatrixAlgEquiv + - Matrix.toLin_finTwoProd_apply + - Algebra.toMatrix_lmul' + - Matrix.toLin_self + - LinearMap.toMatrixAlgEquiv_reindexRange + - LinearMap.toMatrixAlgEquiv_id + - LinearMap.toMatrixAlgEquiv_comp + - LinearMap.toMatrix_algebraMap + - Matrix.mulVecLin_one + - LinearMap.toMatrix_pow + - Matrix.range_mulVecLin + - Matrix.toLin_one + - Algebra.leftMulMatrix_injective + - LinearMap.toMatrix'_apply + - LinearMap.toMatrix_prodMap + - Matrix.toLinearMapRight'_one + - LinearMap.toMatrix_comp + - path: Mathlib/LinearAlgebra/Matrix/Adjugate.lean + theorems: + - Matrix.adjugate_mul_distrib_aux + - Matrix.adjugate_pow + - Matrix.det_eq_sum_mul_adjugate_col + - Matrix.adjugate_fin_succ_eq_det_submatrix + - Matrix.det_eq_sum_mul_adjugate_row + - RingHom.map_adjugate + - Matrix.det_smul_adjugate_adjugate + - Matrix.cramer_transpose_row_self + - Matrix.cramer_row_self + - Matrix.cramer_zero + - Matrix.cramer_transpose_apply + - Matrix.mul_adjugate + - Matrix.cramer_one + - Matrix.adjugate_adjugate + - Matrix.isRegular_of_isLeftRegular_det + - Matrix.adjugate_submatrix_equiv_self + - Matrix.adjugate_mul_distrib + - Matrix.cramer_is_linear + - Matrix.adjugate_fin_three + - Matrix.adjugate_conjTranspose + - Matrix.mul_adjugate_apply + - Matrix.adjugate_apply + - Matrix.cramer_submatrix_equiv + - Matrix.adjugate_mul + - Matrix.sum_cramer_apply + - Matrix.adjugate_fin_two + - Matrix.adjugate_smul + - Matrix.cramer_subsingleton_apply + - Matrix.adjugate_one + - Matrix.det_adjugate + - Matrix.adjugate_subsingleton + - Matrix.adjugate_zero + - Matrix.cramer_eq_adjugate_mulVec + - Matrix.mulVec_cramer + - path: Mathlib/Combinatorics/SetFamily/FourFunctions.lean + theorems: + - fkg + - holley + - ineq + - Finset.card_le_card_diffs + - collapse_eq + - Finset.le_card_infs_mul_card_sups + - le_collapse_of_mem + - four_functions_theorem + - sum_collapse + - filter_collapse_eq + - le_collapse_of_insert_mem + - collapse_of_mem + - erase_eq_iff + - four_functions_theorem_univ + - four_functions_theorem_aux + - Finset.le_card_diffs_mul_card_diffs + - Finset.four_functions_theorem + - path: Mathlib/Topology/ContinuousOn.lean + theorems: + - mem_nhdsWithin_insert + - eventually_mem_of_tendsto_nhdsWithin + - insert_mem_nhds_iff + - nhdsWithin_inter' + - nhdsWithin_eq_nhdsWithin' + - ContinuousWithinAt.preimage_mem_nhdsWithin'' + - continuousOn_prod_of_discrete_left + - ContinuousOn.if' + - nhdsWithin_insert + - ContinuousOn.preimage_isClosed_of_isClosed + - diff_mem_nhdsWithin_diff + - eventually_nhdsWithin_nhdsWithin + - "ContinuousAt.comp\u2082_continuousWithinAt_of_eq" + - continuousOn_iff_continuous_restrict + - continuousWithinAt_insert_self + - continuousOn_open_iff + - continuousWithinAt_inter + - continuousWithinAt_congr_nhds + - continuous_if + - Function.LeftInverse.map_nhds_eq + - continuousWithinAt_compl_self + - mem_nhdsWithin_subtype + - ite_inter_closure_compl_eq_of_inter_frontier_eq + - IsOpenMap.continuousOn_image_of_leftInvOn + - continuous_prod_of_discrete_right + - continuousWithinAt_of_not_mem_closure + - continuousOn_of_locally_continuousOn + - Set.LeftInvOn.map_nhdsWithin_eq + - nhdsWithin_singleton + - nhdsWithin_prod + - preimage_coe_mem_nhds_subtype + - mem_closure_pi + - Filter.Tendsto.piecewise_nhdsWithin + - continuous_if' + - continuousAt_prod_of_discrete_left + - ContinuousOn.congr_mono + - continuous_iff_continuousOn_univ + - continuousOn_singleton + - ContinuousOn.if + - preimage_nhdsWithin_coinduced' + - Inducing.continuousOn_iff + - continuousWithinAt_iff_continuousAt + - nhdsWithin_sUnion + - OpenEmbedding.map_nhdsWithin_preimage_eq + - nhdsWithin_empty + - Continuous.continuousOn + - frontier_inter_open_inter + - isOpenMap_prod_of_discrete_right + - continuousWithinAt_inter' + - continuousOn_iff + - isOpenMap_prod_of_discrete_left + - continuousWithinAt_prod_of_discrete_right + - nhdsWithin_inter + - nhdsWithin_eq_nhdsWithin + - continuous_if_const + - continuousOn_iff' + - nhdsWithin_univ + - IsOpenMap.continuousOn_range_of_leftInverse + - nhdsWithin_pi_eq_bot + - IsOpen.ite' + - continuousAt_update_same + - ContinuousWithinAt.prod_map + - continuousOn_piecewise_ite' + - nhdsWithin_pi_neBot + - nhdsWithin_union + - mem_closure_ne_iff_frequently_within + - Continuous.mulIndicator + - continuousWithinAt_singleton + - continuousAt_prod_of_discrete_right + - mem_nhdsWithin_iff_eventuallyEq + - nhdsWithin_inter_of_mem + - preimage_nhds_within_coinduced + - Filter.EventuallyEq.congr_continuousWithinAt + - nhds_of_nhdsWithin_of_nhds + - tendsto_nhdsWithin_iff_subtype + - Inducing.continuousWithinAt_iff + - nhdsWithin_compl_singleton_sup_pure + - ContinuousOn.isOpen_preimage + - continuousWithinAt_univ + - ContinuousOn.preimage_interior_subset_interior_preimage + - dense_pi + - nhdsWithin_le_nhds + - continuousWithinAt_prod_of_discrete_left + - ContinuousOn.comp_continuous + - ite_inter_closure_eq_of_inter_frontier_eq + - continuousOn_to_generateFrom_iff + - continuousOn_prod_of_discrete_right + - mem_nhds_subtype_iff_nhdsWithin + - frequently_nhdsWithin_iff + - continuousOn_iff_isClosed + - IsOpen.ite + - continuous_of_cover_nhds + - insert_mem_nhdsWithin_insert + - nhdsWithin_biUnion + - principal_subtype + - nhdsWithin_inter_of_mem' + - continuousWithinAt_union + - nhdsWithin_iUnion + - ContinuousWithinAt.diff_iff + - continuous_prod_of_discrete_left + - nhdsWithin_pi_eq + - mem_nhdsWithin + - Embedding.map_nhdsWithin_eq + - path: Mathlib/Order/Basic.lean + theorems: + - Preorder.toLE_injective + - lt_of_strongLT + - denselyOrdered_orderDual + - Pi.lt_def + - lt_update_self_iff + - min_def_lt + - PartialOrder.ext + - ltByCases_eq_iff + - dite_lt_one + - ltTrichotomy_eq_iff + - eq_or_eq_or_eq_of_forall_not_lt_lt + - Prod.lt_iff + - le_update_self_iff + - Function.const_le_const + - update_lt_self_iff + - associative_of_commutative_of_le + - ite_lt_one + - update_le_update_iff + - max_def' + - dite_le_one + - Preorder.ext + - one_le_dite + - update_le_update_iff' + - max_def_lt + - ltByCases_comm + - ite_le_one + - compare_of_injective_eq_compareOfLessAndEq + - ltByCases_congr + - eq_iff_not_lt_of_le + - min_def' + - LinearOrder.toPartialOrder_injective + - eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt + - Function.const_lt_const + - one_le_ite + - PartialOrder.toPreorder_injective + - update_le_self_iff + - one_lt_ite + - LinearOrder.ext + - one_lt_dite + - path: Mathlib/FieldTheory/Subfield.lean + theorems: + - Subfield.coe_sInf + - RingHom.map_fieldRange + - SubfieldClass.rat_smul_mem + - Subfield.sInf_toSubring + - RingHom.fieldRange_eq_map + - Subfield.coe_iSup_of_directed + - Subfield.mem_closure_iff + - Subfield.zpow_mem + - Subfield.mem_iSup_of_directed + - SubfieldClass.coe_rat_mem + - Subfield.isGLB_sInf + - Subfield.closure_induction + - Subfield.mem_map + - Subfield.mem_sSup_of_directedOn + - Subfield.coe_sSup_of_directedOn + - path: Mathlib/GroupTheory/Subsemigroup/Basic.lean + theorems: + - Subsemigroup.mem_iSup + - Subsemigroup.iSup_eq_closure + - Subsemigroup.closure_singleton_le_iff_mem + - Subsemigroup.subsingleton_of_subsingleton + - Subsemigroup.coe_iInf + - Subsemigroup.dense_induction + - Subsemigroup.mem_iInf + - Subsemigroup.closure_induction' + - path: Mathlib/AlgebraicGeometry/OpenImmersion.lean + theorems: + - AlgebraicGeometry.IsOpenImmersion.app_eq_inv_app_app_of_comp_eq_aux + - AlgebraicGeometry.Scheme.OpenCover.iSup_opensRange + - AlgebraicGeometry.Scheme.image_basicOpen + - AlgebraicGeometry.Scheme.OpenCover.compactSpace + - AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_eq_of_locallyRingedSpace_eq + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme + - AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right + - AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left + - AlgebraicGeometry.Scheme.OpenCover.iUnion_range + - AlgebraicGeometry.Scheme.affineBasisCover_map_range + - AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right + - AlgebraicGeometry.isIso_iff_stalk_iso + - AlgebraicGeometry.Scheme.affineBasisCover_is_basis + - AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq + - path: Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean + theorems: + - CategoryTheory.Limits.pullbackZeroZeroIso_inv_fst + - CategoryTheory.Limits.inr_coprodZeroIso_hom + - CategoryTheory.Limits.pullbackZeroZeroIso_hom_snd + - CategoryTheory.Limits.inl_pushoutZeroZeroIso_inv + - CategoryTheory.Limits.inr_pushoutZeroZeroIso_inv + - CategoryTheory.Limits.inl_pushoutZeroZeroIso_hom + - CategoryTheory.Limits.pullbackZeroZeroIso_hom_fst + - CategoryTheory.Limits.pullbackZeroZeroIso_inv_snd + - CategoryTheory.Limits.prodZeroIso_iso_inv_snd + - CategoryTheory.Limits.zeroProdIso_inv_snd + - CategoryTheory.Limits.inr_zeroCoprodIso_hom + - CategoryTheory.Limits.inr_pushoutZeroZeroIso_hom + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean + theorems: + - ContinuousLinearMap.opNorm_smul_le + - ContinuousLinearMap.opNorm_comp_le + - ContinuousLinearMap.opNorm_subsingleton + - ContinuousLinearMap.opNorm_le_of_ball + - ContinuousLinearMap.uniformity_eq_seminorm + - ContinuousLinearMap.opNorm_le_of_lipschitz + - LinearIsometry.norm_toContinuousLinearMap_le + - ContinuousLinearMap.opNorm_le_of_unit_norm + - ContinuousLinearMap.norm_id_le + - ContinuousLinearMap.opNorm_zero + - ContinuousLinearMap.opNorm_le_bound' + - ContinuousLinearMap.dist_le_opNorm + - norm_image_of_norm_zero + - ContinuousLinearMap.opNorm_neg + - ContinuousLinearMap.norm_id_of_nontrivial_seminorm + - ContinuousLinearMap.isLeast_opNorm + - ContinuousLinearMap.opNorm_le_of_shell' + - path: Mathlib/Topology/GDelta.lean + theorems: + - IsClosed.isNowhereDense_iff + - "isG\u03B4_iff_eq_iInter_nat" + - "Set.Countable.isG\u03B4_compl" + - "residual_of_dense_G\u03B4" + - "IsG\u03B4.iInter_of_isOpen" + - isNowhereDense_empty + - "IsG\u03B4.sUnion" + - isMeagre_iUnion + - "IsG\u03B4.iInter" + - isMeagre_iff_countable_union_isNowhereDense + - isClosed_isNowhereDense_iff_compl + - "IsG\u03B4.biInter" + - "IsG\u03B4.setOf_continuousAt" + - "IsG\u03B4.sInter" + - mem_residual_iff + - "IsG\u03B4.biInter_of_isOpen" + - meagre_empty + - "IsG\u03B4.inter" + - "IsG\u03B4.union" + - "IsG\u03B4.singleton" + - "IsG\u03B4.biUnion" + - "IsClosed.isG\u03B4" + - "IsOpen.isG\u03B4" + - IsNowhereDense.closure + - path: Mathlib/Data/Finset/Powerset.lean + theorems: + - Finset.coe_powerset + - Finset.powerset_eq_singleton_empty + - Finset.powerset_insert + - Finset.not_mem_of_mem_powerset_of_not_mem + - Finset.powersetCard_card_add + - Finset.powersetCard_nonempty + - Finset.powersetCard_zero + - Finset.map_val_val_powersetCard + - Finset.powersetCard_eq_filter + - Finset.empty_mem_ssubsets + - Finset.mem_powersetCard + - Finset.powerset_card_disjiUnion + - Finset.powersetCard_eq_empty + - Finset.powerset_card_biUnion + - Finset.powersetCard_succ_insert + - Finset.powersetCard_sup + - Finset.powersetCard_one + - Finset.powersetCard_self + - Finset.powersetCard_map + - Finset.powersetCard_empty_subsingleton + - Finset.mem_ssubsets + - Finset.mem_powerset + - path: Mathlib/Algebra/BigOperators/Multiset/Basic.lean + theorems: + - Multiset.prod_eq_foldl + - Multiset.le_prod_of_submultiplicative + - Multiset.prod_nat_mod + - Multiset.prod_eq_one + - Multiset.sum_map_singleton + - Multiset.single_le_prod + - Multiset.prod_singleton + - Multiset.sum_map_mul_right + - Multiset.prod_int_mod + - Multiset.prod_map_eq_pow_single + - Multiset.prod_induction + - Multiset.prod_nonneg + - Multiset.sum_nat_mod + - Multiset.prod_lt_prod_of_nonempty' + - Multiset.prod_hom + - Multiset.prod_le_pow_card + - Multiset.prod_pair + - Multiset.prod_dvd_prod_of_le + - Multiset.prod_le_prod_of_rel_le + - Multiset.prod_replicate + - Multiset.prod_map_one + - Multiset.prod_map_inv + - Multiset.prod_map_prod_map + - Multiset.le_prod_nonempty_of_submultiplicative_on_pred + - Multiset.prod_lt_prod' + - Multiset.prod_hom' + - Multiset.prod_nsmul + - Multiset.dvd_sum + - "Multiset.prod_hom\u2082" + - Multiset.prod_eq_zero + - Multiset.le_prod_of_mem + - Multiset.prod_eq_foldr + - Multiset.prod_hom_rel + - Multiset.all_one_of_le_one_le_of_prod_eq_one + - Multiset.prod_eq_pow_single + - Multiset.prod_toList + - Multiset.prod_map_zpow + - Multiset.le_prod_of_submultiplicative_on_pred + - Multiset.pow_card_le_prod + - Multiset.prod_add + - Multiset.prod_map_erase + - Multiset.prod_eq_zero_iff + - Multiset.prod_dvd_prod_of_dvd + - Multiset.sum_map_mul_left + - Multiset.prod_erase + - Multiset.prod_map_neg + - Multiset.pow_count + - Multiset.le_prod_nonempty_of_submultiplicative + - path: Mathlib/Analysis/Calculus/Deriv/Mul.lean + theorems: + - HasDerivAt.clm_comp + - deriv_mul_const_field + - HasDerivAt.clm_apply + - HasStrictDerivAt.clm_comp + - HasDerivAt.smul + - HasDerivWithinAt.clm_apply + - HasDerivAt.const_mul + - HasStrictDerivAt.const_smul + - HasStrictDerivAt.clm_apply + - HasDerivAt.finset_prod + - HasDerivWithinAt.smul_const + - HasDerivAt.mul + - HasStrictDerivAt.const_mul + - deriv_const_mul_field + - derivWithin_div_const + - deriv_div_const + - HasStrictDerivAt.mul + - HasDerivAt.mul_const + - HasDerivWithinAt.mul + - HasStrictDerivAt.smul + - HasStrictDerivAt.div_const + - HasStrictDerivAt.mul_const + - HasStrictDerivAt.finset_prod + - HasDerivAt.div_const + - HasDerivWithinAt.div_const + - HasDerivWithinAt.finset_prod + - HasDerivAt.smul_const + - HasDerivWithinAt.mul_const + - HasDerivWithinAt.clm_comp + - HasDerivWithinAt.smul + - HasDerivWithinAt.const_mul + - HasStrictDerivAt.smul_const + - deriv_const_smul' + - hasDerivAt_mul_const + - path: Mathlib/Logic/Equiv/TransferInstance.lean + theorems: + - Equiv.algebraMap_def + - Equiv.ringEquiv_symm_apply + - Finite.exists_type_zero_nonempty_mulEquiv + - Equiv.algEquiv_symm_apply + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean + theorems: + - Real.sin_add_pi_div_two + - Real.cos_antiperiodic + - Real.sin_pi_div_thirty_two + - Real.cos_pi_div_eight + - Real.sin_sq_pi_over_two_pow + - Real.range_sin_infinite + - Real.tan_pi_div_three + - Real.sin_nonneg_of_mem_Icc + - Real.cos_lt_cos_of_nonneg_of_le_pi + - Real.tan_nonneg_of_nonneg_of_le_pi_div_two + - Complex.continuous_cosh + - Real.pi_le_four + - Real.sin_pi_div_four + - Real.sqrtTwoAddSeries_one + - Complex.sin_add_pi_div_two + - Real.cos_pi_div_six + - Real.sin_sq_pi_over_two_pow_succ + - Real.sin_pi_over_two_pow_succ + - Complex.exp_mul_I_antiperiodic + - Complex.continuous_cos + - Complex.tan_periodic + - Complex.sin_eq_zero_iff_cos_eq + - Real.tendsto_cos_neg_pi_div_two + - Real.sin_pi + - Real.cos_pi_div_three + - Real.two_pi_pos + - Real.cos_pi_div_two_sub + - Real.sin_eq_zero_iff_cos_eq + - Real.tan_periodic + - Real.one_le_pi_div_two + - Real.cos_pos_of_mem_Ioo + - Real.abs_sin_half + - Real.sq_cos_pi_div_six + - Real.tan_pos_of_pos_of_lt_pi_div_two + - Real.cos_eq_sqrt_one_sub_sin_sq + - Real.cos_half + - Real.sin_pi_div_sixteen + - Real.sqrtTwoAddSeries_monotone_left + - Real.cos_sub_pi_div_two + - Real.sin_sub_pi_div_two + - Complex.abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le + - Real.pi_pos + - Real.sin_antiperiodic + - Real.cos_nonpos_of_pi_div_two_le_of_le + - Real.sqrtTwoAddSeries_succ + - Complex.cos_pi_div_two_sub + - Complex.cos_antiperiodic + - Real.tan_pi_div_six + - Real.tan_pi_div_four + - Complex.cos_int_mul_two_pi_sub_pi + - Real.cos_pi_over_two_pow + - Real.cos_eq_one_iff + - Complex.sin_pi_div_two + - Real.sin_pi_div_three + - Complex.sin_pi_div_two_sub + - Real.tan_neg_of_neg_of_pi_div_two_lt + - Real.cos_int_mul_two_pi_add_pi + - Real.tendsto_tan_pi_div_two + - Real.cos_nonneg_of_mem_Icc + - Real.surjOn_cos + - Complex.cos_add_pi_div_two + - Complex.continuous_sin + - Complex.sin_antiperiodic + - Real.cos_nat_mul_two_pi_sub_pi + - Real.cos_two_pi + - Real.tendsto_sin_neg_pi_div_two + - Complex.cos_nat_mul_two_pi_sub_pi + - Real.sin_pi_div_two + - Complex.cos_int_mul_two_pi_add_pi + - Complex.exp_antiperiodic + - Complex.cos_pi + - Real.range_cos_infinite + - Real.tan_pi_div_two + - Complex.cos_nat_mul_two_pi_add_pi + - Real.sin_eq_zero_iff + - Real.cos_add_pi_div_two + - Real.sqrtTwoAddSeries_two + - Real.sin_pi_div_six + - Complex.continuous_sinh + - Real.two_le_pi + - Real.sin_half_eq_neg_sqrt + - Complex.cos_sub_pi_div_two + - Real.pi_div_two_le_two + - Real.tan_pi + - Real.tan_lt_tan_of_nonneg_of_lt_pi_div_two + - Real.tan_nonpos_of_nonpos_of_neg_pi_div_two_le + - Real.sin_ne_zero_iff + - Real.cos_pi_div_two + - Complex.sin_two_pi + - Complex.sin_sub_pi_div_two + - Real.cos_eq_one_iff_of_lt_of_lt + - Complex.cos_two_pi + - Real.surjOn_sin + - Real.cos_int_mul_two_pi_sub_pi + - Real.sin_pos_of_pos_of_lt_pi + - Complex.sin_pi + - Real.cos_pi_div_thirty_two + - Real.cos_neg_of_pi_div_two_lt_of_lt + - Real.sqrtTwoAddSeries_lt_two + - Real.sin_pi_div_eight + - Real.tendsto_tan_neg_pi_div_two + - Real.cos_nat_mul_two_pi_add_pi + - Real.exists_cos_eq_zero + - Real.sqrtTwoAddSeries_zero + - Real.tendsto_sin_pi_div_two + - Real.tendsto_cos_pi_div_two + - Real.sin_half_eq_sqrt + - Real.sin_two_pi + - Real.strictMonoOn_tan + - Real.tan_pi_div_two_sub + - Complex.tan_pi_div_two_sub + - Real.sq_sin_pi_div_three + - Real.cos_lt_cos_of_nonneg_of_le_pi_div_two + - Real.cos_pi_div_sixteen + - Real.sin_eq_zero_iff_of_lt_of_lt + - path: Mathlib/Analysis/NormedSpace/Banach.lean + theorems: + - ContinuousLinearEquiv.coe_ofBijective + - ContinuousLinearMap.exists_nonlinearRightInverse_of_surjective + - ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot + - LinearMap.continuous_of_isClosed_graph + - ContinuousLinearMap.coe_ofSeqClosedGraph + - LinearEquiv.continuous_symm + - ContinuousLinearMap.nonlinearRightInverseOfSurjective_nnnorm_pos + - ContinuousLinearMap.coe_ofIsClosedGraph + - ContinuousLinearMap.exists_preimage_norm_le + - ContinuousLinearMap.exists_approx_preimage_norm_le + - ContinuousLinearMap.isOpenMap + - path: Mathlib/Algebra/Order/Ring/Defs.lean + theorems: + - nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nonneg + - le_mul_of_le_one_left + - pow_pos + - mul_nonneg_iff_pos_imp_nonneg + - mul_nonpos_iff + - sq_nonneg + - mul_le_of_one_le_right + - zero_lt_bit0 + - Decidable.mul_lt_mul'' + - mul_le_mul_of_nonpos_left + - mul_pos_of_neg_of_neg + - bit0_le_bit0 + - mul_lt_mul_of_neg_right + - max_mul_mul_le_max_mul_max + - mul_nonneg_iff_neg_imp_nonpos + - mul_nonneg_of_three + - two_mul_le_add_sq + - mul_lt_mul'' + - mul_le_of_one_le_left + - add_le_mul_two_add + - bit1_pos' + - mul_nonneg_iff_of_pos_right + - mul_nonneg_of_nonpos_of_nonpos + - le_iff_exists_nonneg_add + - bit0_lt_bit0 + - mul_nonpos_iff_neg_imp_nonneg + - mul_lt_of_one_lt_left + - mul_nonpos_iff_pos_imp_nonpos + - mul_nonneg_iff_of_pos_left + - lt_mul_left + - mul_neg_iff + - mul_add_mul_le_mul_add_mul' + - one_lt_bit1 + - pow_le_pow_of_le_one + - lt_mul_of_lt_one_right + - sign_cases_of_C_mul_pow_nonneg + - mul_self_pos + - lt_mul_right + - mul_add_mul_lt_mul_add_mul' + - mul_add_mul_lt_mul_add_mul + - mul_self_add_mul_self_eq_zero + - pow_nonneg + - mul_add_mul_le_mul_add_mul + - mul_lt_mul_of_neg_left + - lt_mul_of_lt_one_left + - zero_le_bit0 + - mul_self_nonneg + - mul_le_mul_of_nonpos_right + - mul_lt_of_one_lt_right + - path: Mathlib/Data/List/Func.lean + theorems: + - List.Func.get_neg + - List.Func.length_neg + - List.Func.get_map' + - List.Func.map_add_map + - List.Func.get_set + - List.Func.get_pointwise + - List.Func.forall_val_of_forall_mem + - List.Func.get_nil + - List.Func.sub_nil + - List.Func.get_sub + - List.Func.add_nil + - List.Func.nil_add + - List.Func.equiv_of_eq + - List.Func.eq_of_equiv + - List.Func.length_set + - List.Func.mem_get_of_le + - List.Func.nil_pointwise + - List.Func.length_pointwise + - List.Func.pointwise_nil + - List.Func.get_set_eq_of_ne + - List.Func.get_add + - List.Func.mem_get_of_ne_zero + - List.Func.get_eq_default_of_le + - List.Func.nil_sub + - List.Func.eq_get_of_mem + - List.Func.get_map + - path: Mathlib/Analysis/Calculus/FDeriv/Equiv.lean + theorems: + - ContinuousLinearEquiv.uniqueDiffOn_preimage_iff + - ContinuousLinearEquiv.comp_right_fderivWithin + - HasFDerivAt.of_local_left_inverse + - ContinuousLinearEquiv.comp_differentiable_iff + - HasFDerivWithinAt.uniqueDiffWithinAt + - ContinuousLinearEquiv.comp_right_fderiv + - ContinuousLinearEquiv.comp_right_hasFDerivAt_iff' + - ContinuousLinearEquiv.comp_fderivWithin + - ContinuousLinearEquiv.comp_right_differentiableAt_iff + - HasFDerivWithinAt.eventually_ne + - has_fderiv_at_filter_real_equiv + - ContinuousLinearEquiv.comp_right_differentiable_iff + - ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff' + - ContinuousLinearEquiv.comp_right_differentiableWithinAt_iff + - ContinuousLinearEquiv.comp_differentiableAt_iff + - ContinuousLinearEquiv.comp_hasFDerivAt_iff + - ContinuousLinearEquiv.comp_hasStrictFDerivAt_iff + - HasFDerivAt.lim_real + - ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff + - ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff + - ContinuousLinearEquiv.comp_right_hasFDerivAt_iff + - ContinuousLinearEquiv.comp_hasFDerivAt_iff' + - HasFDerivWithinAt.mapsTo_tangent_cone + - ContinuousLinearEquiv.comp_differentiableWithinAt_iff + - HasStrictFDerivAt.of_local_left_inverse + - ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff' + - ContinuousLinearEquiv.comp_fderiv + - HasFDerivAt.eventually_ne + - ContinuousLinearEquiv.comp_differentiableOn_iff + - ContinuousLinearEquiv.comp_right_differentiableOn_iff + - path: Mathlib/Data/List/Cycle.lean + theorems: + - Cycle.toMultiset_eq_nil + - Cycle.chain_iff_pairwise + - Cycle.chain_ne_nil + - List.mem_of_nextOr_ne + - Cycle.Subsingleton.congr + - Cycle.chain_range_succ + - List.isRotated_next_eq + - Cycle.mem_lists_iff_coe_eq + - Cycle.prev_mem + - List.prev_cons_cons_eq' + - List.prev_next + - List.next_ne_head_ne_getLast + - List.prev_ne_cons_cons + - Cycle.toFinset_eq_nil + - List.next_cons_concat + - List.prev_cons_cons_of_ne' + - List.next_cons_cons_eq' + - List.nextOr_cons_of_ne + - Cycle.forall_eq_of_chain + - Cycle.chain_singleton + - List.pmap_next_eq_rotate_one + - Cycle.length_nontrivial + - Cycle.reverse_reverse + - Cycle.Chain.imp + - List.prev_getLast_cons' + - List.next_get + - Cycle.subsingleton_reverse_iff + - List.nextOr_concat + - List.prev_nthLe + - Cycle.next_reverse_eq_prev + - Cycle.chain_of_pairwise + - Cycle.Chain.eq_nil_of_irrefl + - Cycle.lists_nil + - List.next_getLast_cons + - Cycle.nontrivial_coe_nodup_iff + - List.prev_mem + - Cycle.nontrivial_reverse_iff + - Cycle.next_mem + - Cycle.mem_map + - Cycle.Chain.nil + - Cycle.coe_cons_eq_coe_append + - Cycle.chain_map + - List.nextOr_eq_nextOr_of_mem_of_ne + - Cycle.Nodup.nontrivial_iff + - List.nextOr_mem + - List.isRotated_prev_eq + - Cycle.map_eq_nil + - Cycle.Subsingleton.nodup + - Cycle.next_reverse_eq_prev' + - Cycle.card_toMultiset + - List.prev_reverse_eq_next + - List.next_prev + - List.pmap_prev_eq_rotate_length_sub_one + - Cycle.induction_on + - List.next_reverse_eq_prev + - path: Mathlib/Data/Fin/Interval.lean + theorems: + - Fin.Ici_eq_finset_subtype + - Fin.map_valEmbedding_Iic + - Fin.card_Ici + - Fin.map_valEmbedding_Ioi + - Fin.card_fintypeIco + - Fin.card_Iio + - Fin.card_fintypeIcc + - Fin.map_valEmbedding_Icc + - Fin.map_valEmbedding_Ico + - Fin.card_fintypeIio + - Fin.card_Icc + - Fin.card_uIcc + - Fin.card_Ioo + - Fin.card_fintypeIic + - Fin.map_valEmbedding_Ici + - Fin.map_valEmbedding_Ioo + - Fin.Ioi_eq_finset_subtype + - Fin.card_fintypeIoo + - Fin.card_fintypeIoc + - Fin.card_Ico + - Fin.card_Ioc + - Fin.card_Ioi + - Fin.card_fintype_uIcc + - Fin.card_fintypeIci + - Fin.card_fintypeIoi + - Fin.map_valEmbedding_Iio + - Fin.map_valEmbedding_Ioc + - Fin.card_Iic + - path: Mathlib/Algebra/Ring/Equiv.lean + theorems: + - RingEquiv.symm_toRingHom_comp_toRingHom + - MulEquiv.isField + - RingEquiv.coe_nonUnitalRingHom_inj_iff + - RingEquiv.symm_toNonUnitalRingHom_comp_toNonUnitalRingHom + - RingEquiv.toRingHom_comp_symm_toRingHom + - RingEquiv.toNonUnitalRingHomm_comp_symm_toNonUnitalRingHom + - RingEquiv.coe_ringHom_inj_iff + - RingEquiv.map_eq_neg_one_iff + - path: .lake/packages/std/Std/Data/Int/Order.lean + theorems: + - Int.mul_neg_of_neg_of_pos + - Int.lt_add_of_sub_right_lt + - Int.nonneg_of_neg_nonpos + - Int.neg_le_sub_left_of_le_add + - Int.neg_lt_of_neg_lt + - Int.mul_pos_of_neg_of_neg + - Int.neg_of_neg_pos + - Int.nonpos_of_neg_nonneg + - Int.lt_sub_right_of_add_lt + - Int.neg_le_of_neg_le + - Int.neg_add_lt_right_of_lt_add + - Int.sign_sign + - Int.mul_lt_mul_of_neg_right + - Int.add_lt_of_lt_neg_add + - Int.le_add_of_sub_right_le + - Int.sub_right_lt_of_lt_add + - Int.negSucc_not_pos + - Int.neg_add_le_of_le_add + - Int.mul_lt_mul_of_neg_left + - Int.mul_neg_of_pos_of_neg + - Int.natAbs_eq_iff + - Int.natAbs_sign_of_nonzero + - Int.sub_right_le_of_le_add + - Int.mul_nonneg_of_nonpos_of_nonpos + - Int.lt_neg_add_of_add_lt + - Int.neg_add_le_left_of_le_add + - Int.lt_sub_left_of_add_lt + - Int.lt_add_of_pos_left + - Int.le_add_of_sub_left_le + - Int.sub_neg_of_lt + - Int.add_lt_of_lt_sub_right + - Int.sign_nonneg + - Int.neg_add_lt_of_lt_add + - Int.sign_mul + - Int.eq_natAbs_iff_mul_eq_zero + - Int.neg_lt_sub_left_of_lt_add + - Int.le_sub_right_of_add_le + - Int.le_sub_left_of_add_le + - Int.natAbs_mul_self' + - Int.lt_of_sub_neg + - Int.sign_eq_zero_iff_zero + - Int.le_of_sub_nonpos + - Int.lt_add_of_neg_add_lt_right + - Int.add_lt_of_lt_sub_left + - Int.lt_neg_of_lt_neg + - Int.add_le_of_le_sub_right + - Int.sign_neg + - Int.lt_of_add_lt_add_right + - Int.le_add_of_neg_add_le_right + - Int.negSucc_eq' + - Int.add_le_of_le_sub_left + - Int.lt_add_of_sub_left_lt + - Int.le_add_of_neg_add_le + - Int.lt_of_add_lt_add_left + - Int.natAbs_mul_natAbs_eq + - Int.sub_left_lt_of_lt_add + - Int.le_add_of_neg_add_le_left + - Int.natAbs_sub_le + - Int.mem_toNat' + - Int.sub_nonpos_of_le + - Int.pos_of_neg_neg + - Int.le_neg_add_of_add_le + - Int.lt_add_of_neg_add_lt_left + - Int.lt_add_of_neg_add_lt + - Int.neg_add_lt_left_of_lt_add + - Int.add_le_of_le_neg_add + - Int.lt_add_of_pos_right + - Int.le_neg_of_le_neg + - Int.neg_add_le_right_of_le_add + - path: Mathlib/Data/Sym/Basic.lean + theorems: + - Sym.erase_mk + - Sym.count_coe_fill_self_of_not_mem + - Sym.map_mk + - Sym.fill_filterNe + - SymOptionSuccEquiv.encode_decode + - Sym.cons_of_coe_eq + - Sym.filter_ne_fill + - Sym.ofVector_cons + - SymOptionSuccEquiv.decode_encode + - Sym.map_id + - Sym.exists_eq_cons_of_succ + - Sym.count_coe_fill_of_ne + - Sym.exists_cons_of_mem + - Sym.append_comm + - Sym.eq_replicate_iff + - Sym.mem_fill_iff + - Sym.sigma_sub_ext + - Sym.map_map + - path: Mathlib/Analysis/LocallyConvex/Bounded.lean + theorems: + - Bornology.IsVonNBounded.neg + - Bornology.isVonNBounded_iff_smul_tendsto_zero + - Bornology.IsVonNBounded.sub + - Filter.HasBasis.isVonNBounded_iff + - NormedSpace.isBounded_iff_subset_smul_closedBall + - Bornology.isVonNBounded_of_smul_tendsto_zero + - Bornology.isVonNBounded_vadd + - NormedSpace.isVonNBounded_iff + - NormedSpace.image_isVonNBounded_iff + - NormedSpace.vonNBornology_eq + - NormedSpace.isVonNBounded_closedBall + - Bornology.isVonNBounded_add_self + - Bornology.isVonNBounded_sub_of_nonempty + - Bornology.IsVonNBounded.image + - NormedSpace.isVonNBounded_ball + - Bornology.isVonNBounded_add + - NormedSpace.isBounded_iff_subset_smul_ball + - TotallyBounded.isVonNBounded + - NormedSpace.isVonNBounded_iff' + - Bornology.IsVonNBounded.vadd + - Bornology.IsVonNBounded.add + - Bornology.isVonNBounded_sub + - Bornology.isVonNBounded_pi_iff + - Bornology.isVonNBounded_iff_tendsto_smallSets_nhds + - path: Mathlib/RingTheory/FractionalIdeal/Basic.lean + theorems: + - FractionalIdeal.mul_self_le_self + - FractionalIdeal.isFractional_of_le_one + - FractionalIdeal.le_self_mul_self + - FractionalIdeal.eq_zero_iff + - FractionalIdeal.num_zero_eq + - FractionalIdeal.mul_def + - FractionalIdeal.le_one_iff_exists_coeIdeal + - FractionalIdeal.coeToSubmodule_eq_bot + - FractionalIdeal.coe_one + - FractionalIdeal.mem_zero_iff + - IsFractional.sup + - FractionalIdeal.mul_left_mono + - FractionalIdeal.one_le + - FractionalIdeal.mul_right_mono + - IsFractional.inf_right + - IsFractional.nsmul + - FractionalIdeal.den_mul_self_eq_num + - FractionalIdeal.zero_le + - FractionalIdeal.equivNum_apply + - IsFractional.mul + - FractionalIdeal.mul_mem_mul + - FractionalIdeal.mul_induction_on + - FractionalIdeal.coe_nat_cast + - FractionalIdeal.isFractional_of_le + - FractionalIdeal.zero_of_num_eq_bot + - FractionalIdeal.mul_le + - FractionalIdeal.coe_mul + - path: Mathlib/Data/Int/GCD.lean + theorems: + - Int.gcd_mul_left + - Commute.pow_eq_pow_iff_of_coprime + - Nat.xgcdAux_rec + - Nat.xgcdAux_P + - Int.lcm_mul_left + - Int.gcd_mul_lcm + - Nat.xgcd_zero_left + - Int.gcd_least_linear + - Nat.gcdB_zero_left + - Nat.xgcdAux_val + - Int.gcd_mul_right + - Int.lcm_one_right + - Nat.exists_mul_emod_eq_gcd + - Nat.gcd_eq_gcd_ab + - pow_mem_range_pow_of_coprime + - Int.lcm_zero_left + - Nat.xgcd_val + - pow_gcd_eq_one + - Int.pow_dvd_pow_iff + - Int.gcd_eq_right + - Int.gcd_div_gcd_div_gcd + - Nat.gcdB_zero_right + - Int.gcd_eq_zero_iff + - Int.gcd_dvd_iff + - Int.dvd_of_dvd_mul_right_of_gcd_one + - Int.lcm_dvd + - Int.lcm_assoc + - Int.natAbs_ediv + - Int.lcm_comm + - Int.dvd_of_dvd_mul_left_of_gcd_one + - Int.dvd_of_mul_dvd_mul_left + - Nat.gcdA_zero_left + - Int.gcd_self + - Nat.gcdA_zero_right + - Int.lcm_mul_right + - Int.gcd_eq_gcd_ab + - Int.lcm_one_left + - Int.gcd_zero_left + - Nat.xgcdAux_fst + - Int.dvd_of_mul_dvd_mul_right + - Int.gcd_div + - Int.gcd_zero_right + - path: Mathlib/Algebra/DirectSum/Basic.lean + theorems: + - DirectSum.fromAddMonoid_of + - DirectSum.toAddMonoid.unique + - DirectSum.coe_of_apply + - DirectSum.induction_on + - DirectSum.fromAddMonoid_of_apply + - DirectSum.equivCongrLeft_apply + - DirectSum.IsInternal.addSubmonoid_iSup_eq_top + - path: Mathlib/FieldTheory/PolynomialGaloisGroup.lean + theorems: + - Polynomial.Gal.restrictDvd_surjective + - Polynomial.Gal.prime_degree_dvd_card + - Polynomial.Gal.restrictDvd_def + - Polynomial.Gal.ext + - Polynomial.Gal.galAction_isPretransitive + - Polynomial.Gal.mul_splits_in_splittingField_of_mul + - Polynomial.Gal.splits_in_splittingField_of_comp + - Polynomial.Gal.mapRoots_bijective + - Polynomial.Gal.restrictComp_surjective + - Polynomial.Gal.restrict_smul + - Polynomial.Gal.galActionHom_injective + - Polynomial.Gal.restrictProd_injective + - path: Mathlib/Analysis/Convex/Slope.lean + theorems: + - StrictConcaveOn.secant_strict_mono + - ConvexOn.secant_mono_aux1 + - StrictConcaveOn.slope_anti_adjacent + - ConvexOn.secant_mono_aux2 + - ConvexOn.secant_mono + - ConvexOn.secant_mono_aux3 + - ConvexOn.strict_mono_of_lt + - convexOn_of_slope_mono_adjacent + - strictConcaveOn_of_slope_strict_anti_adjacent + - StrictConvexOn.secant_strict_mono_aux1 + - StrictConvexOn.secant_strict_mono_aux2 + - StrictConvexOn.slope_strict_mono_adjacent + - StrictConvexOn.secant_strict_mono_aux3 + - ConvexOn.slope_mono_adjacent + - concaveOn_of_slope_anti_adjacent + - strictConvexOn_of_slope_strict_mono_adjacent + - ConcaveOn.slope_anti_adjacent + - StrictConvexOn.secant_strict_mono + - path: Mathlib/Geometry/Manifold/Complex.lean + theorems: + - MDifferentiableOn.apply_eq_of_isPreconnected_isCompact_isOpen + - MDifferentiableOn.eqOn_of_isPreconnected_of_isMaxOn_norm + - MDifferentiableOn.norm_eqOn_of_isPreconnected_of_isMaxOn + - Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax + - path: Mathlib/Algebra/BigOperators/Finsupp.lean + theorems: + - Finsupp.prod_add_index + - Finsupp.sum_mul + - Finsupp.prod_of_support_subset + - Finsupp.sum_ite_self_eq' + - Finsupp.univ_sum_single_apply' + - Finsupp.sum_ite_self_eq + - Finsupp.prod_mapRange_index + - Finsupp.prod_single_index + - Finsupp.mul_sum + - Finsupp.prod_eq_single + - Finsupp.prod_dvd_prod_of_subset_of_dvd + - Nat.prod_pow_pos_of_zero_not_mem_support + - Finsupp.sum_cons' + - Finsupp.prod_ite_eq' + - Finsupp.prod_embDomain + - Finsupp.prod_add_index' + - Finsupp.multiset_sum_sum_index + - Finsupp.single_multiset_sum + - Finsupp.mul_prod_erase' + - Finsupp.prod_ite_eq + - Finsupp.prod_add_index_of_disjoint + - Finsupp.equivFunOnFinite_symm_eq_sum + - Finsupp.support_sum + - Finsupp.single_finset_sum + - Finsupp.support_sum_eq_biUnion + - Finsupp.sum_cons + - Finsupp.sum_ite_self_eq_aux + - Finsupp.support_finset_sum + - Finsupp.onFinset_prod + - Finsupp.univ_sum_single + - Finsupp.prod_finset_sum_index + - Finsupp.comp_liftAddHom + - Finsupp.mul_prod_erase + - Finsupp.indicator_eq_sum_attach_single + - Finsupp.prod_indicator_index_eq_prod_attach + - Finsupp.univ_sum_single_apply + - Finsupp.sum_sum_index' + - path: Mathlib/Topology/EMetricSpace/Lipschitz.lean + theorems: + - LipschitzWith.mul_edist_le + - LipschitzWith.const' + - LipschitzWith.of_edist_le + - lipschitzOnWith_iff_restrict + - LocallyLipschitz.iterate + - continuous_prod_of_dense_continuous_lipschitzWith + - LipschitzWith.prod_mk_left + - LipschitzWith.const + - LipschitzWith.prod + - LipschitzWith.pow_end + - LocallyLipschitz.prod + - LipschitzWith.uncurry + - LocallyLipschitz.comp + - LipschitzWith.edist_iterate_succ_le_geometric + - LipschitzWith.comp + - LocallyLipschitz.pow_end + - LipschitzWith.prod_mk_right + - LipschitzWith.eval + - lipschitzOn_univ + - LocallyLipschitz.continuous + - LipschitzOnWith.ediam_image2_le + - LipschitzWith.list_prod + - LipschitzWith.iterate + - LipschitzOnWith.prod + - LipschitzWith.ediam_image_le + - path: Mathlib/GroupTheory/HNNExtension.lean + theorems: + - HNNExtension.induction_on + - HNNExtension.NormalWord.t_smul_eq_unitsSMul + - HNNExtension.ReducedWord.map_fst_eq_and_of_prod_eq + - HNNExtension.inv_t_mul_of + - HNNExtension.NormalWord.ext + - HNNExtension.NormalWord.prod_cons + - HNNExtension.equiv_eq_conj + - HNNExtension.of_mul_t + - HNNExtension.NormalWord.of_smul_eq_smul + - HNNExtension.equiv_symm_eq_conj + - HNNExtension.NormalWord.not_cancels_of_cons_hyp + - HNNExtension.NormalWord.prod_unitsSMul + - HNNExtension.ReducedWord.exists_normalWord_prod_eq + - HNNExtension.NormalWord.unitsSMul_one_group_smul + - HNNExtension.ReducedWord.toList_eq_nil_of_mem_of_range + - HNNExtension.NormalWord.prod_group_smul + - HNNExtension.NormalWord.prod_empty + - HNNExtension.NormalWord.t_pow_smul_eq_unitsSMul + - HNNExtension.NormalWord.unitsSMul_cancels_iff + - HNNExtension.t_mul_of + - HNNExtension.NormalWord.prod_smul + - HNNExtension.of_mul_inv_t + - HNNExtension.NormalWord.unitsSMulGroup_snd + - HNNExtension.toSubgroupEquiv_neg_apply + - HNNExtension.NormalWord.prod_smul_empty + - HNNExtension.NormalWord.unitsSMul_neg + - HNNExtension.of_injective + - HNNExtension.lift_of + - path: Mathlib/Geometry/Euclidean/MongePoint.lean + theorems: + - Affine.Simplex.mongePoint_eq_of_range_eq + - Affine.Simplex.mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub + - Affine.Simplex.affineSpan_pair_eq_altitude_iff + - Affine.Simplex.finrank_direction_altitude + - EuclideanGeometry.exists_dist_eq_circumradius_of_subset_insert_orthocenter + - Affine.Simplex.mongePoint_vsub_face_centroid_eq_weightedVSub_of_pointsWithCircumcenter + - Affine.Simplex.sum_mongePointVSubFaceCentroidWeightsWithCircumcenter + - Affine.Simplex.direction_mongePlane + - Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub + - Affine.Simplex.vectorSpan_isOrtho_altitude_direction + - EuclideanGeometry.OrthocentricSystem.eq_insert_orthocenter + - EuclideanGeometry.OrthocentricSystem.exists_circumradius_eq + - Affine.Triangle.orthocenter_replace_orthocenter_eq_point + - Affine.Triangle.affineSpan_orthocenter_point_le_altitude + - EuclideanGeometry.OrthocentricSystem.affineIndependent + - Affine.Simplex.mongePlane_comm + - Affine.Simplex.mongePoint_mem_mongePlane + - Affine.Triangle.orthocenter_eq_smul_vsub_vadd_circumcenter + - EuclideanGeometry.exists_of_range_subset_orthocentricSystem + - Affine.Triangle.eq_orthocenter_of_forall_mem_altitude + - Affine.Simplex.eq_mongePoint_of_forall_mem_mongePlane + - Affine.Simplex.sum_mongePointWeightsWithCircumcenter + - Affine.Triangle.orthocenter_mem_altitude + - Affine.Triangle.altitude_replace_orthocenter_eq_affineSpan + - Affine.Simplex.direction_altitude + - Affine.Triangle.dist_orthocenter_reflection_circumcenter + - Affine.Simplex.mongePoint_eq_affineCombination_of_pointsWithCircumcenter + - EuclideanGeometry.affineSpan_of_orthocentricSystem + - Affine.Triangle.dist_orthocenter_reflection_circumcenter_finset + - Affine.Triangle.altitude_eq_mongePlane + - path: Mathlib/Data/QPF/Univariate/Basic.lean + theorems: + - QPF.comp_map + - QPF.Fix.ind_rec + - QPF.suppPreservation_iff_liftpPreservation + - QPF.Fix.dest_mk + - QPF.Wequiv.abs' + - QPF.Fix.rec_unique + - QPF.supp_eq_of_isUniform + - QPF.Cofix.bisim_aux + - QPF.supp_eq + - QPF.liftpPreservation_iff_uniform + - QPF.liftp_iff' + - QPF.id_map + - QPF.Cofix.bisim + - QPF.Fix.ind_aux + - QPF.liftp_iff + - QPF.Cofix.bisim_rel + - QPF.liftr_iff + - QPF.corecF_eq + - QPF.has_good_supp_iff + - QPF.liftp_iff_of_isUniform + - QPF.suppPreservation_iff_uniform + - QPF.supp_map + - QPF.Wequiv.symm + - QPF.mem_supp + - QPF.recF_eq + - QPF.Wequiv.refl + - QPF.Cofix.bisim' + - QPF.Wrepr_equiv + - QPF.Fix.mk_dest + - QPF.recF_eq_of_Wequiv + - path: Mathlib/Algebra/EuclideanDomain/Basic.lean + theorems: + - EuclideanDomain.xgcdAux_P + - EuclideanDomain.gcd_mul_lcm + - EuclideanDomain.xgcdAux_fst + - EuclideanDomain.div_self + - EuclideanDomain.dvd_mod_iff + - EuclideanDomain.mul_div_cancel_left + - EuclideanDomain.eq_div_of_mul_eq_left + - EuclideanDomain.mul_div_cancel' + - EuclideanDomain.gcd_zero_right + - EuclideanDomain.gcd_val + - EuclideanDomain.div_dvd_of_dvd + - EuclideanDomain.lcm_zero_left + - EuclideanDomain.gcd_eq_left + - EuclideanDomain.lcm_eq_zero_iff + - EuclideanDomain.dvd_lcm_left + - EuclideanDomain.dvd_div_of_mul_dvd + - EuclideanDomain.mul_div_assoc + - EuclideanDomain.zero_div + - EuclideanDomain.gcd_eq_gcd_ab + - EuclideanDomain.mul_div_cancel + - EuclideanDomain.mul_div_mul_comm_of_dvd_dvd + - EuclideanDomain.xgcdAux_val + - EuclideanDomain.lcm_dvd + - EuclideanDomain.gcd_dvd + - EuclideanDomain.gcd_eq_zero_iff + - EuclideanDomain.dvd_gcd + - EuclideanDomain.mod_eq_zero + - EuclideanDomain.dvd_lcm_right + - EuclideanDomain.mul_div_mul_cancel + - EuclideanDomain.lcm_zero_right + - path: Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean + theorems: + - AffineMap.lineMap_apply_zero + - AffineMap.lineMap_eq_right_iff + - AffineMap.decomp' + - AffineMap.lineMap_eq_left_iff + - AffineMap.lineMap_apply_module + - AffineMap.lineMap_symm + - AffineMap.right_vsub_lineMap + - AffineMap.lineMap_eq_lineMap_iff + - Convex.combo_affine_apply + - AffineMap.lineMap_apply_one + - AffineMap.lineMap_apply_one_sub + - AffineMap.apply_lineMap + - AffineMap.linear_eq_zero_iff_exists_const + - AffineMap.homothety_zero + - AffineMap.homothety_one + - AffineMap.linear_surjective_iff + - AffineMap.linearMap_vsub + - AffineMap.lineMap_same_apply + - AffineMap.decomp + - AffineMap.lineMap_vsub_right + - AffineMap.homothety_add + - AffineMap.lineMap_vadd_apply + - AffineMap.left_vsub_lineMap + - AffineMap.homothety_mul_apply + - AffineMap.linear_injective_iff + - AffineMap.image_uIcc + - path: Mathlib/Combinatorics/Colex.lean + theorems: + - Finset.geomSum_ofColex_strictMono + - Finset.Colex.toColex_sdiff_le_toColex_sdiff' + - Finset.Colex.lt_iff_max'_mem + - Finset.Colex.forall_le_mono + - Finset.Colex.toColex_le_toColex_iff_max'_mem + - Finset.toColex_inj + - Finset.Colex.max_mem_aux + - Finset.Colex.toColex_sdiff_le_toColex_sdiff + - Finset.Colex.toColex_lt_toColex_iff_exists_forall_lt + - Finset.Colex.IsInitSeg.exists_initSeg + - Finset.Colex.toColex_lt_singleton + - Finset.ofColex_inj + - Finset.Colex.IsInitSeg.total + - Finset.Colex.singleton_le_toColex + - Finset.Colex.mem_initSeg_self + - Finset.Colex.toColex_image_le_toColex_image + - Finset.Colex.singleton_le_singleton + - Finset.Colex.antisymm_aux + - Finset.Colex.toColex_lt_toColex + - Finset.Colex.isInitSeg_empty + - Finset.Colex.isInitSeg_initSeg + - Finset.Colex.mem_initSeg + - Finset.Colex.singleton_lt_singleton + - Finset.Colex.toColex_lt_toColex_iff_max'_mem + - Finset.Colex.trans_aux + - Finset.Colex.forall_lt_mono + - Finset.ofColex_ne_ofColex + - Finset.toColex_ne_toColex + - Finset.Colex.isInitSeg_iff_exists_initSeg + - Finset.Colex.le_iff_sdiff_subset_lowerClosure + - Finset.Colex.toColex_le_singleton + - path: Mathlib/Topology/Filter.lean + theorems: + - Filter.mem_closure + - Filter.nhds_top + - Filter.nhds_iInf + - Filter.tendsto_nhds_atTop_iff + - Filter.nhds_nhds + - Filter.HasBasis.nhds' + - Filter.closure_singleton + - Filter.specializes_iff_le + - Filter.nhds_inf + - Filter.isTopologicalBasis_Iic_principal + - Filter.mem_interior + - Filter.inducing_nhds + - Filter.nhds_mono + - Filter.isOpen_setOf_mem + - Filter.tendsto_nhds + - Filter.nhds_atTop + - Filter.nhds_pure + - Filter.HasBasis.nhds + - Filter.tendsto_pure_self + - path: Mathlib/Topology/Instances/Rat.lean + theorems: + - Nat.dist_cast_rat + - Rat.uniformContinuous_abs + - Int.uniformEmbedding_coe_rat + - Nat.closedEmbedding_coe_rat + - Int.dist_cast_rat + - Rat.uniformContinuous_neg + - Nat.uniformEmbedding_coe_rat + - Int.closedEmbedding_coe_rat + - Rat.uniformContinuous_add + - Rat.totallyBounded_Icc + - path: Mathlib/RingTheory/WittVector/StructurePolynomial.lean + theorems: + - "C_p_pow_dvd_bind\u2081_rename_wittPolynomial_sub_sum" + - constantCoeff_wittStructureRat + - wittStructureRat_prop + - witt_structure_prop + - wittStructureRat_rec + - constantCoeff_wittStructureInt_zero + - wittStructureInt_vars + - "bind\u2081_rename_expand_wittPolynomial" + - wittStructureRat_existsUnique + - wittStructureRat_rec_aux + - wittStructureInt_prop + - eq_wittStructureInt + - map_wittStructureInt + - constantCoeff_wittStructureInt + - wittStructureRat_vars + - constantCoeff_wittStructureRat_zero + - wittStructureInt_rename + - path: Mathlib/Algebra/Order/CauSeq/Basic.lean + theorems: + - CauSeq.neg_limZero + - rat_add_continuous_lemma + - CauSeq.inf_eq_right + - CauSeq.pow_equiv_pow + - CauSeq.trichotomy + - CauSeq.exists_gt + - CauSeq.smul_equiv_smul + - CauSeq.mul_inv_cancel + - CauSeq.mul_limZero_right + - CauSeq.sup_eq_left + - CauSeq.sup_lt + - CauSeq.mul_equiv_zero + - CauSeq.sub_limZero + - CauSeq.const_le + - CauSeq.le_inf + - CauSeq.sup_equiv_sup + - CauSeq.add_equiv_add + - CauSeq.of_near + - CauSeq.const_equiv + - CauSeq.inf_limZero + - CauSeq.mul_equiv_mul + - CauSeq.mul_not_equiv_zero + - CauSeq.zero_limZero + - rat_inv_continuous_lemma + - CauSeq.lt_irrefl + - CauSeq.one_not_equiv_zero + - CauSeq.pos_add_limZero + - rat_mul_continuous_lemma + - CauSeq.lt_of_lt_of_eq + - CauSeq.sup_le + - CauSeq.lt_inf + - CauSeq.not_limZero_of_not_congr_zero + - CauSeq.inf_eq_left + - CauSeq.sup_limZero + - CauSeq.inf_equiv_inf + - CauSeq.inv_mul_cancel + - IsCauSeq.bounded + - "CauSeq.equiv_def\u2083" + - CauSeq.lt_trans + - IsCauSeq.const + - CauSeq.mul_limZero_left + - CauSeq.limZero_sub_rev + - CauSeq.const_inv + - CauSeq.sub_equiv_sub + - CauSeq.neg_equiv_neg + - CauSeq.exists_lt + - CauSeq.add_limZero + - CauSeq.lt_total + - CauSeq.const_lt + - CauSeq.limZero_congr + - CauSeq.lt_of_eq_of_lt + - CauSeq.sup_eq_right + - isCauSeq_neg + - CauSeq.abv_pos_of_not_limZero + - CauSeq.mul_equiv_zero' + - "IsCauSeq.cauchy\u2082" + - path: Mathlib/CategoryTheory/Adjunction/Evaluation.lean + theorems: + - CategoryTheory.NatTrans.mono_iff_mono_app + - CategoryTheory.NatTrans.epi_iff_epi_app + - path: Mathlib/NumberTheory/Liouville/Residual.lean + theorems: + - setOf_liouville_eq_iInter_iUnion + - eventually_residual_liouville + - setOf_liouville_eq_irrational_inter_iInter_iUnion + - "IsG\u03B4.setOf_liouville" + - path: Mathlib/CategoryTheory/Generator.lean + theorems: + - CategoryTheory.isCoseparator_pi_of_isCoseparator + - CategoryTheory.isCoseparating_iff_mono + - CategoryTheory.IsCodetecting.isCoseparating + - CategoryTheory.isDetecting_op_iff + - CategoryTheory.isCoseparator_prod + - CategoryTheory.isCoseparating_op_iff + - CategoryTheory.isDetector_iff_reflectsIsomorphisms_coyoneda_obj + - CategoryTheory.isSeparator_coprod_of_isSeparator_right + - CategoryTheory.isSeparator_op_iff + - CategoryTheory.wellPowered_of_isDetecting + - CategoryTheory.isSeparator_coprod + - CategoryTheory.isCoseparator_op_iff + - CategoryTheory.isCoseparating_unop_iff + - CategoryTheory.isCoseparator_pi + - CategoryTheory.isCoseparator_iff_mono + - CategoryTheory.isCodetector_iff_reflectsIsomorphisms_yoneda_obj + - CategoryTheory.isSeparator_sigma + - CategoryTheory.isCodetector_unop_iff + - CategoryTheory.isCodetecting_op_iff + - CategoryTheory.isDetector_unop_iff + - CategoryTheory.isCodetecting_unop_iff + - CategoryTheory.isSeparator_sigma_of_isSeparator + - CategoryTheory.isSeparating_iff_epi + - CategoryTheory.isSeparator_iff_epi + - CategoryTheory.StructuredArrow.isCoseparating_proj_preimage + - CategoryTheory.isDetector_def + - CategoryTheory.isCoseparator_def + - CategoryTheory.isCodetector_op_iff + - CategoryTheory.IsCoseparating.isCodetecting + - CategoryTheory.isCoseparator_prod_of_isCoseparator_left + - CategoryTheory.Subobject.eq_of_le_of_isDetecting + - CategoryTheory.isSeparator_def + - CategoryTheory.hasTerminal_of_isSeparating + - CategoryTheory.CostructuredArrow.isSeparating_proj_preimage + - CategoryTheory.isSeparator_unop_iff + - CategoryTheory.isSeparating_op_iff + - CategoryTheory.isSeparator_coprod_of_isSeparator_left + - CategoryTheory.isDetecting_unop_iff + - CategoryTheory.isSeparating_unop_iff + - CategoryTheory.isDetector_op_iff + - CategoryTheory.IsSeparating.isDetecting + - CategoryTheory.isCoseparator_prod_of_isCoseparator_right + - CategoryTheory.hasInitial_of_isCoseparating + - CategoryTheory.isCoseparator_unop_iff + - CategoryTheory.isCodetector_def + - path: Mathlib/Data/Finset/Image.lean + theorems: + - Finset.subtype_map_of_mem + - Finset.filterMap_mono + - Finset.property_of_mem_map_subtype + - Finset.image_nonempty + - Finset.image_union + - Finset.map_cons + - Finset.map_refl + - Finset.subtype_map + - Finset.image_singleton + - Finset.subtype_eq_empty + - Finset.image_subset_image + - Finset.map_cast_heq + - Finset.map_toFinset + - Finset.image_inter_of_injOn + - Finset.coe_filterMap + - Finset.map_singleton + - Finset.filterMap_some + - Finset.attach_map_val + - Finset.mem_fin + - Finset.map_nonempty + - Finset.image_sdiff + - Finset.fin_mono + - Finset.disjoint_image + - Finset.image_erase + - Finset.image_toFinset + - Finset.subset_image_iff + - Finset.mem_subtype + - Finset.range_add + - Equiv.finsetCongr_trans + - Finset.disjoint_range_addRightEmbedding + - Finset.coe_map + - Finset.disjoint_range_addLeftEmbedding + - Finset.map_insert + - Finset.map_subtype_subset + - Finset.biUnion_image + - Finset.subset_map_symm + - Equiv.finsetCongr_refl + - Finset.disjoint_map + - Finset.image_congr + - Finset.map_filter + - Finset.map_map + - Finset.forall_mem_map + - Finset.coe_image + - Finset.range_add_one' + - Finset.disjiUnion_map + - Finset.image_id + - Finset.mem_image + - Finset.mem_range_iff_mem_finset_range_of_mod_eq' + - Finset.image_symmDiff + - Finset.image_injective + - Finset.mem_map_equiv + - Finset.image_insert + - Finset.image_ssubset_image + - Finset.fin_map + - Finset.image_subset_iff + - Finset.image_subset_image_iff + - Finset.map_symm_subset + - Finset.mem_range_iff_mem_finset_range_of_mod_eq + - Finset.fiber_nonempty_iff_mem_image + - Finset.range_sdiff_zero + - Finset.forall_image + - Finset.filter_mem_image_eq_image + - Finset.map_inter + - Finset.mem_image_const + - Finset.image_const + - Finset.map_comm + - Finset.biUnion_singleton + - Finset.erase_image_subset_image_erase + - Finset.filter_image + - Function.Injective.mem_finset_image + - Finset.map_union + - Finset.image_comm + - Finset.attach_image_val + - Finset.image_image + - Finset.map_erase + - path: Mathlib/AlgebraicTopology/SimplicialObject.lean + theorems: + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_of_le" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03B4_self" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_self'" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_of_le" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_of_gt" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03B4_self'" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03B4'" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_self" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_succ'" + - CategoryTheory.SimplicialObject.eqToIso_refl + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03B4" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_of_gt'" + - CategoryTheory.SimplicialObject.augment_hom_zero + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_self" + - "CategoryTheory.SimplicialObject.\u03C3_comp_\u03C3" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_succ'" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_succ" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_self'" + - "CategoryTheory.SimplicialObject.Augmented.w\u2080" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03B4_self'" + - CategoryTheory.CosimplicialObject.augment_hom_zero + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03B4" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03B4''" + - "CategoryTheory.CosimplicialObject.\u03C3_comp_\u03C3" + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_of_gt'" + - CategoryTheory.CosimplicialObject.hom_ext + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03C3_succ" + - CategoryTheory.SimplicialObject.hom_ext + - "CategoryTheory.CosimplicialObject.\u03B4_comp_\u03C3_of_gt" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03B4_self" + - CategoryTheory.CosimplicialObject.eqToIso_refl + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03B4'" + - "CategoryTheory.SimplicialObject.\u03B4_comp_\u03B4''" + - path: Mathlib/Analysis/SpecialFunctions/CompareExp.lean + theorems: + - Complex.IsExpCmpFilter.isTheta_cpow_exp_re_mul_log + - Complex.IsExpCmpFilter.isLittleO_exp_cpow + - Complex.IsExpCmpFilter.of_boundedUnder_abs_im + - Complex.IsExpCmpFilter.of_isBigO_im_re_rpow + - Complex.IsExpCmpFilter.abs_im_pow_eventuallyLE_exp_re + - Complex.IsExpCmpFilter.isLittleO_cpow_mul_exp + - Complex.IsExpCmpFilter.isLittleO_pow_mul_exp + - Complex.IsExpCmpFilter.isLittleO_cpow_exp + - Complex.IsExpCmpFilter.isLittleO_log_abs_re + - Complex.IsExpCmpFilter.isLittleO_im_pow_exp_re + - Complex.IsExpCmpFilter.isLittleO_zpow_mul_exp + - path: Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean + theorems: + - Besicovitch.SatelliteConfig.exists_normalized_aux3 + - Besicovitch.SatelliteConfig.exists_normalized_aux1 + - Besicovitch.SatelliteConfig.exists_normalized + - Besicovitch.card_le_of_separated + - Besicovitch.card_le_multiplicity + - "Besicovitch.one_lt_good\u03C4" + - Besicovitch.multiplicity_le + - Besicovitch.SatelliteConfig.exists_normalized_aux2 + - Besicovitch.isEmpty_satelliteConfig_multiplicity + - Besicovitch.SatelliteConfig.centerAndRescale_radius + - "Besicovitch.le_multiplicity_of_\u03B4_of_fin" + - "Besicovitch.exists_good\u03B4" + - Besicovitch.SatelliteConfig.centerAndRescale_center + - path: Mathlib/Data/List/Infix.lean + theorems: + - List.prefix_take_le_iff + - List.cons_prefix_iff + - List.tails_eq_inits + - List.insert_eq_ite + - List.tails_cons + - List.tails_reverse + - List.mem_tails + - List.dropSlice_sublist + - List.nth_le_tails + - List.tail_suffix + - List.tails_append + - List.IsPrefix.filter_map + - List.prefix_concat + - List.suffix_iff_eq_append + - List.inits_reverse + - List.length_tails + - List.inits_eq_tails + - List.inits_append + - List.nth_le_inits + - List.dropLast_prefix + - List.inits_cons + - List.length_inits + - List.IsPrefix.map + - List.mem_inits + - List.map_reverse_tails + - List.suffix_insert + - List.prefix_iff_eq_append + - path: Mathlib/Analysis/SpecialFunctions/Gaussian.lean + theorems: + - integrable_cexp_quadratic' + - rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg + - GaussianFourier.integral_cexp_neg_sum_mul_add + - integrableOn_rpow_mul_exp_neg_rpow + - GaussianFourier.integral_cexp_neg_mul_sum_add + - fourierIntegral_gaussian_innerProductSpace + - GaussianFourier.integrable_cexp_neg_mul_sum_add + - integrable_rpow_mul_exp_neg_mul_sq + - cexp_neg_quadratic_isLittleO_abs_rpow_cocompact + - integral_cexp_quadratic + - integral_mul_cexp_neg_mul_sq + - GaussianFourier.integral_rexp_neg_mul_sq_norm + - GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I' + - Real.Gamma_one_half_eq + - integral_gaussian_complex_Ioi + - fourierIntegral_gaussian_innerProductSpace' + - rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg + - fourierIntegral_gaussian_pi + - GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace + - GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I + - integrable_cexp_quadratic + - integrableOn_Ioi_exp_neg_mul_sq_iff + - integrable_cexp_neg_mul_sq + - isLittleO_exp_neg_mul_sq_cocompact + - cexp_neg_quadratic_isLittleO_rpow_atTop + - continuousAt_gaussian_integral + - GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I + - GaussianFourier.integral_cexp_neg_mul_sq_norm + - fourierIntegral_gaussian_pi' + - GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I + - GaussianFourier.integral_cexp_neg_mul_sq_norm_add + - integrable_mul_exp_neg_mul_sq + - GaussianFourier.verticalIntegral_norm_le + - Real.tsum_exp_neg_mul_int_sq + - integral_gaussian_Ioi + - tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact + - integrableOn_rpow_mul_exp_neg_mul_rpow + - integrableOn_rpow_mul_exp_neg_mul_sq + - Complex.Gamma_one_half_eq + - integral_gaussian + - Complex.tsum_exp_neg_quadratic + - fourierIntegral_gaussian + - Complex.tsum_exp_neg_mul_int_sq + - integral_gaussian_complex + - GaussianFourier.integrable_cexp_neg_sum_mul_add + - rexp_neg_quadratic_isLittleO_rpow_atTop + - GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace + - GaussianFourier.tendsto_verticalIntegral + - integral_gaussian_sq_complex + - exp_neg_mul_sq_isLittleO_exp_neg + - exp_neg_mul_rpow_isLittleO_exp_neg + - integrable_exp_neg_mul_sq + - path: Mathlib/Data/Set/Subset.lean + theorems: + - Set.preimage_val_sInter_eq_sInter + - Set.image_val_compl + - Set.eq_of_preimage_val_eq_of_subset + - Set.subset_preimage_val_image_val_iff + - Set.preimage_val_sInter + - Set.preimage_val_sUnion + - Set.preimage_val_eq_univ_of_subset + - Set.image_val_sInter + - path: Mathlib/Topology/Compactness/Compact.lean + theorems: + - isCompact_iff_isCompact_univ + - isCompact_iff_ultrafilter_le_nhds + - Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact + - ClosedEmbedding.compactSpace + - Inducing.isCompact_preimage_iff + - Set.Infinite.exists_accPt_cofinite_inf_principal + - Filter.hasBasis_coclosedCompact + - generalized_tube_lemma + - Filter.mem_coclosedCompact_iff + - Ultrafilter.le_nhds_lim + - IsCompact.inter_iInter_nonempty + - LocallyFinite.finite_nonempty_inter_compact + - IsCompact.elim_finite_subcover_image + - IsCompact.le_nhds_of_unique_clusterPt + - IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed + - Set.Finite.isCompact_sUnion + - IsCompact.disjoint_nhdsSet_right + - Inducing.isCompact_iff + - IsCompact.eventually_forall_of_forall_eventually + - Set.Finite.isCompact_biUnion + - IsCompact.elim_nhds_subcover_nhdsSet + - IsCompact.elim_directed_family_closed + - isCompact_iff_ultrafilter_le_nhds' + - Filter.comap_cocompact_le + - isCompact_singleton + - Pi.isCompact_iff_of_isClosed + - Filter.mem_coclosedCompact + - IsCompact.elim_finite_subfamily_closed + - IsCompact.compl_mem_sets + - IsCompact.nhdsSet_prod_eq + - IsCompact.finite_of_discrete + - "Filter.coprod\u1D62_cocompact" + - IsCompact.image_of_continuousOn + - IsCompact.mem_nhdsSet_prod_of_forall + - IsClosed.exists_minimal_nonempty_closed_subset + - Filter.Tendsto.isCompact_insert_range_of_cocompact + - exists_subset_nhds_of_isCompact' + - CompactSpace.elim_nhds_subcover + - IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed + - IsCompact.disjoint_nhdsSet_left + - Prod.noncompactSpace_iff + - compactSpace_of_finite_subfamily_closed + - IsCompact.elim_nhds_subcover_nhdsSet' + - Filter.mem_coclosed_compact' + - IsCompact.compl_mem_sets_of_nhdsWithin + - isCompact_pi_infinite + - isCompact_of_finite_subfamily_closed + - exists_clusterPt_of_compactSpace + - IsCompact.mem_prod_nhdsSet_of_forall + - LocallyFinite.finite_of_compact + - Pi.exists_compact_superset_iff + - isClosedMap_snd_of_compactSpace + - Bornology.inCompact.isBounded_iff + - Filter.coprod_cocompact + - Inducing.isCompact_preimage + - Filter.cocompact_eq_cofinite + - IsCompact.prod_nhdsSet_eq_biSup + - LocallyFinite.finite_nonempty_of_compact + - isCompact_of_finite_subcover + - IsCompact.prod + - isCompact_range + - isCompact_univ_pi + - isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis + - IsCompact.induction_on + - Filter.Tendsto.isCompact_insert_range_of_cofinite + - Filter.compl_mem_coclosedCompact + - exists_nhds_ne_neBot + - path: Mathlib/CategoryTheory/Sites/Over.lean + theorems: + - CategoryTheory.Sieve.overEquiv_symm_iff + - CategoryTheory.Sieve.functorPushforward_over_map + - CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving + - CategoryTheory.GrothendieckTopology.overEquiv_symm_mem_over + - CategoryTheory.Sieve.overEquiv_iff + - CategoryTheory.GrothendieckTopology.over_forget_compatiblePreserving + - CategoryTheory.Sieve.overEquiv_pullback + - CategoryTheory.Sieve.overEquiv_top + - CategoryTheory.GrothendieckTopology.over_map_coverPreserving + - CategoryTheory.Sieve.overEquiv_symm_top + - CategoryTheory.GrothendieckTopology.mem_over_iff + - path: Mathlib/RingTheory/DedekindDomain/AdicValuation.lean + theorems: + - IsDedekindDomain.HeightOneSpectrum.IntValuation.map_mul' + - IsDedekindDomain.HeightOneSpectrum.int_valuation_lt_one_iff_dvd + - IsDedekindDomain.HeightOneSpectrum.IntValuation.map_add_le_max' + - IsDedekindDomain.HeightOneSpectrum.int_valuation_zero_le + - IsDedekindDomain.HeightOneSpectrum.valuation_le_one + - IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_dvd + - IsDedekindDomain.HeightOneSpectrum.valuation_exists_uniformizer + - IsDedekindDomain.HeightOneSpectrum.int_valuation_le_pow_iff_dvd + - IsDedekindDomain.HeightOneSpectrum.int_valuation_le_one + - IsDedekindDomain.HeightOneSpectrum.IntValuation.le_max_iff_min_le + - IsDedekindDomain.HeightOneSpectrum.int_valuation_ne_zero + - IsDedekindDomain.HeightOneSpectrum.int_valuation_exists_uniformizer + - IsDedekindDomain.HeightOneSpectrum.IntValuation.map_one' + - IsDedekindDomain.HeightOneSpectrum.valuation_of_mk' + - IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap + - path: Mathlib/MeasureTheory/Measure/MeasureSpace.lean + theorems: + - MeasurableEquiv.comap_symm + - MeasureTheory.liminf_ae_eq_of_forall_ae_eq + - "MeasureTheory.Measure.map\u2097_mk_apply_of_aemeasurable" + - MeasurableEquiv.map_symm + - MeasureTheory.Measure.sum_add_sum_compl + - MeasureTheory.Measure.sum_comm + - MeasureTheory.Measure.sum_sum + - MeasureTheory.measure_iUnion_eq_iSup + - MeasureTheory.Measure.lt_iff + - MeasurableEquiv.map_apply_eq_iff_map_symm_apply_eq + - MeasureTheory.measure_diff' + - MeasurableEquiv.map_map_symm + - MeasureTheory.tendsto_measure_iInter + - MeasureTheory.Measure.map_mono + - MeasureTheory.Measure.sum_apply_eq_zero + - MeasureTheory.Measure.measure_toMeasurable_add_inter_right + - MeasureTheory.ae_eq_bot + - MeasureTheory.measure_union_add_inter + - MeasureTheory.Measure.toOuterMeasure_top + - MeasureTheory.Measure.sum_comp_equiv + - MeasureTheory.measure_inter_conull' + - MeasureTheory.measure_if + - MeasureTheory.Measure.AbsolutelyContinuous.add_right + - "MeasureTheory.Measure.sum_apply\u2080" + - MeasureTheory.measure_iUnion_congr_of_subset + - MeasureTheory.Measure.sum_apply_eq_zero' + - MeasureTheory.Measure.sum_apply_of_countable + - MeasureTheory.Measure.QuasiMeasurePreserving.image_zpow_ae_eq + - MeasureTheory.Measure.comap_preimage + - MeasurableEmbedding.map_apply + - MeasureTheory.Measure.lt_iff' + - MeasureTheory.Measure.sum_bool + - MeasureTheory.tendsto_measure_iUnion + - MeasureTheory.Measure.le_intro + - MeasureTheory.mem_ae_map_iff + - MeasureTheory.measure_sUnion + - MeasureTheory.Measure.sum_fintype + - MeasureTheory.Measure.QuasiMeasurePreserving.smul_ae_eq_of_ae_eq + - MeasureTheory.measure_diff_lt_of_lt_add + - MeasureTheory.Measure.AbsolutelyContinuous.zero + - MeasureTheory.measure_diff_le_iff_le_add + - MeasureTheory.Measure.sum_extend_zero + - MeasureTheory.Measure.measure_toMeasurable_add_inter_left + - MeasureTheory.Measure.NullMeasurableSet.image + - MeasureTheory.measure_biUnion_eq_iSup + - MeasureTheory.Measure.nnreal_smul_coe_apply + - MeasureTheory.measure_preimage_eq_zero_iff_of_countable + - MeasureTheory.Measure.map_zero + - MeasureTheory.Iio_ae_eq_Iic' + - MeasureTheory.measure_iInter_eq_iInf + - MeasureTheory.measure_symmDiff_eq + - MeasureTheory.Measure.le_comap_apply + - MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap + - MeasureTheory.Measure.QuasiMeasurePreserving.comp + - MeasurableEquiv.map_symm_map + - "MeasureTheory.toMeasure_apply\u2080" + - MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_sum_measure + - MeasureTheory.tsum_measure_preimage_singleton + - MeasureTheory.Measure.map_of_not_aemeasurable + - MeasureTheory.union_ae_eq_right_iff_ae_subset + - "MeasureTheory.Measure.map\u2097_apply_of_measurable" + - MeasureTheory.union_ae_eq_left_iff_ae_subset + - MeasureTheory.ae_eq_of_ae_subset_of_measure_ge + - MeasureTheory.measure_eq_measure_of_between_null_diff + - "MeasureTheory.Measure.comap\u2097_apply" + - MeasureTheory.Measure.finset_sum_apply + - MeasurableEmbedding.comap_add + - MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq + - MeasureTheory.tendsto_measure_Ico_atTop + - MeasureTheory.biSup_measure_Iic + - MeasureTheory.Measure.sum_add_sum + - MeasureTheory.measure_union_congr_of_subset + - MeasureTheory.OuterMeasure.toMeasure_top + - MeasureTheory.nonempty_inter_of_measure_lt_add + - MeasureTheory.Measure.measure_inter_eq_of_measure_eq + - MeasureTheory.measure_liminf_eq_zero + - "MeasureTheory.Measure.comap_apply\u2080" + - MeasureTheory.tendsto_measure_Ioc_atBot + - MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_tsum_measure + - MeasureTheory.Measure.map_smul + - MeasureTheory.measure_union_add_inter' + - "MeasureTheory.measure_compl\u2080" + - MeasureTheory.Measure.map_toOuterMeasure + - MeasureTheory.Measure.map_add + - MeasureTheory.Measure.map_congr + - MeasureTheory.Measure.sum_of_empty + - "MeasureTheory.tsum_meas_le_meas_iUnion_of_disjoint\u2080" + - MeasureTheory.Measure.AbsolutelyContinuous.add + - MeasureTheory.measure_inter_conull + - MeasureTheory.Measure.measure_eq_right_of_subset_of_measure_add_eq + - MeasureTheory.Measure.compl_mem_cofinite + - MeasureTheory.Measure.QuasiMeasurePreserving.preimage_iterate_ae_eq + - MeasureTheory.sum_measure_le_measure_univ + - MeasureTheory.Measure.AbsolutelyContinuous.map + - MeasureTheory.ae_uIoc_iff + - MeasureTheory.Measure.QuasiMeasurePreserving.limsup_preimage_iterate_ae_eq + - MeasureTheory.OuterMeasure.toMeasure_zero + - MeasureTheory.Measure.absolutelyContinuous_smul + - MeasureTheory.limsup_ae_eq_of_forall_ae_eq + - MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one + - MeasureTheory.Measure.QuasiMeasurePreserving.liminf_preimage_iterate_ae_eq + - MeasureTheory.measure_add_diff + - MeasureTheory.measure_diff + - MeasureTheory.tendsto_measure_biInter_gt + - MeasureTheory.Measure.map_map + - "MeasureTheory.Measure.map_apply\u2080" + - MeasurableEquiv.quasiMeasurePreserving_symm + - MeasureTheory.Measure.sum_coe_finset + - MeasureTheory.Measure.AbsolutelyContinuous.mk + - MeasureTheory.nonempty_inter_of_measure_lt_add' + - MeasureTheory.tsum_measure_le_measure_univ + - MeasureTheory.Measure.ae_le_iff_absolutelyContinuous + - MeasurableEquiv.map_measurableEquiv_injective + - "MeasureTheory.measure_biUnion_finset\u2080" + - MeasureTheory.Measure.apply_eq_zero_of_isEmpty + - "MeasureTheory.Measure.map\u2097_eq_zero_iff" + - MeasureTheory.Measure.sInf_caratheodory + - MeasureTheory.tendsto_measure_Iic_atTop + - MeasureTheory.Measure.le_sum + - MeasurableEquiv.map_ae + - MeasureTheory.Measure.measure_sInf_le + - MeasureTheory.Measure.map_eq_zero_iff + - "MeasureTheory.measure_biUnion\u2080" + - MeasureTheory.Measure.le_iff + - MeasureTheory.Measure.ae_smul_measure_iff + - "MeasureTheory.Measure.map\u2097_congr" + - MeasureTheory.measure_biUnion_toMeasurable + - MeasureTheory.measure_limsup_eq_zero + - path: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean + theorems: + - NNReal.rpow_self_rpow_inv + - NNReal.le_rpow_one_div_iff + - ENNReal.mul_rpow_eq_ite + - ENNReal.rpow_eq_top_iff + - ENNReal.rpow_nat_cast + - ENNReal.rpow_pos_of_nonneg + - ENNReal.rpow_mul_intCast + - ENNReal.rpow_zero + - ENNReal.ofReal_rpow_of_pos + - ENNReal.rpow_le_rpow_of_exponent_ge + - NNReal.eq_rpow_one_div_iff + - ENNReal.div_rpow_of_nonneg + - ENNReal.toNNReal_rpow + - ENNReal.rpow_lt_rpow_of_exponent_gt + - ENNReal.prod_rpow_of_nonneg + - ENNReal.prod_coe_rpow + - ENNReal.mul_rpow_of_ne_zero + - ENNReal.one_rpow + - NNReal.rpow_rpow_inv + - ENNReal.orderIsoRpow_symm_apply + - ENNReal.rpow_le_self_of_le_one + - ENNReal.rpow_sub + - ENNReal.rpow_eq_zero_iff_of_pos + - ENNReal.strictMono_rpow_of_pos + - ENNReal.rpow_eq_top_iff_of_pos + - NNReal.rpow_left_surjective + - ENNReal.coe_rpow_of_nonneg + - ENNReal.rpow_inv_rpow + - NNReal.rpow_left_injective + - NNReal.rpow_eq_zero_iff + - ENNReal.rpow_neg_one + - ENNReal.rpow_pos + - Real.list_prod_map_rpow' + - ENNReal.zero_rpow_of_neg + - ENNReal.le_rpow_one_div_iff + - ENNReal.prod_rpow_of_ne_top + - NNReal.rpow_one_div_eq_iff + - NNReal.strictMono_rpow_of_pos + - ENNReal.rpow_le_rpow_of_exponent_le + - NNReal.rpow_pos + - NNReal.rpow_of_add_eq + - ENNReal.le_rpow_self_of_one_le + - NNReal.rpow_one_div_le_iff + - ENNReal.rpow_le_one + - ENNReal.rpow_inv_natCast_pow + - ENNReal.rpow_lt_one + - NNReal.monotone_rpow_of_nonneg + - ENNReal.rpow_lt_top_iff_of_pos + - NNReal.pow_rpow_inv_natCast + - ENNReal.zero_rpow_mul_self + - ENNReal.one_lt_rpow + - ENNReal.zero_rpow_def + - NNReal.rpow_le_self_of_le_one + - ENNReal.rpow_eq_top_of_nonneg + - ENNReal.ofReal_rpow_of_nonneg + - ENNReal.rpow_int_cast + - ENNReal.rpow_neg + - ENNReal.rpow_le_one_of_one_le_of_neg + - NNReal.list_prod_map_rpow' + - Real.list_prod_map_rpow + - ENNReal.rpow_mul_natCast + - Real.toNNReal_rpow_of_nonneg + - ENNReal.pow_rpow_inv_natCast + - NNReal.rpow_inv_rpow_self + - NNReal.rpow_nat_cast + - ENNReal.mul_rpow_of_ne_top + - ENNReal.rpow_lt_one_of_one_lt_of_neg + - ENNReal.rpow_natCast_mul + - ENNReal.one_le_rpow_of_pos_of_le_one_of_neg + - ENNReal.monotone_rpow_of_nonneg + - ENNReal.one_le_rpow + - ENNReal.inv_rpow + - NNReal.sqrt_eq_rpow + - Real.multiset_prod_map_rpow + - ENNReal.rpow_rpow_inv + - ENNReal.zero_rpow_of_pos + - ENNReal.rpow_intCast_mul + - ENNReal.lt_rpow_one_div_iff + - ENNReal.rpow_one + - ENNReal.rpow_lt_rpow_of_exponent_lt + - ENNReal.rpow_add + - ENNReal.one_lt_rpow_of_pos_of_lt_one_of_neg + - ENNReal.rpow_eq_zero_iff + - ENNReal.rpow_mul + - NNReal.rpow_inv_natCast_pow + - ENNReal.mul_rpow_of_nonneg + - ENNReal.rpow_one_div_le_iff + - NNReal.rpow_inv_rpow + - ENNReal.toReal_rpow + - ENNReal.coe_rpow_of_ne_zero + - path: Mathlib/Algebra/CharP/Basic.lean + theorems: + - CharP.neg_one_pow_char_pow + - CharP.char_is_prime_of_two_le + - add_pow_char_pow_of_commute + - charP_of_prime_pow_injective + - CharP.neg_one_ne_one + - CharP.neg_one_pow_char + - CharP.ringChar_ne_one + - CharP.cast_eq_mod + - add_pow_char_of_commute + - CharP.intCast_eq_intCast + - ringChar.spec + - CharP.exists + - CharP.natCast_eq_natCast + - CharP.natCast_eq_natCast' + - CharZero.charZero_iff_forall_prime_ne_zero + - Commute.add_pow_prime_pow_eq + - CharP.nontrivial_of_char_ne_one + - NeZero.not_char_dvd + - CharP.ringChar_zero_iff_CharZero + - CharP.int_cast_eq_zero_iff + - CharP.char_ne_one + - CharP.exists' + - RingHom.charP_iff_charP + - CharP.char_ne_zero_of_finite + - Commute.add_pow_prime_eq + - sub_pow_char_of_commute + - ringChar.Nat.cast_ringChar + - CharP.addOrderOf_one + - CharP.cast_card_eq_zero + - charP_of_ne_zero + - Int.cast_injOn_of_ringChar_ne_two + - sub_pow_char_pow_of_commute + - path: Mathlib/Data/Nat/Sqrt.lean + theorems: + - Nat.le_sqrt' + - Nat.exists_mul_self + - Nat.exists_mul_self' + - Nat.sqrt_add_eq + - Nat.sqrt_succ_le_succ_sqrt + - Nat.not_exists_sq + - Nat.sqrt_eq_zero + - Nat.not_exists_sq' + - Nat.eq_sqrt' + - Nat.sqrt_isSqrt + - path: Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean + theorems: + - MeasurableSpace.measurableSet_succ_memPartition + - MeasurableSpace.CountablyGenerated.sup + - MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated + - MeasurableSpace.generateFrom_iUnion_countablePartition + - MeasurableSpace.generateFrom_iUnion_memPartition + - MeasurableSpace.measurableSet_countableGeneratingSet + - MeasurableSpace.CountablyGenerated.comap + - MeasurableSpace.measurableSet_generateFrom_memPartition + - MeasurableSpace.generateFrom_iUnion_memPartition_le + - MeasurableSpace.measurableSet_generateFrom_memPartition_iff + - MeasurableSpace.generateFrom_memPartition_le_range + - path: Mathlib/Algebra/Group/Pi/Basic.lean + theorems: + - Function.comp_eq_one_iff + - Pi.mulSingle_comm + - Sum.elim_mul_mul + - Pi.apply_mulSingle + - Function.extend_inv + - Function.extend_one + - Function.extend_mul + - Function.extend_div + - Subsingleton.pi_mulSingle_eq + - "Pi.apply_mulSingle\u2082" + - Sum.elim_mulSingle_one + - Sum.elim_div_div + - Sum.elim_one_mulSingle + - path: Mathlib/CategoryTheory/Subobject/Lattice.lean + theorems: + - CategoryTheory.Subobject.bot_eq_zero + - CategoryTheory.Subobject.le_sInf + - CategoryTheory.Subobject.mk_eq_bot_iff_zero + - CategoryTheory.Subobject.inf_map + - CategoryTheory.Subobject.inf_eq_map_pullback' + - CategoryTheory.Subobject.prod_eq_inf + - CategoryTheory.Subobject.inf_factors + - CategoryTheory.Subobject.inf_pullback + - CategoryTheory.Subobject.inf_arrow_factors_left + - CategoryTheory.Subobject.inf_arrow_factors_right + - CategoryTheory.Subobject.le_sSup + - CategoryTheory.Subobject.finset_inf_factors + - CategoryTheory.Subobject.sInf_le + - CategoryTheory.Subobject.isIso_iff_mk_eq_top + - CategoryTheory.Subobject.symm_apply_mem_iff_mem_image + - CategoryTheory.Subobject.le_inf + - CategoryTheory.Subobject.finset_sup_factors + - CategoryTheory.Subobject.sSup_le + - CategoryTheory.Subobject.bot_factors_iff_zero + - CategoryTheory.Subobject.isIso_arrow_iff_eq_top + - CategoryTheory.Subobject.underlyingIso_top_hom + - CategoryTheory.Subobject.finset_inf_arrow_factors + - path: Mathlib/CategoryTheory/Monoidal/Bimod.lean + theorems: + - Bimod.LeftUnitorBimod.inv_hom_id + - Bimod.whiskerRight_id_bimod + - Bimod.id_whiskerLeft_bimod + - "Bimod.TensorBimod.\u03C0_tensor_id_actRight" + - Bimod.triangle_bimod + - Bimod.RightUnitorBimod.inv_hom_id + - Bimod.LeftUnitorBimod.hom_inv_id + - Bimod.TensorBimod.one_act_left' + - Bimod.pentagon_bimod + - Bimod.whiskerLeft_id_bimod + - Bimod.whiskerLeft_comp_bimod + - Bimod.RightUnitorBimod.hom_left_act_hom' + - Bimod.whisker_assoc_bimod + - Bimod.TensorBimod.actRight_one' + - Bimod.TensorBimod.left_assoc' + - Bimod.AssociatorBimod.inv_hom_id + - Bimod.comp_whiskerRight_bimod + - Bimod.TensorBimod.middle_assoc' + - Bimod.whiskerRight_comp_bimod + - Bimod.LeftUnitorBimod.hom_right_act_hom' + - Bimod.AssociatorBimod.hom_left_act_hom' + - Bimod.RightUnitorBimod.hom_right_act_hom' + - Bimod.whisker_exchange_bimod + - "Bimod.TensorBimod.whiskerLeft_\u03C0_actLeft" + - Bimod.comp_whiskerLeft_bimod + - Bimod.AssociatorBimod.hom_inv_id + - Bimod.AssociatorBimod.hom_right_act_hom' + - Bimod.RightUnitorBimod.hom_inv_id + - Bimod.LeftUnitorBimod.hom_left_act_hom' + - Bimod.TensorBimod.right_assoc' + - Bimod.id_whiskerRight_bimod + - path: Mathlib/Analysis/Calculus/FDeriv/Comp.lean + theorems: + - HasStrictFDerivAt.iterate + - HasFDerivWithinAt.iterate + - HasFDerivAt.iterate + - "fderivWithin.comp\u2083" + - HasFDerivAtFilter.comp + - fderivWithin_fderivWithin + - HasStrictFDerivAt.comp + - HasFDerivAtFilter.iterate + - path: Mathlib/Analysis/InnerProductSpace/Projection.lean + theorems: + - orthogonalProjection_orthogonal_val + - OrthogonalFamily.isInternal_iff_of_isComplete + - orthogonalProjection_norm_le + - norm_sq_eq_add_norm_sq_projection + - orthogonalProjection_eq_zero_iff + - eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero + - ker_orthogonalProjection + - orthogonalProjection_tendsto_self + - inner_orthogonalProjection_eq_of_mem_right + - orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero + - orthogonalProjection_unit_singleton + - orthogonalProjection_singleton + - LinearIsometry.map_orthogonalProjection + - orthogonalProjection_tendsto_closure_iSup + - Submodule.finrank_add_finrank_orthogonal' + - id_eq_sum_orthogonalProjection_self_orthogonalComplement + - orthogonalProjection_eq_self_iff + - reflection_sub + - reflection_orthogonal + - Submodule.sup_orthogonal_inf_of_completeSpace + - reflection_mem_subspace_orthogonalComplement_eq_neg + - LinearIsometryEquiv.reflections_generate_dim_aux + - reflection_bot + - orthogonalProjectionFn_norm_sq + - eq_orthogonalProjection_of_mem_orthogonal' + - Submodule.orthogonal_orthogonal + - Submodule.finrank_add_inf_finrank_orthogonal + - exists_norm_eq_iInf_of_complete_subspace + - LinearIsometryEquiv.reflections_generate + - Submodule.triorthogonal_eq_orthogonal + - reflection_map_apply + - reflection_eq_self_iff + - reflection_singleton_apply + - Submodule.finrank_add_inf_finrank_orthogonal' + - orthogonalProjection_mem_subspace_eq_self + - orthogonalProjection_eq_linear_proj + - orthogonalProjection_orthogonalProjection_of_le + - norm_eq_iInf_iff_real_inner_le_zero + - Submodule.finrank_add_finrank_orthogonal + - orthogonalProjection_comp_subtypeL_eq_zero_iff + - Submodule.orthogonal_eq_bot_iff + - Submodule.exists_add_mem_mem_orthogonal + - orthogonalProjection_bot + - maximal_orthonormal_iff_orthogonalComplement_eq_bot + - reflection_orthogonal_apply + - norm_eq_iInf_iff_inner_eq_zero + - inner_orthogonalProjection_eq_of_mem_left + - inner_orthogonalProjection_left_eq_right + - orthogonalProjection_minimal + - Submodule.orthogonal_orthogonal_eq_closure + - OrthogonalFamily.sum_projection_of_mem_iSup + - Submodule.topologicalClosure_eq_top_iff + - Submodule.sup_orthogonal_of_completeSpace + - orthogonalProjection_add_orthogonalProjection_orthogonal + - maximal_orthonormal_iff_basis_of_finiteDimensional + - norm_eq_iInf_iff_real_inner_eq_zero + - eq_orthogonalProjection_of_eq_submodule + - Dense.eq_zero_of_inner_right + - finrank_orthogonal_span_singleton + - Dense.eq_zero_of_inner_left + - OrthogonalFamily.projection_directSum_coeAddHom + - orthogonalProjection_map_apply + - path: Mathlib/GroupTheory/DoubleCoset.lean + theorems: + - Doset.mk_eq_of_doset_eq + - Doset.left_bot_eq_left_quot + - Doset.right_bot_eq_right_quot + - Doset.disjoint_out' + - Doset.mem_doset + - Doset.doset_union_rightCoset + - Doset.doset_eq_of_mem + - Doset.eq_of_not_disjoint + - Doset.union_quotToDoset + - Doset.bot_rel_eq_leftRel + - Doset.mk_out'_eq_mul + - Doset.doset_union_leftCoset + - Doset.mem_doset_of_not_disjoint + - Doset.rel_bot_eq_right_group_rel + - Doset.doset_eq_image2 + - path: Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean + theorems: + - NonUnitalStarSubalgebra.unitization_range + - StarSubalgebra.toNonUnitalStarSubalgebra_toStarSubalgebra + - Unitization.starLift_range_le + - AlgHomClass.unitization_injective' + - NonUnitalStarSubalgebra.unitization_injective + - Unitization.lift_range_le + - Unitization.starLift_range + - NonUnitalStarSubalgebra.toStarSubalgebra_toNonUnitalStarSubalgebra + - NonUnitalSubring.toSubring_toNonUnitalSubring + - Subalgebra.toNonUnitalSubalgebra_toSubalgebra + - NonUnitalSubalgebra.unitization_range + - NonUnitalSubsemiring.toSubsemiring_toNonUnitalSubsemiring + - Unitization.lift_range + - Subsemiring.toNonUnitalSubsemiring_toSubsemiring + - Subring.toNonUnitalSubring_toSubring + - NonUnitalSubring.unitization_range + - NonUnitalSubsemiring.unitization_range + - NonUnitalSubalgebra.unitization_injective + - AlgHomClass.unitization_injective + - path: Mathlib/CategoryTheory/Functor/Basic.lean + theorems: + - CategoryTheory.Functor.comp_id + - CategoryTheory.Functor.id_comp + - CategoryTheory.Functor.map_comp_assoc + - CategoryTheory.Functor.map_dite + - path: Mathlib/Combinatorics/SimpleGraph/Finite.lean + theorems: + - SimpleGraph.card_incidenceFinset_eq_degree + - SimpleGraph.maxDegree_lt_card_verts + - SimpleGraph.edgeFinset_deleteEdges + - SimpleGraph.degree_pos_iff_exists_adj + - SimpleGraph.edgeFinset_subset_edgeFinset + - SimpleGraph.edgeFinset_ssubset_edgeFinset + - SimpleGraph.incidenceFinset_eq_filter + - SimpleGraph.neighborFinset_compl + - SimpleGraph.not_mem_neighborFinset_self + - SimpleGraph.edgeFinset_sup + - SimpleGraph.deleteFar_iff + - SimpleGraph.exists_maximal_degree_vertex + - SimpleGraph.edgeFinset_inj + - SimpleGraph.degree_compl + - SimpleGraph.bot_degree + - SimpleGraph.degree_le_maxDegree + - SimpleGraph.IsRegularOfDegree.compl + - SimpleGraph.neighborFinset_eq_filter + - SimpleGraph.degree_lt_card_verts + - SimpleGraph.complete_graph_degree + - SimpleGraph.card_edgeFinset_le_card_choose_two + - SimpleGraph.edgeFinset_bot + - SimpleGraph.edgeFinset_sdiff + - SimpleGraph.card_incidenceSet_eq_degree + - SimpleGraph.card_commonNeighbors_le_degree_left + - SimpleGraph.Adj.card_commonNeighbors_lt_degree + - SimpleGraph.card_edgeFinset_top_eq_card_choose_two + - SimpleGraph.edgeFinset_inf + - SimpleGraph.edgeFinset_top + - SimpleGraph.maxDegree_le_of_forall_degree_le + - SimpleGraph.exists_minimal_degree_vertex + - SimpleGraph.card_commonNeighbors_le_degree_right + - SimpleGraph.IsRegularOfDegree.top + - SimpleGraph.le_minDegree_of_forall_le_degree + - SimpleGraph.minDegree_le_degree + - SimpleGraph.card_commonNeighbors_top + - path: Mathlib/Data/PFun.lean + theorems: + - PFun.coe_comp + - PFun.prodLift_fst_comp_snd_comp + - PFun.mem_core_res + - PFun.mem_prodLift + - PFun.fix_fwd + - PFun.core_restrict + - PFun.mem_dom + - PFun.fixInduction'_fwd + - PFun.fix_fwd_eq + - PFun.fixInduction'_stop + - PFun.comp_assoc + - PFun.prodMap_id_id + - PFun.preimage_asSubtype + - PFun.mem_prodMap + - PFun.id_comp + - PFun.comp_id + - PFun.lift_graph + - PFun.fix_stop + - PFun.mem_restrict + - PFun.preimage_comp + - PFun.core_eq + - PFun.fixInduction_spec + - PFun.dom_of_mem_fix + - PFun.mem_res + - PFun.mem_toSubtype_iff + - PFun.mem_fix_iff + - PFun.core_res + - PFun.preimage_univ + - PFun.Part.bind_comp + - PFun.dom_comp + - path: Mathlib/GroupTheory/PGroup.lean + theorems: + - IsPGroup.cyclic_center_quotient_of_card_eq_prime_sq + - IsPGroup.bot_lt_center + - IsPGroup.powEquiv_symm_apply + - IsPGroup.index + - IsPGroup.to_sup_of_normal_right + - IsPGroup.card_orbit + - IsPGroup.comap_of_ker_isPGroup + - IsPGroup.nonempty_fixed_point_of_prime_not_dvd_card + - IsPGroup.of_card + - IsPGroup.iff_orderOf + - IsPGroup.of_surjective + - IsPGroup.card_eq_or_dvd + - IsPGroup.iff_card + - IsPGroup.card_center_eq_prime_pow + - IsPGroup.map + - IsPGroup.exists_fixed_point_of_prime_dvd_card_of_fixed_point + - IsPGroup.to_le + - IsPGroup.nontrivial_iff_card + - IsPGroup.card_modEq_card_fixedPoints + - IsPGroup.coprime_card_of_ne + - IsPGroup.center_nontrivial + - IsPGroup.disjoint_of_ne + - IsPGroup.of_injective + - path: Mathlib/Algebra/BigOperators/Finprod.lean + theorems: + - finprod_mem_one + - finprod_congr_Prop + - finprod_mem_pair + - finprod_cond_eq_prod_of_cond_iff + - finprod_mem_union'' + - finprod_mem_eq_toFinset_prod + - finprod_mem_finset_product' + - smul_finsum + - finprod_mem_div_distrib + - finprod_mem_eq_prod_filter + - finprod_eq_prod_plift_of_mulSupport_subset + - finprod_cond_ne + - finprod_mem_eq_prod_of_inter_mulSupport_eq + - finprod_mem_mul_distrib' + - finprod_mem_empty + - finprod_prod_comm + - finprod_eq_mulIndicator_apply + - finprod_mem_inter_mulSupport_eq + - finprod_of_isEmpty + - finprod_mem_inter_mulSupport_eq' + - finprod_mem_eq_prod + - finprod_induction + - finprod_eq_finset_prod_of_mulSupport_subset + - exists_ne_one_of_finprod_mem_ne_one + - finprod_mem_comm + - finprod_mem_sUnion + - finprod_def + - finprod_eq_zero + - finprod_mem_union_inter + - "finprod_mem_finset_product\u2083" + - "finprod_curry\u2083" + - finprod_mem_dvd + - finprod_set_coe_eq_finprod_mem + - Finset.mulSupport_of_fiberwise_prod_subset_image + - finprod_eq_prod_plift_of_mulSupport_toFinset_subset + - MonoidHom.map_finprod_mem' + - finprod_eq_single + - finprod_mem_insert_of_eq_one_if_not_mem + - finprod_mem_finset_product + - finprod_div_distrib + - finprod_eq_of_bijective + - finprod_eq_one_of_forall_eq_one + - finprod_mem_union_inter' + - finprod_mem_image' + - finprod_emb_domain' + - finprod_mem_eq_of_bijOn + - finprod_mem_inter_mul_diff' + - MonoidHom.map_finprod_plift + - finprod_mul_distrib + - finprod_cond_eq_right + - finprod_mem_singleton + - finprod_mem_union' + - finprod_one + - finprod_of_infinite_mulSupport + - single_le_finprod + - finprod_mem_mul_diff' + - finprod_mem_eq_finite_toFinset_prod + - finprod_mem_eq_one_of_forall_eq_one + - finprod_mem_biUnion + - finprod_curry + - finprod_mem_mulSupport + - MonoidHom.map_finprod_of_preimage_one + - mul_finprod_cond_ne + - finprod_eq_prod + - finprod_mem_inter_mulSupport + - finsum_smul + - finprod_eq_prod_of_mulSupport_subset + - one_lt_finprod' + - finprod_mem_range' + - finprod_mem_iUnion + - finprod_mem_of_eqOn_one + - path: Mathlib/Data/Rel.lean + theorems: + - Rel.comp_right_bot + - Equiv.graph_inv + - Rel.comp_right_id + - Rel.image_inter_dom_eq + - Function.graph_comp + - Rel.inv_inv + - Rel.preimage_univ + - Rel.preimage_inter_codom_eq + - Rel.comp_left_id + - Rel.image_comp + - Rel.preimage_bot + - Rel.dom_inv + - Rel.core_inter + - Rel.comp_right_top + - Rel.image_id + - Function.graph_id + - Set.preimage_eq + - Rel.image_preimage_subset_inter_codom + - Rel.inter_dom_subset_preimage_image + - Set.image_eq + - Rel.core_univ + - Rel.inv_id + - Rel.comp_assoc + - Rel.image_eq_dom_of_codomain_subset + - Rel.core_id + - Rel.core_comp + - Rel.image_univ + - Rel.image_bot + - Rel.inv_bot + - Rel.inv_comp + - Rel.inv_top + - Rel.preimage_comp + - Rel.preimage_inv + - Rel.preimage_top + - Rel.preimage_id + - Rel.image_empty + - Rel.preimage_eq_codom_of_domain_subset + - Rel.comp_left_bot + - Set.preimage_eq_core + - Rel.codom_inv + - Rel.preimage_empty + - Relation.is_graph_iff + - Rel.comp_left_top + - Rel.image_top + - path: Mathlib/MeasureTheory/Integral/Bochner.lean + theorems: + - MeasureTheory.continuousWithinAt_of_dominated + - MeasureTheory.ofReal_integral_norm_eq_lintegral_nnnorm + - MeasureTheory.L1.integral_smul + - MeasureTheory.integral_eq_iff_of_ae_le + - MeasureTheory.integral_mono_measure + - MeasureTheory.lintegral_coe_le_coe_iff_integral_le + - MeasureTheory.L1.norm_of_fun_eq_integral_norm + - MeasureTheory.integral_coe_le_of_lintegral_coe_le + - MeasureTheory.ofReal_integral_eq_lintegral_ofReal + - MeasureTheory.integral_countable' + - MeasureTheory.integral_norm_eq_lintegral_nnnorm + - MeasureTheory.L1.integral_eq_norm_posPart_sub + - MeasureTheory.SimpleFunc.norm_integral_le_integral_norm + - MeasureTheory.norm_integral_le_of_norm_le_const + - MeasureTheory.SimpleFunc.negPart_map_norm + - MeasureTheory.integral_trim + - MeasureTheory.integral_toReal + - MeasureTheory.L1.integral_sub + - MeasureTheory.weightedSMul_smul + - MeasureTheory.L1.SimpleFunc.posPart_toSimpleFunc + - MeasureTheory.integral_trim_simpleFunc + - MeasureTheory.L1.SimpleFunc.norm_integral_le_norm + - MeasureTheory.integral_countable + - MeasureTheory.weightedSMul_zero_measure + - MeasureTheory.SimpleFunc.integral_eq_integral + - MeasureTheory.integral_smul + - MeasureTheory.L1.norm_integral_le + - MeasureTheory.integral_const + - MeasureTheory.integral_eq_zero_of_ae + - MeasureTheory.set_integral_dirac' + - MeasureTheory.weightedSMul_empty + - MeasureTheory.L1.integral_neg + - MeasureTheory.SimpleFunc.integral_eq + - MeasureTheory.SimpleFunc.integral_eq_sum_of_subset + - MeasureTheory.integral_eq_lintegral_of_nonneg_ae + - MeasureTheory.weightedSMul_nonneg + - MeasureTheory.SimpleFunc.integral_add_measure + - MeasureTheory.L1.SimpleFunc.integral_eq_lintegral + - MeasureTheory.nndist_integral_add_measure_le_lintegral + - MeasureTheory.SimpleFunc.integral_eq_lintegral' + - MeasureTheory.tendsto_integral_of_L1 + - MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part + - MeasureTheory.L1.integral_eq + - MeasureTheory.integral_fintype + - MeasureTheory.integral_map_of_stronglyMeasurable + - MeasureTheory.integral_pos_iff_support_of_nonneg_ae + - MeasureTheory.L1.continuous_integral + - MeasureTheory.ae_eq_trim_of_stronglyMeasurable + - MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg + - MeasureTheory.integral_simpleFunc_larger_space + - MeasureTheory.integral_map + - MeasureTheory.integral_eq_integral_pos_part_sub_integral_neg_part + - MeasureTheory.integral_finset + - MeasureTheory.norm_integral_le_lintegral_norm + - MeasureTheory.integral_add + - MeasureTheory.weightedSMul_null + - MeasureTheory.L1.integral_eq_setToL1 + - MeasureTheory.L1.integral_zero + - "MeasureTheory.Mem\u2112p.snorm_eq_integral_rpow_norm" + - MeasureTheory.integral_zero + - MeasureTheory.weightedSMul_union' + - MeasureTheory.integral_finset_sum_measure + - MeasureTheory.integral_eq_setToFun + - MeasureTheory.integral_subtype_comap + - MeasureTheory.tendsto_integral_norm_approxOn_sub + - MeasureTheory.HasFiniteIntegral.tendsto_set_integral_nhds_zero + - MeasureTheory.SimpleFunc.integral_eq_sum + - MeasureTheory.integral_nonneg_of_ae + - MeasureTheory.integral_unique + - MeasureTheory.integral_neg + - MeasureTheory.L1.integral_of_fun_eq_integral + - MeasureTheory.L1.SimpleFunc.integral_eq_norm_posPart_sub + - MeasureTheory.integral_dirac' + - MeasureTheory.SimpleFunc.posPart_map_norm + - MeasureTheory.integral_singleton' + - MeasureTheory.integral_nonpos_of_ae + - MeasureTheory.L1.integral_eq_integral + - MeasureTheory.continuousOn_of_dominated + - MeasureTheory.weightedSMul_congr + - MeasureTheory.integral_eq + - MeasureTheory.tendsto_integral_approxOn_of_measurable + - MeasureTheory.integral_sub + - MeasureTheory.integral_dirac + - MeasureTheory.integral_trim_ae + - MeasureTheory.weightedSMul_smul_measure + - MeasureTheory.integral_zero_measure + - MeasureTheory.integral_tendsto_of_tendsto_of_antitone + - MeasureTheory.tendsto_of_integral_tendsto_of_monotone + - MeasureTheory.SimpleFunc.integral_eq_lintegral + - MeasureTheory.integral_div + - MeasureTheory.integral_mono_of_nonneg + - MeasureTheory.integral_exp_pos + - MeasureTheory.integral_smul_measure + - MeasureTheory.mul_meas_ge_le_integral_of_nonneg + - MeasureTheory.integral_congr_ae + - MeasureTheory.L1.SimpleFunc.norm_Integral_le_one + - MeasureTheory.set_integral_dirac + - MeasurableEmbedding.integral_map + - MeasureTheory.integral_eq_zero_iff_of_nonneg_ae + - MeasureTheory.tendsto_integral_of_L1' + - MeasureTheory.continuous_of_dominated + - MeasureTheory.tendsto_integral_approxOn_of_measurable_of_range_subset + - MeasureTheory.integral_add_measure + - MeasureTheory.L1.SimpleFunc.integral_L1_eq_integral + - MeasureTheory.SimpleFunc.integral_const + - MeasureTheory.continuousAt_of_dominated + - MeasureTheory.tendsto_of_integral_tendsto_of_antitone + - MeasureTheory.ennnorm_integral_le_lintegral_ennnorm + - MeasureTheory.SimpleFunc.integral_piecewise_zero + - MeasureTheory.integral_mul_norm_le_Lp_mul_Lq + - MeasureTheory.lintegral_coe_eq_integral + - MeasureTheory.L1.dist_eq_integral_dist + - MeasureTheory.integral_mono_ae + - MeasureTheory.snorm_one_le_of_le + - MeasureTheory.SimpleFunc.posPart_sub_negPart + - MeasureTheory.snorm_one_le_of_le' + - MeasureTheory.weightedSMul_add_measure + - MeasureTheory.L1.SimpleFunc.negPart_toSimpleFunc + - MeasureTheory.norm_integral_le_integral_norm + - MeasureTheory.tendsto_set_integral_of_L1 + - MeasureTheory.integral_singleton + - MeasureTheory.integral_tendsto_of_tendsto_of_monotone + - MeasureTheory.L1.integral_add + - MeasureTheory.continuous_integral + - MeasureTheory.weightedSMul_apply + - MeasureTheory.tendsto_set_integral_of_L1' + - MeasureTheory.integral_mul_right + - MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_integral_norm + - MeasureTheory.integral_finset_sum + - MeasureTheory.L1.norm_eq_integral_norm + - MeasureTheory.hasSum_integral_measure + - MeasureTheory.L1.SimpleFunc.norm_eq_integral + - path: Mathlib/Data/Set/Finite.lean + theorems: + - Equiv.set_finite_iff + - Set.Finite.toFinset_subset_toFinset + - Set.exists_lower_bound_image + - Set.Finite.surjOn_iff_bijOn_of_mapsTo + - Set.finite_subset_iUnion + - Set.union_finset_finite_of_range_finite + - Set.card_ne_eq + - Set.Finite.toFinset_image + - Set.Finite.induction_to + - Finite.Set.finite_biUnion + - Set.Finite.toFinset_ssubset + - Set.Finite.toFinset_inter + - Set.Finite.exists_finset_coe + - Finite.of_forall_not_lt_lt + - Set.Finite.induction_to_univ + - Set.finite_of_forall_not_lt_lt + - Set.Finite.toFinset_eq_toFinset + - Set.Finite.biUnion' + - Set.Finite.toFinset_compl + - Set.Finite.union + - Set.Finite.induction_on' + - Set.infinite_univ_iff + - Set.infinite_range_of_injective + - Set.infinite_of_injective_forall_mem + - Set.Finite.pi + - Set.Finite.toFinset_offDiag + - Set.exists_max_image + - Set.card_insert + - Set.Finite.toFinset_prod + - Set.Finite.toFinset_setOf + - Set.finite_image2 + - Set.seq_of_forall_finite_exists + - Set.Finite.induction_on + - Set.Finite.exists_maximal_wrt + - Set.Finite.toFinset_subset + - Set.card_fintypeInsertOfNotMem + - Set.Finite.toFinset_diff + - Set.infinite_of_finite_compl + - Set.Finite.exists_finset + - Set.Finite.toFinset_empty + - Set.Finite.subset + - Set.Finite.ssubset_toFinset + - Set.Finite.prod + - Set.Finite.iUnion + - Finset.exists_card_eq + - Set.Finite.exists_not_mem + - Set.finite_diff_iUnion_Ioo' + - Set.eq_finite_iUnion_of_finite_subset_iUnion + - Set.Finite.toFinset_univ + - Set.Finite.subset_toFinset + - Set.Finite.finite_of_compl + - Set.not_injOn_infinite_finite_image + - Set.Finite.iSup_biInf_of_monotone + - Set.Infinite.exists_subset_card_eq + - Set.Finite.pi' + - Set.card_image_of_inj_on + - Finite.Set.subset + - Set.Finite.image + - Set.Finite.sUnion + - Set.Finite.exists_lt_map_eq_of_forall_mem + - Set.Finite.finite_subsets + - Set.Finite.bddAbove_biUnion + - Set.Infinite.exists_not_mem_finite + - Set.Finite.toFinset_union + - Set.Finite.toFinset_insert + - Set.infinite_image2 + - Set.Infinite.nonempty + - Multiset.finite_toSet_toFinset + - Set.Finite.toFinset_symmDiff + - Set.Finite.exists_maximal_wrt' + - Set.infinite_of_forall_exists_gt + - Set.infinite_prod + - Set.Finite.dependent_image + - Set.Finite.toFinset_ssubset_toFinset + - Set.empty_card' + - Set.infinite_union + - Set.iUnion_pi_of_monotone + - Set.Finite.image2 + - Set.Finite.fin_embedding + - Set.iSup_iInf_of_monotone + - Set.Finite.coeSort_toFinset + - Set.Finite.toFinset_nonempty + - Set.Infinite.exists_ne_map_eq_of_mapsTo + - Finset.finite_toSet_toFinset + - Set.Finite.injOn_iff_bijOn_of_mapsTo + - Multiset.finite_toSet + - Set.Finite.toFinset_range + - Set.finite_of_finite_preimage + - Set.exists_min_image + - Set.finite_prod + - Set.Finite.infinite_compl + - path: Mathlib/Algebra/DirectSum/LinearMap.lean + theorems: + - LinearMap.toMatrix_directSum_collectedBasis_eq_blockDiagonal' + - LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup + - LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero + - LinearMap.diag_toMatrix_directSum_collectedBasis_eq_zero_of_mapsTo_ne + - LinearMap.trace_eq_sum_trace_restrict' + - LinearMap.trace_eq_sum_trace_restrict + - LinearMap.mapsTo_biSup_of_mapsTo + - LinearMap.trace_eq_zero_of_mapsTo_ne + - path: Mathlib/RingTheory/Localization/Basic.lean + theorems: + - Localization.mk_int_cast + - Localization.algEquiv_symm_mk + - IsLocalization.smul_mk'_one + - IsLocalization.map_map + - IsLocalization.noZeroDivisors_of_le_nonZeroDivisors + - IsLocalization.at_units + - IsLocalization.eq_zero_of_fst_eq_zero + - IsLocalization.mk'_mul_mk'_eq_one + - IsLocalization.algebraMap_mk' + - IsLocalization.ringEquivOfRingEquiv_eq + - IsLocalization.algHom_subsingleton + - IsLocalization.eq_iff_exists + - IsLocalization.injective + - IsLocalization.smul_mk' + - IsLocalization.mk'_eq_zero_iff + - Localization.add_mk + - IsLocalization.ext + - IsLocalization.algEquiv_mk' + - IsLocalization.map_smul + - IsLocalization.mk'_add_eq_iff_add_mul_eq_mul + - IsLocalization.lift_spec_mul_add + - Localization.sub_mk + - IsLocalization.sec_fst_ne_zero + - IsLocalization.isDomain_of_le_nonZeroDivisors + - IsLocalization.nonZeroDivisors_le_comap + - Localization.mk_nat_cast + - Localization.mk_algebraMap + - IsLocalization.algEquiv_symm_mk' + - IsLocalization.smul_mk'_self + - IsLocalization.mul_add_inv_left + - IsLocalization.mk'_zero + - IsLocalization.ringEquivOfRingEquiv_mk' + - IsLocalization.of_le + - Localization.neg_mk + - IsLocalization.isLocalization_of_base_ringEquiv + - Localization.add_mk_self + - IsLocalization.map_units_map_submonoid + - IsLocalization.to_map_eq_zero_iff + - IsLocalization.isLocalization_iff_of_base_ringEquiv + - IsLocalization.mk'_add + - IsLocalization.ne_zero_of_mk'_ne_zero + - IsLocalization.isLocalization_of_algEquiv + - IsField.localization_map_bijective + - Localization.algEquiv_mk + - IsLocalization.sec_spec' + - Localization.mk_eq_mk'_apply + - IsLocalization.mk'_mem_iff + - path: Mathlib/Data/Stream/Init.lean + theorems: + - Stream'.cycle_eq + - Stream'.inits_core_eq + - Stream'.mem_cons_of_mem + - Stream'.iterate_eq + - Stream'.mem_interleave_right + - Stream'.tail_iterate + - Stream'.zip_eq + - Stream'.tail_drop + - Stream'.corec_eq + - Stream'.drop_drop + - Stream'.append_stream_head_tail + - Stream'.cycle_singleton + - Stream'.tail_interleave + - Stream'.map_append_stream + - Stream'.head_drop + - Stream'.get_unfolds_head_tail + - Stream'.cons_get_inits_core + - Stream'.get_even + - Stream'.get_succ_iterate + - Stream'.tail_even + - Stream'.get_interleave_right + - Stream'.unfolds_eq + - Stream'.even_cons_cons + - Stream'.eq_or_mem_of_mem_cons + - Stream'.append_take_drop + - Stream'.tail_const + - Stream'.drop_append_stream + - Stream'.take_theorem + - Stream'.even_interleave + - Stream'.interleave_tail_tail + - Stream'.append_append_stream + - Stream'.mem_of_mem_even + - Stream'.inits_eq + - Stream'.map_iterate + - Stream'.mem_cycle + - Stream'.interleave_even_odd + - Stream'.get?_take + - Stream'.const_eq + - Stream'.nats_eq + - Stream'.length_take + - Stream'.eta + - Stream'.iterate_id + - Stream'.take_take + - Stream'.mem_of_mem_odd + - Stream'.get_interleave_left + - Stream'.interleave_eq + - Stream'.map_eq + - Stream'.get_inits + - Stream'.mem_map + - Stream'.take_succ' + - Stream'.cons_injective2 + - Stream'.dropLast_take + - Stream'.tail_inits + - Stream'.zip_inits_tails + - Stream'.map_cons + - Stream'.corec_id_id_eq_const + - Stream'.tail_drop' + - Stream'.tails_eq + - Stream'.get_tails + - Stream'.bisim_simple + - Stream'.get_odd + - path: Mathlib/GroupTheory/Submonoid/Membership.lean + theorems: + - IsScalarTower.of_mclosure_eq_top + - Submonoid.closure_induction_left + - Submonoid.exists_multiset_of_mem_closure + - Submonoid.mem_sup + - Submonoid.mem_sSup_of_directedOn + - Submonoid.noncommProd_mem + - Submonoid.closure_eq_image_prod + - Submonoid.closure_singleton_one + - multiset_prod_mem + - Submonoid.mem_closure_pair + - Submonoid.multiset_prod_mem + - MulMemClass.mul_right_mem_add_closure + - FreeMonoid.mrange_lift + - Submonoid.closure_induction_right + - Submonoid.map_powers + - Submonoid.induction_of_closure_eq_top_right + - Submonoid.mem_iSup_of_directed + - Submonoid.powers_eq_closure + - Submonoid.multiset_noncommProd_mem + - Submonoid.mem_sup_left + - Submonoid.exists_list_of_mem_closure + - ofAdd_image_multiples_eq_powers_ofAdd + - Submonoid.card_le_one_iff_eq_bot + - Submonoid.list_prod_mem + - Submonoid.mem_iSup_of_mem + - Submonoid.eq_bot_iff_card + - SMulCommClass.of_mclosure_eq_top + - Submonoid.mem_sSup_of_mem + - Submonoid.powers_le + - list_prod_mem + - ofMul_image_powers_eq_multiples_ofMul + - Submonoid.iSup_induction + - Submonoid.coe_iSup_of_directed + - Submonoid.sup_eq_range + - Submonoid.closure_eq_mrange + - Submonoid.iSup_induction' + - Submonoid.mem_sup_right + - Submonoid.induction_of_closure_eq_top_left + - Submonoid.coe_sSup_of_directedOn + - AddSubmonoid.mem_closure_singleton + - AddSubmonoid.closure_singleton_zero + - path: Mathlib/Analysis/Complex/ReImTopology.lean + theorems: + - Complex.frontier_setOf_re_le + - Complex.closure_setOf_im_lt + - Complex.interior_setOf_re_le + - Complex.frontier_setOf_re_lt + - Complex.interior_setOf_im_le + - Complex.frontier_setOf_le_im + - Complex.frontier_setOf_im_le + - Complex.closure_setOf_re_lt + - Complex.frontier_setOf_lt_re + - Complex.interior_setOf_le_re + - Complex.frontier_setOf_lt_im + - Complex.frontier_setOf_le_re_and_im_le + - Complex.interior_reProdIm + - Complex.frontier_setOf_le_re + - Complex.closure_setOf_lt_re + - Complex.closure_reProdIm + - Complex.frontier_setOf_im_lt + - Complex.frontier_setOf_le_re_and_le_im + - Complex.interior_setOf_le_im + - Complex.frontier_reProdIm + - Complex.closure_setOf_lt_im + - path: Mathlib/Data/Set/Intervals/OrdConnected.lean + theorems: + - OrderEmbedding.image_Ico + - Set.image_subtype_val_Ico + - Set.ordConnected_of_uIcc_subset_left + - Set.ordConnected_range + - Set.ordConnected_image + - IsAntichain.ordConnected + - Set.image_subtype_val_Ioo + - Set.ordConnected_dual + - Set.ordConnected_singleton + - Set.image_subtype_val_Icc + - Set.dual_ordConnected_iff + - Set.ordConnected_of_Ioo + - OrderEmbedding.image_Icc + - OrderEmbedding.image_Ioc + - Set.image_subtype_val_Ioc + - Set.ordConnected_iff_uIcc_subset_right + - path: Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean + theorems: + - uniformCauchySeqOnFilter_of_fderiv + - hasDerivAt_of_tendstoUniformlyOnFilter + - uniformCauchySeqOn_ball_of_fderiv + - cauchy_map_of_uniformCauchySeqOn_fderiv + - hasFDerivAt_of_tendsto_locally_uniformly_on' + - UniformCauchySeqOnFilter.one_smulRight + - hasFDerivAt_of_tendstoUniformly + - hasDerivAt_of_tendstoUniformly + - hasDerivAt_of_tendstoLocallyUniformlyOn + - hasFDerivAt_of_tendstoUniformlyOnFilter + - hasFDerivAt_of_tendstoLocallyUniformlyOn + - hasDerivAt_of_tendsto_locally_uniformly_on' + - uniformCauchySeqOnFilter_of_deriv + - uniformCauchySeqOn_ball_of_deriv + - difference_quotients_converge_uniformly + - path: Mathlib/RingTheory/Ideal/Basic.lean + theorems: + - Ideal.span_singleton_neg + - Ideal.IsMaximal.exists_inv + - Ideal.eq_top_of_unit_mem + - Ideal.span_pair_add_mul_left + - Ideal.factors_decreasing + - Ideal.isMaximal_iff + - exists_max_ideal_of_mem_nonunits + - Ideal.not_isPrime_iff + - Ideal.span_pair_comm + - Ring.ne_bot_of_isMaximal_of_not_isField + - Ideal.IsMaximal.coprime_of_ne + - Ideal.span_singleton_mul_left_unit + - Ideal.mem_span_singleton_sup + - Ideal.span_zero + - Ideal.span_pow_eq_top + - Ideal.span_singleton_lt_span_singleton + - Ideal.unit_mul_mem_iff_mem + - Ideal.bot_prime + - Ring.exists_not_isUnit_of_not_isField + - Ideal.bot_isMaximal + - Ideal.pow_mem_of_mem + - Ideal.IsPrime.mul_mem_iff_mem_or_mem + - Ideal.eq_top_iff_one + - Ideal.span_pair_add_mul_right + - Ideal.maximal_of_no_maximal + - Ideal.span_singleton_prime + - Ideal.span_singleton_eq_top + - Ideal.mul_sub_mul_mem + - Ideal.span_one + - Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top + - Ideal.span_eq_top_iff_finite + - Ideal.isPrime_of_maximally_disjoint + - Ideal.span_singleton_eq_span_singleton + - Ideal.IsMaximal.isPrime + - Ideal.sum_pow_mem_span_pow + - Ideal.bot_lt_of_maximal + - Ideal.isCompactElement_top + - Ideal.IsPrime.mem_of_pow_mem + - Ideal.sInf_isPrime_of_isChain + - Ideal.pow_multiset_sum_mem_span_pow + - Ring.isField_iff_isSimpleOrder_ideal + - Ideal.span_singleton_mul_right_unit + - path: Mathlib/CategoryTheory/LiftingProperties/Basic.lean + theorems: + - CategoryTheory.HasLiftingProperty.op + - CategoryTheory.HasLiftingProperty.iff_of_arrow_iso_right + - CategoryTheory.HasLiftingProperty.of_arrow_iso_right + - CategoryTheory.HasLiftingProperty.unop + - CategoryTheory.HasLiftingProperty.iff_of_arrow_iso_left + - CategoryTheory.HasLiftingProperty.of_arrow_iso_left + - path: Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean + theorems: + - "CliffordAlgebra.ofEven_\u03B9" + - CliffordAlgebra.EquivEven.v_sq_scalar + - "CliffordAlgebra.EquivEven.\u03B9_eq_v_add_smul_e0" + - CliffordAlgebra.EquivEven.e0_mul_e0 + - CliffordAlgebra.EquivEven.neg_v_mul_e0 + - "CliffordAlgebra.toEven_\u03B9" + - CliffordAlgebra.EquivEven.neg_e0_mul_v + - CliffordAlgebra.EquivEven.e0_mul_v_mul_e0 + - CliffordAlgebra.toEven_comp_ofEven + - CliffordAlgebra.evenToNeg_comp_evenToNeg + - CliffordAlgebra.coe_toEven_reverse_involute + - path: Mathlib/Dynamics/PeriodicPts.lean + theorems: + - MulAction.zpow_period_add_smul + - MulAction.zpow_smul_mod_minimalPeriod + - MulAction.pow_smul_mod_minimalPeriod + - Function.minimalPeriod_apply_iterate + - MulAction.zpow_smul_eq_iff_period_dvd + - Function.IsPeriodicPt.add + - Function.IsPeriodicPt.mul_const + - Function.isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate + - Function.nodup_periodicOrbit + - Function.minimalPeriod_eq_minimalPeriod_iff + - Function.IsPeriodicPt.iterate + - Function.periodicOrbit_length + - MulAction.pow_period_smul + - MulAction.pow_smul_eq_iff_minimalPeriod_dvd + - Function.periodicOrbit_chain' + - Function.minimalPeriod_prod_map + - Function.IsPeriodicPt.left_of_add + - Function.IsPeriodicPt.sub + - Function.minimalPeriod_eq_zero_of_nmem_periodicPts + - Function.isPeriodicPt_prod_map + - Function.IsPeriodicPt.minimalPeriod_le + - Function.periodicOrbit_apply_iterate_eq + - Function.IsPeriodicPt.comp + - Function.minimalPeriod_eq_zero_iff_nmem_periodicPts + - MulAction.pow_mod_period_smul + - Function.isPeriodicPt_minimalPeriod + - MulAction.zpow_add_period_smul + - Function.Commute.minimalPeriod_of_comp_dvd_lcm + - Function.bijOn_ptsOfPeriod + - Function.IsPeriodicPt.left_of_comp + - Function.minimalPeriod_fst_dvd + - MulAction.pow_period_add_smul + - Function.IsPeriodicPt.right_of_add + - MulAction.pow_smul_eq_iff_period_dvd + - Function.mem_periodicOrbit_iff + - Function.minimalPeriod_iterate_eq_div_gcd_aux + - Function.iterate_add_minimalPeriod_eq + - Function.minimalPeriod_eq_one_iff_isFixedPt + - Function.Commute.minimalPeriod_of_comp_eq_mul_of_coprime + - Function.IsPeriodicPt.iterate_mod_apply + - Function.bUnion_ptsOfPeriod + - Function.periodicOrbit_chain + - Function.periodicOrbit_eq_nil_iff_not_periodic_pt + - Function.minimalPeriod_pos_iff_mem_periodicPts + - Function.IsPeriodicPt.const_mul + - MulAction.isPeriodicPt_smul_iff + - MulAction.zpow_mod_period_smul + - Function.IsPeriodicPt.gcd + - Function.minimalPeriod_pos_of_mem_periodicPts + - MulAction.zpow_smul_eq_iff_minimalPeriod_dvd + - Function.iterate_prod_map + - Function.le_of_lt_minimalPeriod_of_iterate_eq + - Function.IsPeriodicPt.eq_of_apply_eq_same + - Function.minimalPeriod_snd_dvd + - Function.minimalPeriod_eq_prime_pow + - path: Mathlib/RepresentationTheory/Basic.lean + theorems: + - Representation.ofModule_asAlgebraHom_apply_apply + - Representation.smul_one_tprod_asModule + - Representation.asModuleEquiv_symm_map_rho + - Representation.dualTensorHom_comm + - Representation.asAlgebraHom_of + - Representation.ofMulAction_self_smul_eq_mul + - Representation.asModuleEquiv_symm_map_smul + - Representation.asGroupHom_apply + - Representation.ofMulAction_apply + - Representation.smul_ofModule_asModule + - Representation.asAlgebraHom_single_one + - Representation.smul_tprod_one_asModule + - Representation.ofModule_asModule_act + - path: Mathlib/Data/Finset/NAry.lean + theorems: + - "Finset.card_le_card_image\u2082_right" + - "Finset.sup'_image\u2082_right" + - "Finset.image_image\u2082_antidistrib_left" + - "Finset.mem_image\u2082_iff" + - "Finset.sup_image\u2082_right" + - "Finset.image\u2082_inter_subset_right" + - "Finset.card_dvd_card_image\u2082_right" + - Set.toFinset_image2 + - "Finset.image\u2082_subset_iff_left" + - "Finset.image\u2082_union_inter_subset" + - "Finset.image\u2082_congr" + - "Finset.image_image\u2082_antidistrib" + - "Finset.image\u2082_singleton_inter" + - "Finset.sup'_image\u2082_le" + - "Finset.image\u2082_inter_singleton" + - "Finset.image\u2082_empty_right" + - "Fintype.piFinset_image\u2082" + - "Finset.image\u2082_right" + - "Finset.image\u2082_left_identity" + - "Finset.image\u2082_left" + - "Finset.image\u2082_distrib_subset_left" + - "Finset.image_image\u2082_distrib_left" + - "Finset.image\u2082_union_right" + - "Finset.image\u2082_distrib_subset_right" + - "Finset.image\u2082_insert_right" + - "Finset.image\u2082_singleton_right" + - Finset.biUnion_image_left + - Set.Finite.toFinset_image2 + - "Finset.card_dvd_card_image\u2082_left" + - "Finset.image\u2082_subset" + - "Finset.image\u2082_right_comm" + - "Finset.image\u2082_subset_iff_right" + - "Finset.sup_image\u2082_left" + - "Finset.image\u2082_image_right" + - "Finset.image\u2082_inter_union_subset_union" + - "Finset.image\u2082_empty_left" + - "Finset.image\u2082_singleton" + - "Finset.image\u2082_union_left" + - "Finset.mem_image\u2082" + - "Finset.image\u2082_assoc" + - "Finset.image\u2082_inter_subset_left" + - "Finset.image_image\u2082" + - "Finset.card_image\u2082_iff" + - "Finset.le_inf'_image\u2082" + - "Finset.card_image\u2082_singleton_right" + - "Finset.image\u2082_singleton_left" + - "Finset.image\u2082_image\u2082_image\u2082_comm" + - "Finset.image_image\u2082_antidistrib_right" + - "Finset.image\u2082_mk_eq_product" + - "Finset.image\u2082_inter_right" + - "Finset.image\u2082_inter_left" + - "Finset.sup_image\u2082_le" + - "Finset.image\u2082_union_inter_subset_union" + - "Finset.image\u2082_eq_empty_iff" + - "Finset.image\u2082_comm" + - "Finset.card_image\u2082_singleton_left" + - "Finset.image\u2082_swap" + - "Finset.sup'_image\u2082_left" + - "Finset.image\u2082_inter_union_subset" + - "Finset.image\u2082_right_identity" + - Finset.biUnion_image_right + - "Finset.subset_image\u2082" + - "Finset.image_image\u2082_distrib_right" + - "Finset.card_le_card_image\u2082_left" + - "Finset.image\u2082_nonempty_iff" + - "Finset.image\u2082_image_left" + - "Finset.image\u2082_insert_left" + - path: Mathlib/Data/Nat/Fib/Basic.lean + theorems: + - Nat.fib_lt_fib + - Nat.fast_fib_eq + - Nat.gcd_fib_add_self + - Nat.fast_fib_aux_bit_tt + - Nat.fast_fib_aux_bit_ff + - Nat.fib_add + - Nat.fib_bit1_succ + - Nat.gcd_fib_add_mul_self + - Nat.fib_add_two + - Nat.fib_succ_eq_sum_choose + - Nat.le_fib_self + - Nat.fib_succ_eq_succ_sum + - Nat.fib_lt_fib_succ + - Nat.fib_add_two_sub_fib_add_one + - Nat.fib_gcd + - Nat.fib_two_mul_add_one + - Nat.fib_add_two_strictMono + - Nat.fib_eq_zero + - Nat.fib_le_fib_succ + - Nat.fib_bit1 + - Nat.fib_pos + - Nat.fib_dvd + - Nat.fib_two_mul + - Nat.fast_fib_aux_eq + - Nat.fib_coprime_fib_succ + - Nat.fib_bit0 + - path: Mathlib/Analysis/BoxIntegral/Partition/Split.lean + theorems: + - BoxIntegral.Box.splitLower_def + - BoxIntegral.Prepartition.inf_splitMany + - BoxIntegral.Box.coe_splitLower + - BoxIntegral.Prepartition.compl_congr + - BoxIntegral.Prepartition.exists_iUnion_eq_diff + - BoxIntegral.Prepartition.sum_split_boxes + - BoxIntegral.Box.disjoint_splitLower_splitUpper + - BoxIntegral.Prepartition.IsPartition.compl_eq_bot + - BoxIntegral.Box.splitUpper_def + - BoxIntegral.Box.splitUpper_le + - BoxIntegral.Prepartition.restrict_split + - BoxIntegral.Box.splitLower_ne_splitUpper + - BoxIntegral.Prepartition.eventually_splitMany_inf_eq_filter + - BoxIntegral.Prepartition.eventually_not_disjoint_imp_le_of_mem_splitMany + - BoxIntegral.Box.splitUpper_eq_self + - BoxIntegral.Prepartition.split_of_not_mem_Ioo + - BoxIntegral.Box.splitLower_le + - BoxIntegral.Box.splitUpper_eq_bot + - BoxIntegral.Prepartition.splitMany_insert + - BoxIntegral.Prepartition.iUnion_split + - BoxIntegral.Prepartition.mem_split_iff' + - BoxIntegral.Prepartition.mem_split_iff + - BoxIntegral.Prepartition.coe_eq_of_mem_split_of_mem_le + - BoxIntegral.Prepartition.isPartition_splitMany + - BoxIntegral.Prepartition.coe_eq_of_mem_split_of_lt_mem + - BoxIntegral.Box.splitLower_eq_bot + - path: Mathlib/Order/BooleanAlgebra.lean + theorems: + - sdiff_sdiff_sup_sdiff + - sup_lt_of_lt_sdiff_right + - le_iff_eq_sup_sdiff + - sup_inf_inf_sdiff + - sdiff_eq_self_iff_disjoint' + - sup_eq_sdiff_sup_sdiff_sup_inf + - disjoint_compl_right_iff + - sup_lt_of_lt_sdiff_left + - compl_le_self + - sdiff_sdiff_sdiff_cancel_right + - Disjoint.sdiff_eq_of_sup_eq + - sdiff_eq_right + - sdiff_lt + - sdiff_sup_self' + - sdiff_eq_sdiff_iff_inf_eq_inf + - sdiff_sdiff_eq_self + - sdiff_sdiff_right' + - sdiff_eq_symm + - inf_sdiff_self_right + - compl_le_compl_iff_le + - sdiff_sdiff_left' + - inf_sdiff_right_comm + - sdiff_sup + - Disjoint.sdiff_unique + - sdiff_compl + - sup_inf_inf_compl + - inf_sdiff_distrib_right + - himp_eq_left + - compl_eq_iff_isCompl + - eq_compl_iff_isCompl + - sdiff_sdiff_sup_sdiff' + - sup_sdiff_inf + - sdiff_sdiff_eq_sdiff_sup + - disjoint_compl_left_iff + - sdiff_inf_sdiff + - eq_compl_comm + - compl_sdiff + - le_sdiff + - sdiff_unique + - sdiff_sdiff_right + - compl_eq_comm + - sdiff_sdiff_right_self + - inf_sdiff_assoc + - inf_sdiff_inf + - eq_of_sdiff_eq_sdiff + - compl_sdiff_compl + - sdiff_eq_self_iff_disjoint + - disjoint_sdiff_comm + - compl_sup_eq_top + - inf_sdiff + - disjoint_sdiff_iff_le + - compl_lt_self + - compl_le_of_compl_le + - inf_sdiff_self_left + - inf_sdiff_distrib_left + - path: Mathlib/Data/Set/Basic.lean + theorems: + - Set.eq_singleton_or_nontrivial + - Set.inter_compl_nonempty_iff + - Set.ite_inter_compl_self + - Set.subset_ite + - Set.inclusion_self + - Set.mem_dite_univ_left + - Set.diff_singleton_subset_iff + - Set.ite_eq_of_subset_left + - Set.subsingleton_or_nontrivial + - Set.subsingleton_coe + - Set.sep_ext_iff + - Eq.subset + - Set.insert_diff_of_mem + - Set.ne_univ_iff_exists_not_mem + - Set.Nonempty.eq_univ + - Set.sep_eq_self_iff_mem_true + - Set.inter_nonempty_iff_exists_right + - Set.sep_eq_empty_iff_mem_false + - Prop.compl_singleton + - Set.ssubset_singleton_iff + - Set.isEmpty_coe_sort + - Set.inclusion_right + - Set.insert_subset_insert_iff + - Set.not_nonempty_iff_eq_empty' + - Set.monotoneOn_iff_monotone + - Set.ite_inter + - Set.not_monotoneOn_not_antitoneOn_iff_exists_le_le + - Set.nonempty_compl_of_nontrivial + - Set.singleton_subset_singleton + - Set.inter_insert_of_mem + - Set.not_nontrivial_singleton + - Set.Nontrivial.exists_ne + - Set.diff_eq_compl_inter + - Set.Subsingleton.induction_on + - Set.not_subsingleton_iff + - Set.mem_ite_empty_left + - Set.ite_empty_right + - Set.disjoint_singleton_left + - Set.ssubset_iff_sdiff_singleton + - Set.insert_diff_self_of_not_mem + - Set.insert_diff_eq_singleton + - Set.univ_eq_true_false + - Set.union_empty_iff + - Set.ite_inter_inter + - Set.diff_union_inter + - Set.not_subset + - Set.disjoint_iff_forall_ne + - Set.not_subset_iff_exists_mem_not_mem + - Set.mem_dite_univ_right + - Set.strictAntiOn_iff_strictAnti + - Set.forall_in_swap + - Set.ite_left + - Set.subset_insert_diff_singleton + - Set.compl_union_self + - Set.ite_empty + - Set.powerset_singleton + - Set.inter_singleton_eq_empty + - Set.singleton_inter_nonempty + - Set.ite_eq_of_subset_right + - Set.antitoneOn_iff_antitone + - Set.not_nonempty_iff_eq_empty + - Set.not_monotoneOn_not_antitoneOn_iff_exists_lt_lt + - Function.Injective.nonempty_apply_iff + - Set.Nontrivial.ne_singleton + - Set.inter_nonempty_iff_exists_left + - Set.insert_diff_singleton + - Set.diff_singleton_eq_self + - Set.exists_eq_singleton_iff_nonempty_subsingleton + - Set.strictMonoOn_iff_strictMono + - Set.inter_singleton_nonempty + - Set.nontrivial_coe_sort + - Set.disjoint_right + - Set.ite_inter_self + - Set.bex_insert_iff + - Set.subset_singleton_iff_eq + - Set.insert_diff_singleton_comm + - Set.mem_dite_empty_left + - Set.insert_diff_of_not_mem + - Set.ssubset_iff_insert + - Set.ite_univ + - Set.ite_inter_of_inter_eq + - Set.pair_eq_pair_iff + - Set.diff_singleton_sSubset + - Set.eq_of_inclusion_surjective + - Set.ite_empty_left + - Set.insert_inter_of_mem + - Set.ext_iff + - Set.disjoint_singleton + - Set.insert_subset_iff + - Set.mem_dite + - Set.inclusion_inclusion + - Set.mem_ite_empty_right + - path: Mathlib/Topology/Support.lean + theorems: + - isCompact_range_of_mulSupport_subset_isCompact + - HasCompactSupport.smul_right + - hasCompactMulSupport_comp_left + - HasCompactMulSupport.comp_closedEmbedding + - hasCompactMulSupport_def + - HasCompactMulSupport.continuous_extend_one + - LocallyFinite.smul_left + - HasCompactSupport.mul_right + - HasCompactMulSupport.isCompact_preimage + - LocallyFinite.smul_right + - exists_compact_iff_hasCompactMulSupport + - HasCompactSupport.mul_left + - not_mem_mulTSupport_iff_eventuallyEq + - HasCompactSupport.smul_left + - LocallyFinite.exists_finset_nhd_mulSupport_subset + - HasCompactSupport.smul_left' + - mulTSupport_eq_empty_iff + - HasCompactMulSupport.intro' + - "HasCompactMulSupport.comp\u2082_left" + - path: Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean + theorems: + - Complex.GammaIntegral_eq_mellin + - Complex.Gamma_integrand_deriv_integrable_B + - Complex.differentiableAt_GammaAux + - Complex.tendsto_self_mul_Gamma_nhds_zero + - Complex.integral_cpow_mul_exp_neg_mul_Ioi + - Real.integral_rpow_mul_exp_neg_mul_Ioi + - Real.Gamma_eq_zero_iff + - Complex.GammaAux_recurrence1 + - Real.differentiableAt_Gamma + - Complex.Gamma_zero + - Complex.GammaIntegral_add_one + - Complex.GammaIntegral_conj + - Complex.GammaIntegral_ofReal + - Complex.hasDerivAt_GammaIntegral + - Real.Gamma_nonneg_of_nonneg + - Complex.differentiableAt_Gamma + - Real.Gamma_neg_nat_eq_zero + - Real.Gamma_zero + - Complex.Gamma_integrand_interval_integrable + - Complex.Gamma_eq_integral + - Complex.Gamma_integrand_deriv_integrable_A + - Real.Gamma_add_one + - Complex.Gamma_one + - Real.Gamma_pos_of_pos + - Complex.GammaIntegral_convergent + - Complex.Gamma_nat_eq_factorial + - Complex.Gamma_conj + - Complex.Gamma_neg_nat_eq_zero + - Real.Gamma_eq_integral + - Complex.partialGamma_add_one + - Complex.GammaIntegral_one + - Complex.GammaAux_recurrence2 + - Complex.Gamma_add_one + - Real.Gamma_integrand_isLittleO + - Real.Gamma_nat_eq_factorial + - Real.GammaIntegral_convergent + - Real.Gamma_ne_zero + - path: Mathlib/MeasureTheory/OuterMeasure/Basic.lean + theorems: + - MeasureTheory.OuterMeasure.f_iUnion + - MeasureTheory.OuterMeasure.comap_iInf + - MeasureTheory.OuterMeasure.smul_boundedBy + - MeasureTheory.OuterMeasure.le_smul_caratheodory + - MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat + - MeasureTheory.OuterMeasure.isCaratheodory_sum + - MeasureTheory.OuterMeasure.comap_iSup + - MeasureTheory.OuterMeasure.sup_apply + - MeasureTheory.OuterMeasure.map_iInf + - MeasureTheory.OuterMeasure.map_comap + - MeasureTheory.OuterMeasure.comap_map + - MeasureTheory.extend_eq + - MeasureTheory.OuterMeasure.isCaratheodory_empty + - MeasureTheory.OuterMeasure.boundedBy_zero + - MeasureTheory.OuterMeasure.restrict_trim + - MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top + - MeasureTheory.OuterMeasure.map_top_of_surjective + - MeasureTheory.OuterMeasure.boundedBy_apply + - MeasureTheory.OuterMeasure.top_caratheodory + - MeasureTheory.OuterMeasure.smul_iSup + - MeasureTheory.OuterMeasure.isCaratheodory_inter + - MeasureTheory.extend_iUnion_nat + - MeasureTheory.OuterMeasure.null_of_locally_null + - MeasureTheory.extend_eq_top + - MeasureTheory.OuterMeasure.sInf_eq_boundedBy_sInfGen + - MeasureTheory.OuterMeasure.trim_iSup + - MeasureTheory.OuterMeasure.map_sup + - MeasureTheory.le_extend + - MeasureTheory.OuterMeasure.iSup_sInfGen_nonempty + - MeasureTheory.OuterMeasure.smul_dirac_apply + - MeasureTheory.OuterMeasure.restrict_ofFunction + - MeasureTheory.OuterMeasure.iUnion_null_iff + - MeasureTheory.OuterMeasure.restrict_univ + - MeasureTheory.OuterMeasure.boundedBy_caratheodory + - MeasureTheory.OuterMeasure.restrict_biInf + - MeasureTheory.OuterMeasure.top_apply + - MeasureTheory.OuterMeasure.comap_top + - MeasureTheory.OuterMeasure.le_inter_add_diff + - MeasureTheory.inducedOuterMeasure_caratheodory + - MeasureTheory.OuterMeasure.map_ofFunction_le + - MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero + - MeasureTheory.inducedOuterMeasure_eq_iInf + - MeasureTheory.OuterMeasure.ofFunction_caratheodory + - MeasureTheory.OuterMeasure.trim_binop + - MeasureTheory.OuterMeasure.biInf_apply' + - MeasureTheory.OuterMeasure.map_ofFunction + - MeasureTheory.OuterMeasure.iInf_apply + - MeasureTheory.OuterMeasure.iInf_apply' + - MeasureTheory.OuterMeasure.le_sum_caratheodory + - MeasureTheory.OuterMeasure.isCaratheodory_compl + - MeasureTheory.extend_mono + - MeasureTheory.OuterMeasure.trim_eq_iInf' + - MeasureTheory.OuterMeasure.isCaratheodory_union + - MeasureTheory.OuterMeasure.isCaratheodory_compl_iff + - MeasureTheory.OuterMeasure.ofFunction_le + - MeasureTheory.OuterMeasure.comap_boundedBy + - MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos + - MeasureTheory.inducedOuterMeasure_preimage + - MeasureTheory.OuterMeasure.dirac_caratheodory + - MeasureTheory.OuterMeasure.biInf_apply + - MeasureTheory.OuterMeasure.exists_measurable_superset_forall_eq_trim + - MeasureTheory.OuterMeasure.map_iInf_comap + - MeasureTheory.extend_iUnion_le_tsum_nat + - MeasureTheory.OuterMeasure.diff_null + - MeasureTheory.OuterMeasure.exists_measurable_superset_of_trim_eq_zero + - MeasureTheory.OuterMeasure.trim_sum_ge + - MeasureTheory.OuterMeasure.le_trim_iff + - MeasureTheory.OuterMeasure.trim_eq_iInf + - MeasureTheory.OuterMeasure.trim_congr + - MeasureTheory.OuterMeasure.le_boundedBy + - MeasureTheory.OuterMeasure.coe_iSup + - MeasureTheory.OuterMeasure.biUnion_null_iff + - MeasureTheory.extend_mono' + - MeasureTheory.OuterMeasure.restrict_iInf_restrict + - MeasureTheory.OuterMeasure.map_top + - MeasureTheory.OuterMeasure.restrict_empty + - MeasureTheory.OuterMeasure.ofFunction_union_of_top_of_nonempty_inter + - MeasureTheory.inducedOuterMeasure_exists_set + - MeasureTheory.OuterMeasure.boundedBy_top + - MeasureTheory.OuterMeasure.top_apply' + - MeasureTheory.OuterMeasure.smul_ofFunction + - MeasureTheory.OuterMeasure.map_comap_of_surjective + - MeasureTheory.OuterMeasure.sInf_apply + - MeasureTheory.OuterMeasure.restrict_iSup + - MeasureTheory.OuterMeasure.map_le_restrict_range + - MeasureTheory.smul_extend + - MeasureTheory.extend_iUnion + - MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim + - MeasureTheory.OuterMeasure.le_boundedBy' + - MeasureTheory.OuterMeasure.sInf_apply' + - MeasureTheory.OuterMeasure.map_biInf_comap + - MeasureTheory.OuterMeasure.trim_le_trim_iff + - MeasureTheory.OuterMeasure.boundedBy_eq + - MeasureTheory.OuterMeasure.boundedBy_eq_ofFunction + - MeasureTheory.OuterMeasure.restrict_sInf_eq_sInf_restrict + - MeasureTheory.OuterMeasure.trim_eq_trim_iff + - MeasureTheory.extend_iUnion_le_tsum_nat' + - MeasureTheory.OuterMeasure.map_iSup + - MeasureTheory.extend_union + - MeasureTheory.OuterMeasure.le_add_caratheodory + - MeasureTheory.OuterMeasure.coe_fn_injective + - MeasureTheory.OuterMeasure.comap_ofFunction + - MeasureTheory.OuterMeasure.measure_inter_union + - path: Mathlib/FieldTheory/Separable.lean + theorems: + - Polynomial.separable_map + - Polynomial.Separable.isCoprime + - Polynomial.Separable.map + - Polynomial.exists_separable_of_irreducible + - Polynomial.Separable.inj_of_prod_X_sub_C + - Polynomial.unique_separable_of_irreducible + - Polynomial.separable_prod_X_sub_C_iff + - AlgHom.card_of_powerBasis + - Polynomial.separable_of_subsingleton + - Polynomial.Separable.of_mul_left + - Polynomial.Separable.of_pow' + - Polynomial.separable_iff_derivative_ne_zero + - Associated.separable + - Polynomial.isUnit_of_self_mul_dvd_separable + - Polynomial.card_rootSet_eq_natDegree_iff_of_splits + - Polynomial.separable_X_add_C + - IsSeparable.of_equiv_equiv + - Polynomial.Separable.squarefree + - Polynomial.separable_X_sub_C + - Polynomial.exists_finset_of_splits + - Polynomial.separable_C + - Polynomial.nodup_aroots_iff_of_splits + - Polynomial.card_rootSet_eq_natDegree + - Polynomial.Separable.isIntegral + - Polynomial.count_roots_le_one + - Polynomial.separable_prod_X_sub_C_iff' + - "Polynomial.Separable.eval\u2082_derivative_ne_zero" + - Polynomial.multiplicity_le_one_of_separable + - Polynomial.Separable.of_mul_right + - Polynomial.separable_X + - Polynomial.nodup_roots_iff_of_splits + - Irreducible.separable + - Polynomial.eq_X_sub_C_of_separable_of_root_eq + - Polynomial.nodup_of_separable_prod + - Polynomial.separable_prod' + - IsSeparable.of_algHom + - Polynomial.separable_or + - Polynomial.X_pow_sub_one_separable_iff + - Polynomial.isUnit_or_eq_zero_of_separable_expand + - Polynomial.Separable.mul + - Polynomial.Separable.of_dvd + - Polynomial.separable_X_pow_sub_C_unit + - AlgEquiv.isSeparable + - Polynomial.not_separable_zero + - path: Mathlib/Algebra/Order/CauSeq/Completion.lean + theorems: + - CauSeq.lim_add + - CauSeq.Completion.ofRat_div + - CauSeq.lim_neg + - CauSeq.lim_eq_zero_iff + - CauSeq.lim_mul_lim + - CauSeq.Completion.mk_eq_zero + - CauSeq.Completion.inv_zero + - CauSeq.Completion.inv_mk + - CauSeq.Completion.ofRat_inv + - CauSeq.lim_mul + - CauSeq.Completion.mul_inv_cancel + - CauSeq.Completion.cau_seq_zero_ne_one + - CauSeq.lim_inv + - CauSeq.Completion.inv_mul_cancel + - path: Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean + theorems: + - MeasureTheory.integral_comp_smul_deriv_Ioi + - MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded + - MeasureTheory.AECover.iSup_lintegral_eq_of_countably_generated + - MeasureTheory.AECover.aestronglyMeasurable + - MeasureTheory.integrableOn_Ioi_deriv_of_nonneg + - MeasureTheory.aecover_closedBall + - MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded + - MeasureTheory.integrableOn_Ioi_comp_mul_left_iff + - MeasureTheory.integrableOn_Ioi_deriv_of_nonneg' + - MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded' + - HasCompactSupport.integral_Iic_deriv_eq + - MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto' + - MeasureTheory.integrableOn_Ioi_comp_rpow_iff + - MeasureTheory.AECover.aemeasurable + - MeasureTheory.integrableOn_Ioi_deriv_of_nonpos + - MeasureTheory.intervalIntegral_tendsto_integral_Ioi + - MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto' + - MeasureTheory.integral_comp_mul_left_Ioi + - MeasureTheory.integral_mul_deriv_eq_deriv_mul + - MeasureTheory.intervalIntegral_tendsto_integral + - MeasureTheory.integral_comp_rpow_Ioi + - MeasureTheory.aecover_restrict_of_ae_imp + - MeasureTheory.integrableOn_Ioi_deriv_of_nonpos' + - MeasureTheory.integrableOn_Ioi_comp_mul_right_iff + - MeasureTheory.AECover.biInter_Ici_aecover + - MeasureTheory.integrableOn_Ioi_comp_rpow_iff' + - MeasureTheory.integral_comp_mul_right_Ioi + - MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto + - MeasureTheory.AECover.integral_tendsto_of_countably_generated + - MeasureTheory.integral_comp_mul_deriv_Ioi + - MeasureTheory.AECover.integrable_of_integral_norm_bounded + - MeasureTheory.integral_of_hasDerivAt_of_tendsto + - MeasureTheory.AECover.lintegral_tendsto_of_nat + - MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto + - MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto' + - MeasureTheory.integral_Iic_mul_deriv_eq_deriv_mul + - MeasureTheory.integral_comp_rpow_Ioi_of_pos + - MeasureTheory.aecover_ball + - HasCompactSupport.integral_Ioi_deriv_eq + - MeasureTheory.integral_Iic_deriv_mul_eq_sub + - MeasureTheory.integral_Ioi_mul_deriv_eq_deriv_mul + - MeasureTheory.intervalIntegral_tendsto_integral_Iic + - MeasureTheory.integral_Ioi_deriv_mul_eq_sub + - MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded + - path: Mathlib/SetTheory/Game/PGame.lean + theorems: + - SetTheory.PGame.zero_le_lf + - SetTheory.PGame.neg_fuzzy_neg_iff + - SetTheory.PGame.fuzzy_congr + - SetTheory.PGame.add_moveRight_inr + - SetTheory.PGame.neg_equiv_neg_iff + - SetTheory.PGame.Subsequent.moveRight_mk_left + - SetTheory.PGame.star_fuzzy_zero + - SetTheory.PGame.moveRight_neg_symm + - SetTheory.PGame.neg_lt_zero_iff + - SetTheory.PGame.bddBelow_range_of_small + - SetTheory.PGame.lf_zero + - SetTheory.PGame.zero_fuzzy_neg_iff + - SetTheory.PGame.zero_le + - SetTheory.PGame.lf_iff_lt_or_fuzzy + - SetTheory.PGame.moveRight_neg_symm' + - SetTheory.PGame.zero_lf_neg_iff + - SetTheory.PGame.neg_le_iff + - SetTheory.PGame.lf_of_lf_of_le + - SetTheory.PGame.neg_fuzzy_iff + - SetTheory.PGame.relabel_moveLeft + - SetTheory.PGame.lf_def + - SetTheory.PGame.isOption_neg_neg + - SetTheory.PGame.neg_lt_iff + - SetTheory.PGame.Subsequent.mk_right' + - SetTheory.PGame.moveRight_neg' + - SetTheory.PGame.lt_or_equiv_or_gt_or_fuzzy + - SetTheory.PGame.bddAbove_of_small + - SetTheory.PGame.lt_neg_iff + - SetTheory.PGame.relabel_moveRight + - SetTheory.PGame.neg_equiv_zero_iff + - SetTheory.PGame.lt_or_equiv_or_gf + - SetTheory.PGame.neg_ofLists + - SetTheory.PGame.moveLeft_neg' + - SetTheory.PGame.neg_equiv_iff + - SetTheory.PGame.rightMoves_add_cases + - SetTheory.PGame.zero_lt_neg_iff + - SetTheory.PGame.moveLeft_neg + - SetTheory.PGame.equiv_of_mk_equiv + - SetTheory.PGame.zero_equiv_neg_iff + - SetTheory.PGame.lf_zero_le + - SetTheory.PGame.Subsequent.moveLeft_mk_right + - SetTheory.PGame.sub_zero + - SetTheory.PGame.neg_lf_iff + - SetTheory.PGame.lf_iff_exists_le + - SetTheory.PGame.le_zero + - SetTheory.PGame.lf_or_equiv_or_gf + - SetTheory.PGame.wf_isOption + - SetTheory.PGame.zero_le_add_left_neg + - SetTheory.PGame.lf_neg_iff + - SetTheory.PGame.neg_lf_zero_iff + - SetTheory.PGame.add_lf_add_right + - SetTheory.PGame.lf_of_le_of_lf + - SetTheory.PGame.neg_le_zero_iff + - SetTheory.PGame.add_moveRight_inl + - SetTheory.PGame.moveRight_neg + - SetTheory.PGame.add_moveLeft_inl + - SetTheory.PGame.add_left_neg_le_zero + - SetTheory.PGame.Subsequent.moveRight_mk_right + - SetTheory.PGame.add_moveLeft_inr + - SetTheory.PGame.zero_lf_le + - SetTheory.PGame.add_lf_add_left + - SetTheory.PGame.neg_le_lf_neg_iff + - SetTheory.PGame.le_zero_lf + - SetTheory.PGame.equiv_of_eq + - SetTheory.PGame.bddBelow_of_small + - SetTheory.PGame.neg_fuzzy_zero_iff + - SetTheory.PGame.zero_lf + - SetTheory.PGame.le_iff_forall_lf + - SetTheory.PGame.zero_le_neg_iff + - SetTheory.PGame.le_def + - SetTheory.PGame.moveLeft_neg_symm' + - SetTheory.PGame.le_or_gf + - SetTheory.PGame.Subsequent.moveLeft_mk_left + - SetTheory.PGame.neg_lt_neg_iff + - SetTheory.PGame.neg_star + - path: Mathlib/Topology/Order/NhdsSet.lean + theorems: + - nhdsSet_Ioc + - nhdsSet_Icc + - Ioi_mem_nhdsSet_Ioc + - Iio_mem_nhdsSet_Ico + - hasBasis_nhdsSet_Iic_Iic + - hasBasis_nhdsSet_Iic_Iio + - nhdsSet_Ico + - Ioi_mem_nhdsSet_Ici_iff + - path: Mathlib/Topology/ContinuousFunction/NonUnitalFunctionalCalculus.lean + theorems: + - "cfc\u2099_star" + - "cfc\u2099_zero" + - "cfc\u2099Hom_eq_of_continuous_of_map_id" + - "cfc\u2099_comp_star" + - "cfc\u2099_sub" + - "eqOn_of_cfc\u2099_eq_cfc\u2099" + - "cfc\u2099_smul_id" + - "cfc\u2099_comp" + - NonUnitalStarAlgHom.ext_continuousMap + - eq_zero_of_quasispectrum_eq_zero + - "cfc\u2099_apply" + - "cfc\u2099_mul" + - "cfc\u2099_star_id" + - "cfc\u2099_congr" + - "cfc\u2099_smul" + - "cfc\u2099_comp_neg" + - "cfc\u2099_neg_id" + - "cfc\u2099_apply_of_not_continuousOn" + - "cfc\u2099_add" + - cfc_map_quasispectrum + - "cfc\u2099_apply_of_not_predicate" + - "cfc\u2099_apply_of_not_and_and" + - "cfc\u2099_neg" + - "cfc\u2099_apply_of_not_map_zero" + - path: Mathlib/Data/Finset/LocallyFinite/Basic.lean + theorems: + - Finset.filter_lt_le_eq_Ioc + - Finset.Icc_eq_cons_Ioc + - Finset.Ioc_subset_Icc_self + - Finset.Ioo_insert_right + - Finset.nonempty_Ico + - Finset.uIcc_subset_Icc + - Finset.Ioi_subset_Ici_self + - Finset.Ico_inter_Ico + - Finset.Ioo_filter_lt + - Finset.Icc_subset_Ici_self + - Finset.Ico_subset_Icc_self + - Finset.nonempty_Ioo + - Finset.Ico_eq_empty_iff + - Finset.Icc_eq_cons_Ico + - Finset.Ico_filter_lt_of_le_right + - Finset.card_Ioo_eq_card_Icc_sub_two + - Finset.image_add_left_Icc + - Finset.Ioc_subset_Ioc + - Finset.nonempty_Ioc + - Finset.Ico_diff_Ico_left + - Finset.uIcc_of_le + - Finset.card_Iio_eq_card_Iic_sub_one + - Finset.Icc_subset_Iic_self + - Finset.Icc_self + - Finset.Ico_filter_le_left + - Finset.Icc_diff_Ioo_self + - transGen_covBy_of_lt + - Finset.uIcc_subset_uIcc + - Finset.image_add_right_Icc + - Finset.image_add_left_Ioc + - Finset.Ioc_subset_Ioi_self + - Finset.Icc_subset_Ico_right + - Finset.image_add_right_Ico + - Finset.filter_le_lt_eq_Ico + - Finset.Icc_ssubset_Icc_right + - Finset.Ioc_diff_Ioo_self + - Finset.Ico_union_Ico + - Finset.uIcc_injective_right + - Finset.Ico_insert_right + - Finset.Ioc_eq_empty_iff + - Finset.Ico_subset_Ico_union_Ico + - Finset.nonempty_Iio + - Finset.right_mem_Icc + - Finset.Iic_eq_cons_Iio + - Finset.Ico_subset_Ico_iff + - Finset.Ico_union_Ico_eq_Ico + - Finset.Icc_erase_left + - Finset.Icc_eq_singleton_iff + - Finset.Ioc_union_Ioc_eq_Ioc + - Finset.card_Ioo_eq_card_Ico_sub_one + - Finset.Ioc_subset_Iic_self + - Finset.Ioc_erase_right + - Finset.Ioo_subset_Ioi_self + - Finset.Ico_diff_Ioo_self + - Finset.uIcc_eq_union + - Finset.Ioo_subset_Ico_self + - Finset.nonempty_Icc + - Finset.Icc_diff_Ico_self + - Finset.Ici_erase + - le_iff_reflTransGen_covBy + - Finset.filter_le_eq_Ici + - Finset.map_add_right_Ico + - Finset.Ioi_disjUnion_Iio + - Finset.Icc_ssubset_Icc_left + - Finset.not_mem_uIcc_of_lt + - Finset.Ico_diff_Ico_right + - Finset.Icc_subset_Icc_iff + - Finset.uIcc_subset_uIcc_union_uIcc + - Finset.Ioo_subset_Ioo + - Finset.filter_lt_eq_Ioi + - Finset.uIcc_comm + - strictMono_iff_forall_covBy + - Finset.Iio_insert + - Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag + - Finset.Ioc_eq_cons_Ioo + - Finset.Ico_subset_Ioo_left + - monotone_iff_forall_wcovBy + - Finset.image_add_right_Ioo + - Finset.Iio_subset_Iic_self + - Finset.Ioo_subset_Ioc_self + - Finset.eq_of_mem_uIcc_of_mem_uIcc + - Finset.card_Ioi_eq_card_Ici_sub_one + - Finset.uIcc_injective_left + - Finset.image_add_left_Ico + - monotone_iff_forall_covBy + - Finset.image_add_left_Ioo + - Finset.Icc_diff_both + - Finset.left_mem_Ico + - Finset.nonempty_Ioi + - Finset.Ioo_subset_Iio_self + - Finset.Iic_erase + - Finset.map_add_left_Icc + - Finset.Icc_eq_empty_iff + - Finset.Ioi_insert + - Finset.Ico_filter_lt + - Finset.uIcc_of_ge + - Finset.Ioo_eq_empty_iff + - Finset.Ico_subset_Iio_self + - Finset.filter_gt_eq_Iio + - Finset.Icc_diff_Ioc_self + - Finset.filter_lt_lt_eq_Ioo + - Finset.map_add_left_Ico + - Finset.Ico_eq_cons_Ioo + - Finset.filter_le_le_eq_Icc + - le_iff_transGen_wcovBy + - Finset.Ici_eq_cons_Ioi + - Finset.Icc_subset_Ico_iff + - Finset.Ioo_insert_left + - Finset.filter_ge_eq_Iic + - Finset.map_add_right_Ioc + - Finset.card_Ico_eq_card_Icc_sub_one + - transGen_wcovBy_of_le + - Finset.Ico_subset_Ico + - Finset.map_add_right_Icc + - Finset.left_mem_Icc + - Finset.map_add_right_Ioo + - Finset.Ico_subset_Ici_self + - Finset.Icc_subset_Icc + - Finset.right_mem_Ioc + - Finset.Ico_filter_le + - Finset.Icc_subset_Ioo_iff + - Finset.mem_uIcc' + - Finset.Ico_filter_le_of_left_le + - Finset.Ico_union_Ico' + - Finset.Iio_filter_lt + - Finset.eq_of_mem_uIcc_of_mem_uIcc' + - Finset.image_add_right_Ioc + - Finset.uIcc_self + - lt_iff_transGen_covBy + - Finset.Ico_erase_left + - Finset.not_mem_uIcc_of_gt + - path: Mathlib/Algebra/Ring/Idempotents.lean + theorems: + - IsIdempotentElem.mul_of_commute + - IsIdempotentElem.one_sub + - IsIdempotentElem.pow_succ_eq + - IsIdempotentElem.iff_eq_zero_or_one + - IsIdempotentElem.pow + - path: Mathlib/Analysis/Normed/Order/UpperLower.lean + theorems: + - IsUpperSet.exists_subset_ball + - IsLowerSet.cthickening' + - IsLowerSet.mem_interior_of_forall_lt + - IsLowerSet.exists_subset_ball + - IsUpperSet.cthickening' + - IsUpperSet.mem_interior_of_forall_lt + - IsUpperSet.thickening' + - IsLowerSet.thickening' + - path: Mathlib/Analysis/Convex/Gauge.lean + theorems: + - gauge_lt_eq + - gauge_pos + - gauge_eq_zero + - gauge_le_eq + - gauge_ball + - Convex.lipschitzWith_gauge + - Balanced.starConvex + - mul_gauge_le_norm + - tendsto_gauge_nhds_zero' + - gauge_of_subset_zero + - gauge_smul_left_of_nonneg + - gauge_def' + - gauge_add_le + - gauge_closure_zero + - one_le_gauge_of_not_mem + - comap_gauge_nhds_zero_le + - le_gauge_of_subset_closedBall + - gauge_smul_left + - gauge_lt_of_mem_smul + - exists_lt_of_gauge_lt + - gauge_le_of_mem + - gauge_le_one_of_mem + - interior_subset_gauge_lt_one + - gauge_smul_of_nonneg + - gaugeSeminorm_ball_one + - gauge_smul + - gauge_norm_smul + - gauge_closedBall + - gauge_zero' + - gauge_eq_one_iff_mem_frontier + - gauge_zero + - le_gauge_of_not_mem + - mem_closure_of_gauge_le_one + - gauge_lt_one_eq_interior + - gauge_neg_set_neg + - gauge_lt_eq' + - gauge_neg_set_eq_gauge_neg + - mem_openSegment_of_gauge_lt_one + - gauge_empty + - continuousAt_gauge_zero + - Convex.gauge_le + - Seminorm.gauge_ball + - gauge_lt_one_of_mem_of_isOpen + - gauge_lt_one_eq_self_of_isOpen + - gauge_unit_ball + - path: Mathlib/Data/Int/CharZero.lean + theorems: + - Int.cast_eq_zero + - Int.cast_div_ofNat_charZero + - Int.cast_inj + - Int.cast_div_charZero + - path: Mathlib/Data/ENNReal/Operations.lean + theorems: + - ENNReal.sub_mul + - ENNReal.lt_top_of_mul_ne_top_right + - ENNReal.top_pow + - ENNReal.lt_add_right + - ENNReal.mul_ne_top + - ENNReal.exists_le_of_sum_le + - ENNReal.addLECancellable_iff_ne + - ENNReal.toReal_sum + - ENNReal.sum_lt_sum_of_nonempty + - ENNReal.nat_cast_sub + - ENNReal.mul_left_strictMono + - ENNReal.add_ne_top + - ENNReal.add_lt_add + - ENNReal.smul_top + - ENNReal.toNNReal_sum + - ENNReal.not_lt_top + - ENNReal.mul_lt_mul + - ENNReal.pow_ne_zero + - ENNReal.coe_smul + - ENNReal.mul_lt_top_iff + - ENNReal.pow_strictMono + - ENNReal.lt_add_of_sub_lt_left + - ENNReal.not_lt_zero + - ENNReal.pow_eq_top_iff + - ENNReal.pow_lt_top + - ENNReal.mul_top' + - ENNReal.ofReal_sum_of_nonneg + - ENNReal.mul_sub + - ENNReal.top_mul' + - ENNReal.toNNReal_add + - ENNReal.mul_self_lt_top_iff + - path: Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean + theorems: + - LeftInvariantDerivation.toDerivation_injective + - LeftInvariantDerivation.comp_L + - LeftInvariantDerivation.evalAt_mul + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean + theorems: + - Matrix.reverse_charpoly + - Matrix.det_one_add_smul + - Matrix.matPolyEquiv_eval_eq_map + - Matrix.matPolyEquiv_eval + - Matrix.eval_charpolyRev + - Matrix.eval_det + - Matrix.coeff_charpoly_mem_ideal_pow + - Matrix.charpoly_degree_eq_dim + - Matrix.coeff_charpolyRev_eq_neg_trace + - Matrix.matPolyEquiv_symm_map_eval + - Matrix.isNilpotent_trace_of_isNilpotent + - matPolyEquiv_eq_X_pow_sub_C + - Matrix.charmatrix_apply_natDegree + - Matrix.charmatrix_apply_natDegree_le + - Matrix.isUnit_charpolyRev_of_isNilpotent + - Matrix.det_eq_sign_charpoly_coeff + - Matrix.derivative_det_one_add_X_smul + - Matrix.det_of_card_zero + - Matrix.charpoly_sub_diagonal_degree_lt + - Matrix.derivative_det_one_add_X_smul_aux + - Matrix.charpoly_monic + - Matrix.pow_eq_aeval_mod_charpoly + - Matrix.eval_det_add_X_smul + - Matrix.isNilpotent_charpoly_sub_pow_of_isNilpotent + - Matrix.trace_eq_neg_charpoly_coeff + - Matrix.det_one_add_X_smul + - Matrix.charpoly_coeff_eq_prod_coeff_of_le + - path: Mathlib/LinearAlgebra/FreeModule/Norm.lean + theorems: + - finrank_quotient_span_eq_natDegree_norm + - associated_norm_prod_smith + - path: Mathlib/MeasureTheory/Measure/Haar/Basic.lean + theorems: + - MeasureTheory.Measure.haar.is_left_invariant_chaar + - MeasureTheory.Measure.haar.prehaar_empty + - MeasureTheory.Measure.haar.nonempty_iInter_clPrehaar + - MeasureTheory.Measure.haar.is_left_invariant_prehaar + - MeasureTheory.Measure.haar.is_left_invariant_index + - MeasureTheory.Measure.haar.index_union_eq + - MeasureTheory.Measure.haar.is_left_invariant_haarContent + - MeasureTheory.Measure.haar.chaar_empty + - MeasureTheory.Measure.haar.haarContent_self + - MeasureTheory.Measure.haar.chaar_sup_eq + - MeasureTheory.Measure.haarMeasure_closure_self + - MeasureTheory.Measure.haar.mem_prehaar_empty + - MeasureTheory.Measure.haar.index_mono + - MeasureTheory.Measure.haar.index_pos + - MeasureTheory.Measure.haar.prehaar_sup_le + - MeasureTheory.Measure.haar.index_empty + - MeasureTheory.Measure.haar.chaar_mono + - MeasureTheory.Measure.haar.index_defined + - MeasureTheory.Measure.haar.le_index_mul + - MeasureTheory.Measure.haar.index_union_le + - MeasureTheory.Measure.regular_of_isMulLeftInvariant + - MeasureTheory.Measure.haar.chaar_mem_clPrehaar + - MeasureTheory.Measure.haar.prehaar_pos + - MeasureTheory.Measure.haar.mul_left_index_le + - MeasureTheory.Measure.haarMeasure_self + - MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos + - MeasureTheory.Measure.haar.prehaar_mono + - MeasureTheory.Measure.haar.prehaar_le_index + - MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos + - MeasureTheory.Measure.haar.chaar_sup_le + - MeasureTheory.Measure.haar.prehaar_mem_haarProduct + - MeasureTheory.Measure.haar.prehaar_sup_eq + - MeasureTheory.Measure.haar.chaar_self + - MeasureTheory.Measure.haar.prehaar_nonneg + - MeasureTheory.Measure.haar.chaar_nonneg + - MeasureTheory.Measure.haarMeasure_eq_iff + - MeasureTheory.Measure.haarMeasure_apply + - MeasureTheory.Measure.haarMeasure_unique + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean + theorems: + - Orientation.rotation_pi_apply + - Orientation.rotation_neg_orientation_eq_neg + - Orientation.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two + - Orientation.eq_rotation_self_iff + - Orientation.rotation_map + - Orientation.oangle_rotation_left + - Orientation.kahler_rotation_left' + - Orientation.rotation_oangle_eq_iff_norm_eq + - Orientation.oangle_rotation_oangle_right + - Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_or_eq_zero + - Complex.rotation + - Orientation.oangle_rotation_right + - Orientation.oangle_rotation_self_right + - Orientation.inner_smul_rotation_pi_div_two_left + - Orientation.oangle_rotation + - Orientation.eq_rotation_self_iff_angle_eq_zero + - Orientation.inner_rotation_pi_div_two_right + - Orientation.inner_smul_rotation_pi_div_two_smul_left + - Orientation.rotation_zero + - Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero + - Orientation.rotation_pi_div_two + - Orientation.oangle_rotation_self_left + - Orientation.rotation_eq_self_iff + - Orientation.rotation_map_complex + - Orientation.linearEquiv_det_rotation + - Orientation.neg_rotation + - Orientation.rotation_symm + - Orientation.rotation_pi + - Orientation.inner_rotation_pi_div_two_right_smul + - Orientation.rotation_rotation + - Orientation.oangle_rotation_oangle_left + - Orientation.rotation_eq_matrix_toLin + - Orientation.inner_smul_rotation_pi_div_two_right + - Orientation.kahler_rotation_right + - Orientation.inner_rotation_pi_div_two_left_smul + - Orientation.neg_rotation_neg_pi_div_two + - Orientation.exists_linearIsometryEquiv_eq_of_det_pos + - Orientation.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero + - Orientation.rotation_trans + - Orientation.inner_rotation_pi_div_two_left + - Orientation.rotation_eq_self_iff_angle_eq_zero + - Orientation.inner_smul_rotation_pi_div_two_smul_right + - Orientation.oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero + - Orientation.kahler_rotation_left + - path: Mathlib/RingTheory/GradedAlgebra/Radical.lean + theorems: + - Ideal.IsPrime.homogeneousCore + - Ideal.IsHomogeneous.radical + - Ideal.IsHomogeneous.radical_eq + - Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem + - path: Mathlib/Data/Set/Lattice.lean + theorems: + - Set.sigmaToiUnion_surjective + - Set.preimage_iInter + - Set.biUnion_and + - Set.sUnion_prod_const + - Set.sUnion_iUnion + - Set.sUnion_eq_univ_iff + - Set.iUnion_eq_range_sigma + - Set.biUnion_range_preimage_singleton + - Set.image_sInter_subset + - "Set.preimage_iUnion\u2082" + - Set.sInter_eq_biInter + - Set.kernImage_compl + - Set.kernImage_union_preimage + - sSup_iUnion + - Set.image2_eq_iUnion + - Set.compl_range_subset_kernImage + - Set.iInter_eq_compl_iUnion_compl + - Set.biUnion_pair + - Set.compl_sInter + - Set.seq_def + - Set.preimage_sInter + - Set.iUnion_of_singleton + - Set.prod_eq_seq + - Set.sUnion_eq_iUnion + - Set.image_projection_prod + - "Set.iUnion\u2082_eq_univ_iff" + - Set.compl_sUnion + - Set.bijective_iff_bijective_of_iUnion_eq_univ + - Set.image2_iUnion_left + - Set.sInter_iUnion + - Set.iUnion_image_left + - Set.surjective_iff_surjective_of_iUnion_eq_univ + - "Set.image2_iInter\u2082_subset_left" + - sSup_sUnion + - Set.inj_on_iUnion_of_directed + - Set.sInter_prod_sInter + - Set.nonempty_sInter + - "Set.iUnion\u2082_subset_iff" + - Set.image2_iInter_subset_left + - "Set.prod_iUnion\u2082" + - Set.biInter_inter + - Set.biUnion_diff_biUnion_subset + - "Set.union_distrib_iInter\u2082_right" + - Set.injective_iff_injective_of_iUnion_eq_univ + - Set.inter_empty_of_inter_sUnion_empty + - Set.range_eq_iUnion + - Set.biUnion_of_singleton + - Set.prod_eq_biUnion_right + - Set.range_sigma_eq_iUnion_range + - Set.Ici_iSup + - "Set.image2_iUnion\u2082_left" + - Set.nonempty_of_nonempty_iUnion_eq_univ + - Set.inter_biInter + - "Set.Ici_iSup\u2082" + - Set.iUnion_image_preimage_sigma_mk_eq_self + - Set.InjOn.image_iInter_eq + - Set.iUnion_image_right + - "Set.iInter\u2082_eq_empty_iff" + - Set.prod_iInter + - Set.sUnion_eq_compl_sInter_compl + - Set.prod_sUnion + - Set.nonempty_iInter_Iic_iff + - exists_sUnion + - Set.iUnion_prod_const + - Set.biInter_and' + - "Set.nonempty_iInter\u2082" + - "Set.union_iInter\u2082" + - Set.nonempty_iInter + - Set.image_eq_iUnion + - Set.iUnion_singleton_eq_range + - "Set.compl_iUnion\u2082" + - Set.sInter_prod + - "Set.mem_iInter\u2082" + - Set.image2_eq_seq + - "Set.union_distrib_iInter\u2082_left" + - Set.univ_pi_eq_iInter + - Set.nonempty_iUnion + - Set.prod_sInter + - Set.preimage_iUnion + - Set.nonempty_biUnion + - forall_sUnion + - "Set.image2_iInter\u2082_subset_right" + - Set.Ici_sSup + - Set.image2_iInter_subset_right + - Set.iInter_iInter_eq' + - "Set.iInter\u2082_union" + - Set.prod_eq_biUnion_left + - Set.kernImage_preimage_union + - "Set.mapsTo_iInter\u2082" + - iInf_sUnion + - "Set.inter_iUnion\u2082" + - Set.biInter_pair + - Set.image_iUnion + - Set.biInter_and + - Set.univ_subtype + - Set.image_iInter + - Set.diff_iUnion + - iSup_iUnion + - Set.iUnion_eq_compl_iInter_compl + - "Set.compl_iInter\u2082" + - "Set.preimage_iInter\u2082" + - Set.preimage_sUnion + - Set.iUnion_iUnion_eq' + - Set.iUnion_eq_univ_iff + - "Set.image_iUnion\u2082" + - Set.sigmaToiUnion_injective + - Set.nonempty_of_nonempty_iUnion + - Set.biUnion_insert + - Set.iInter_setOf + - Set.iUnion_iUnion_eq_or_left + - "Set.Iic_iInf\u2082" + - Set.sInter_eq_empty_iff + - Set.iUnion_range_eq_iUnion + - Set.image_seq + - Set.exists_set_mem_of_union_eq_top + - Set.iUnion_prod_of_monotone + - Set.iUnion_range_eq_sUnion + - Set.sInter_eq_iInter + - "Set.image2_iUnion\u2082_right" + - Set.nonempty_of_union_eq_top_of_nonempty + - Set.surjOn_iInter + - Set.iUnion_nonempty_self + - Set.pi_diff_pi_subset + - Set.pi_def + - Set.diff_iInter + - "Set.mem_iUnion\u2082" + - Set.sInter_eq_compl_sUnion_compl + - Set.nonempty_sUnion + - Set.sUnion_mem_empty_univ + - Set.seq_seq + - Set.kernImage_empty + - Set.iUnion_univ_pi + - Set.InjOn.image_biInter_eq + - Set.biUnion_and' + - iSup_sUnion + - Set.iUnion_eq_range_psigma + - Set.kernImage_preimage_eq_iff + - "Set.subset_iInter\u2082_iff" + - Set.Iic_sInf + - Set.image2_iUnion_right + - Set.biUnion_preimage_singleton + - Set.directed_on_iUnion + - Set.Iic_iInf + - Set.iUnion_prod + - "Set.image_iInter\u2082" + - Set.biInter_insert + - "Set.iUnion\u2082_inter" + - Set.sUnion_eq_biUnion + - Set.prod_iUnion + - Set.iInter_iInter_eq_or_left + - Set.prod_image_seq_comm + - "Set.iUnion\u2082_prod_const" + - path: Mathlib/RingTheory/Algebraic.lean + theorems: + - isAlgebraic_iff_isIntegral + - Subalgebra.isAlgebraic_iff + - isAlgebraic_algHom_iff + - exists_integral_multiple + - IsAlgebraic.tower_top_of_injective + - IsAlgebraic.invOf + - isAlgebraic_rat + - inv_eq_of_aeval_divX_ne_zero + - isAlgebraic_one + - Algebra.isAlgebraic_iff + - Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero + - isAlgebraic_algebraMap + - isAlgebraic_iff_not_injective + - Algebra.IsAlgebraic.trans + - isAlgebraic_nat + - inv_eq_of_root_of_coeff_zero_ne_zero + - IsIntegralClosure.exists_smul_eq_mul + - Subalgebra.inv_mem_of_algebraic + - IsAlgebraic.of_pow + - isAlgebraic_int + - IsAlgebraic.inv_iff + - Algebra.IsAlgebraic.algHom_bijective + - IsAlgebraic.algebraMap + - Subalgebra.isField_of_algebraic + - path: Mathlib/Algebra/GroupPower/Basic.lean + theorems: + - pow_sub_mul_pow + - pow_eq_pow_mod + - zpow_add_one + - pow_one + - one_nsmul + - pow_right_comm + - zpow_sub_one + - pow_bit1' + - pow_add + - one_zpow + - pow_mul_pow_eq_one + - nsmul_zero + - zpow_two + - Commute.mul_pow + - inv_zpow' + - one_pow + - one_div_zpow + - zpow_eq_zpow_emod + - div_pow + - pow_two + - zpow_mul + - eq_zero_or_one_of_sq_eq_self + - zpow_one + - pow_boole + - pow_bit1 + - mul_self_zpow + - Commute.mul_zpow + - zpow_induction_left + - zpow_mul' + - zpow_sub + - zpow_bit0 + - zpow_eq_zpow_emod' + - pow_sub + - pow_mul + - zpow_bit1 + - mul_zpow_neg_one + - zpow_one_add + - add_nsmul + - zpow_induction_right + - pow_mul' + - zpow_add + - one_div_pow + - pow_three + - pow_mul_pow_sub + - inv_pow + - inv_zpow + - zpow_neg + - inv_pow_sub + - pow_three' + - SemiconjBy.zpow_right + - zpow_mul_comm + - div_zpow + - path: Mathlib/Algebra/NeZero.lean + theorems: + - not_neZero + - path: Mathlib/Data/Real/Pi/Leibniz.lean + theorems: + - Real.tendsto_sum_pi_div_four + - path: Mathlib/MeasureTheory/Measure/Stieltjes.lean + theorems: + - StieltjesFunction.length_Ioc + - StieltjesFunction.measure_Icc + - StieltjesFunction.measure_univ + - StieltjesFunction.eq_of_measure_of_tendsto_atBot + - StieltjesFunction.length_empty + - StieltjesFunction.countable_leftLim_ne + - StieltjesFunction.isProbabilityMeasure + - StieltjesFunction.iInf_rat_gt_eq + - StieltjesFunction.ext + - StieltjesFunction.measure_Ico + - StieltjesFunction.eq_of_measure_of_eq + - StieltjesFunction.measurableSet_Ioi + - StieltjesFunction.isFiniteMeasure + - StieltjesFunction.measure_Iic + - StieltjesFunction.outer_Ioc + - StieltjesFunction.borel_le_measurable + - StieltjesFunction.rightLim_eq + - StieltjesFunction.length_subadditive_Icc_Ioo + - StieltjesFunction.outer_trim + - StieltjesFunction.measure_Ici + - path: Mathlib/NumberTheory/Cyclotomic/Discriminant.lean + theorems: + - IsCyclotomicExtension.discr_prime_pow_ne_two' + - IsCyclotomicExtension.discr_odd_prime + - IsCyclotomicExtension.discr_prime_pow + - IsCyclotomicExtension.discr_prime_pow_eq_unit_mul_pow + - IsCyclotomicExtension.discr_prime_pow_ne_two + - IsPrimitiveRoot.discr_zeta_eq_discr_zeta_sub_one + - path: Mathlib/Analysis/InnerProductSpace/Spectrum.lean + theorems: + - eigenvalue_nonneg_of_nonneg + - LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces + - inner_product_apply_eigenvector + - LinearMap.IsSymmetric.conj_eigenvalue_eq_self + - LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot + - LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace + - LinearMap.IsSymmetric.eigenvectorBasis_apply_self_apply + - LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot' + - LinearMap.IsSymmetric.orthogonalFamily_eigenspaces + - LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_invariant + - eigenvalue_pos_of_pos + - LinearMap.IsSymmetric.diagonalization_apply_self_apply + - LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis + - path: Mathlib/Probability/Distributions/Exponential.lean + theorems: + - ProbabilityTheory.lintegral_exponentialPDF_eq_antiDeriv + - ProbabilityTheory.exp_neg_integrableOn_Ioc + - ProbabilityTheory.hasDerivAt_neg_exp_mul_exp + - ProbabilityTheory.exponentialCDFReal_eq + - ProbabilityTheory.exponentialPDF_eq + - ProbabilityTheory.exponentialPDF_of_nonneg + - path: Mathlib/CategoryTheory/Equivalence.lean + theorems: + - CategoryTheory.Equivalence.functor_asEquivalence + - CategoryTheory.Equivalence.counit_app_functor + - CategoryTheory.IsEquivalence.ofIso_refl + - CategoryTheory.Equivalence.funInvIdAssoc_inv_app + - CategoryTheory.Equivalence.cancel_counitInv_right_assoc + - CategoryTheory.Equivalence.cancel_unitInv_right + - CategoryTheory.Equivalence.cancel_counitInv_right_assoc' + - CategoryTheory.Equivalence.cancel_unit_right + - CategoryTheory.Equivalence.cancel_unit_right_assoc + - CategoryTheory.IsEquivalence.fun_inv_map + - CategoryTheory.Equivalence.counitInv_functor_comp + - "CategoryTheory.Equivalence.adjointify_\u03B7_\u03B5" + - CategoryTheory.IsEquivalence.inv_fun_map + - CategoryTheory.Equivalence.invFunIdAssoc_inv_app + - CategoryTheory.Equivalence.cancel_counit_right + - CategoryTheory.Equivalence.invFunIdAssoc_hom_app + - CategoryTheory.Equivalence.unitInv_app_inverse + - CategoryTheory.Equivalence.cancel_unit_right_assoc' + - CategoryTheory.Equivalence.unit_inverse_comp + - CategoryTheory.Equivalence.counitInv_app_functor + - CategoryTheory.Equivalence.cancel_counitInv_right + - CategoryTheory.Equivalence.inverse_counitInv_comp + - CategoryTheory.IsEquivalence.ofIso_trans + - CategoryTheory.Equivalence.funInvIdAssoc_hom_app + - CategoryTheory.Equivalence.unit_app_inverse + - CategoryTheory.Equivalence.inverse_asEquivalence + - path: Mathlib/Algebra/Algebra/NonUnitalHom.lean + theorems: + - NonUnitalAlgHom.to_distribMulActionHom_injective + - NonUnitalAlgHom.coe_injective + - NonUnitalAlgHom.mk_coe + - NonUnitalAlgHom.to_mulHom_injective + - NonUnitalAlgHom.ext_iff + - NonUnitalAlgHom.snd_prod + - NonUnitalAlgHom.fst_prod + - NonUnitalAlgHom.coe_mulHom_mk + - NonUnitalAlgHom.coe_distribMulActionHom_mk + - path: Mathlib/Analysis/NormedSpace/ENorm.lean + theorems: + - ENorm.map_sub_rev + - ENorm.map_zero + - ENorm.map_sub_le + - ENorm.coeFn_injective + - ENorm.map_smul + - ENorm.map_neg + - path: Mathlib/Topology/MetricSpace/Gluing.lean + theorems: + - Metric.Sigma.dist_same + - Metric.Sum.mem_uniformity + - Metric.glueDist_triangle_inl_inr_inr + - Metric.Sum.one_le_dist_inr_inl + - Metric.Sum.mem_uniformity_iff_glueDist + - Metric.toGlue_commute + - Metric.eq_of_glueDist_eq_zero + - Metric.le_glueDist_inr_inl + - Metric.toInductiveLimit_commute + - Metric.Sum.dist_comm + - Metric.inductiveLimitDist_eq_dist + - Metric.Sigma.isOpen_iff + - Metric.Sigma.dist_triangle + - Metric.Sigma.fst_eq_of_dist_lt_one + - Metric.Sigma.completeSpace + - Metric.glueDist_glued_points + - Metric.Sum.dist_eq_glueDist + - Metric.glueDist_triangle_inl_inr_inl + - Metric.Sigma.isometry_mk + - Metric.toInductiveLimit_isometry + - Metric.glueDist_triangle + - Metric.glueDist_swap + - Metric.Sigma.one_le_dist_of_ne + - path: Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean + theorems: + - "CategoryTheory.ShortComplex.LeftHomologyData.leftHomology\u03C0_comp_leftHomologyIso_hom" + - CategoryTheory.ShortComplex.isIso_cyclesMap'_of_isIso_of_mono + - CategoryTheory.ShortComplex.LeftHomologyMapData.leftHomologyMap_comm + - "CategoryTheory.ShortComplex.LeftHomologyMapData.congr_\u03C6H" + - CategoryTheory.ShortComplex.HasLeftHomology.hasCokernel + - CategoryTheory.ShortComplex.liftCycles_comp_cyclesMap + - CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_hom_comp_i + - CategoryTheory.ShortComplex.LeftHomologyMapData.cyclesMap_comm + - CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_f' + - "CategoryTheory.ShortComplex.LeftHomologyData.isIso_\u03C0" + - CategoryTheory.ShortComplex.LeftHomologyData.ofZeros_f' + - CategoryTheory.ShortComplex.LeftHomologyData.ofIsLimitKernelFork_f' + - "CategoryTheory.ShortComplex.liftCycles_leftHomology\u03C0_eq_zero_of_boundary" + - CategoryTheory.ShortComplex.cycles_ext_iff + - "CategoryTheory.ShortComplex.LeftHomologyData.\u03C4\u2081_ofEpiOfIsIsoOfMono_f'" + - CategoryTheory.ShortComplex.leftHomology_ext_iff + - CategoryTheory.ShortComplex.LeftHomologyData.ofIsColimitCokernelCofork_f' + - CategoryTheory.ShortComplex.LeftHomologyData.isIso_i + - CategoryTheory.ShortComplex.LeftHomologyData.cyclesIso_inv_comp_iCycles + - CategoryTheory.ShortComplex.leftHomology_ext + - "CategoryTheory.ShortComplex.LeftHomologyMapData.congr_\u03C6K" + - CategoryTheory.ShortComplex.leftHomologyMap'_comp + - CategoryTheory.ShortComplex.cyclesMap'_comp + - CategoryTheory.ShortComplex.LeftHomologyMapData.leftHomologyMap_eq + - "CategoryTheory.ShortComplex.LeftHomologyData.\u03C0_comp_leftHomologyIso_inv" + - "CategoryTheory.ShortComplex.LeftHomologyData.liftK_\u03C0_eq_zero_of_boundary" + - CategoryTheory.ShortComplex.LeftHomologyData.lift_K_comp_cyclesIso_inv + - CategoryTheory.ShortComplex.comp_liftCycles + - CategoryTheory.ShortComplex.cycles_ext + - path: Mathlib/Algebra/Order/Monoid/NatCast.lean + theorems: + - zero_le_three + - zero_lt_three + - zero_lt_four + - zero_le_two + - zero_le_four + - one_lt_two + - path: Mathlib/Control/Traversable/Instances.lean + theorems: + - Sum.traverse_eq_map_id + - Option.naturality + - Sum.map_traverse + - Option.comp_traverse + - List.traverse_eq_map_id + - List.mem_traverse + - Option.id_traverse + - Sum.traverse_map + - Option.traverse_eq_map_id + - List.comp_traverse + - Sum.id_traverse + - List.traverse_append + - List.naturality + - Sum.comp_traverse + - Sum.naturality + - List.id_traverse + - path: Mathlib/Analysis/Calculus/FDeriv/Prod.lean + theorems: + - hasFDerivWithinAt_apply + - hasFDerivAtFilter_pi' + - differentiableAt_apply + - differentiable_apply + - differentiableOn_apply + - differentiableWithinAt_apply + - hasStrictFDerivAt_apply + - hasFDerivAt_apply + - hasStrictFDerivAt_pi' + - path: Mathlib/Algebra/Field/Basic.lean + theorems: + - div_neg_self + - one_div_mul_sub_mul_one_div_eq_one_div_add_one_div + - same_sub_div + - div_sub_div_same + - same_add_div + - div_sub_same + - neg_div_self + - inv_sub_inv + - inv_sub_inv' + - add_div + - Commute.div_add_div + - inv_neg_one + - neg_div' + - one_div_neg_one_eq_neg_one + - add_div' + - inv_neg + - div_add' + - neg_inv + - Commute.div_sub_div + - Commute.inv_add_inv + - add_div_eq_mul_add_div + - neg_div_neg_eq + - div_add_same + - div_neg + - Commute.one_div_add_one_div + - one_div_neg_eq_neg_one_div + - div_sub' + - neg_div + - div_neg_eq_neg_div + - sub_div' + - path: Mathlib/Algebra/Ring/Hom/Defs.lean + theorems: + - RingHom.domain_nontrivial + - RingHom.cancel_left + - RingHom.codomain_trivial_iff_range_trivial + - NonUnitalRingHom.cancel_left + - RingHom.toMonoidWithZeroHom_eq_coe + - NonUnitalRingHom.comp_zero + - RingHom.map_ite_zero_one + - AddMonoidHom.coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero + - RingHom.codomain_trivial_iff_map_one_eq_zero + - NonUnitalRingHom.zero_comp + - map_bit1 + - path: Mathlib/AlgebraicGeometry/AffineScheme.lean + theorems: + - AlgebraicGeometry.isAffineOfIso + - AlgebraicGeometry.topIsAffineOpen + - AlgebraicGeometry.exists_basicOpen_le_affine_inter + - AlgebraicGeometry.IsAffineOpen.exists_basicOpen_le + - AlgebraicGeometry.rangeIsAffineOpenOfOpenImmersion + - AlgebraicGeometry.IsAffineOpen.basicOpen_fromSpec_app + - AlgebraicGeometry.IsAffineOpen.opensFunctor_map_basicOpen + - AlgebraicGeometry.IsAffineOpen.isLocalization_basicOpen + - AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine + - AlgebraicGeometry.IsAffineOpen.basicOpen_basicOpen_is_basicOpen + - AlgebraicGeometry.of_affine_open_cover + - AlgebraicGeometry.isBasis_basicOpen + - AlgebraicGeometry.IsAffineOpen.fromSpec_map_basicOpen' + - "AlgebraicGeometry.IsAffineOpen.Spec\u0393Identity_hom_app_fromSpec" + - AlgebraicGeometry.IsAffineOpen.self_le_basicOpen_union_iff + - AlgebraicGeometry.IsAffineOpen.isLocalization_of_eq_basicOpen + - AlgebraicGeometry.IsAffineOpen.isLocalization_stalk' + - AlgebraicGeometry.IsAffineOpen.basicOpenIsAffine + - AlgebraicGeometry.IsAffineOpen.fromSpec_base_preimage + - AlgebraicGeometry.IsAffineOpen.isLocalization_stalk + - AlgebraicGeometry.IsAffineOpen.fromSpec_map_basicOpen + - AlgebraicGeometry.IsAffineOpen.fromSpec_primeIdealOf + - AlgebraicGeometry.isBasis_affine_open + - AlgebraicGeometry.IsAffineOpen.isCompact + - AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion + - AlgebraicGeometry.IsAffineOpen.fromSpec_app_self + - AlgebraicGeometry.IsAffineOpen.fromSpec_range + - AlgebraicGeometry.IsAffineOpen.imageIsOpenImmersion + - AlgebraicGeometry.IsAffineOpen.basicOpen_union_eq_self_iff + - AlgebraicGeometry.IsAffineOpen.mapRestrictBasicOpen + - AlgebraicGeometry.IsAffineOpen.fromSpec_image_top + - path: Mathlib/Data/List/TFAE.lean + theorems: + - List.tfae_cons_self + - List.tfae_singleton + - List.exists_tfae + - List.forall_tfae + - List.tfae_not_iff + - List.tfae_cons_of_mem + - List.tfae_of_cycle + - path: Mathlib/Algebra/GroupWithZero/Divisibility.lean + theorems: + - IsPrimal.mul + - zero_dvd_iff + - dvdNotUnit_of_dvd_of_not_dvd + - IsRelPrime.ne_zero_or_ne_zero + - dvd_zero + - dvd_and_not_dvd_iff + - mul_dvd_mul_iff_right + - mul_dvd_mul_iff_left + - ne_zero_of_dvd_ne_zero + - dvd_antisymm + - isRelPrime_of_no_nonunits_factors + - path: Mathlib/Analysis/NormedSpace/MStructure.lean + theorems: + - IsLprojection.mul_compl_self + - IsLprojection.mul + - IsLprojection.distrib_lattice_lemma + - IsLprojection.join + - IsLprojection.commute + - IsLprojection.Lcomplement + - path: Mathlib/Data/Vector3.lean + theorems: + - Vector3.insert_fz + - exists_vector_zero + - Vector3.append_insert + - Vector3.insert_fs + - Vector3.append_add + - exists_vector_succ + - vectorAllP_iff_forall + - path: Mathlib/Data/Finsupp/NeLocus.lean + theorems: + - Finsupp.neLocus_eq_empty + - Finsupp.neLocus_self_sub_right + - Finsupp.neLocus_comm + - Finsupp.neLocus_sub_left + - Finsupp.subset_mapRange_neLocus + - Finsupp.mem_neLocus + - Finsupp.neLocus_zero_right + - Finsupp.neLocus_eq_support_sub + - Finsupp.neLocus_self_add_right + - Finsupp.neLocus_self_add_left + - Finsupp.zipWith_neLocus_eq_left + - Finsupp.neLocus_sub_right + - Finsupp.neLocus_neg + - Finsupp.coe_neLocus + - Finsupp.mapRange_neLocus_eq + - Finsupp.neLocus_self_sub_left + - Finsupp.zipWith_neLocus_eq_right + - path: Mathlib/LinearAlgebra/BilinearForm/Basic.lean + theorems: + - BilinForm.neg_left + - BilinForm.smul_left_of_tower + - BilinForm.smul_right_of_tower + - BilinForm.sub_left + - BilinForm.neg_right + - BilinForm.flip_flip + - BilinForm.ext + - BilinForm.sub_right + - BilinForm.zero_left + - BilinForm.zero_right + - path: Mathlib/LinearAlgebra/BilinearForm/TensorProduct.lean + theorems: + - LinearMap.IsSymm.tmul + - LinearMap.BilinForm.tensorDistribEquiv_toLinearMap + - path: Mathlib/Data/Polynomial/Inductions.lean + theorems: + - Polynomial.X_mul_divX_add + - Polynomial.divX_C_mul_X_pow + - Polynomial.natDegree_ne_zero_induction_on + - Polynomial.divX_C_mul + - Polynomial.degree_divX_lt + - Polynomial.divX_one + - Polynomial.degree_pos_induction_on + - Polynomial.coeff_divX + - Polynomial.divX_C + - Polynomial.natDegree_divX_eq_natDegree_tsub_one + - Polynomial.divX_add + - Polynomial.divX_eq_zero_iff + - Polynomial.divX_mul_X_add + - path: Mathlib/Order/WithBot.lean + theorems: + - WithBot.le_coe_iff + - WithBot.coe_le_iff + - WithTop.top_le_iff + - WithBot.le_bot_iff + - WithTop.strictMono_map_iff + - WithBot.map_le_iff + - WithBot.lt_iff_exists_coe + - WithBot.unbot'_eq_self_iff + - WithTop.monotone_map_iff + - "WithTop.map\u2082_coe_right" + - WithTop.coe_lt_top + - "WithTop.map\u2082_top_right" + - WithBot.unbot'_le_iff + - WithBot.bot_lt_iff_ne_bot + - WithTop.coe_le_coe + - WithTop.map_le_iff + - WithBot.le_unbot_iff + - WithBot.lt_coe_bot + - WithTop.not_none_lt + - WithBot.unbot_le_iff + - WithTop.lt_iff_exists_coe + - WithBot.strictMono_map_iff + - WithBot.unbot'_eq_unbot'_iff + - WithBot.monotone_map_iff + - "WithBot.map\u2082_coe_right" + - WithBot.unbot'_lt_iff + - WithBot.unbot'_eq_iff + - WithBot.some_lt_some + - WithBot.lt_coe_iff + - WithTop.coe_lt_coe + - IsMin.withTop + - "WithBot.map\u2082_bot_right" + - path: Mathlib/Algebra/FreeMonoid/Basic.lean + theorems: + - FreeMonoid.ofList_join + - FreeMonoid.hom_eq + - FreeMonoid.toList_prod + - FreeMonoid.comp_lift + - path: Mathlib/LinearAlgebra/Dimension/Constructions.lean + theorems: + - subalgebra_top_finrank_eq_submodule_top_finrank + - FiniteDimensional.finrank_matrix + - FiniteDimensional.finrank_finsupp + - rank_fun + - rank_pi + - FiniteDimensional.finrank_tensorProduct + - subalgebra_top_rank_eq_submodule_top_rank + - FiniteDimensional.finrank_pi + - rank_tensorProduct' + - FiniteDimensional.finrank_finsupp_self + - rank_span_finset_le + - rank_tensorProduct + - rank_span_le + - finrank_range_le_card + - Submodule.lt_of_le_of_finrank_lt_finrank + - rank_matrix + - finrank_span_set_eq_card + - rank_fin_fun + - rank_finsupp + - rank_matrix' + - finrank_span_le_card + - Subalgebra.rank_top + - rank_fun_eq_lift_mul + - FiniteDimensional.finrank_fin_fun + - finrank_ulift + - FiniteDimensional.finrank_directSum + - rank_prod' + - FiniteDimensional.finrank_prod + - finrank_span_finset_eq_card + - rank_finsupp' + - finrank_span_eq_card + - rank_finsupp_self' + - rank_directSum + - rank_quotient_eq_of_le_torsion + - lift_rank_add_lift_rank_le_rank_prod + - finrank_span_finset_le_card + - Subalgebra.finrank_bot + - rank_fun' + - Submodule.lt_top_of_finrank_lt_finrank + - Subalgebra.rank_bot + - rank_quotient_add_rank_le + - rank_matrix'' + - rank_prod + - rank_finsupp_self + - rank_add_rank_le_rank_prod + - FiniteDimensional.finrank_pi_fintype + - path: Mathlib/Algebra/MonoidAlgebra/Basic.lean + theorems: + - MonoidAlgebra.mapDomain_mul + - MonoidAlgebra.opRingEquiv_single + - MonoidAlgebra.opRingEquiv_symm_single + - MonoidAlgebra.mul_apply + - MonoidAlgebra.single_pow + - MonoidAlgebra.lift_single + - AddMonoidAlgebra.algHom_ext_iff + - AddMonoidAlgebra.lift_apply + - MonoidAlgebra.ringHom_ext + - AddMonoidAlgebra.mul_single_zero_apply + - MonoidAlgebra.lift_of + - AddMonoidAlgebra.prod_single + - AddMonoidAlgebra.single_pow + - MonoidAlgebra.liftNC_one + - AddMonoidAlgebra.opRingEquiv_symm_single + - MonoidAlgebra.mul_single_apply_of_not_exists_mul + - MonoidAlgebra.single_one_mul_apply + - MonoidAlgebra.mul_single_one_apply + - MonoidAlgebra.of_injective + - AddMonoidAlgebra.mapDomain_mul + - MonoidAlgebra.lift_apply + - MonoidAlgebra.mul_apply_antidiagonal + - MonoidAlgebra.single_eq_algebraMap_mul_of + - MonoidAlgebra.induction_on + - AddMonoidAlgebra.opRingEquiv_single + - MonoidAlgebra.single_algebraMap_eq_algebraMap_mul_of + - MonoidAlgebra.mul_apply_right + - MonoidAlgebra.mul_apply_left + - MonoidAlgebra.mul_single_apply_aux + - MonoidAlgebra.liftNC_mul + - MonoidAlgebra.lift_unique + - AddMonoidAlgebra.mapDomain_algebraMap + - MonoidAlgebra.single_mul_apply_aux + - AddMonoidAlgebra.mapDomain_one + - MonoidAlgebra.liftNC_smul + - MonoidAlgebra.prod_single + - MonoidAlgebra.single_mul_apply_of_not_exists_mul + - AddMonoidAlgebra.lift_unique + - MonoidAlgebra.mapDomain_one + - AddMonoidAlgebra.induction_on + - MonoidAlgebra.smul_of + - AddMonoidAlgebra.single_zero_mul_apply + - MonoidAlgebra.single_mul_single + - path: Mathlib/FieldTheory/SeparableDegree.lean + theorems: + - Polynomial.natSepDegree_map + - Field.finSepDegree_mul_finSepDegree_of_isAlgebraic + - Polynomial.Monic.eq_X_pow_char_pow_sub_C_of_natSepDegree_eq_one_of_irreducible + - Polynomial.natSepDegree_le_of_dvd + - IntermediateField.finSepDegree_adjoin_simple_eq_natSepDegree + - Polynomial.natSepDegree_C_mul + - Polynomial.natSepDegree_eq_zero + - Polynomial.natSepDegree_eq_of_splits + - Irreducible.natSepDegree_dvd_natDegree + - Polynomial.natSepDegree_C_mul_X_sub_C_pow + - IntermediateField.isSeparable_adjoin_pair_of_separable + - Field.finSepDegree_eq_finrank_of_isSeparable + - minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_pow + - Polynomial.natSepDegree_X_pow + - Polynomial.natSepDegree_mul_eq_iff + - Polynomial.Monic.eq_X_sub_C_pow_of_natSepDegree_eq_one_of_splits + - Field.finSepDegree_eq_finrank_iff + - Polynomial.natSepDegree_le_natDegree + - Polynomial.natSepDegree_one + - minpoly.natSepDegree_eq_one_iff_eq_expand_X_sub_C + - Polynomial.Monic.eq_X_pow_char_pow_sub_C_pow_of_natSepDegree_eq_one + - minpoly.natSepDegree_eq_one_iff_pow_mem + - IntermediateField.finSepDegree_adjoin_simple_dvd_finrank + - Field.finSepDegree_self + - Polynomial.Separable.comap_minpoly_of_isSeparable + - Polynomial.natSepDegree_smul_nonzero + - Polynomial.natSepDegree_eq_natDegree_iff + - Polynomial.natSepDegree_mul + - perfectField_iff_splits_of_natSepDegree_eq_one + - Polynomial.natSepDegree_pow_of_ne_zero + - minpoly.natSepDegree_eq_one_iff_eq_X_pow_sub_C + - Field.separable_algebraMap + - IntermediateField.isSeparable_adjoin_simple_iff_separable + - Field.finSepDegree_dvd_finrank + - Irreducible.natSepDegree_eq_one_iff_of_monic' + - Polynomial.natSepDegree_zero + - Polynomial.natSepDegree_X + - Polynomial.natSepDegree_expand + - IntermediateField.finSepDegree_adjoin_simple_le_finrank + - Irreducible.natSepDegree_eq_one_iff_of_monic + - IntermediateField.separable_of_mem_isSeparable + - Polynomial.natSepDegree_X_pow_char_pow_sub_C + - IntermediateField.finSepDegree_adjoin_simple_eq_finrank_iff + - Polynomial.natSepDegree_pow + - Polynomial.natSepDegree_X_sub_C_pow + - Polynomial.natSepDegree_X_sub_C + - Polynomial.Monic.natSepDegree_eq_one_iff + - Polynomial.natSepDegree_ne_zero + - path: Mathlib/GroupTheory/Index.lean + theorems: + - Subgroup.relindex_mul_relindex + - Subgroup.relindex_comap + - Subgroup.mul_mem_iff_of_index_two + - Subgroup.card_dvd_of_surjective + - Subgroup.relindex_inf_ne_zero + - Subgroup.index_eq_two_iff + - Subgroup.index_center_le_pow + - Subgroup.index_dvd_card + - Subgroup.index_comap + - Subgroup.card_mul_index + - Subgroup.index_ne_zero_of_finite + - Subgroup.index_iInf_ne_zero + - Subgroup.relindex_bot_left + - Subgroup.relindex_sup_left + - Subgroup.relindex_inf_mul_relindex + - Subgroup.index_iInf_le + - Subgroup.index_inf_ne_zero + - Subgroup.index_map_dvd + - Subgroup.relindex_ker + - Subgroup.index_map + - Subgroup.nat_card_dvd_of_surjective + - Subgroup.index_comap_of_surjective + - Subgroup.mul_self_mem_of_index_two + - Subgroup.nat_card_dvd_of_injective + - Subgroup.index_mul_card + - Subgroup.dvd_index_map + - Subgroup.index_ker + - Subgroup.index_inf_le + - Subgroup.inf_relindex_right + - Subgroup.relindex_top_right + - Subgroup.relindex_inf_le + - path: Mathlib/GroupTheory/Finiteness.lean + theorems: + - Submonoid.FG.map + - Subgroup.rank_closure_finset_le_card + - Group.rank_le_of_surjective + - Monoid.fg_of_surjective + - Subgroup.fg_iff_submonoid_fg + - AddSubmonoid.fg_iff_mul_fg + - Subgroup.rank_congr + - AddSubgroup.fg_iff_mul_fg + - Subgroup.rank_closure_finite_le_nat_card + - Monoid.fg_iff_submonoid_fg + - Submonoid.fg_iff_add_fg + - Submonoid.fg_iff + - Subgroup.fg_iff_add_fg + - Subgroup.fg_iff + - Submonoid.FG.map_injective + - path: Mathlib/Algebra/DirectSum/Internal.lean + theorems: + - DirectSum.coe_of_mul_apply_of_not_le + - SetLike.algebraMap_mem_graded + - DirectSum.coe_mul_of_apply_aux + - DirectSum.coe_mul_of_apply_of_not_le + - DirectSum.coe_mul_apply + - SetLike.nat_cast_mem_graded + - SetLike.int_cast_mem_graded + - DirectSum.coe_of_mul_apply_aux + - DirectSum.coeAlgHom_of + - DirectSum.coe_mul_of_apply + - DirectSum.coe_of_mul_apply + - DirectSum.coe_of_mul_apply_of_le + - DirectSum.coe_mul_apply_eq_dfinsupp_sum + - path: Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean + theorems: + - "MeasureTheory.WithDensity\u1D65Eq.congr_ae" + - "MeasureTheory.withDensity\u1D65_smul_eq_withDensity\u1D65_withDensity'" + - "MeasureTheory.Integrable.withDensity\u1D65_trim_eq_integral" + - "MeasureTheory.withDensity\u1D65_neg" + - "MeasureTheory.withDensity\u1D65_apply" + - "MeasureTheory.withDensity\u1D65_eq_withDensity_pos_part_sub_withDensity_neg_part" + - "MeasureTheory.withDensity\u1D65_add" + - "MeasureTheory.withDensity\u1D65_zero" + - "MeasureTheory.Integrable.withDensity\u1D65_trim_absolutelyContinuous" + - "MeasureTheory.withDensity\u1D65_smul_eq_withDensity\u1D65_withDensity" + - "MeasureTheory.withDensity\u1D65_toReal" + - "MeasureTheory.Measure.withDensity\u1D65_absolutelyContinuous" + - "MeasureTheory.withDensity\u1D65_smul" + - "MeasureTheory.withDensity\u1D65_sub" + - "MeasureTheory.Integrable.ae_eq_of_withDensity\u1D65_eq" + - path: Mathlib/SetTheory/Cardinal/Ordinal.lean + theorems: + - Cardinal.aleph'_limit + - Cardinal.mk_equiv_eq_zero_iff_lift_ne + - Cardinal.mk_surjective_eq_zero_iff_lift + - Cardinal.mul_ciSup + - Cardinal.mk_equiv_eq_arrow_of_lift_eq + - Cardinal.aleph'_pos + - Cardinal.add_eq_left_iff + - Ordinal.Cardinal.mk_iUnion_Ordinal_le_of_le + - Cardinal.mk_arrow_eq_zero_iff + - Cardinal.eq_aleph_of_eq_card_ord + - Cardinal.mul_le_max_of_aleph0_le_left + - Cardinal.mk_perm_eq_two_power + - Cardinal.mul_eq_right + - Cardinal.mk_compl_eq_mk_compl_finite + - Cardinal.prod_eq_two_power + - Cardinal.powerlt_aleph0_le + - Cardinal.mk_embedding_eq_arrow_of_lift_le + - Cardinal.mk_equiv_of_eq + - Cardinal.mk_bounded_set_le_of_infinite + - Cardinal.beth_normal + - Cardinal.power_nat_le_max + - Cardinal.power_self_eq + - Cardinal.aleph'_le_of_limit + - Cardinal.aleph0_lt_aleph_one + - Cardinal.ord_card_unbounded + - Cardinal.aleph_limit + - Cardinal.le_mul_right + - Cardinal.add_eq_left + - Cardinal.mk_compl_eq_mk_compl_infinite + - Cardinal.mk_list_eq_max_mk_aleph0 + - Cardinal.mk_multiset_of_countable + - Cardinal.add_eq_max' + - Cardinal.mk_multiset_of_infinite + - Cardinal.mul_lt_of_lt + - Cardinal.le_mul_left + - Cardinal.mul_le_max_of_aleph0_le_right + - Cardinal.eq_of_add_eq_of_aleph0_le + - Cardinal.mk_compl_finset_of_infinite + - Cardinal.aleph'_succ + - Cardinal.mk_compl_of_infinite + - Cardinal.aleph'_nat + - Cardinal.aleph0_le_aleph + - Cardinal.extend_function_of_lt + - Cardinal.ciSup_mul + - Cardinal.aleph_zero + - Cardinal.ord_aleph_eq_enum_card + - Cardinal.mk_embedding_eq_zero_iff_lt + - Cardinal.add_eq_right + - Cardinal.aleph'_isNormal + - Cardinal.eq_of_add_eq_add_left + - Cardinal.principal_add_ord + - Cardinal.mul_eq_left_iff + - Cardinal.mk_embedding_eq_zero_iff_lift_lt + - Cardinal.alephIdx_le + - Cardinal.aleph0_le_aleph' + - Cardinal.add_eq_right_iff + - Cardinal.eq_aleph'_of_eq_card_ord + - Cardinal.ciSup_add + - Cardinal.mk_bounded_subset_le + - Cardinal.beth_strictMono + - Cardinal.mul_eq_max + - Cardinal.mul_eq_left + - Cardinal.exists_aleph + - Cardinal.nat_power_eq + - Cardinal.mul_eq_max' + - Cardinal.mk_equiv_of_lift_eq + - Cardinal.mk_list_eq_mk + - Cardinal.extend_function + - Cardinal.mk_surjective_eq_arrow_of_lift_le + - Cardinal.nat_add_eq + - Cardinal.mul_eq_self + - Cardinal.ord_isLimit + - Cardinal.add_eq_self + - Cardinal.add_le_max + - Cardinal.aleph_le_beth + - Cardinal.add_le_add_iff_of_lt_aleph0 + - Cardinal.add_ciSup + - Cardinal.mk_surjective_eq_zero_iff + - Cardinal.extend_function_finite + - Cardinal.mk_compl_eq_mk_compl_finite_lift + - Cardinal.mk_finsupp_of_infinite' + - Cardinal.add_lt_of_lt + - Cardinal.aleph_add_aleph + - Cardinal.ciSup_add_ciSup + - Cardinal.mk_finsupp_lift_of_infinite + - Cardinal.mul_eq_max_of_aleph0_le_right + - Cardinal.mk_bounded_set_le + - Cardinal.mul_eq_max_of_aleph0_le_left + - Cardinal.powerlt_aleph0 + - Cardinal.ord_aleph'_eq_enum_card + - Cardinal.mk_equiv_eq_zero_iff_ne + - Cardinal.aleph'_omega + - Cardinal.mk_perm_eq_self_power + - Cardinal.eq_of_add_eq_add_right + - Cardinal.pow_le + - Cardinal.mul_le_max + - Cardinal.countable_iff_lt_aleph_one + - Cardinal.mk_finsupp_lift_of_infinite' + - Cardinal.aleph_mul_aleph + - Cardinal.mk_list_le_max + - Cardinal.aleph_succ + - Cardinal.succ_aleph0 + - Cardinal.type_cardinal + - Cardinal.mk_finsupp_nat + - Cardinal.mk_multiset_of_isEmpty + - Cardinal.max_aleph_eq + - Cardinal.aleph'_zero + - path: Mathlib/Data/Set/Function.lean + theorems: + - Set.exists_eq_graphOn_image_fst + - Function.Semiconj.injOn_preimage + - Set.EqOn.image_eq_self + - Function.Semiconj.bijOn_range + - Set.piecewise_univ + - Set.range_extend_subset + - Set.piecewise_insert_of_ne + - Set.EqOn.piecewise_ite' + - Function.invFunOn_image_image_subset + - Set.image_restrict + - Set.injective_piecewise_iff + - Set.univ_pi_piecewise_univ + - Set.SurjOn.inter_inter + - Set.LeftInvOn.image_inter + - Set.preimage_invFun_of_mem + - Set.eqOn_singleton + - Set.MapsTo.iterate_restrict + - Set.maps_range_to + - Set.SurjOn.leftInvOn_of_rightInvOn + - Function.monotoneOn_of_rightInvOn_of_mapsTo + - Function.invFunOn_injOn_image + - Set.surjective_iff_surjOn_univ + - Set.injOn_insert + - Set.bijOn_singleton + - Set.piecewise_le + - Set.SurjOn.comp_right + - Set.preimage_invFun_of_not_mem + - Set.apply_piecewise + - Set.piecewise_same + - Set.piecewise_compl + - Set.range_restrictPreimage + - Set.EqOn.inter_preimage_eq + - Function.Semiconj.surjOn_image + - Disjoint.image + - Set.MapsTo.extendDomain + - Set.range_piecewise + - Set.image_fst_graphOn + - Set.LeftInvOn.image_image + - Set.surjOn_iff_exists_map_subtype + - Set.restrict_extend_compl_range + - Function.Semiconj.surjOn_range + - Function.Injective.injOn_range + - Set.range_extend + - MonotoneOn.congr + - Set.eqOn_univ + - Set.LeftInvOn.extendDomain + - Set.surjOn_id + - Set.surjOn_iff_exists_bijOn_subset + - Set.piecewise_empty + - Set.piecewise_singleton + - Set.SurjOn.comp_left + - Set.SurjOn.invOn_invFunOn + - Set.piecewise_mem_pi + - Function.Semiconj.mapsTo_preimage + - Set.SurjOn.congr + - Set.restrict_extend_range + - Set.MapsTo.restrict_inj + - Set.piecewise_insert_self + - Function.leftInvOn_invFunOn_of_subset_image_image + - Set.LeftInvOn.image_inter' + - Set.piecewise_insert + - Set.injective_codRestrict + - Set.mapsTo_iff_exists_map_subtype + - Set.SurjOn.extendDomain + - Set.MapsTo.piecewise_ite + - Set.RightInvOn.extendDomain + - StrictMonoOn.congr + - Set.MapsTo.coe_iterate_restrict + - Set.image_eq_iff_surjOn_mapsTo + - Function.Semiconj.injOn_range + - strictMono_restrict + - Set.SurjOn.bijOn_subset + - Equiv.bijOn + - Set.piecewise_le_piecewise + - Set.LeftInvOn.mapsTo + - Set.piecewise_preimage + - Set.injOn_pair + - Set.EqOn.cancel_right + - Set.exists_eq_graphOn + - Set.injOn_union + - Set.InjOn.image_inter + - Set.MapsTo.restrict_surjective_iff + - Function.invFunOn_neg + - Equiv.bijOn_swap + - Set.univ_pi_piecewise + - Function.Semiconj.injOn_image + - "Set.apply_piecewise\u2082" + - Function.invFunOn_pos + - Set.exists_injOn_iff_injective + - Set.eqOn_piecewise + - path: Mathlib/SetTheory/Ordinal/FixedPoint.lean + theorems: + - Ordinal.nfp_add_eq_mul_omega + - Ordinal.apply_le_nfpFamily + - Ordinal.sup_iterate_eq_nfp + - Ordinal.deriv_eq_id_of_nfp_eq_id + - Ordinal.fp_bfamily_unbounded + - Ordinal.derivFamily_isNormal + - Ordinal.lt_nfp + - Ordinal.nfpFamily_eq_self + - Ordinal.nfp_le_iff + - Ordinal.add_eq_right_iff_mul_omega_le + - Ordinal.deriv_id_of_nfp_id + - Ordinal.nfp_id + - Ordinal.fp_unbounded + - Ordinal.nfp_mul_opow_omega_add + - Ordinal.derivBFamily_eq_enumOrd + - Ordinal.deriv_add_eq_mul_omega_add + - Ordinal.eq_zero_or_opow_omega_le_of_mul_eq_right + - Ordinal.derivFamily_eq_enumOrd + - Ordinal.IsNormal.apply_le_nfp + - Ordinal.nfpFamily_le_fp + - Ordinal.nfp_mul_zero + - Ordinal.apply_lt_nfpBFamily_iff + - Ordinal.le_iff_derivBFamily + - Ordinal.le_iff_derivFamily + - Ordinal.derivBFamily_fp + - Ordinal.nfpFamily_fp + - Ordinal.derivFamily_fp + - Ordinal.nfpBFamily_fp + - Ordinal.apply_le_nfpBFamily + - Ordinal.IsNormal.apply_lt_nfp + - Ordinal.nfpFamily_le_apply + - Ordinal.deriv_mul_eq_opow_omega_mul + - Ordinal.add_le_right_iff_mul_omega_le + - Ordinal.nfp_mul_eq_opow_omega + - Ordinal.nfpBFamily_le_apply + - Ordinal.fp_iff_derivBFamily + - Ordinal.IsNormal.le_iff_deriv + - Ordinal.nfp_mul_one + - Ordinal.nfp_add_zero + - Ordinal.mul_le_right_iff_opow_omega_dvd + - Ordinal.mul_eq_right_iff_opow_omega_dvd + - Ordinal.nfp_zero_mul + - Ordinal.IsNormal.fp_iff_deriv + - Ordinal.deriv_eq_enumOrd + - Ordinal.fp_family_unbounded + - Ordinal.iterate_le_nfp + - path: Mathlib/Algebra/Module/Submodule/Lattice.lean + theorems: + - Submodule.disjoint_def + - Submodule.mem_sSup_of_mem + - Submodule.mem_finset_inf + - Submodule.subsingleton_iff_eq_bot + - Submodule.sub_mem_sup + - Submodule.mem_iInf + - Submodule.ne_bot_iff + - Submodule.nontrivial_iff_ne_bot + - Submodule.bot_ext + - Submodule.toAddSubmonoid_sSup + - Submodule.mem_sup_right + - Submodule.iInf_coe + - Submodule.mem_sup_left + - Submodule.subsingleton_iff + - Submodule.finset_inf_coe + - path: Mathlib/Algebra/Lie/Basic.lean + theorems: + - LieModuleHom.coe_mk + - neg_lie + - "LieModuleHom.map_lie\u2082" + - LieEquiv.symm_symm + - lie_skew + - LieModuleEquiv.symm_symm + - lie_neg + - lie_sub + - LieModuleHom.coe_injective + - LieHom.comp_id + - LieEquiv.coe_linearEquiv_injective + - LieModuleHom.ext_iff + - lie_lie + - LieModuleHom.mk_coe + - sub_lie + - LieModuleEquiv.toEquiv_injective + - LieHom.coe_injective + - LieHom.id_comp + - LieHom.mk_coe + - LieModuleHom.coe_linear_mk + - LieModule.compLieHom + - lie_jacobi + - path: Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean + theorems: + - Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_norm + - Asymptotics.superpolynomialDecay_mul_const_iff + - Asymptotics.SuperpolynomialDecay.mul + - Asymptotics.superpolynomialDecay_iff_zpow_tendsto_zero + - Asymptotics.superpolynomialDecay_mul_param_pow_iff + - Asymptotics.superpolynomialDecay_iff_isLittleO + - Asymptotics.superpolynomialDecay_param_pow_mul_iff + - Asymptotics.SuperpolynomialDecay.inv_param_mul + - Asymptotics.SuperpolynomialDecay.polynomial_mul + - Asymptotics.SuperpolynomialDecay.param_pow_mul + - Asymptotics.SuperpolynomialDecay.param_zpow_mul + - Asymptotics.superpolynomialDecay_iff_superpolynomialDecay_abs + - Asymptotics.SuperpolynomialDecay.mul_const + - Asymptotics.SuperpolynomialDecay.trans_eventually_abs_le + - Asymptotics.superpolynomialDecay_iff_abs_isBoundedUnder + - Asymptotics.superpolynomialDecay_mul_param_iff + - Asymptotics.SuperpolynomialDecay.add + - Asymptotics.superpolynomialDecay_zero + - Asymptotics.superpolynomialDecay_param_mul_iff + - Asymptotics.superpolynomialDecay_iff_isBigO + - Asymptotics.superpolynomialDecay_const_mul_iff + - Asymptotics.SuperpolynomialDecay.param_mul + - path: Mathlib/Analysis/Analytic/Inverse.lean + theorems: + - FormalMultilinearSeries.comp_rightInv_aux2 + - FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1 + - FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos + - FormalMultilinearSeries.rightInv_coeff + - FormalMultilinearSeries.leftInv_coeff_one + - FormalMultilinearSeries.leftInv_comp + - FormalMultilinearSeries.comp_rightInv + - FormalMultilinearSeries.rightInv_coeff_one + - FormalMultilinearSeries.rightInv_coeff_zero + - FormalMultilinearSeries.leftInv_removeZero + - FormalMultilinearSeries.leftInv_coeff_zero + - FormalMultilinearSeries.rightInv_removeZero + - FormalMultilinearSeries.leftInv_eq_rightInv + - FormalMultilinearSeries.leftInv_eq_rightInv_aux + - FormalMultilinearSeries.radius_rightInv_pos_of_radius_pos_aux2 + - path: Mathlib/Algebra/Lie/Submodule.lean + theorems: + - LieHom.range_eq_top + - LieHom.isIdealMorphism_of_surjective + - LieSubmodule.lieSpan_mono + - Submodule.exists_lieSubmodule_coe_eq_iff + - LieIdeal.map_comap_le + - LieHom.idealRange_eq_map + - LieSubmodule.map_injective_of_injective + - LieIdeal.incl_isIdealMorphism + - LieSubmodule.submodule_span_le_lieSpan + - LieSubmodule.lieSpan_le + - LieSubmodule.map_comp + - LieSubmodule.iSup_induction' + - LieIdeal.map_eq_bot_iff + - LieSubmodule.iSup_eq_top_iff_coe_toSubmodule + - LieSubmodule.lieSpan_eq_bot_iff + - lie_mem_left + - LieModuleEquiv.range_coe + - LieSubmodule.range_incl + - LieIdeal.bot_of_map_eq_bot + - LieIdeal.coe_map_of_surjective + - LieHom.idealRange_eq_top_of_surjective + - LieSubmodule.comap_incl_eq_bot + - LieSubmodule.subset_lieSpan + - LieSubmodule.sSup_image_lieSpan_singleton + - LieSubmodule.comap_incl_self + - LieSubmodule.nontrivial_iff_ne_bot + - LieSubmodule.map_incl_top + - LieSubmodule.coeSubmodule_eq_bot_iff + - LieSubmodule.mem_sup + - LieHom.mem_ker + - LieModuleHom.le_ker_iff_map + - LieSubmodule.mem_iInf + - LieIdeal.comap_incl_self + - LieIdeal.map_mono + - LieHom.mem_idealRange_iff + - LieSubmodule.sInf_coe_toSubmodule' + - LieSubmodule.map_id + - LieSubmodule.sSup_coe_toSubmodule' + - LieSubmodule.coeSubmodule_injective + - LieSubmodule.mem_inf + - LieSubmodule.coe_lieSpan_submodule_eq_iff + - LieSubmodule.ker_incl + - LieSubalgebra.exists_lieIdeal_coe_eq_iff + - LieSubmodule.subsingleton_iff + - LieHom.ker_eq_bot + - LieSubmodule.disjoint_iff_coe_toSubmodule + - LieIdeal.comap_map_le + - LieHom.map_le_idealRange + - LieHom.le_ker_iff + - LieModuleHom.comp_ker_incl + - LieSubmodule.iInf_coe_toSubmodule + - LieSubmodule.map_bot + - LieSubmodule.map_incl_le + - LieSubmodule.iSup_coe_toSubmodule + - LieSubmodule.iSup_induction + - LieModuleHom.map_top + - LieIdeal.map_coeSubmodule + - LieIdeal.map_le_iff_le_comap + - LieSubmodule.isCompactElement_lieSpan_singleton + - LieIdeal.mem_map_of_surjective + - LieSubmodule.codisjoint_iff_coe_toSubmodule + - LieHom.isIdealMorphism_iff + - LieIdeal.inclusion_injective + - LieSubmodule.sup_coe_toSubmodule + - LieSubalgebra.exists_nested_lieIdeal_coe_eq_iff + - LieSubmodule.eq_bot_iff + - LieModuleHom.ker_eq_bot + - LieIdeal.map_of_image + - LieIdeal.comap_mono + - LieModuleHom.range_eq_top + - LieSubmodule.mem_lieSpan + - LieIdeal.ker_incl + - LieIdeal.comap_map_eq + - LieIdeal.incl_idealRange + - LieSubmodule.map_le_range + - LieSubmodule.map_incl_lt_iff_lt_top + - LieSubmodule.inclusion_injective + - LieIdeal.mem_map + - LieSubmodule.comap_incl_eq_top + - LieIdeal.map_sup_ker_eq_map' + - LieSubmodule.coeSubmodule_eq_top_iff + - LieSubmodule.independent_iff_coe_toSubmodule + - LieSubmodule.sInf_coe + - LieIdeal.map_comap_eq + - LieIdeal.map_sup_ker_eq_map + - path: Mathlib/Analysis/Fourier/PoissonSummation.lean + theorems: + - Real.tsum_eq_tsum_fourierIntegral + - isBigO_norm_restrict_cocompact + - isBigO_norm_Icc_restrict_atBot + - SchwartzMap.tsum_eq_tsum_fourierIntegral + - isBigO_norm_Icc_restrict_atTop + - Real.fourierCoeff_tsum_comp_add + - path: Mathlib/GroupTheory/SpecificGroups/Dihedral.lean + theorems: + - DihedralGroup.card_conjClasses_odd + - DihedralGroup.orderOf_r + - DihedralGroup.nat_card + - DihedralGroup.r_one_pow_n + - DihedralGroup.exponent + - DihedralGroup.sr_mul_self + - DihedralGroup.card_commute_odd + - DihedralGroup.orderOf_sr + - DihedralGroup.card + - path: Mathlib/Data/Complex/Abs.lean + theorems: + - Complex.abs_intCast + - Complex.AbsTheory.abs_mul + - Complex.abs_ofReal + - Complex.normSq_eq_abs + - Complex.abs_re_div_abs_le_one + - Complex.abs_I + - Complex.range_abs + - Complex.sq_abs_sub_sq_im + - Complex.abs_im_lt_abs + - Complex.abs_im_le_abs + - Complex.abs_re_le_abs + - Complex.isCauSeq_re + - Complex.abs_re_lt_abs + - Complex.equiv_limAux + - Complex.isCauSeq_conj + - Complex.abs_le_sqrt_two_mul_max + - Complex.AbsTheory.abs_conj + - Complex.range_normSq + - Complex.sq_abs_sub_sq_re + - Complex.AbsTheory.abs_re_le_abs + - Complex.isCauSeq_im + - Complex.abs_add_mul_I + - Complex.abs_im_div_abs_le_one + - Complex.re_neg_ne_zero_of_one_lt_re + - Complex.lim_eq_lim_im_add_lim_re + - Complex.lim_re + - Complex.abs_eq_sqrt_sq_add_sq + - Complex.AbsTheory.abs_add + - Complex.lim_conj + - path: Mathlib/Algebra/Order/Kleene.lean + theorems: + - pow_le_kstar + - add_eq_right_iff_le + - kstar_le_of_mul_le_right + - add_idem + - kstar_eq_one + - add_le_iff + - nsmul_eq_self + - add_eq_left_iff_le + - kstar_le_of_mul_le_left + - kstar_eq_self + - path: Mathlib/Data/Fintype/Card.lean + theorems: + - Function.Embedding.equiv_of_fintype_self_embedding_to_embedding + - Finite.injective_iff_surjective_of_equiv + - List.exists_pw_disjoint_with_card + - Function.Embedding.exists_of_card_le_finset + - Finset.exists_minimal + - Fintype.card_lt_of_injective_of_not_mem + - Finite.wellFounded_of_trans_of_irrefl + - Function.Embedding.nonempty_of_card_le + - Finset.card_fin + - Fintype.card_le_one_iff + - Finset.eq_univ_of_card + - Fintype.card_subtype + - Infinite.of_injective_to_set + - Infinite.nonempty + - Finite.surjective_of_injective + - Fintype.card_congr' + - Finite.injective_iff_bijective + - Fintype.card_fin_lt_of_le + - Finset.card_add_card_compl + - card_finset_fin_le + - Fintype.card_eq_zero_iff + - Finite.exists_univ_list + - List.Nodup.length_le_card + - Finite.surjective_iff_bijective + - Fintype.induction_subsingleton_or_nontrivial + - Fintype.card_subtype_lt + - Finset.card_compl_add_card + - univ_eq_singleton_of_card_one + - Fintype.card_congr + - Fintype.card_range_le + - Fintype.card_subtype_compl + - set_fintype_card_eq_univ_iff + - Fintype.exists_unique_iff_card_one + - Infinite.natEmbeddingAux_injective + - Fin.cast_eq_cast' + - Finite.exists_infinite_fiber + - Fintype.card_compl_eq_card_compl + - Fintype.card_eq_one_iff + - Fintype.card_compl_set + - Fintype.two_lt_card_iff + - Infinite.exists_superset_card_eq + - Finite.of_injective + - Fintype.card_of_subtype + - Finite.exists_ne_map_eq_of_infinite + - Infinite.exists_subset_card_eq + - Fintype.card_of_finset' + - Finset.card_eq_iff_eq_univ + - Finset.univ_map_embedding + - Fintype.one_lt_card_iff_nontrivial + - path: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean + theorems: + - MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite + - MeasureTheory.ae_eq_zero_of_forall_set_integral_isClosed_eq_zero + - MeasureTheory.Integrable.ae_eq_zero_of_forall_set_integral_eq_zero + - MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg_inter + - "MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite\u2080" + - MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero + - MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg_of_stronglyMeasurable + - MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_set_integral_nonneg + - MeasureTheory.ae_eq_of_forall_set_integral_eq_of_sigmaFinite + - MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_of_forall_set_integral_eq_zero + - MeasureTheory.withDensity_eq_iff + - MeasureTheory.ae_nonneg_restrict_of_forall_set_integral_nonneg + - MeasureTheory.ae_eq_zero_of_forall_set_integral_isCompact_eq_zero + - MeasureTheory.ae_eq_zero_of_forall_dual_of_isSeparable + - MeasureTheory.ae_eq_zero_of_forall_set_integral_isCompact_eq_zero' + - MeasureTheory.ae_eq_zero_restrict_of_forall_set_integral_eq_zero_real + - MeasureTheory.ae_nonneg_of_forall_set_integral_nonneg + - MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_sigmaFinite + - MeasureTheory.ae_eq_zero_of_forall_inner + - MeasureTheory.Integrable.ae_eq_of_forall_set_integral_eq + - MeasureTheory.AEMeasurable.ae_eq_of_forall_set_lintegral_eq + - MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq + - "MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite\u2080" + - MeasureTheory.ae_eq_restrict_of_forall_set_integral_eq + - MeasureTheory.withDensity_eq_iff_of_sigmaFinite + - MeasureTheory.ae_le_of_forall_set_lintegral_le_of_sigmaFinite + - MeasureTheory.ae_le_of_forall_set_integral_le + - MeasureTheory.ae_const_le_iff_forall_lt_measure_zero + - MeasureTheory.ae_eq_zero_of_forall_set_integral_eq_of_finStronglyMeasurable_trim + - path: Mathlib/Algebra/Module/Injective.lean + theorems: + - Module.Baer.extensionOfMax_adjoin.aux1 + - Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_eq + - Module.Baer.ExtensionOf.ext + - Module.Baer.chain_linearPMap_of_chain_extensionOf + - Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd' + - Module.injective_of_ulift_injective + - Module.Baer.extension_property + - Module.Baer.extensionOfMax_to_submodule_eq_top + - Module.Baer.of_injective + - Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd + - Module.injective_module_of_injective_object + - Module.Baer.extensionOfMax_le + - Module.Baer.ExtensionOfMaxAdjoin.extensionToFun_wd + - path: Mathlib/Data/List/ToFinsupp.lean + theorems: + - List.toFinsupp_cons_eq_single_add_embDomain + - List.toFinsupp_concat_eq_toFinsupp_add_single + - List.toFinsupp_append + - List.toFinsupp_nil + - List.toFinsupp_singleton + - List.toFinsupp_eq_sum_map_enum_single + - path: Mathlib/Topology/Instances/EReal.lean + theorems: + - EReal.tendsto_toReal + - EReal.embedding_coe + - EReal.continuousAt_add_top_top + - EReal.mem_nhds_top_iff + - EReal.nhds_bot + - EReal.nhds_top + - EReal.nhds_bot_basis + - EReal.tendsto_nhds_top_iff_real + - EReal.continuousAt_add_bot_bot + - EReal.openEmbedding_coe + - EReal.embedding_coe_ennreal + - EReal.mem_nhds_bot_iff + - EReal.continuousAt_add + - EReal.tendsto_nhds_bot_iff_real + - EReal.continuousAt_add_top_coe + - EReal.nhds_top_basis + - EReal.continuousAt_add_bot_coe + - EReal.closedEmbedding_coe_ennreal + - path: Mathlib/RingTheory/DedekindDomain/Dvr.lean + theorems: + - IsLocalization.AtPrime.not_isField + - IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain + - IsLocalization.isDedekindDomain + - Ring.DimensionLEOne.localization + - path: Mathlib/Data/Polynomial/Eval.lean + theorems: + - "Polynomial.eval\u2082_nat_cast" + - "Polynomial.eval\u2082_list_prod_noncomm" + - "Polynomial.eval\u2082_mul_eq_zero_of_left" + - "Polynomial.iterate_comp_eval\u2082" + - Polynomial.support_map_of_injective + - Polynomial.map_ofNat + - Polynomial.comp_assoc + - Polynomial.bit0_comp + - Polynomial.coeff_zero_eq_eval_zero + - Polynomial.leadingCoeff_map_of_leadingCoeff_ne_zero + - Polynomial.IsRoot.of_map + - Polynomial.isRoot_prod + - "Polynomial.eval\u2082_X_pow" + - Polynomial.eval_nat_cast + - Polynomial.cast_int_comp + - Polynomial.zero_isRoot_of_coeff_zero_eq_zero + - "Polynomial.eval\u2082_at_ofNat" + - Polynomial.map_mul + - "Polynomial.eval\u2082_bit0" + - Polynomial.map_map + - Polynomial.map_comp + - Polynomial.comp_eq_sum_left + - "Polynomial.eval\u2082_mul_X" + - Polynomial.mem_map_rangeS + - "Polynomial.eval\u2082_ofNat" + - "Polynomial.eval\u2082_eq_sum_range'" + - "Polynomial.eval\u2082_mul_C'" + - Polynomial.eval_int_cast + - "Polynomial.eval\u2082_smul" + - Polynomial.degree_map_eq_of_leadingCoeff_ne_zero + - Polynomial.nat_cast_comp + - "Polynomial.eval\u2082_C_X" + - Polynomial.root_mul_left_of_isRoot + - Polynomial.map_smul + - "Polynomial.eval\u2082_at_int_cast" + - "Polynomial.eval\u2082_C" + - Polynomial.map_surjective + - Polynomial.eval_int_cast_map + - "Polynomial.eval\u2082_eq_sum_range" + - "Polynomial.eval\u2082_zero" + - Polynomial.nat_cast_mul_comp + - Polynomial.comp_one + - Polynomial.X_pow_comp + - "Polynomial.eval\u2082_sub" + - "Polynomial.eval\u2082_bit1" + - "Polynomial.eval\u2082_sum" + - Polynomial.map_monic_eq_zero_iff + - Polynomial.eval_C_mul + - Polynomial.eval_monomial_one_add_sub + - Polynomial.isRoot_comp + - "Polynomial.eval\u2082_at_nat_cast" + - Polynomial.C_mul_comp + - "Polynomial.eval\u2082_congr" + - "Polynomial.eval\u2082_at_one" + - "Polynomial.eval\u2082_eq_sum" + - Polynomial.eval_eq_sum_range + - Polynomial.mul_X_comp + - "Polynomial.eval\u2082_X_mul" + - Polynomial.eval_eq_sum_range' + - Polynomial.IsRoot.map + - Polynomial.eval_zero_map + - "Polynomial.eval\u2082_comp" + - Polynomial.map_monomial + - "Polynomial.eval\u2082_add" + - Polynomial.degree_map_le + - Polynomial.bit1_comp + - Polynomial.eval_smul + - Polynomial.mul_X_pow_comp + - Polynomial.eval_mul_X + - "Polynomial.eval\u2082_neg" + - Polynomial.root_X_sub_C + - "Polynomial.eval\u2082_ofFinsupp" + - Polynomial.eval_nat_cast_mul + - Polynomial.comp_X + - "Polynomial.eval\u2082_mul_eq_zero_of_right" + - Polynomial.eval_eq_sum + - "Polynomial.eval\u2082_one" + - Polynomial.eval_comp + - "Polynomial.eval\u2082_X" + - Polynomial.coeff_map + - "Polynomial.hom_eval\u2082" + - "Polynomial.eval\u2082_eq_eval_map" + - Polynomial.eval_one_map + - Polynomial.eval_nat_cast_map + - Polynomial.mul_X_add_nat_cast_comp + - "Polynomial.eval\u2082_mul_noncomm" + - Polynomial.map_injective + - Polynomial.mul_X_sub_int_cast_comp + - Polynomial.eval_geom_sum + - Polynomial.map_id + - Polynomial.root_mul_right_of_isRoot + - "Polynomial.eval\u2082_map" + - Polynomial.comp_C + - "Polynomial.eval\u2082_comp'" + - Polynomial.eval_mul_X_pow + - Polynomial.comp_zero + - Polynomial.zero_comp + - Polynomial.support_map_subset + - "Polynomial.eval\u2082_at_apply" + - "Polynomial.eval\u2082_pow'" + - Polynomial.not_isRoot_C + - Polynomial.one_comp + - Polynomial.smul_comp + - Polynomial.IsRoot.dvd + - path: Mathlib/Analysis/InnerProductSpace/Orientation.lean + theorems: + - OrthonormalBasis.same_orientation_iff_det_eq_det + - Orientation.abs_volumeForm_apply_le + - OrthonormalBasis.abs_det_adjustToOrientation + - OrthonormalBasis.orientation_adjustToOrientation + - Orientation.abs_volumeForm_apply_of_orthonormal + - OrthonormalBasis.orthonormal_adjustToOrientation + - OrthonormalBasis.det_adjustToOrientation + - OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation + - OrthonormalBasis.det_eq_neg_det_of_opposite_orientation + - Orientation.volumeForm_robust_neg + - Orientation.volumeForm_zero_pos + - Orientation.volumeForm_robust' + - Orientation.volumeForm_comp_linearIsometryEquiv + - Orientation.volumeForm_zero_neg + - Orientation.volumeForm_map + - Orientation.finOrthonormalBasis_orientation + - Orientation.volumeForm_neg_orientation + - OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation + - OrthonormalBasis.adjustToOrientation_apply_eq_or_eq_neg + - Orientation.volumeForm_robust + - path: Mathlib/GroupTheory/Subsemigroup/Center.lean + theorems: + - Set.zero_mem_center + - Subsemigroup.mem_center_iff + - Set.natCast_mem_center + - Set.div_mem_center + - IsMulCentral.right_comm + - Set.invOf_mem_center + - Set.mul_mem_center + - Set.inv_mem_center + - "Set.div_mem_center\u2080" + - "Set.inv_mem_center\u2080" + - Semigroup.mem_center_iff + - Set.subset_center_units + - Set.one_mem_center + - IsMulCentral.left_comm + - Set.center_units_subset + - Set.units_inv_mem_center + - Set.intCast_mem_center + - Set.add_mem_center + - path: Mathlib/LinearAlgebra/QuadraticForm/Prod.lean + theorems: + - QuadraticForm.pi_apply_single + - QuadraticForm.posDef_pi_iff + - QuadraticForm.nonneg_pi_iff + - QuadraticForm.Ring.polar_pi + - QuadraticForm.anisotropic_of_prod + - QuadraticForm.polar_prod + - QuadraticForm.associated_prod + - QuadraticForm.isOrtho_inr_inr_iff + - QuadraticForm.Ring.associated_pi + - QuadraticForm.nonneg_prod_iff + - QuadraticForm.isOrtho_inl_inl_iff + - QuadraticForm.anisotropic_of_pi + - QuadraticForm.Ring.polarBilin_pi + - path: Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean + theorems: + - MeasureTheory.tendsto_lintegral_thickenedIndicator_of_isClosed + - MeasureTheory.measure_of_cont_bdd_of_tendsto_indicator + - MeasureTheory.ext_of_forall_lintegral_eq_of_IsFiniteMeasure + - HasOuterApproxClosed.measure_le_lintegral + - MeasureTheory.measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure + - MeasureTheory.measure_of_cont_bdd_of_tendsto_filter_indicator + - HasOuterApproxClosed.indicator_le_apprSeq + - MeasureTheory.tendsto_lintegral_nn_filter_of_le_const + - path: Mathlib/Computability/TuringMachine.lean + theorems: + - Turing.TM2.step_supports + - Turing.TM1.step_supports + - Turing.TM2to1.tr_eval + - Turing.ListBlank.append_assoc + - Turing.ListBlank.append_mk + - Turing.TM1to1.exists_enc_dec + - Turing.BlankRel.trans + - Turing.ListBlank.cons_head_tail + - Turing.tr_eval + - Turing.TM2to1.addBottom_head_fst + - Turing.TM2to1.tr_respects + - Turing.TM1to1.trTape'_move_left + - "Turing.reaches\u2081_eq" + - Turing.ListBlank.ext + - Turing.TM1to1.supportsStmt_read + - Turing.TM1to1.trTape'_move_right + - "Turing.reaches\u2081_fwd" + - Turing.TM1to0.tr_eval + - Turing.TM1.stmts_supportsStmt + - Turing.Tape.mk'_nth_nat + - Turing.Tape.map_mk' + - Turing.BlankExtends.above_of_le + - Turing.ListBlank.head_map + - Turing.mem_eval + - Turing.ListBlank.map_modifyNth + - Turing.TM2to1.trNormal_run + - Turing.eval_maximal + - Turing.reaches_eval + - Turing.TM1to1.supportsStmt_write + - Turing.TM1to1.trTape_mk' + - Turing.TM2.stmts_supportsStmt + - Turing.Tape.move_right_left + - Turing.ListBlank.tail_map + - "Turing.TM1.stmts\u2081_self" + - "Turing.TM2to1.trStmts\u2081_run" + - Turing.TM1.stmts_trans + - Turing.TM2to1.tr_supports + - "Turing.Tape.map_mk\u2082" + - Turing.TM1to1.stepAux_read + - Turing.TM1to1.stepAux_move + - Turing.PointedMap.headI_map + - Turing.ListBlank.map_cons + - Turing.TM2to1.stk_nth_val + - Turing.Tape.move_left_right + - Turing.proj_map_nth + - Turing.ListBlank.nth_zero + - Turing.Tape.move_left_nth + - Turing.Tape.move_right_n_head + - Turing.tr_reaches_rev + - Turing.TM1to0.tr_respects + - Turing.Tape.map_write + - Turing.Tape.move_right_nth + - "Turing.Tape.right\u2080_nth" + - Turing.TM2to1.supports_run + - Turing.tr_eval_rev + - Turing.Tape.write_mk' + - Turing.TM0.Machine.map_respects + - Turing.tr_reaches + - Turing.ListBlank.nth_succ + - Turing.TM2to1.trCfg_init + - Turing.TM1to1.stepAux_write + - Turing.Tape.move_right_mk' + - "Turing.TM2.stmts\u2081_self" + - Turing.Tape.write_move_right_n + - Turing.TM2to1.addBottom_nth_snd + - Turing.TM1to0.tr_supports + - Turing.Tape.move_left_mk' + - Turing.BlankExtends.trans + - Turing.ListBlank.nth_modifyNth + - "Turing.Tape.mk'_left_right\u2080" + - Turing.Tape.write_self + - Turing.TM2to1.addBottom_map + - "Turing.TM1.stmts\u2081_supportsStmt_mono" + - Turing.TM2to1.tr_respects_aux + - Turing.tr_eval' + - Turing.fun_respects + - "Turing.tr_reaches\u2081" + - Turing.BlankExtends.refl + - "Turing.TM1.stmts\u2081_trans" + - Turing.ListBlank.cons_bind + - Turing.TM1to1.tr_respects + - Turing.TM0.step_supports + - Turing.TM1to1.supportsStmt_move + - "Turing.TM2to1.tr_respects_aux\u2082" + - "Turing.TM2.stmts\u2081_supportsStmt_mono" + - "Turing.TM2to1.tr_respects_aux\u2081" + - Turing.TM0.Machine.map_step + - "Turing.TM2to1.tr_respects_aux\u2083" + - Turing.TM0to1.tr_respects + - Turing.TM2to1.addBottom_nth_succ_fst + - Turing.TM2to1.step_run + - Turing.frespects_eq + - Turing.ListBlank.nth_map + - Turing.TM2.stmts_trans + - Turing.Tape.map_fst + - "Turing.TM2.stmts\u2081_trans" + - Turing.Tape.map_move + - path: Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean + theorems: + - simply_connected_iff_paths_homotopic' + - simply_connected_iff_unique_homotopic + - simply_connected_iff_paths_homotopic + - path: Mathlib/Topology/Instances/ENNReal.lean + theorems: + - ENNReal.tendsto_finset_prod_of_ne_top + - ENNReal.tendsto_cofinite_zero_of_tsum_ne_top + - ENNReal.tendsto_atTop_zero + - ENNReal.Tendsto.mul_const + - ENNReal.hasSum_iff_tendsto_nat + - NNReal.indicator_summable + - ENNReal.iInf_mul_left' + - ENNReal.tsum_eq_iSup_sum' + - ENNReal.isOpen_Ico_zero + - NNReal.tsum_eq_add_tsum_ite + - ENNReal.tendsto_atTop + - NNReal.hasSum_lt + - Real.diam_eq + - ENNReal.tendsto_nat_tsum + - ENNReal.tsum_iSup_eq + - Real.diam_Ico + - Real.ediam_Ioo + - Metric.diam_closure + - ENNReal.lt_top_of_tsum_ne_top + - ENNReal.biSup_add_biSup_le' + - summable_of_sum_range_le + - ENNReal.biSup_add' + - ENNReal.hasBasis_nhds_of_ne_top' + - Summable.countable_support_nnreal + - ENNReal.biInf_le_nhds + - NNReal.summable_iff_not_tendsto_nat_atTop + - ENNReal.hasSum_toReal + - ENNReal.exists_frequently_lt_of_liminf_ne_top' + - ENNReal.openEmbedding_coe + - ENNReal.Tendsto.const_mul + - ENNReal.mul_iSup + - Summable.countable_support_ennreal + - ENNReal.tendsto_tsum_compl_atTop_zero + - ENNReal.nhds_top + - ENNReal.tsum_add_one_eq_top + - ENNReal.exists_frequently_lt_of_liminf_ne_top + - ENNReal.eventuallyEq_of_toReal_eventuallyEq + - NNReal.tsum_pos + - continuous_edist + - ENNReal.continuous_pow + - ENNReal.coe_tsum + - Real.ediam_Icc + - ENNReal.tendsto_sum_nat_add + - ENNReal.monotone_truncateToReal + - ENNReal.tsum_biUnion_le_tsum + - cauchySeq_of_edist_le_of_tsum_ne_top + - ENNReal.smul_iSup + - ENNReal.tendsto_coe_nhds_top + - isClosed_setOf_lipschitzOnWith + - ENNReal.sSup_add + - NNReal.tsum_eq_toNNReal_tsum + - ENNReal.tendsto_nhds_coe_iff + - ENNReal.exists_upcrossings_of_not_bounded_under + - ENNReal.hasBasis_nhds_of_ne_top + - ENNReal.exists_lt_add_of_lt_add + - EMetric.diam_closure + - Real.diam_Ioc + - ENNReal.embedding_coe + - ENNReal.tendsto_mul + - ENNReal.tsum_le_tsum_comp_of_surjective + - ENNReal.tendsto_toNNReal + - ENNReal.tsum_coe_ne_top_iff_summable_coe + - ENNReal.add_biSup' + - ENNReal.Tendsto.div + - ENNReal.tendsto_atTop_zero_of_tsum_ne_top + - ENNReal.ofReal_tsum_of_nonneg + - edist_ne_top_of_mem_ball + - ENNReal.tsum_const_eq_top_of_ne_zero + - summable_iff_not_tendsto_nat_atTop_of_nonneg + - EMetric.cauchySeq_iff_le_tendsto_0 + - ENNReal.iSup_mul + - ENNReal.tendsto_sub + - NNReal.exists_le_hasSum_of_le + - ENNReal.hasSum_coe + - ENNReal.truncateToReal_eq_toReal + - ENNReal.finset_card_const_le_le_of_tsum_le + - ENNReal.tsum_coe_ne_top_iff_summable + - ENNReal.continuous_sub_right + - ENNReal.tsum_toReal_eq + - summable_sigma_of_nonneg + - ENNReal.continuous_truncateToReal + - Real.ediam_eq + - ENNReal.summable_toNNReal_of_tsum_ne_top + - ENNReal.hasSum_lt + - ENNReal.tsum_sub + - ENNReal.exists_countable_dense_no_zero_top + - Summable.of_nonneg_of_le + - ENNReal.mul_sSup + - tsum_comp_le_tsum_of_inj + - "edist_le_tsum_of_edist_le_of_tendsto\u2080" + - ENNReal.continuousOn_sub + - Summable.toNNReal + - ENNReal.finset_sum_iSup_nat + - ENNReal.tendsto_nhds + - NNReal.hasSum_iff_tendsto_nat + - ENNReal.tsum_iUnion_le + - continuous_of_le_add_edist + - ENNReal.iInf_mul_right' + - ENNReal.liminf_toReal_eq + - NNReal.summable_of_sum_range_le + - ENNReal.Tendsto.div_const + - hasSum_iff_tendsto_nat_of_nonneg + - ENNReal.tendsto_nhds_top_iff_nat + - edist_le_tsum_of_edist_le_of_tendsto + - not_summable_iff_tendsto_nat_atTop_of_nonneg + - ENNReal.tsum_const_smul + - ENNReal.continuous_div_const + - ENNReal.iSup_add_iSup + - ENNReal.Tendsto.const_div + - ENNReal.tendsto_toReal_iff + - ENNReal.continuous_sub_left + - ENNReal.ofReal_cinfi + - ENNReal.nhds_zero + - ENNReal.smul_sSup + - ENNReal.add_iSup + - ENNReal.limsup_toReal_eq + - NNReal.not_summable_iff_tendsto_nat_atTop + - tendsto_iff_edist_tendsto_0 + - isClosed_setOf_lipschitzWith + - ENNReal.tsum_mul_right + - ENNReal.tsum_mul_left + - ENNReal.le_of_forall_lt_one_mul_le + - ENNReal.finite_const_le_of_tsum_ne_top + - NNReal.summable_sigma + - Real.diam_Icc + - ENNReal.iSup_add_iSup_le + - Real.diam_Ioo + - ENNReal.truncateToReal_le + - ENNReal.tendsto_nhds_of_Icc + - NNReal.tendsto_sum_nat_add + - ENNReal.tendsto_inv_iff + - path: Mathlib/RingTheory/FractionalIdeal/Operations.lean + theorems: + - FractionalIdeal.coeIdeal_eq_one + - FractionalIdeal.map_mul + - IsFractional.div_of_nonzero + - FractionalIdeal.canonicalEquiv_canonicalEquiv + - FractionalIdeal.num_le + - FractionalIdeal.mem_span_mul_finite_of_mem_mul + - FractionalIdeal.div_one + - FractionalIdeal.isFractional_span_iff + - FractionalIdeal.le_spanSingleton_mul_iff + - FractionalIdeal.mem_canonicalEquiv_apply + - FractionalIdeal.canonicalEquiv_flip + - FractionalIdeal.isNoetherian + - FractionalIdeal.spanSingleton_mul_spanSingleton + - FractionalIdeal.spanSingleton_pow + - FractionalIdeal.mem_spanSingleton + - FractionalIdeal.mul_one_div_le_one + - FractionalIdeal.map_one_div + - FractionalIdeal.spanSingleton_one + - IsFractional.map + - FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mul + - FractionalIdeal.canonicalEquiv_spanSingleton + - FractionalIdeal.den_mul_self_eq_num' + - FractionalIdeal.map_map_symm + - FractionalIdeal.canonicalEquiv_self + - FractionalIdeal.eq_spanSingleton_mul + - FractionalIdeal.canonicalEquiv_coeIdeal + - FractionalIdeal.spanFinset_ne_zero + - FractionalIdeal.eq_spanSingleton_of_principal + - FractionalIdeal.spanFinset_eq_zero + - FractionalIdeal.spanSingleton_mul_le_iff + - FractionalIdeal.mul_div_self_cancel_iff + - FractionalIdeal.coeIdeal_span_singleton + - FractionalIdeal.spanSingleton_mul_coeIdeal_eq_coeIdeal + - FractionalIdeal.mapEquiv_refl + - FractionalIdeal.eq_zero_or_one + - FractionalIdeal.isNoetherian_zero + - Ideal.fg_of_isUnit + - FractionalIdeal.coe_spanSingleton + - FractionalIdeal.canonicalEquiv_symm + - FractionalIdeal.le_self_mul_one_div + - FractionalIdeal.map_ne_zero + - FractionalIdeal.exists_eq_spanSingleton_mul + - FractionalIdeal.map_coeIdeal + - FractionalIdeal.map_div + - FractionalIdeal.div_spanSingleton + - FractionalIdeal.isFractional_of_fg + - FractionalIdeal.spanSingleton_zero + - FractionalIdeal.isNoetherian_coeIdeal + - FractionalIdeal.one_div_spanSingleton + - FractionalIdeal.spanSingleton_le_iff_mem + - FractionalIdeal.mem_div_iff_of_nonzero + - FractionalIdeal.spanSingleton_eq_zero_iff + - FractionalIdeal.ne_zero_of_mul_eq_one + - FractionalIdeal.eq_one_div_of_mul_eq_one_right + - FractionalIdeal.exists_ne_zero_mem_isInteger + - FractionalIdeal.le_div_iff_mul_le + - FractionalIdeal.mem_singleton_mul + - FractionalIdeal.mk'_mul_coeIdeal_eq_coeIdeal + - path: Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean + theorems: + - EuclideanSpace.volume_preserving_measurableEquiv + - Orientation.measure_orthonormalBasis + - OrthonormalBasis.addHaar_eq_volume + - OrthonormalBasis.measurePreserving_measurableEquiv + - LinearIsometryEquiv.measurePreserving + - Orientation.measure_eq_volume + - OrthonormalBasis.volume_parallelepiped + - volume_euclideanSpace_eq_dirac + - path: Mathlib/RingTheory/MvPowerSeries/Inverse.lean + theorems: + - MvPowerSeries.coeff_inv_aux + - MvPowerSeries.inv_eq_zero + - MvPowerSeries.invOfUnit_eq' + - MvPowerSeries.coeff_invOfUnit + - MvPowerSeries.eq_mul_inv_iff_mul_eq + - MvPowerSeries.inv_eq_iff_mul_eq_one + - MvPowerSeries.X_inv + - MvPowerSeries.mul_invOfUnit + - MvPowerSeries.C_inv + - MvPowerSeries.zero_inv + - MvPowerSeries.mul_inv_rev + - MvPowerSeries.smul_inv + - MvPowerSeries.inv_mul_cancel + - MvPowerSeries.eq_inv_iff_mul_eq_one + - MvPowerSeries.constantCoeff_inv + - MvPowerSeries.mul_inv_cancel + - MvPowerSeries.constantCoeff_invOfUnit + - path: Mathlib/Combinatorics/Young/YoungDiagram.lean + theorems: + - YoungDiagram.mk_mem_row_iff + - YoungDiagram.mem_row_iff + - YoungDiagram.transpose_le_iff + - YoungDiagram.mem_cellsOfRowLens + - YoungDiagram.mem_iff_lt_rowLen + - YoungDiagram.rowLen_ofRowLens + - YoungDiagram.colLen_anti + - YoungDiagram.colLen_eq_card + - YoungDiagram.length_rowLens + - YoungDiagram.transpose_eq_iff_eq_transpose + - YoungDiagram.exists_not_mem_col + - YoungDiagram.transpose_eq_iff + - YoungDiagram.ofRowLens_to_rowLens_eq_self + - YoungDiagram.get_rowLens + - YoungDiagram.rowLen_eq_card + - YoungDiagram.rowLen_transpose + - YoungDiagram.coe_bot + - YoungDiagram.colLen_transpose + - YoungDiagram.mem_iff_lt_colLen + - YoungDiagram.rowLens_length_ofRowLens + - YoungDiagram.pos_of_mem_rowLens + - YoungDiagram.mk_mem_col_iff + - YoungDiagram.mem_transpose + - YoungDiagram.transpose_transpose + - YoungDiagram.row_eq_prod + - YoungDiagram.col_eq_prod + - YoungDiagram.rowLen_anti + - YoungDiagram.le_of_transpose_le + - YoungDiagram.mem_col_iff + - path: Mathlib/Probability/Process/Stopping.lean + theorems: + - MeasureTheory.IsStoppingTime.measurableSet_lt' + - MeasureTheory.IsStoppingTime.piecewise_of_le + - MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range' + - "MeasureTheory.mem\u2112p_stoppedValue_of_mem_finset" + - MeasureTheory.stoppedProcess_eq'' + - MeasureTheory.IsStoppingTime.measurableSet_le_stopping_time + - MeasureTheory.IsStoppingTime.measurableSet_eq' + - MeasureTheory.IsStoppingTime.measurableSet_inter_eq_iff + - MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range' + - MeasureTheory.IsStoppingTime.measurableSet_min_iff + - MeasureTheory.IsStoppingTime.measurableSet_le' + - MeasureTheory.isStoppingTime_const + - MeasureTheory.progMeasurable_min_stopping_time + - MeasureTheory.IsStoppingTime.measurableSpace_le' + - "MeasureTheory.mem\u2112p_stoppedProcess_of_mem_finset" + - MeasureTheory.integrable_stoppedProcess_of_mem_finset + - MeasureTheory.stoppedProcess_eq + - MeasureTheory.IsStoppingTime.measurableSet_inter_le_const_iff + - MeasureTheory.IsStoppingTime.min + - MeasureTheory.IsStoppingTime.measurableSet_stopping_time_le + - MeasureTheory.IsStoppingTime.measurableSet_min_const_iff + - MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range + - MeasureTheory.stoppedProcess_eq_of_le + - MeasureTheory.IsStoppingTime.measurableSpace_min + - MeasureTheory.measurable_stoppedValue + - MeasureTheory.IsStoppingTime.max + - MeasureTheory.stoppedValue_sub_eq_sum + - MeasureTheory.IsStoppingTime.measurableSpace_le + - MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable + - MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time_of_countable + - MeasureTheory.IsStoppingTime.add_const_nat + - MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range' + - MeasureTheory.stoppedValue_piecewise_const' + - MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff + - MeasureTheory.IsStoppingTime.measurableSet_lt_of_isLUB + - MeasureTheory.IsStoppingTime.measurableSpace_mono + - MeasureTheory.IsStoppingTime.add_const + - MeasureTheory.IsStoppingTime.add + - MeasureTheory.isStoppingTime_of_measurableSet_eq + - MeasureTheory.IsStoppingTime.measurableSet_gt + - MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq_of_countable_range + - MeasureTheory.stoppedProcess_eq' + - MeasureTheory.IsStoppingTime.measurableSet_eq_stopping_time + - MeasureTheory.IsStoppingTime.measurableSet_lt_of_pred + - MeasureTheory.integrable_stoppedValue_of_mem_finset + - MeasureTheory.IsStoppingTime.measurableSet_eq + - MeasureTheory.stoppedProcess_eq_of_ge + - MeasureTheory.stoppedProcess_eq_of_mem_finset + - MeasureTheory.stoppedValue_piecewise_const + - MeasureTheory.IsStoppingTime.measurableSpace_le_of_countable_range + - MeasureTheory.stoppedValue_eq_of_mem_finset + - MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range + - MeasureTheory.IsStoppingTime.measurableSet_gt' + - MeasureTheory.IsStoppingTime.measurableSet_inter_le + - MeasureTheory.IsStoppingTime.measurableSet_ge + - MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range + - MeasureTheory.IsStoppingTime.measurableSpace_const + - MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le_const + - MeasureTheory.IsStoppingTime.measurableSet_ge' + - MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq + - MeasureTheory.IsStoppingTime.measurableSpace_min_const + - MeasureTheory.stoppedValue_sub_eq_sum' + - MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le + - path: Mathlib/GroupTheory/PushoutI.lean + theorems: + - Monoid.PushoutI.NormalWord.ext_smul + - Monoid.PushoutI.of_injective + - Monoid.PushoutI.NormalWord.base_smul_eq_smul + - Monoid.PushoutI.NormalWord.base_smul_def + - Monoid.PushoutI.ofCoprodI_of + - Monoid.PushoutI.induction_on + - Monoid.PushoutI.NormalWord.of_smul_eq_smul + - Monoid.PushoutI.NormalWord.summand_smul_def + - Monoid.PushoutI.lift_base + - Monoid.PushoutI.of_comp_eq_base + - Monoid.PushoutI.Reduced.exists_normalWord_prod_eq + - Monoid.PushoutI.NormalWord.prod_smul + - Monoid.PushoutI.NormalWord.ext_iff + - Monoid.PushoutI.of_apply_eq_base + - Monoid.PushoutI.Reduced.eq_empty_of_mem_range + - Monoid.PushoutI.lift_of + - Monoid.PushoutI.NormalWord.prod_empty + - Monoid.PushoutI.hom_ext_nonempty + - Monoid.PushoutI.NormalWord.prod_summand_smul + - Monoid.PushoutI.NormalWord.rcons_injective + - Monoid.PushoutI.NormalWord.prod_injective + - Monoid.PushoutI.NormalWord.cons_eq_smul + - Monoid.PushoutI.base_injective + - Monoid.PushoutI.NormalWord.prod_cons + - Monoid.PushoutI.NormalWord.ext + - Monoid.PushoutI.inf_of_range_eq_base_range + - Monoid.PushoutI.NormalWord.prod_base_smul + - Monoid.PushoutI.NormalWord.transversal_nonempty + - Monoid.PushoutI.NormalWord.prod_smul_empty + - path: Mathlib/Analysis/Calculus/MeanValue.lean + theorems: + - strictAntiOn_of_deriv_neg + - Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' + - strictMono_of_hasDerivAt_pos + - strictMonoOn_of_deriv_pos + - antitoneOn_of_hasDerivWithinAt_nonpos + - is_const_of_deriv_eq_zero + - exists_ratio_hasDerivAt_eq_ratio_slope + - antitoneOn_of_deriv_nonpos + - constant_of_derivWithin_zero + - strictMonoOn_of_hasDerivWithinAt_pos + - Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le + - is_const_of_fderiv_eq_zero + - Convex.is_const_of_fderivWithin_eq_zero + - norm_image_sub_le_of_norm_deriv_right_le_segment + - Convex.eqOn_of_fderivWithin_eq + - domain_mvt + - monotoneOn_of_hasDerivWithinAt_nonneg + - eq_of_fderiv_eq + - Convex.image_sub_le_mul_sub_of_deriv_le + - lipschitzWith_of_nnnorm_fderiv_le + - image_le_of_liminf_slope_right_lt_deriv_boundary' + - Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt + - exists_ratio_hasDerivAt_eq_ratio_slope' + - eq_of_derivWithin_eq + - Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le + - norm_image_sub_le_of_norm_deriv_le_segment_01' + - Convex.mul_sub_le_image_sub_of_le_deriv + - Convex.mul_sub_lt_image_sub_of_lt_deriv + - Convex.norm_image_sub_le_of_norm_hasDerivWithin_le + - strictAnti_of_hasDerivAt_neg + - exists_hasDerivAt_eq_slope + - Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le + - eq_of_has_deriv_right_eq + - norm_image_sub_le_of_norm_deriv_le_segment + - image_le_of_liminf_slope_right_le_deriv_boundary + - norm_image_sub_le_of_norm_deriv_le_segment' + - hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt + - monotoneOn_of_deriv_nonneg + - norm_image_sub_le_of_norm_deriv_le_segment_01 + - Convex.image_sub_lt_mul_sub_of_deriv_lt + - antitone_of_hasDerivAt_nonpos + - constant_of_has_deriv_right_zero + - monotone_of_hasDerivAt_nonneg + - strictAntiOn_of_hasDerivWithinAt_neg + - path: Mathlib/Analysis/MellinTransform.lean + theorems: + - hasMellin_sub + - mellin_convergent_iff_norm + - MellinConvergent.cpow_smul + - mellin_convergent_top_of_isBigO + - mellin_const_smul + - mellin_comp_mul_left + - mellin_hasDerivAt_of_isBigO_rpow + - mellin_comp_mul_right + - mellin_comp_rpow + - mellin_convergent_zero_of_isBigO + - MellinConvergent.comp_rpow + - hasMellin_one_Ioc + - MellinConvergent.comp_mul_left + - mellin_comp_inv + - mellin_convergent_of_isBigO_scalar + - hasMellin_cpow_Ioc + - mellinConvergent_of_isBigO_rpow + - Complex.cpow_mul_ofReal_nonneg + - mellin_div_const + - MellinConvergent.div_const + - MellinConvergent.const_smul + - isBigO_rpow_zero_log_smul + - isBigO_rpow_top_log_smul + - hasMellin_add + - mellin_cpow_smul + - path: Mathlib/Analysis/Calculus/Gradient/Basic.lean + theorems: + - HasGradientAtFilter.hasDerivAtFilter + - hasGradientAtFilter_const + - gradient_const + - Filter.EventuallyEq.gradient_eq + - HasDerivAtFilter.hasGradientAtFilter + - DifferentiableWithinAt.hasGradientWithinAt + - DifferentiableAt.hasGradientAt + - Filter.EventuallyEq.hasGradientAtFilter_iff + - HasGradientAt.hasDerivAt + - HasDerivAt.hasGradientAt + - gradient_eq_deriv + - hasFDerivWithinAt_iff_hasGradientWithinAt + - HasGradientAtFilter.congr_of_eventuallyEq + - hasFDerivAt_iff_hasGradientAt + - gradient_eq_zero_of_not_differentiableAt + - hasGradientWithinAt_univ + - path: Mathlib/Data/Finset/Card.lean + theorems: + - Finset.exists_mem_ne + - Finset.card_image_le + - Finset.exists_of_one_lt_card_pi + - Finset.card_insert_of_mem + - Finset.exists_eq_insert_iff + - Finset.inj_on_of_surj_on_of_card_le + - Finset.card_congr + - Finset.strongInduction_eq + - Finset.length_toList + - Finset.card_insert_eq_ite + - Finset.card_pair + - Finset.cast_card_inter + - Finset.one_lt_card_iff + - Finset.card_le_one + - Finset.card_filter_le_iff + - Finset.card_inter + - Finset.cast_card_erase_of_mem + - Finset.filter_card_add_filter_neg_card_eq_card + - Finset.cast_card_sdiff + - Finset.card_eq_of_bijective + - Finset.two_lt_card_iff + - Finset.card_union_eq_card_add_card + - Finset.card_image_of_injOn + - Finset.strongDownwardInduction_eq + - Finset.card_sdiff_add_card_inter + - Finset.filter_card_eq + - Finset.card_eq_one + - Finset.card_eq_three + - Finset.card_insert_of_not_mem + - Finset.exists_subset_or_subset_of_two_mul_lt_card + - Finset.two_lt_card + - Multiset.toFinset_card_eq_one_iff + - Finset.one_lt_card + - Finset.card_sdiff_add_card + - Finset.exists_smaller_set + - Finset.card_le_card_of_inj_on + - Finset.card_union_add_card_inter + - Finset.le_card_iff_exists_subset_card + - Finset.card_inter_add_card_sdiff + - Finset.card_le_one_iff_subset_singleton + - Finset.fiber_card_ne_zero_iff_mem_image + - Finset.card_insert_le + - Finset.card_eq_two + - Finset.card_mono + - Finset.one_lt_card_iff_nontrivial + - Finset.card_sdiff_comm + - Finset.strongDownwardInductionOn_eq + - Finset.card_singleton_inter + - Finset.card_inter_add_card_union + - Finset.le_card_sdiff + - Finset.card_eq_succ + - Finset.pred_card_le_card_erase + - Finset.card_le_one_iff + - Finset.exists_ne_of_one_lt_card + - Finset.Nonempty.strong_induction + - Finset.card_subtype + - Finset.card_union + - Finset.exists_intermediate_set + - Finset.strongInductionOn_eq + - Finset.card_sdiff + - Finset.exists_ne_map_eq_of_card_lt_of_maps_to + - Finset.surj_on_of_inj_on_of_card_le + - Finset.cast_card_union + - path: Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean + theorems: + - AffineSubspace.map_pointwise_vadd + - AffineSubspace.pointwise_vadd_direction + - AffineSubspace.pointwise_vadd_bot + - path: Mathlib/Algebra/Ring/Divisibility/Lemmas.lean + theorems: + - Commute.pow_dvd_pow_of_add_pow_eq_zero + - Commute.pow_dvd_sub_pow_of_pow_eq_zero_left + - Commute.pow_dvd_add_pow_of_pow_eq_zero_right + - Commute.pow_dvd_sub_pow_of_pow_eq_zero_right + - dvd_smul_of_dvd + - Commute.pow_dvd_pow_of_sub_pow_eq_zero + - Commute.add_pow_dvd_pow_of_pow_eq_zero_right + - path: Mathlib/Algebra/Category/ModuleCat/Free.lean + theorems: + - ModuleCat.free_shortExact_rank_add + - ModuleCat.free_shortExact_finrank_add + - ModuleCat.linearIndependent_shortExact + - ModuleCat.linearIndependent_leftExact + - ModuleCat.span_exact + - ModuleCat.span_rightExact + - ModuleCat.disjoint_span_sum + - path: Mathlib/Algebra/GroupWithZero/Power.lean + theorems: + - "zpow_one_add\u2080" + - zpow_neg_mul_zpow_self + - "pow_sub\u2080" + - "zpow_add_one\u2080" + - zpow_add' + - div_sq_cancel + - "zpow_add\u2080" + - zpow_ne_zero_of_ne_zero + - "zpow_sub_one\u2080" + - "zpow_sub\u2080" + - pow_sub_of_lt + - "inv_pow_sub\u2080" + - inv_pow_sub_of_lt + - "SemiconjBy.zpow_right\u2080" + - zero_zpow + - zero_zpow_eq + - path: Mathlib/Probability/Kernel/Basic.lean + theorems: + - ProbabilityTheory.kernel.set_lintegral_deterministic' + - ProbabilityTheory.kernel.sum_apply' + - ProbabilityTheory.kernel.set_integral_restrict + - ProbabilityTheory.kernel.lintegral_piecewise + - ProbabilityTheory.kernel.comapRight_apply' + - ProbabilityTheory.kernel.piecewise_apply' + - ProbabilityTheory.kernel.integral_const + - ProbabilityTheory.kernel.set_lintegral_piecewise + - ProbabilityTheory.kernel.isSFiniteKernel_sum_of_denumerable + - ProbabilityTheory.kernel.sum_comm + - ProbabilityTheory.kernel.sum_const + - ProbabilityTheory.kernel.const_add + - ProbabilityTheory.kernel.lintegral_const + - ProbabilityTheory.kernel.set_integral_piecewise + - ProbabilityTheory.kernel.measure_sum_seq + - ProbabilityTheory.kernel.set_lintegral_const + - ProbabilityTheory.kernel.const_zero + - ProbabilityTheory.isFiniteKernel_of_le + - ProbabilityTheory.kernel.set_integral_const + - ProbabilityTheory.kernel.integral_deterministic' + - ProbabilityTheory.kernel.set_integral_deterministic + - ProbabilityTheory.kernel.finset_sum_apply' + - ProbabilityTheory.kernel.deterministic_apply' + - ProbabilityTheory.kernel.ext_fun + - ProbabilityTheory.kernel.finset_sum_apply + - ProbabilityTheory.kernel.lintegral_deterministic + - ProbabilityTheory.kernel.set_lintegral_deterministic + - ProbabilityTheory.kernel.comapRight_id + - ProbabilityTheory.kernel.set_integral_deterministic' + - ProbabilityTheory.kernel.ext_iff' + - ProbabilityTheory.kernel.restrict_univ + - ProbabilityTheory.kernel.IsFiniteKernel.integrable + - ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum + - ProbabilityTheory.kernel.IsMarkovKernel.comapRight + - ProbabilityTheory.kernel.sum_zero + - ProbabilityTheory.kernel.sum_fintype + - ProbabilityTheory.kernel.isSFiniteKernel_sum + - ProbabilityTheory.kernel.ext_fun_iff + - ProbabilityTheory.kernel.set_lintegral_restrict + - ProbabilityTheory.kernel.lintegral_deterministic' + - ProbabilityTheory.kernel.lintegral_restrict + - path: Mathlib/Data/Polynomial/Derivative.lean + theorems: + - Polynomial.iterate_derivative_smul + - Polynomial.derivative_X_pow + - Polynomial.derivative_comp_one_sub_X + - Polynomial.iterate_derivative_int_cast_mul + - Polynomial.iterate_derivative_sub + - Polynomial.derivative_X_add_C_sq + - Polynomial.natDegree_derivative_le + - Polynomial.derivative_X_add_C + - Polynomial.derivative_of_natDegree_zero + - Polynomial.dvd_iterate_derivative_pow + - Polynomial.iterate_derivative_C_mul + - Polynomial.derivative_nat_cast + - Polynomial.derivative_bit1 + - Polynomial.derivative_monomial + - Polynomial.derivative_C_mul_X + - Polynomial.iterate_derivative_X_sub_pow_self + - Polynomial.eval_multiset_prod_X_sub_C_derivative + - Polynomial.iterate_derivative_eq_zero + - Polynomial.derivative_X_sub_C + - Polynomial.derivative_sq + - Polynomial.derivative_pow_succ + - Polynomial.iterate_derivative_X_sub_pow + - Polynomial.derivative_eval + - Polynomial.natDegree_derivative_lt + - Polynomial.derivative_C_mul_X_pow + - Polynomial.iterate_derivative_zero + - Polynomial.derivative_C + - Polynomial.iterate_derivative_nat_cast_mul + - Polynomial.natDegree_iterate_derivative + - Polynomial.derivative_mul + - Polynomial.derivative_bit0 + - Polynomial.derivative_X + - Polynomial.derivative_pow + - "Polynomial.derivative_eval\u2082_C" + - Polynomial.derivative_comp + - Polynomial.degree_derivative_eq + - Polynomial.iterate_derivative_mul + - Polynomial.iterate_derivative_X_pow_eq_nat_cast_mul + - Polynomial.natDegree_eq_zero_of_derivative_eq_zero + - Polynomial.derivative_nat_cast_mul + - Polynomial.iterate_derivative_map + - Polynomial.derivative_X_sub_C_pow + - Polynomial.derivative_map + - Polynomial.degree_derivative_le + - Polynomial.pow_sub_one_dvd_derivative_of_pow_dvd + - Polynomial.derivative_X_sub_C_sq + - Polynomial.derivative_X_sq + - Polynomial.iterate_derivative_comp_one_sub_X + - Polynomial.pow_sub_dvd_iterate_derivative_of_pow_dvd + - Polynomial.coeff_iterate_derivative + - Polynomial.mem_support_derivative + - Polynomial.of_mem_support_derivative + - Polynomial.derivative_int_cast + - Polynomial.derivative_X_add_C_pow + - Polynomial.derivative_prod + - Polynomial.iterate_derivative_X_pow_eq_C_mul + - Polynomial.derivative_int_cast_mul + - Polynomial.coeff_derivative + - Polynomial.iterate_derivative_X_add_pow + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean + theorems: + - Finpartition.equitabilise_aux + - Finpartition.exists_equipartition_card_eq + - Finpartition.card_parts_equitabilise + - Finpartition.card_filter_equitabilise_small + - path: Mathlib/Analysis/SpecialFunctions/PolarCoord.lean + theorems: + - Complex.polarCoord_apply + - polarCoord_source_ae_eq_univ + - hasFDerivAt_polarCoord_symm + - Complex.polarCoord_symm_apply + - Complex.integral_comp_polarCoord_symm + - Complex.polardCoord_symm_abs + - integral_comp_polarCoord_symm + - path: Mathlib/Topology/Category/Stonean/Basic.lean + theorems: + - Profinite.projective_of_extrDisc + - Stonean.epi_iff_surjective + - CompHaus.Gleason + - CompHaus.lift_lifts + - path: Mathlib/Analysis/SpecialFunctions/Integrals.lean + theorems: + - integral_cos_pow_three + - integral_sqrt_one_sub_sq + - integral_cos_sq + - integral_sin_pow_succ_le + - integral_one_div + - integral_zpow + - integral_inv_one_add_sq + - integral_inv + - integral_sin_pow_odd + - integral_sin_pow_odd_mul_cos_pow + - intervalIntegral.intervalIntegrable_rpow' + - integral_sin_sq_mul_cos_sq + - integral_mul_cpow_one_add_sq + - integral_cos_pow_aux + - integral_const_on_unit_interval + - integral_sin_pow_even_mul_cos_pow_even + - "integral_sin_mul_cos\u2081" + - integral_sin_pow_mul_cos_pow_odd + - integral_exp_mul_complex + - integral_one_div_of_pos + - integral_cos_pow + - integral_sin + - integral_sin_sq_mul_cos + - integral_mul_rpow_one_add_sq + - integral_id + - integral_cos_mul_complex + - intervalIntegral.intervalIntegrable_cpow' + - integral_sin_pow_pos + - integral_one + - integral_sin_sq + - intervalIntegral.intervalIntegrable_cpow + - integral_exp + - integral_one_div_one_add_sq + - integral_pow_abs_sub_uIoc + - "integral_sin_mul_cos\u2082" + - intervalIntegral.intervalIntegrable_inv + - integral_cos_sq_sub_sin_sq + - intervalIntegral.integrableOn_Ioo_rpow_iff + - integral_log + - integral_cos + - integral_cpow + - integral_pow + - integral_sin_mul_cos_sq + - integral_sin_pow_three + - integral_sin_pow + - intervalIntegral.intervalIntegrable_inv_one_add_sq + - intervalIntegral.integrableOn_Ioo_cpow_iff + - integral_sin_pow_even + - integral_one_div_of_neg + - intervalIntegral.intervalIntegrable_one_div_one_add_sq + - integral_rpow + - path: Mathlib/LinearAlgebra/LinearIndependent.lean + theorems: + - LinearIndependent.union + - exists_linearIndependent_extension + - linearIndependent_iUnion_finite_subtype + - LinearIndependent.map_of_surjective_injective + - linearIndependent_insert + - linearIndependent_fin_succ + - linearIndependent_iff_not_mem_span + - LinearIndependent.repr_ker + - LinearIndependent.restrict_scalars + - surjective_of_linearIndependent_of_span + - linearIndependent_monoidHom + - exists_maximal_independent' + - LinearIndependent.ne_zero + - linearIndependent_option + - Fintype.linearIndependent_iff' + - linearIndependent_fin2 + - linearIndependent_comp_subtype + - LinearIndependent.repr_range + - linearIndependent_bounded_of_finset_linearIndependent_bounded + - linearIndependent_algHom_toLinearMap + - Fintype.linearIndependent_iff + - linearIndependent_fin_snoc + - LinearIndependent.eq_zero_of_pair' + - exists_maximal_independent + - LinearIndependent.not_mem_span_image + - LinearIndependent.independent_span_singleton + - linearIndependent_subtype_disjoint + - LinearIndependent.comp + - linearIndependent_iff_totalOn + - LinearIndependent.image + - Fintype.not_linearIndependent_iff + - linearIndependent_algHom_toLinearMap' + - exists_of_linearIndependent_of_finite_span + - exists_finite_card_le_of_finite_of_linearIndependent_of_span + - LinearIndependent.eq_of_smul_apply_eq_smul_apply + - linearIndependent_biUnion_of_directed + - linearIndependent_unique_iff + - LinearIndependent.repr_eq + - linearIndependent_comp_subtype_disjoint + - linearIndependent_insert' + - linearDependent_comp_subtype' + - LinearIndependent.mono + - LinearIndependent.map_of_injective_injective + - LinearIndependent.eq_of_pair + - LinearMap.linearIndependent_iff + - linearIndependent_fin_succ' + - LinearIndependent.image_subtype + - linearIndependent_inl_union_inr' + - mem_span_insert_exchange + - linearIndependent_empty + - linearIndependent_iff' + - linearIndependent_iff_not_smul_mem_span + - LinearIndependent.group_smul + - LinearIndependent.total_ne_of_not_mem_support + - LinearIndependent.fin_cons' + - linearIndependent_option' + - LinearIndependent.eq_zero_of_pair + - linearIndependent_finset_map_embedding_subtype + - LinearIndependent.span_repr_eq + - LinearIndependent.inl_union_inr + - linearIndependent_iff'' + - exists_linearIndependent + - LinearIndependent.map + - LinearIndependent.insert + - LinearIndependent.to_subtype_range + - LinearIndependent.units_smul + - LinearIndependent.injective + - linearIndependent_iUnion_finite + - linearIndependent_fin_cons + - linearIndependent_subtype + - Submodule.range_ker_disjoint + - linearIndependent_iUnion_of_directed + - not_linearIndependent_iff + - LinearIndependent.coe_range + - LinearIndependent.linear_combination_pair_of_det_ne_zero + - LinearIndependent.image_of_comp + - linearIndependent_sUnion_of_directed + - le_of_span_le_span + - eq_of_linearIndependent_of_span_subtype + - LinearIndependent.repr_eq_single + - LinearIndependent.of_comp + - LinearIndependent.pair_iff + - path: Mathlib/CategoryTheory/Triangulated/Opposite.lean + theorems: + - CategoryTheory.Pretriangulated.Opposite.complete_distinguished_triangle_morphism + - CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff' + - CategoryTheory.Pretriangulated.Opposite.isomorphic_distinguished + - CategoryTheory.Pretriangulated.Opposite.distinguished_cocone_triangle + - CategoryTheory.Pretriangulated.op_distinguished + - CategoryTheory.Pretriangulated.shiftFunctorZero_op_inv_app + - CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_hom_app + - CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff + - CategoryTheory.Pretriangulated.shiftFunctorZero_op_hom_app + - CategoryTheory.Pretriangulated.shiftFunctorAdd'_op_inv_app + - CategoryTheory.Pretriangulated.Opposite.rotate_distinguished_triangle + - CategoryTheory.Pretriangulated.Opposite.contractible_distinguished + - CategoryTheory.Pretriangulated.mem_distTriang_op_iff + - path: Mathlib/Analysis/NormedSpace/FiniteDimension.lean + theorems: + - ProperSpace.of_locallyCompact_module + - LinearMap.exists_antilipschitzWith + - lipschitzExtensionConstant_pos + - ContinuousLinearMap.isOpen_injective + - LinearMap.injective_iff_antilipschitz + - Basis.opNNNorm_le + - ContinuousLinearMap.continuous_det + - exists_seq_norm_le_one_le_norm_sub + - continuous_clm_apply + - continuousOn_clm_apply + - summable_norm_iff + - LipschitzOnWith.extend_finite_dimension + - "FiniteDimensional.of_isCompact_closedBall\u2080" + - LinearIndependent.eventually + - exists_mem_frontier_infDist_compl_eq_dist + - FiniteDimensional.proper + - exists_seq_norm_le_one_le_norm_sub' + - ProperSpace.of_locallyCompactSpace + - IsCompact.exists_mem_frontier_infDist_compl_eq_dist + - Basis.exists_opNorm_le + - FiniteDimensional.of_isCompact_closedBall + - Basis.exists_opNNNorm_le + - isOpen_setOf_nat_le_rank + - exists_norm_le_le_norm_sub_of_finset + - path: Mathlib/Algebra/Order/Archimedean.lean + theorems: + - exists_pos_rat_lt + - exists_nat_gt + - exists_int_gt + - exists_floor + - exists_int_lt + - existsUnique_zsmul_near_of_pos + - exists_nat_one_div_lt + - archimedean_iff_rat_lt + - existsUnique_add_zsmul_mem_Ioc + - archimedean_iff_int_lt + - exists_nat_ge + - archimedean_iff_nat_lt + - exists_nat_pow_near + - existsUnique_add_zsmul_mem_Ico + - add_one_pow_unbounded_of_pos + - exists_rat_btwn + - existsUnique_zsmul_near_of_pos' + - exists_pow_lt_of_lt_one + - exists_nat_pow_near_of_lt_one + - exists_mem_Ico_zpow + - exists_rat_lt + - existsUnique_sub_zsmul_mem_Ioc + - exists_rat_gt + - exists_mem_Ioc_zpow + - pow_unbounded_of_one_lt + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean + theorems: + - Orientation.oangle_neg_neg + - Orientation.oangle_sign_smul_add_smul_smul_add_smul + - Orientation.oangle_sign_neg_right + - Orientation.cos_oangle_eq_inner_div_norm_mul_norm + - Orientation.oangle_map + - Orientation.oangle_smul_right_self_of_nonneg + - Orientation.oangle_smul_left_self_of_nonneg + - Orientation.oangle_eq_zero_iff_oangle_rev_eq_zero + - Orientation.oangle_neg_left + - Orientation.two_zsmul_oangle_smul_smul_self + - Orientation.oangle_sign_add_left + - Orientation.two_zsmul_oangle_neg_self_right + - Orientation.oangle_eq_pi_iff_sameRay_neg + - Orientation.two_zsmul_oangle_smul_right_self + - Orientation.oangle_neg_self_right + - Orientation.oangle_sign_sub_left_eq_neg + - Orientation.oangle_smul_right_of_pos + - Orientation.oangle_sign_sub_left_swap + - Orientation.oangle_ne_zero_and_ne_pi_iff_linearIndependent + - Orientation.oangle_sign_smul_right + - Orientation.oangle_add_oangle_rev_neg_left + - Orientation.oangle_sub_right + - Orientation.oangle_eq_neg_angle_of_sign_eq_neg_one + - Orientation.oangle_sub_eq_oangle_sub_rev_of_norm_eq + - Orientation.oangle_sign_sub_right_swap + - Orientation.oangle_rev + - Orientation.oangle_neg_orientation_eq_neg + - Orientation.ne_of_oangle_sign_eq_one + - Orientation.angle_eq_abs_oangle_toReal + - Orientation.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq + - Orientation.oangle_eq_pi_iff_oangle_rev_eq_pi + - Orientation.inner_rev_eq_zero_of_oangle_eq_neg_pi_div_two + - Orientation.left_ne_zero_of_oangle_sign_eq_one + - Orientation.oangle_sign_smul_add_smul_right + - Orientation.oangle_sign_neg_left + - Orientation.oangle_sign_smul_add_right + - Orientation.eq_zero_or_oangle_eq_iff_inner_eq_zero + - Orientation.two_zsmul_oangle_right_of_span_eq + - Orientation.oangle_sign_smul_sub_right + - Orientation.oangle_add_swap + - Orientation.oangle_map_complex + - Orientation.oangle_neg_right + - Orientation.angle_eq_iff_oangle_eq_of_sign_eq + - Orientation.oangle_sign_add_smul_left + - Orientation.cos_oangle_eq_cos_angle + - Orientation.oangle_smul_add_right_eq_zero_or_eq_pi_iff + - Orientation.left_ne_zero_of_oangle_sign_eq_neg_one + - Orientation.oangle_sign_smul_left + - Orientation.two_zsmul_oangle_neg_left + - Orientation.oangle_add_cyc3_neg_right + - Orientation.oangle_eq_of_angle_eq_of_sign_eq + - Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smul + - Orientation.oangle_sign_sub_smul_left + - Orientation.left_ne_zero_of_oangle_ne_zero + - Orientation.inner_eq_norm_mul_norm_mul_cos_oangle + - Orientation.oangle_add_oangle_rev + - Orientation.right_ne_zero_of_oangle_sign_eq_neg_one + - Orientation.oangle_smul_right_of_neg + - Orientation.oangle_zero_left + - Orientation.two_zsmul_oangle_left_of_span_eq + - Orientation.ne_of_oangle_ne_zero + - Orientation.abs_oangle_sub_left_toReal_lt_pi_div_two + - Orientation.oangle_eq_angle_of_sign_eq_one + - Orientation.oangle_neg_left_eq_neg_right + - Orientation.oangle_zero_right + - Orientation.two_zsmul_oangle_neg_self_left + - Orientation.oangle_eq_pi_iff_angle_eq_pi + - Orientation.oangle_sign_smul_sub_left + - Orientation.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero + - Orientation.oangle_add + - Orientation.oangle_add_cyc3 + - Orientation.inner_rev_eq_zero_of_oangle_eq_pi_div_two + - Orientation.oangle_smul_smul_self_of_nonneg + - Orientation.oangle_smul_left_of_pos + - Orientation.oangle_sign_add_right + - Orientation.right_ne_zero_of_oangle_sign_eq_one + - Orientation.two_zsmul_oangle_neg_right + - Orientation.eq_iff_norm_eq_and_oangle_eq_zero + - Orientation.oangle_sub_left + - Orientation.oangle_sign_sub_left + - Orientation.oangle_sign_smul_add_smul_left + - Orientation.two_zsmul_oangle_smul_left_self + - Orientation.oangle_neg_self_left + - Orientation.oangle_add_oangle_rev_neg_right + - Complex.oangle + - Orientation.oangle_add_cyc3_neg_left + - Orientation.oangle_sign_sub_right + - Orientation.oangle_sign_sub_right_eq_neg + - Orientation.oangle_eq_zero_or_eq_pi_iff_not_linearIndependent + - Orientation.right_ne_zero_of_oangle_ne_zero + - Orientation.oangle_smul_left_of_neg + - Orientation.oangle_eq_zero_iff_angle_eq_zero + - Orientation.two_zsmul_oangle_of_span_eq_of_span_eq + - Orientation.ne_of_oangle_sign_eq_neg_one + - Orientation.continuousAt_oangle + - Orientation.two_zsmul_oangle_smul_left_of_ne_zero + - path: Mathlib/LinearAlgebra/Matrix/Determinant.lean + theorems: + - Matrix.det_transpose + - Matrix.det_mul + - Matrix.det_eq_of_eq_det_one_mul + - Matrix.det_smul + - Matrix.det_blockDiagonal + - Matrix.det_units_conj + - Matrix.det_succ_row_zero + - Matrix.det_mul_row + - Matrix.coe_det_isEmpty + - Matrix.det_fin_three + - Matrix.det_updateRow_add_self + - Matrix.det_smul_of_tower + - "Matrix.det_fromBlocks_zero\u2081\u2082" + - Matrix.det_neg_eq_smul + - Matrix.det_eq_zero_of_column_eq_zero + - Matrix.det_succ_row + - RingHom.map_det + - Matrix.det_mul_right_comm + - Matrix.det_mul_left_comm + - Matrix.det_permutation + - Matrix.det_succ_column_zero + - Matrix.det_eq_of_forall_row_eq_smul_add_const_aux + - Matrix.det_neg + - Matrix.det_zero_of_column_eq + - Matrix.det_updateColumn_add + - Matrix.det_mul_comm + - Matrix.det_isEmpty + - Matrix.det_permute + - Matrix.det_updateColumn_add_self + - Matrix.det_eq_of_eq_mul_det_one + - Matrix.det_updateColumn_smul' + - Matrix.det_fin_two + - Matrix.det_updateRow_add_smul_self + - Matrix.det_submatrix_equiv_self + - Matrix.det_succ_column + - Matrix.det_eq_of_forall_row_eq_smul_add_pred_aux + - "Matrix.det_fromBlocks_zero\u2082\u2081" + - Matrix.det_eq_elem_of_subsingleton + - Matrix.det_mul_aux + - Matrix.det_apply' + - Matrix.det_unique + - Matrix.det_diagonal + - Matrix.det_updateColumn_add_smul_self + - Matrix.det_updateColumn_smul + - Matrix.det_one + - path: Mathlib/Analysis/Convolution.lean + theorems: + - BddAbove.continuous_convolution_right_of_integrable + - convolution_tendsto_right + - convolution_integrand_bound_right_of_le_of_subset + - HasCompactSupport.convolutionExists_left_of_continuous_right + - MeasureTheory.Integrable.convolution_integrand + - HasCompactSupport.hasDerivAt_convolution_right + - convolution_zero + - BddAbove.convolutionExistsAt + - contDiffOn_convolution_right_with_param_aux + - integrable_posConvolution + - hasFDerivAt_convolution_right_with_param + - convolution_mono_right_of_nonneg + - contDiffOn_convolution_right_with_param_comp + - posConvolution_eq_convolution_indicator + - convolution_mono_right + - continuousOn_convolution_right_with_param_comp + - HasCompactSupport.hasFDerivAt_convolution_left + - ConvolutionExistsAt.add_distrib + - contDiffOn_convolution_right_with_param + - contDiffOn_convolution_left_with_param + - HasCompactSupport.hasDerivAt_convolution_left + - contDiffOn_convolution_left_with_param_comp + - support_convolution_subset_swap + - zero_convolution + - BddAbove.convolutionExistsAt' + - continuousOn_convolution_right_with_param + - HasCompactSupport.continuous_convolution_right + - ConvolutionExists.distrib_add + - convolutionExistsAt_flip + - HasCompactSupport.contDiff_convolution_right + - smul_convolution + - HasCompactSupport.convolutionExistsAt + - convolution_neg_of_neg_eq + - ConvolutionExistsAt.integrable_swap + - convolution_assoc + - HasCompactSupport.contDiff_convolution_left + - convolution_congr + - dist_convolution_le' + - convolution_eq_right' + - convolution_smul + - ConvolutionExistsAt.ofNorm' + - HasCompactSupport.convolution_integrand_bound_right_of_subset + - integral_posConvolution + - HasCompactSupport.continuous_convolution_left + - HasCompactSupport.convolutionExists_right + - convolution_flip + - convolution_assoc' + - ConvolutionExists.add_distrib + - integral_convolution + - dist_convolution_le + - convolution_precompR_apply + - convolution_eq_swap + - HasCompactSupport.hasFDerivAt_convolution_right + - path: Mathlib/Topology/ContinuousFunction/Bounded.lean + theorems: + - BoundedContinuousFunction.coe_nsmulRec + - BoundedContinuousFunction.norm_eq_of_nonempty + - BoundedContinuousFunction.norm_lt_iff_of_compact + - BoundedContinuousFunction.dist_zero_of_empty + - BoundedContinuousFunction.norm_compContinuous_le + - BoundedContinuousFunction.norm_sub_nonneg + - BoundedContinuousFunction.norm_lt_iff_of_nonempty_compact + - BoundedContinuousFunction.NNReal.upper_bound + - BoundedContinuousFunction.isometry_extend + - BoundedContinuousFunction.nnnorm_eq_iSup_nnnorm + - BoundedContinuousFunction.norm_le_of_nonempty + - BoundedContinuousFunction.norm_le + - BoundedContinuousFunction.dist_set_exists + - "BoundedContinuousFunction.arzela_ascoli\u2081" + - BoundedContinuousFunction.nndist_eq + - BoundedContinuousFunction.abs_self_eq_nnrealPart_add_nnrealPart_neg + - BoundedContinuousFunction.nndist_eq_iSup + - BoundedContinuousFunction.norm_eq_iSup_norm + - BoundedContinuousFunction.norm_normComp + - BoundedContinuousFunction.coe_zsmulRec + - BoundedContinuousFunction.inducing_coeFn + - BoundedContinuousFunction.lipschitz_comp + - BoundedContinuousFunction.algebraMap_apply + - BoundedContinuousFunction.dist_lt_iff_of_compact + - BoundedContinuousFunction.sum_apply + - BoundedContinuousFunction.norm_eq + - BoundedContinuousFunction.coe_npowRec + - "BoundedContinuousFunction.arzela_ascoli\u2082" + - BoundedContinuousFunction.self_eq_nnrealPart_sub_nnrealPart_neg + - BoundedContinuousFunction.dist_eq_iSup + - BoundedContinuousFunction.dist_lt_of_nonempty_compact + - BoundedContinuousFunction.norm_coe_le_norm + - BoundedContinuousFunction.dist_extend_extend + - path: Mathlib/GroupTheory/Coset.lean + theorems: + - mem_own_rightCoset + - QuotientGroup.leftRel_eq + - QuotientGroup.mk_mul_of_mem + - QuotientGroup.eq_class_eq_leftCoset + - leftCoset_rightCoset + - leftCoset_mem_leftCoset + - leftCoset_eq_iff + - Subgroup.card_eq_card_quotient_mul_card_subgroup + - eq_cosets_of_normal + - mem_leftCoset_iff + - leftCoset_assoc + - QuotientGroup.eq + - normal_of_eq_cosets + - one_leftCoset + - rightCoset_one + - QuotientGroup.preimage_image_mk_eq_iUnion_image + - QuotientGroup.leftRel_apply + - QuotientGroup.rightRel_r_eq_rightCosetEquivalence + - mem_own_leftCoset + - Subgroup.card_subgroup_dvd_card + - Subgroup.card_dvd_of_injective + - rightCoset_mem_rightCoset + - QuotientGroup.mk_out'_eq_mul + - QuotientGroup.rightRel_eq + - QuotientGroup.leftRel_r_eq_leftCosetEquivalence + - Subgroup.card_quotient_dvd_card + - mem_rightCoset_rightCoset + - mem_rightCoset_iff + - mem_leftCoset_leftCoset + - rightCoset_assoc + - rightCoset_eq_iff + - QuotientGroup.preimage_image_mk + - QuotientGroup.rightRel_apply + - path: Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean + theorems: + - "Complex.inv_Gamma\u211D_two_sub" + - "Complex.differentiable_Gamma\u211D_inv" + - "Complex.Gamma\u211D_add_two" + - "Complex.Gamma\u2102_one" + - "Complex.Gamma\u211D_one_sub_mul_Gamma\u211D_one_add" + - "Complex.Gamma\u211D_div_Gamma\u211D_one_sub" + - "Complex.Gamma\u211D_residue_zero" + - "Complex.Gamma\u211D_ne_zero_of_re_pos" + - "Complex.Gamma\u211D_eq_zero_iff" + - "Complex.Gamma\u2102_add_one" + - "Complex.Gamma\u211D_one" + - "Complex.inv_Gamma\u211D_one_sub" + - path: Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean + theorems: + - UpperHalfPlane.le_dist_coe + - UpperHalfPlane.isometry_vertical_line + - UpperHalfPlane.im_le_im_mul_exp_dist + - UpperHalfPlane.image_coe_sphere + - UpperHalfPlane.dist_self_center + - UpperHalfPlane.cosh_half_dist + - UpperHalfPlane.dist_log_im_le + - UpperHalfPlane.image_coe_ball + - UpperHalfPlane.sinh_half_dist_add_dist + - UpperHalfPlane.center_zero + - UpperHalfPlane.image_coe_closedBall + - UpperHalfPlane.tanh_half_dist + - UpperHalfPlane.dist_triangle + - UpperHalfPlane.isometry_real_vadd + - UpperHalfPlane.dist_comm + - UpperHalfPlane.dist_of_re_eq + - UpperHalfPlane.sinh_half_dist + - UpperHalfPlane.exp_half_dist + - UpperHalfPlane.dist_coe_center_sq + - UpperHalfPlane.dist_coe_le + - UpperHalfPlane.dist_eq_iff_eq_sq_sinh + - UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_center + - UpperHalfPlane.dist_le_iff_le_sinh + - UpperHalfPlane.dist_le_dist_coe_div_sqrt + - UpperHalfPlane.cosh_dist' + - UpperHalfPlane.dist_coe_center + - UpperHalfPlane.cosh_dist + - UpperHalfPlane.im_pos_of_dist_center_le + - UpperHalfPlane.dist_eq_iff_eq_sinh + - path: Mathlib/Algebra/Order/Hom/Basic.lean + theorems: + - abs_sub_map_le_div + - le_map_div_add_map_div + - map_div_le_add + - le_map_add_map_div' + - le_map_mul_map_div + - map_div_rev + - map_eq_zero_iff_eq_one + - le_map_add_map_div + - le_map_div_mul_map_div + - path: Mathlib/Data/Seq/Seq.lean + theorems: + - Stream'.Seq.join_cons + - Stream'.Seq.corec_eq + - Stream'.Seq.destruct_eq_cons + - Stream'.Seq1.map_id + - Stream'.Seq.join_cons_nil + - Stream'.Seq1.ret_bind + - Stream'.Seq.exists_of_mem_map + - Stream'.Seq.ge_stable + - Stream'.Seq.head_cons + - Stream'.Seq.tail_cons + - Stream'.Seq.of_mem_append + - Stream'.Seq.destruct_eq_nil + - Stream'.Seq.map_tail + - Stream'.Seq.eq_of_bisim + - Stream'.Seq1.bind_ret + - Stream'.Seq.not_mem_nil + - Stream'.Seq.append_assoc + - Stream'.Seq.map_comp + - Stream'.Seq.le_stable + - Stream'.Seq.head_eq_destruct + - Stream'.Seq.ofStream_cons + - Stream'.Seq.nil_append + - Stream'.Seq1.map_join' + - Stream'.Seq.join_append + - Stream'.Seq.append_nil + - Stream'.Seq.enum_cons + - Stream'.Seq.not_terminates_iff + - Stream'.Seq.head_dropn + - Stream'.Seq.mem_rec_on + - Stream'.Seq.coinduction2 + - Stream'.Seq.ofList_cons + - Stream'.Seq.map_append + - Stream'.Seq.map_cons + - Stream'.Seq1.map_join + - Stream'.Seq.map_id + - Stream'.Seq.cons_append + - Stream'.Seq.destruct_cons + - Stream'.Seq.eq_or_mem_of_mem_cons + - Stream'.Seq.join_cons_cons + - Stream'.Seq1.join_cons + - Stream'.Seq.cons_injective2 + - Stream'.Seq1.join_map_ret + - Stream'.Seq.ofList_append + - Stream'.Seq1.join_join + - Stream'.Seq.mem_append_left + - Stream'.Seq.ofStream_append + - Stream'.Seq.mem_cons_iff + - Stream'.Seq1.bind_assoc + - path: Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean + theorems: + - MulChar.one_mul + - MulChar.pow_apply_coe + - MulChar.IsQuadratic.sq_eq_one + - MulChar.isNontrivial_iff + - MulChar.ext_iff + - MulChar.one_apply_coe + - MulChar.inv_mul + - MulChar.IsQuadratic.pow_odd + - MulChar.inv_apply + - MulChar.IsQuadratic.comp + - MulChar.IsNontrivial.sum_eq_zero + - MulChar.ext + - MulChar.mul_one + - MulChar.map_zero + - MulChar.IsNontrivial.comp + - MulChar.IsQuadratic.inv + - MulChar.ext' + - MulChar.map_ringChar + - MulChar.sum_one_eq_card_units + - MulChar.IsQuadratic.pow_char + - MulChar.ofUnitHom_coe + - MulChar.IsQuadratic.pow_even + - path: Mathlib/MeasureTheory/Measure/AddContent.lean + theorems: + - MeasureTheory.addContent_mono + - MeasureTheory.addContent_union_le + - MeasureTheory.le_addContent_diff + - MeasureTheory.addContent_union' + - "MeasureTheory.addContent_eq_add_diffFinset\u2080_of_subset" + - MeasureTheory.addContent_biUnion_le + - MeasureTheory.sum_addContent_le_of_subset + - path: Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean + theorems: + - IsPrimitiveRoot.autToPow_eq_ModularCyclotomicCharacter + - rootsOfUnity.integer_power_of_ringEquiv + - ModularCyclotomicCharacter.spec + - rootsOfUnity.integer_power_of_ringEquiv' + - ModularCyclotomicCharacter.toFun_spec + - ModularCyclotomicCharacter.comp + - ModularCyclotomicCharacter.toFun_unique + - ModularCyclotomicCharacter.unique + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Terminal.lean + theorems: + - CategoryTheory.Limits.hasTerminal_of_hasTerminal_of_preservesLimit + - CategoryTheory.Limits.hasInitial_of_hasInitial_of_preservesColimit + - path: Mathlib/Data/Set/Intervals/Disjoint.lean + theorems: + - Set.Ici_disjoint_Iic + - Set.iUnion_Ioo_left + - Set.iUnion_Icc_right + - Set.biUnion_Ioc_eq_Ioi_self_iff + - Set.iUnion_Icc_left + - IsGLB.biUnion_Ioi_eq + - Set.iUnion_Ioo_right + - Set.iUnion_Ioc_left + - Set.iUnion_Ioc_eq_Ioi_self_iff + - Set.iUnion_Ico_left + - IsGLB.biUnion_Ici_eq_Ioi + - IsGLB.biUnion_Ici_eq_Ici + - Set.Ico_disjoint_Ico + - Set.biUnion_Ico_eq_Iio_self_iff + - Set.iUnion_Ico_eq_Iio_self_iff + - Set.iUnion_Ico_right + - Set.Ioc_disjoint_Ioc + - Set.iUnion_Ioc_right + - Set.eq_of_Ico_disjoint + - iUnion_Ici_eq_Ioi_iInf + - iUnion_Ici_eq_Ici_iInf + - path: Mathlib/Data/FP/Basic.lean + theorems: + - FP.Float.Zero.valid + - path: Mathlib/Combinatorics/SimpleGraph/Basic.lean + theorems: + - SimpleGraph.deleteEdges_le_of_le + - SimpleGraph.incidenceSet_inter_incidenceSet_of_adj + - SimpleGraph.disjoint_fromEdgeSet + - SimpleGraph.edge_other_ne + - SimpleGraph.edgeSet_sup + - SimpleGraph.commonNeighbors_top_eq + - SimpleGraph.adj_iff_exists_edge_coe + - SimpleGraph.fromEdgeSet_edgeSet + - SimpleGraph.sInf_adj_of_nonempty + - SimpleGraph.adj_incidenceSet_inter + - SimpleGraph.fromEdgeSet_empty + - SimpleGraph.fromEdgeSet_univ + - SimpleGraph.deleteEdges_empty_eq + - SimpleGraph.iSup_adj + - SimpleGraph.edgeSet_sdiff_sdiff_isDiag + - SimpleGraph.iInf_adj_of_nonempty + - SimpleGraph.fromEdgeSet_sup + - SimpleGraph.edge_other_incident_set + - SimpleGraph.deleteEdges_eq_self + - SimpleGraph.deleteEdges_eq_inter_edgeSet + - SimpleGraph.incidence_other_prop + - SimpleGraph.fromEdgeSet_inf + - SimpleGraph.ne_of_adj + - SimpleGraph.adj_iff_exists_edge + - SimpleGraph.fromEdgeSet_disjoint + - SimpleGraph.sdiff_eq_deleteEdges + - SimpleGraph.deleteEdges_eq_sdiff_fromEdgeSet + - SimpleGraph.edgeSet_fromEdgeSet + - SimpleGraph.edgeSet_deleteEdges + - SimpleGraph.edgeSet_nonempty + - SimpleGraph.edgeSet_inf + - SimpleGraph.fromEdgeSet_sdiff + - SimpleGraph.neighborSet_union_compl_neighborSet_eq + - SimpleGraph.deleteEdges_deleteEdges + - SimpleGraph.incidenceSet_inter_incidenceSet_of_not_adj + - SimpleGraph.card_neighborSet_union_compl_neighborSet + - SimpleGraph.edgeSet_sdiff + - SimpleGraph.adj_of_mem_incidenceSet + - SimpleGraph.deleteEdges_sdiff_eq_of_le + - SimpleGraph.deleteEdges_univ_eq + - SimpleGraph.mem_incidenceSet + - SimpleGraph.deleteEdges_le + - SimpleGraph.compl_eq_deleteEdges + - SimpleGraph.neighborSet_compl + - SimpleGraph.compl_neighborSet_disjoint + - SimpleGraph.iInf_adj + - SimpleGraph.fromEdgeSet_mono + - SimpleGraph.edgeSet_eq_empty + - path: Mathlib/Tactic/Abel.lean + theorems: + - Mathlib.Tactic.Abel.zero_smul + - Mathlib.Tactic.Abel.term_atom_pf + - Mathlib.Tactic.Abel.term_add_constg + - Mathlib.Tactic.Abel.subst_into_smul_upcast + - Mathlib.Tactic.Abel.subst_into_smul + - Mathlib.Tactic.Abel.term_add_const + - Mathlib.Tactic.Abel.zero_termg + - Mathlib.Tactic.Abel.zero_term + - Mathlib.Tactic.Abel.const_add_term + - Mathlib.Tactic.Abel.term_smulg + - Mathlib.Tactic.Abel.unfold_sub + - Mathlib.Tactic.Abel.term_smul + - Mathlib.Tactic.Abel.term_atom_pfg + - Mathlib.Tactic.Abel.subst_into_negg + - Mathlib.Tactic.Abel.term_neg + - Mathlib.Tactic.Abel.const_add_termg + - Mathlib.Tactic.Abel.subst_into_smulg + - Mathlib.Tactic.Abel.subst_into_add + - Mathlib.Tactic.Abel.subst_into_addg + - Mathlib.Tactic.Abel.term_atomg + - Mathlib.Tactic.Abel.term_atom + - Mathlib.Tactic.Abel.zero_smulg + - Mathlib.Tactic.Abel.term_add_term + - path: Mathlib/Data/Nat/Factorial/SuperFactorial.lean + theorems: + - Nat.prod_Icc_factorial + - Nat.det_vandermonde_id_eq_superFactorial + - Nat.prod_range_succ_factorial + - Nat.superFactorial_dvd_vandermonde_det + - Nat.superFactorial_four_mul + - Nat.matrixOf_eval_descPochhammer_eq_mul_matrixOf_choose + - Nat.superFactorial_two_mul + - path: Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean + theorems: + - TopCat.pullbackIsoProdSubtype_hom_apply + - TopCat.snd_iso_of_left_embedding_range_subset + - TopCat.coequalizer_isOpen_iff + - TopCat.pullback_snd_image_fst_preimage + - TopCat.coinduced_of_isColimit + - TopCat.pullback_fst_range + - TopCat.pullbackIsoProdSubtype_inv_fst + - TopCat.range_pullback_map + - TopCat.embedding_of_pullback_embeddings + - TopCat.snd_openEmbedding_of_left_openEmbedding + - TopCat.range_pullback_to_prod + - TopCat.pullbackIsoProdSubtype_inv_snd + - TopCat.pullback_map_openEmbedding_of_open_embeddings + - TopCat.fst_openEmbedding_of_right_openEmbedding + - TopCat.pullback_snd_range + - TopCat.inducing_pullback_to_prod + - TopCat.fst_iso_of_right_embedding_range_subset + - TopCat.openEmbedding_of_pullback_open_embeddings + - TopCat.pullbackIsoProdSubtype_hom_fst + - TopCat.pullback_map_embedding_of_embeddings + - TopCat.snd_embedding_of_left_embedding + - TopCat.colimit_isOpen_iff + - TopCat.pullbackIsoProdSubtype_hom_snd + - TopCat.pullback_topology + - TopCat.fst_embedding_of_right_embedding + - path: Mathlib/Topology/MetricSpace/HausdorffDistance.lean + theorems: + - Metric.hausdorffEdist_ne_top_of_nonempty_of_bounded + - EMetric.exists_pos_forall_lt_edist + - Metric.infDist_le_infDist_add_hausdorffDist + - EMetric.infEdist_closure_pos_iff_not_mem_closure + - EMetric.infEdist_le_infEdist_add_edist + - Metric.dist_le_infDist_add_diam + - Metric.hausdorffDist_closure + - IsClosed.exists_infDist_eq_dist + - Metric.exists_mem_closure_infDist_eq_dist + - EMetric.hausdorffEdist_closure + - Metric.infEdist_ne_top + - Metric.infDist_le_dist_of_mem + - EMetric.disjoint_closedBall_of_lt_infEdist + - EMetric.hausdorffEdist_triangle + - EMetric.hausdorffEdist_le_of_mem_edist + - IsClosed.mem_iff_infDist_zero + - "EMetric.hausdorffEdist_closure\u2082" + - IsOpen.exists_iUnion_isClosed + - EMetric.infEdist_pos_iff_not_mem_closure + - Metric.exists_dist_lt_of_hausdorffDist_lt + - EMetric.infEdist_le_infEdist_add_hausdorffEdist + - EMetric.hausdorffEdist_self + - EMetric.hausdorffEdist_self_closure + - Metric.hausdorffDist_self_closure + - Metric.continuousAt_inv_infDist_pt + - EMetric.infEdist_closure + - Metric.hausdorffDist_self_zero + - Metric.infDist_zero_of_mem + - Metric.infDist_inter_closedBall_of_mem + - Metric.infDist_zero_of_mem_closure + - Metric.infDist_lt_iff + - EMetric.infEdist_le_hausdorffEdist_of_mem + - EMetric.hausdorffEdist_le_ediam + - EMetric.infEdist_lt_iff + - EMetric.mem_iff_infEdist_zero_of_closed + - Metric.hausdorffDist_empty' + - EMetric.infEdist_le_edist_add_infEdist + - EMetric.hausdorffEdist_zero_iff_closure_eq_closure + - Metric.hausdorffDist_le_of_mem_dist + - Metric.infDist_eq_iInf + - EMetric.hausdorffEdist_image + - Metric.infEdist_eq_top_iff + - Metric.hausdorffDist_comm + - Metric.infDist_empty + - Metric.hausdorffDist_le_of_infDist + - EMetric.hausdorffEdist_zero_iff_eq_of_closed + - IsClosed.not_mem_iff_infDist_pos + - EMetric.exists_real_pos_lt_infEdist_of_not_mem_closure + - Metric.hausdorffDist_nonneg + - Metric.infDist_singleton + - Metric.infDist_le_infDist_add_dist + - "EMetric.hausdorffEdist_closure\u2081" + - IsClosed.hausdorffDist_zero_iff_eq + - Metric.mem_closure_iff_infDist_zero + - EMetric.hausdorffEdist_le_of_infEdist + - Metric.infDist_closure + - IsCompact.exists_infDist_eq_dist + - Metric.hausdorffDist_triangle + - Metric.hausdorffDist_le_diam + - Metric.hausdorffDist_zero_iff_closure_eq_closure + - EMetric.empty_or_nonempty_of_hausdorffEdist_ne_top + - EMetric.infEdist_image + - EMetric.mem_closure_iff_infEdist_zero + - "Metric.hausdorffDist_closure\u2081" + - EMetric.continuous_infEdist + - EMetric.le_infEdist + - IsCompact.exists_infEdist_eq_edist + - "Metric.hausdorffDist_closure\u2082" + - Metric.exists_dist_lt_of_hausdorffDist_lt' + - EMetric.edist_le_infEdist_add_ediam + - Metric.hausdorffDist_empty + - EMetric.hausdorffEdist_comm + - EMetric.hausdorffEdist_empty + - Metric.hausdorffDist_image + - path: Mathlib/CategoryTheory/Shift/Induced.lean + theorems: + - CategoryTheory.shiftFunctorAdd_inv_app_obj_of_induced + - CategoryTheory.HasShift.Induced.zero_hom_app_obj + - CategoryTheory.shiftFunctor_of_induced + - CategoryTheory.HasShift.Induced.zero_inv_app_obj + - CategoryTheory.shiftFunctorZero_hom_app_obj_of_induced + - CategoryTheory.shiftFunctorAdd_hom_app_obj_of_induced + - CategoryTheory.HasShift.Induced.add_hom_app_obj + - CategoryTheory.HasShift.Induced.add_inv_app_obj + - CategoryTheory.shiftFunctorZero_inv_app_obj_of_induced + - path: Mathlib/Analysis/Convex/StrictConvexSpace.lean + theorems: + - StrictConvexSpace.of_norm_combo_ne_one + - strictConvex_closedBall + - StrictConvexSpace.of_strictConvex_closed_unit_ball + - StrictConvexSpace.of_norm_add + - lt_norm_sub_of_not_sameRay + - StrictConvexSpace.of_norm_combo_lt_one + - StrictConvexSpace.of_norm_add_ne_two + - norm_midpoint_lt_iff + - abs_lt_norm_sub_of_not_sameRay + - norm_add_lt_of_not_sameRay + - norm_combo_lt_of_ne + - path: Mathlib/Init/Data/Int/CompLemmas.lean + theorems: + - Int.natAbs_add_neg + - Int.ne_of_nat_ne_nonneg_case + - Int.natAbs_bit1_nonneg + - Int.natAbs_add_nonneg + - Int.zero_le_ofNat + - Int.ne_of_natAbs_ne_natAbs_of_nonneg + - Int.neg_ne_zero_of_ne + - Int.natAbs_bit1_nonneg_step + - Int.neg_ne_of_pos + - path: Mathlib/Order/BoundedOrder.lean + theorems: + - StrictMono.maximal_preimage_top + - StrictMono.minimal_preimage_bot + - sup_eq_bot_iff + - OrderTop.ext_top + - Subtype.coe_eq_bot_iff + - min_eq_bot + - Subtype.coe_eq_top_iff + - OrderBot.ext_bot + - path: Mathlib/GroupTheory/GroupAction/Pi.lean + theorems: + - Pi.faithfulSMul_at + - Function.extend_smul + - path: Mathlib/Algebra/Group/NatPowAssoc.lean + theorems: + - npow_mul' + - npow_mul_assoc + - npow_mul + - npow_mul_comm + - Nat.cast_npow + - Int.cast_npow + - path: Mathlib/GroupTheory/Coprod/Basic.lean + theorems: + - Monoid.Coprod.fst_toProd + - Monoid.Coprod.mrange_lift + - Monoid.Coprod.con_mul_left_inv + - Monoid.Coprod.range_lift + - Monoid.Coprod.induction_on' + - Monoid.Coprod.fst_comp_toProd + - Monoid.Coprod.prod_mk_fst_snd + - Monoid.Coprod.mrange_eq + - Monoid.Coprod.fst_prod_snd + - Monoid.Coprod.mrange_inl_sup_mrange_inr + - Monoid.Coprod.range_inl_sup_range_inr + - Monoid.Coprod.induction_on + - Monoid.Coprod.range_eq + - Monoid.Coprod.toProd_surjective + - Monoid.Coprod.mclosure_range_inl_union_inr + - Monoid.Coprod.snd_toProd + - Monoid.Coprod.comp_lift + - Monoid.Coprod.snd_comp_toProd + - path: Mathlib/MeasureTheory/Function/L2Space.lean + theorems: + - MeasureTheory.Integrable.inner_const + - "MeasureTheory.mem\u2112p_two_iff_integrable_sq_norm" + - integral_eq_zero_of_forall_integral_inner_eq_zero + - "MeasureTheory.Mem\u2112p.inner_const" + - MeasureTheory.L2.inner_indicatorConstLp_one + - MeasureTheory.L2.smul_left' + - MeasureTheory.L2.add_left' + - MeasureTheory.L2.mem_L1_inner + - MeasureTheory.BoundedContinuousFunction.inner_toLp + - MeasureTheory.ContinuousMap.inner_toLp + - "MeasureTheory.Mem\u2112p.integrable_sq" + - MeasureTheory.L2.inner_indicatorConstLp_eq_set_integral_inner + - MeasureTheory.Integrable.const_inner + - MeasureTheory.L2.snorm_rpow_two_norm_lt_top + - MeasureTheory.L2.snorm_inner_lt_top + - MeasureTheory.L2.integral_inner_eq_sq_snorm + - MeasureTheory.L2.norm_sq_eq_inner' + - "MeasureTheory.mem\u2112p_two_iff_integrable_sq" + - MeasureTheory.L2.inner_indicatorConstLp_eq_inner_set_integral + - path: Mathlib/SetTheory/Cardinal/ENat.lean + theorems: + - Cardinal.ofENat_eq_lift + - Cardinal.ofENat_add_aleph0 + - Cardinal.ofENat_le_one + - Cardinal.ofENat_eq_zero + - Cardinal.lift_lt_ofENat + - Cardinal.ofENat_pos + - Cardinal.one_le_ofENat + - Cardinal.aleph0_mul_ofENat + - Cardinal.ofENat_lt_nat + - Cardinal.ofENat_lt_lift + - Cardinal.ofENat_toENat_eq_self + - Cardinal.ofENat_eq_nat + - Cardinal.ofENat_mul_aleph0 + - Cardinal.toENatAux_gc + - Cardinal.one_eq_ofENat + - Cardinal.ofENat_le_nat + - Cardinal.ofENat_eq_one + - Cardinal.lift_le_ofENat + - Cardinal.ofENat_add + - Cardinal.range_ofENat + - Cardinal.toENatAux_le_nat + - Cardinal.ofENat_le_lift + - Cardinal.ofENat_mul + - Cardinal.lift_eq_ofENat + - Cardinal.toENat_strictMonoOn + - Cardinal.one_lt_ofENat + - Cardinal.toENatAux_eq_nat + - Cardinal.zero_eq_ofENat + - Cardinal.toENat_lift + - Cardinal.nat_le_ofENat + - path: Mathlib/RingTheory/IntegralClosure.lean + theorems: + - RingHom.isIntegralElem.of_comp + - IsIntegralClosure.mk'_one + - IsIntegral.algebraMap + - IsIntegralClosure.noZeroSMulDivisors + - integralClosure.isIntegral + - isIntegral_iff_isIntegral_closure_finite + - Submodule.span_range_natDegree_eq_adjoin + - IsIntegral.map_of_comp_eq + - IsIntegral.of_pow + - RingHom.IsIntegralElem.sub + - isField_of_isIntegral_of_isField' + - fg_adjoin_of_finite + - normalizeScaleRoots_monic + - normalizeScaleRoots_degree + - IsIntegralClosure.mk'_zero + - IsIntegral.tmul + - IsIntegralClosure.mk'_algebraMap + - isIntegral_algHom_iff + - RingHom.isIntegralElem_map + - RingHom.IsIntegral.quotient + - normalizeScaleRoots_coeff_mul_leadingCoeff_pow + - IsIntegral.smul + - isIntegral_trans + - IsIntegral.pow + - IsIntegralClosure.mk'_mul + - isIntegral_of_smul_mem_submodule + - IsIntegral.of_mul_unit + - normalizeScaleRoots_support + - leadingCoeff_smul_normalizeScaleRoots + - IsIntegral.det + - IsIntegral.map + - "normalizeScaleRoots_eval\u2082_leadingCoeff_mul" + - adjoin_le_integralClosure + - IsIntegralClosure.mk'_add + - isIntegral_algEquiv + - isField_of_isIntegral_of_isField + - Algebra.isIntegral_sup + - RingHom.isIntegralElem_leadingCoeff_mul + - isIntegral_quotientMap_iff + - Algebra.IsIntegral.finite + - IsIntegral.tower_top + - integralClosure_map_algEquiv + - RingHom.IsIntegralElem.of_mem_closure + - isIntegral_leadingCoeff_smul + - path: Mathlib/Data/Polynomial/AlgebraMap.lean + theorems: + - Polynomial.eval_unique + - Polynomial.aevalTower_id + - Polynomial.adjoin_X + - Polynomial.aevalTower_ofId + - Polynomial.not_isUnit_X_sub_C + - Polynomial.eval_mul_X_sub_C + - Polynomial.coeff_zero_eq_aeval_zero' + - Polynomial.coeff_zero_eq_aeval_zero + - Polynomial.aeval_algHom + - Polynomial.map_aeval_eq_aeval_map + - Polynomial.aeval_eq_zero_of_dvd_aeval_eq_zero + - Polynomial.aeval_mem_adjoin_singleton + - Polynomial.toFinsupp_algebraMap + - Polynomial.dvd_term_of_dvd_eval_of_dvd_terms + - Polynomial.aeval_algHom_apply + - "Polynomial.algHom_eval\u2082_algebraMap" + - Polynomial.aeval_eq_sum_range + - Polynomial.aeval_eq_sum_range' + - "Polynomial.isRoot_of_eval\u2082_map_eq_zero" + - "Polynomial.eval\u2082_algebraMap_X" + - Algebra.adjoin_singleton_eq_range_aeval + - Polynomial.aeval_endomorphism + - path: Mathlib/GroupTheory/NoncommPiCoprod.lean + theorems: + - MonoidHom.noncommPiCoprod_range + - MonoidHom.noncommPiCoprod_mulSingle + - Subgroup.injective_noncommPiCoprod_of_independent + - Subgroup.noncommPiCoprod_mulSingle + - Subgroup.commute_subtype_of_commute + - MonoidHom.independent_range_of_coprime_order + - MonoidHom.injective_noncommPiCoprod_of_independent + - Subgroup.eq_one_of_noncommProd_eq_one_of_independent + - MonoidHom.noncommPiCoprod_mrange + - Subgroup.noncommPiCoprod_range + - Subgroup.independent_of_coprime_order + - path: Mathlib/Topology/Order/Lattice.lean + theorems: + - Filter.Tendsto.finset_sup_nhds_apply + - ContinuousWithinAt.finset_inf' + - ContinuousWithinAt.finset_sup' + - Filter.Tendsto.finset_sup'_nhds_apply + - Filter.Tendsto.finset_sup'_nhds + - ContinuousAt.finset_inf' + - ContinuousAt.finset_sup' + - ContinuousAt.finset_sup + - ContinuousAt.finset_inf + - Filter.Tendsto.finset_sup_nhds + - ContinuousWithinAt.finset_sup + - ContinuousWithinAt.finset_inf + - path: Mathlib/Computability/Partrec.lean + theorems: + - "Partrec\u2082.unpaired" + - Partrec.nat_rec + - Computable.list_ofFn + - Computable.map_decode_iff + - Computable.option_getD + - Nat.mem_rfind + - Computable.nat_strong_rec + - Partrec.sum_casesOn_right + - Primrec.to_comp + - Partrec.option_some_iff + - Nat.Partrec.prec' + - Nat.Partrec.of_primrec + - Partrec.nat_casesOn_right + - Partrec.none + - Partrec.map + - Computable.pair + - "Partrec.bind_decode\u2082_iff" + - Partrec.comp + - Nat.rfindOpt_dom + - Partrec.fix_aux + - Nat.Partrec.none + - Computable.cond + - Computable.option_casesOn + - Computable.sum_casesOn + - Computable.option_map + - Partrec.option_casesOn_right + - Partrec.fix + - Partrec.sum_casesOn_left + - Nat.rfind_min' + - Partrec.rfind + - Nat.Partrec.ppred + - Computable.nat_rec + - Partrec.vector_mOfFn + - Computable.option_bind + - Partrec.nat_iff + - Computable.vector_ofFn + - Partrec.bind + - Computable.bind_decode_iff + - path: Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean + theorems: + - MeasureTheory.Measure.integral_comp_smul + - MeasureTheory.Measure.integral_comp_mul_right + - MeasureTheory.Measure.integral_comp_inv_smul + - MeasureTheory.Measure.integral_comp_inv_mul_right + - MeasureTheory.Measure.integral_comp_inv_smul_of_nonneg + - MeasureTheory.Measure.integral_comp_inv_mul_left + - MeasureTheory.integrable_comp_mul_right_iff + - MeasureTheory.integrable_comp_smul_iff + - MeasureTheory.Measure.set_integral_comp_smul_of_pos + - MeasureTheory.integrable_comp_mul_left_iff + - MeasureTheory.Measure.integral_comp_smul_of_nonneg + - MeasureTheory.Measure.integral_comp_mul_left + - MeasureTheory.Measure.set_integral_comp_smul + - path: Mathlib/Data/MvPolynomial/Funext.lean + theorems: + - MvPolynomial.funext_iff + - MvPolynomial.funext_fin + - MvPolynomial.funext + - path: Mathlib/Algebra/BigOperators/Basic.lean + theorems: + - Finset.prod_pow_eq_pow_sum + - Finset.prod_comp + - Commute.sum_right + - Finset.prod_erase_attach + - Finset.prod_sum_elim + - Finset.prod_filter_mul_prod_filter_not + - Finset.card_biUnion + - Finset.prod_range_succ' + - Finset.prod_ite_irrel + - Finset.prod_sdiff_eq_div + - Finset.prod_dvd_prod_of_dvd + - Finset.prod_dite_eq + - List.prod_toFinset + - Finset.prod_dite_of_false + - ofMul_list_prod + - Finset.eq_of_card_le_one_of_prod_eq + - Finset.prod_dite_eq' + - Finset.prod_eq_zero + - Finset.natCast_card_filter + - Finset.prod_insert_of_eq_one_if_not_mem + - Finset.sum_int_mod + - Finset.prod_ite_of_false + - Finset.prod_ite_one + - Finset.prod_finset_product_right + - Multiset.disjoint_list_sum_left + - Finset.prod_list_count_of_subset + - Finset.prod_union_eq_right + - Finset.prod_union + - Finset.prod_attach + - Finset.prod_pair + - toAdd_multiset_sum + - Finset.prod_mulIndicator_subset_of_eq_one + - Finset.prod_mul_prod_comm + - Finset.prod_subtype_of_mem + - Finset.prod_induction_nonempty + - Finset.mem_sum + - Finset.prod_disjiUnion + - map_prod + - Finset.prod_mulIndicator_eq_prod_filter + - ofAdd_multiset_prod + - Finset.prod_mul_distrib + - Finset.prod_powerset_insert + - Finset.prod_inter_mul_prod_diff + - Units.mk0_prod + - Finset.exists_ne_one_of_prod_ne_one + - Finset.prod_filter + - Fintype.prod_unique + - Finset.eq_prod_range_div + - Finset.prod_multiset_count_of_subset + - Finset.prod_boole + - Finset.prod_attach_univ + - Finset.mul_prod_erase + - Fintype.prod_subsingleton + - Finset.prod_congr_set + - Finset.mulIndicator_biUnion_apply + - Finset.prod_erase_eq_div + - Multiset.add_eq_union_right_of_le + - Finset.prod_erase_lt_of_one_lt + - Finset.prod_nbij + - Finset.prod_of_empty + - Finset.card_biUnion_le + - Multiset.finset_sum_eq_sup_iff_disjoint + - Finset.prod_subtype + - toMul_list_sum + - Finset.prod_range_div + - Finset.prod_ite_mem + - Finset.prod_range_induction + - Finset.prod_cancels_of_partition_cancels + - IsCompl.prod_mul_prod + - Finset.prod_of_injOn + - Finset.prod_range_succ_comm + - Finset.prod_sigma + - Finset.prod_fiberwise_of_maps_to + - Finset.prod_int_mod + - Finset.prod_update_of_not_mem + - Multiset.exists_smul_of_dvd_count + - Finset.prod_multiset_map_count + - Finset.prod_flip + - Finset.prod_eq_prod_diff_singleton_mul + - Finset.pow_eq_prod_const + - Multiset.disjoint_finset_sum_left + - Finset.prod_multiset_count + - Finset.prod_const_one + - Equiv.Perm.prod_comp + - Finset.mulSupport_prod + - Finset.prod_filter_of_ne + - Finset.support_prod + - Finset.prod_partition + - Finset.prod_eq_mul_prod_diff_singleton + - Finset.prod_hom_rel + - Finset.prod_eq_zero_iff + - Finset.prod_dite_irrel + - Finset.prod_sdiff_div_prod_sdiff + - Finset.prod_ne_zero_iff + - Multiset.disjoint_list_sum_right + - toAdd_list_sum + - Finset.prod_eq_single + - Multiset.count_sum' + - Finset.prod_range_add + - Finset.prod_biUnion + - Finset.prod_mulIndicator_eq_prod_inter + - Fintype.prod_fiberwise + - Multiset.sum_count_eq + - Multiset.sup_powerset_len + - Finset.prod_add_prod_eq + - Finset.prod_apply_ite_of_false + - Finset.prod_subset + - Finset.prod_nat_mod + - Finset.prod_range_zero + - Multiset.disjoint_sum_left + - Finset.prod_disjUnion + - Finset.prod_erase + - Finset.prod_fiberwise_of_maps_to' + - Multiset.toFinset_sum_count_nsmul_eq + - ofAdd_list_prod + - Finset.sum_filter_count_eq_countP + - Finset.prod_finset_product + - Finset.prod_to_list + - Finset.card_eq_sum_card_fiberwise + - Finset.prod_map + - Finset.sum_nat_mod + - Finset.prod_univ_pi + - Finset.prod_bij' + - Finset.prod_dite_of_true + - Finset.prod_diag + - Finset.prod_apply_ite_of_true + - Finset.prod_comm' + - Finset.prod_pow_boole + - Finset.sum_range_tsub + - toMul_multiset_sum + - Finset.prod_mem_multiset + - Finset.prod_insert_div + - Finset.sum_multiset_singleton + - Finset.prod_eq_mul + - Finset.prod_equiv + - Finset.prod_range_one + - Finset.prod_powerset_cons + - Multiset.toFinset_prod_dvd_prod + - Finset.prod_update_of_mem + - Finset.prod_sdiff + - Finset.prod_involution + - Finset.prod_list_map_count + - Finset.prod_bij_ne_one + - Finset.prod_ite + - Finset.eventually_constant_prod + - Finset.prod_ite_of_true + - Multiset.disjoint_sum_right + - Finset.prod_dite + - Finset.prod_list_count + - Equiv.Perm.prod_comp' + - Finset.prod_range_succ + - Finset.dvd_prod_of_mem + - Finset.prod_eq_mul_of_mem + - Finset.prod_eq_single_of_mem + - Finset.prod_erase_mul + - Fintype.prod_subtype_mul_prod_subtype + - Finset.prod_subset_one_on_sdiff + - Finset.prod_piecewise + - Finset.eq_prod_range_div' + - Finset.sum_const_nat + - Finset.prod_subtype_eq_prod_filter + - Multiset.disjoint_finset_sum_right + - Finset.prod_disj_sum + - Finset.prod_unique_nonempty + - Finset.prod_val + - Fintype.prod_of_injective + - Finset.prod_range_div' + - Fintype.prod_fiberwise' + - Finset.prod_apply_dite + - Multiset.toFinset_sum_count_eq + - Finset.eq_one_of_prod_eq_one + - Finset.prod_subtype_map_embedding + - Finset.prod_powerset + - Finset.prod_dvd_prod_of_subset + - nat_abs_sum_le + - Fintype.prod_bijective + - Finset.prod_powersetCard + - ofMul_multiset_prod + - Finset.card_eq_sum_ones + - Finset.prod_congr + - path: Mathlib/Tactic/CancelDenoms/Core.lean + theorems: + - CancelDenoms.cancel_factors_ne + - CancelDenoms.cancel_factors_eq_div + - CancelDenoms.sub_subst + - CancelDenoms.cancel_factors_eq + - CancelDenoms.pow_subst + - CancelDenoms.cancel_factors_le + - CancelDenoms.div_subst + - CancelDenoms.mul_subst + - CancelDenoms.inv_subst + - CancelDenoms.cancel_factors_lt + - CancelDenoms.neg_subst + - CancelDenoms.add_subst + - path: Mathlib/LinearAlgebra/Quotient.lean + theorems: + - LinearMap.ker_le_range_iff + - Submodule.Quotient.nontrivial_of_lt_top + - Submodule.Quotient.equiv_trans + - LinearMap.range_eq_top_of_cancel + - Submodule.card_eq_card_quotient_mul_card + - Submodule.mapQ_zero + - Submodule.mapQ_pow + - Submodule.liftQ_mkQ + - Submodule.mapQ_id + - Submodule.Quotient.equiv_refl + - Submodule.mkQ_surjective + - Submodule.span_preimage_eq + - Submodule.comap_map_mkQ + - Submodule.le_comap_mkQ + - Submodule.comap_liftQ + - Submodule.quotientRel_r_def + - Submodule.Quotient.mk_surjective + - Submodule.mkQ_map_self + - Submodule.ker_liftQ_eq_bot + - Submodule.map_mkQ_eq_top + - Submodule.range_liftQ + - Submodule.ker_mkQ + - Submodule.mapQ_mkQ + - Submodule.map_liftQ + - Submodule.mapQ_comp + - Submodule.subsingleton_quotient_iff_eq_top + - LinearMap.range_mkQ_comp + - Submodule.range_mkQ + - Submodule.Quotient.mk_eq_zero + - path: Mathlib/Data/Real/Irrational.lean + theorems: + - irrational_sqrt_of_multiplicity_odd + - irrational_mul_rat_iff + - Irrational.sub_rat + - Irrational.of_zpow + - Irrational.of_mul_int + - one_lt_natDegree_of_irrational_root + - irrational_mul_nat_iff + - Irrational.div_rat + - Nat.Prime.irrational_sqrt + - Irrational.int_sub + - irrational_int_div_iff + - irrational_mul_int_iff + - Irrational.of_sub_rat + - Irrational.add_cases + - Irrational.of_rat_sub + - irrational_nat_div_iff + - Irrational.of_int_mul + - Irrational.int_add + - Irrational.div_int + - irrational_nrt_of_n_not_dvd_multiplicity + - Irrational.div_nat + - irrational_sqrt_rat_iff + - Irrational.mul_rat + - irrational_div_int_iff + - irrational_div_rat_iff + - Irrational.rat_add + - Irrational.mul_cases + - Irrational.ne_int + - irrational_sqrt_two + - irrational_nrt_of_notint_nrt + - Irrational.inv + - Transcendental.irrational + - irrational_div_nat_iff + - irrational_rat_div_iff + - Irrational.of_int_add + - Irrational.of_pow + - Irrational.of_sub_int + - Irrational.of_one_div + - Irrational.ne_one + - Irrational.of_neg + - Irrational.rat_sub + - Irrational.sub_int + - Irrational.mul_int + - irrational_nat_mul_iff + - Irrational.neg + - Irrational.of_int_sub + - irrational_int_mul_iff + - path: Mathlib/CategoryTheory/Galois/Basic.lean + theorems: + - CategoryTheory.PreGaloisCategory.cardFiber_coprod_eq_sum + - CategoryTheory.PreGaloisCategory.isIso_of_mono_of_eqCardFiber + - CategoryTheory.PreGaloisCategory.ltCardFiber_of_mono_of_notIso + - CategoryTheory.PreGaloisCategory.non_zero_card_fiber_of_not_initial + - CategoryTheory.PreGaloisCategory.has_non_trivial_subobject_of_not_isConnected_of_not_initial + - CategoryTheory.PreGaloisCategory.evaluationInjective_of_isConnected + - CategoryTheory.PreGaloisCategory.cardFiber_eq_of_iso + - CategoryTheory.PreGaloisCategory.evaluation_aut_injective_of_isConnected + - CategoryTheory.PreGaloisCategory.not_initial_iff_fiber_nonempty + - CategoryTheory.PreGaloisCategory.initial_iff_fiber_empty + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean + theorems: + - Complex.cos_eq_zero_iff + - Complex.sin_eq_one_iff + - Complex.tan_eq_zero_iff + - Complex.cos_eq_neg_one_iff + - Real.tan_eq_zero_iff' + - Complex.tan_add_mul_I + - Complex.cos_eq_one_iff + - Complex.sin_ne_zero_iff + - Complex.sin_eq_zero_iff + - Complex.cos_surjective + - Complex.tan_two_mul + - Complex.cos_eq_cos_iff + - Complex.cos_eq_iff_quadratic + - Complex.tan_eq_zero_iff' + - Real.mul_lt_sin + - Complex.tan_eq + - Complex.sin_eq_neg_one_iff + - Complex.sin_surjective + - Complex.sin_eq_sin_iff + - Complex.tan_ne_zero_iff + - Real.le_sin_mul + - Real.lt_sin_mul + - Complex.tan_int_mul_pi_div_two + - Complex.tan_add + - Complex.cos_ne_zero_iff + - Real.mul_le_sin + - path: Mathlib/Analysis/Calculus/FDeriv/Mul.lean + theorems: + - fderiv_continuousMultilinear_apply_const_apply + - HasFDerivAt.mul_const + - hasStrictFDerivAt_list_prod_finRange' + - hasStrictFDerivAt_multiset_prod + - HasStrictFDerivAt.finset_prod + - HasStrictFDerivAt.multiset_prod + - HasStrictFDerivAt.list_prod' + - DifferentiableWithinAt.pow + - HasFDerivWithinAt.smul_const + - hasStrictFDerivAt_list_prod_attach' + - HasFDerivAt.list_prod' + - hasFDerivAt_ring_inverse + - HasFDerivAt.multiset_prod + - hasFDerivAt_inv' + - HasStrictFDerivAt.smul_const + - hasStrictFDerivAt_list_prod' + - HasFDerivWithinAt.mul + - HasStrictFDerivAt.mul_const + - HasFDerivWithinAt.multiset_prod + - HasFDerivAt.smul_const + - HasStrictFDerivAt.mul + - HasFDerivAt.mul + - HasFDerivWithinAt.finset_prod + - fderivWithin_continuousMultilinear_apply_const_apply + - HasFDerivWithinAt.list_prod' + - hasStrictFDerivAt_list_prod + - fderivWithin_inv' + - HasFDerivAt.finset_prod + - path: Mathlib/Data/Polynomial/Taylor.lean + theorems: + - Polynomial.taylor_eval + - Polynomial.taylor_coeff + - Polynomial.taylor_monomial + - Polynomial.taylor_taylor + - Polynomial.sum_taylor_eq + - Polynomial.taylor_coeff_zero + - Polynomial.taylor_mul + - Polynomial.natDegree_taylor + - Polynomial.taylor_eval_sub + - Polynomial.taylor_zero + - Polynomial.taylor_X + - Polynomial.taylor_C + - Polynomial.taylor_injective + - Polynomial.taylor_zero' + - Polynomial.taylor_one + - Polynomial.eq_zero_of_hasseDeriv_eq_zero + - Polynomial.taylor_coeff_one + - path: Mathlib/Order/Irreducible.lean + theorems: + - SupPrime.ne_bot + - supPrime_iff_not_isMin + - SupPrime.le_finset_sup + - supPrime_iff_supIrred + - not_supIrred + - InfPrime.ne_top + - infPrime_iff_not_isMax + - not_supPrime + - supIrred_iff_not_isMin + - exists_supIrred_decomposition + - infPrime_iff_infIrred + - InfPrime.infIrred + - SupIrred.ne_bot + - SupPrime.supIrred + - SupIrred.finset_sup_eq + - InfIrred.ne_top + - infIrred_iff_not_isMax + - path: Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean + theorems: + - ContinuousMap.sup_mem_subalgebra_closure + - ContinuousMap.polynomial_comp_attachBound_mem + - ContinuousMap.inf_mem_closed_subalgebra + - ContinuousMap.comp_attachBound_mem_closure + - ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints + - ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints + - ContinuousMap.inf_mem_subalgebra_closure + - ContinuousMap.sublattice_closure_eq_top + - ContinuousMap.starAlgHom_ext_map_X + - ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints + - ContinuousMap.algHom_ext_map_X + - ContinuousMap.abs_mem_subalgebra_closure + - Subalgebra.SeparatesPoints.isROrC_to_real + - ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints + - ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints + - ContinuousMap.sup_mem_closed_subalgebra + - ContinuousMap.polynomial_comp_attachBound + - path: Mathlib/Topology/UniformSpace/Cauchy.lean + theorems: + - cauchySeq_shift + - UniformSpace.secondCountable_of_separable + - totallyBounded_iff_ultrafilter + - CauchySeq.tendsto_uniformity + - CompleteSpace.fst_of_prod + - totallyBounded_of_forall_symm + - cauchy_prod_iff + - CauchySeq.prod_map + - TotallyBounded.exists_subset + - Filter.Tendsto.subseq_mem_entourage + - SequentiallyComplete.setSeq_prod_subset + - isComplete_iff_ultrafilter + - cauchy_iff_le + - cauchy_iff + - SequentiallyComplete.le_nhds_of_seq_tendsto_nhds + - Filter.HasBasis.cauchySeq_iff + - CauchySeq.subseq_subseq_mem + - totallyBounded_iff_filter + - isComplete_iUnion_separated + - cauchySeq_of_controlled + - CauchySeq.totallyBounded_range + - cauchySeq_tendsto_of_isComplete + - CompleteSpace.snd_of_prod + - isComplete_iff_ultrafilter' + - complete_univ + - completeSpace_iff_ultrafilter + - CauchySeq.subseq_mem + - UniformSpace.complete_of_convergent_controlled_sequences + - Filter.HasBasis.cauchy_iff + - Cauchy.ultrafilter_of + - cauchy_iInf_uniformSpace' + - Filter.HasBasis.cauchySeq_iff' + - Function.Bijective.cauchySeq_comp_iff + - Cauchy.prod + - cauchySeq_iff + - cauchy_inf_uniformSpace + - cauchy_iInf_uniformSpace + - le_nhds_of_cauchy_adhp + - TotallyBounded.image + - CauchySeq.mem_entourage + - IsComplete.union + - cauchy_comap_uniformSpace + - le_nhds_of_cauchy_adhp_aux + - cauchy_map_iff + - cauchySeq_iff_tendsto + - path: Mathlib/Algebra/CharZero/Lemmas.lean + theorems: + - bit0_injective + - one_eq_bit1 + - add_halves' + - nat_mul_inj' + - bit1_eq_one + - nat_mul_inj + - bit1_injective + - zero_eq_bit0 + - RingHom.charZero + - sub_half + - half_add_self + - Nat.cast_div_charZero + - Nat.cast_pow_eq_one + - units_ne_neg_self + - RingHom.charZero_iff + - add_self_eq_zero + - half_sub + - path: Mathlib/Analysis/Calculus/FDeriv/Basic.lean + theorems: + - hasFDerivWithinAt_inter + - Filter.EventuallyEq.hasFDerivAtFilter_iff + - fderivWithin_of_mem_nhds + - HasFDerivWithinAt.hasFDerivAt + - HasFDerivAtFilter.tendsto_nhds + - differentiableOn_of_locally_differentiableOn + - fderivWithin_zero_of_not_differentiableWithinAt + - hasFDerivWithinAt_inter' + - HasFDerivWithinAt.union + - hasFDerivWithinAt_univ + - fderivWithin_congr_set' + - hasStrictFDerivAt_id + - fderivWithin_const_apply + - differentiableWithinAt_univ + - HasFDerivAt.lim + - hasFDerivWithinAt_singleton + - fderivWithin_univ + - fderivWithin_inter + - HasStrictFDerivAt.of_nmem_tsupport + - HasStrictFDerivAt.hasFDerivAt + - fderivWithin_eq_fderiv + - DifferentiableAt.hasFDerivAt + - hasFDerivWithinAt_of_nmem_closure + - fderivWithin_id + - HasFDerivAt.le_of_lipschitzOn + - Filter.EventuallyEq.hasStrictFDerivAt_iff + - hasFDerivWithinAt_insert + - support_fderiv_subset + - differentiableWithinAt_inter' + - hasFDerivAt_iff_isLittleO_nhds_zero + - hasFDerivWithinAt_congr_set' + - differentiableOn_univ + - HasFDerivAt.le_of_lip' + - HasFDerivAt.fderiv + - fderiv_zero_of_not_differentiableAt + - fderivWithin_mem_iff + - DifferentiableWithinAt.mono + - Asymptotics.IsBigO.hasFDerivWithinAt + - hasFDerivAtFilter_id + - fderiv_mem_iff + - HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt + - hasStrictFDerivAt_const + - norm_fderiv_le_of_lip' + - hasFDerivAt_of_subsingleton + - differentiableWithinAt_inter + - Asymptotics.IsBigO.hasFDerivAt + - hasFDerivAtFilter_const + - HasFDerivWithinAt.of_nhdsWithin_eq_bot + - fderiv_const + - HasFDerivAt.unique + - fderivWithin_zero_of_nmem_closure + - hasFDerivWithinAt_of_mem_nhds + - norm_fderiv_le_of_lipschitzOn + - Filter.EventuallyEq.fderivWithin_eq + - Filter.EventuallyEq.fderiv_eq + - hasFDerivAtFilter_iff_tendsto + - path: Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean + theorems: + - AddMonoidAlgebra.toDirectSum_mul + - DirectSum.toAddMonoidAlgebra_mul + - path: .lake/packages/std/Std/Data/Range/Lemmas.lean + theorems: + - Std.Range.numElems_stop_le_start + - Std.Range.mem_range'_elems + - Std.Range.forIn'_eq_forIn_range' + - Std.Range.numElems_le_iff + - Std.Range.numElems_step_1 + - path: Mathlib/SetTheory/Ordinal/Notation.lean + theorems: + - ONote.scale_opowAux + - ONote.exists_lt_mul_omega' + - ONote.fastGrowing_one + - ONote.fastGrowing_succ + - ONote.NFBelow.lt + - NONote.cmp_compares + - "ONote.fastGrowing\u03B5\u2080_zero" + - ONote.oadd_lt_oadd_3 + - ONote.add_nfBelow + - ONote.split_eq_scale_split' + - ONote.NFBelow.repr_lt + - ONote.cmp_compares + - ONote.repr_scale + - "ONote.repr_opow_aux\u2081" + - ONote.NF.of_dvd_omega_opow + - "ONote.fastGrowing\u03B5\u2080_two" + - ONote.exists_lt_omega_opow' + - ONote.fastGrowing_def + - ONote.exists_lt_add + - ONote.omega_le_oadd + - ONote.fastGrowing_zero' + - ONote.repr_add + - ONote.oadd_mul_nfBelow + - ONote.zero_lt_one + - ONote.NFBelow.mono + - ONote.nf_repr_split + - ONote.nf_repr_split' + - ONote.repr_opow + - ONote.oadd_lt_oadd_2 + - "ONote.repr_opow_aux\u2082" + - ONote.eq_of_cmp_eq + - ONote.split_add_lt + - ONote.mulNat_eq_mul + - ONote.scale_eq_mul + - ONote.NF.of_dvd_omega + - ONote.fundamentalSequence_has_prop + - ONote.sub_nfBelow + - "ONote.fastGrowing\u03B5\u2080_one" + - ONote.NFBelow.fst + - ONote.fastGrowing_limit + - ONote.fastGrowing_two + - ONote.NF.below_of_lt + - ONote.NF.zero_of_zero + - ONote.repr_mul + - ONote.split_dvd + - ONote.repr_ofNat + - ONote.NFBelow.snd + - path: Mathlib/Algebra/Free.lean + theorems: + - FreeSemigroup.traverse_mul + - FreeSemigroup.hom_ext + - FreeSemigroup.length_mul + - FreeSemigroup.lift_of_mul + - FreeMagma.toFreeSemigroup_comp_map + - FreeSemigroup.length_map + - FreeMagma.hom_ext + - path: Mathlib/CategoryTheory/Monoidal/Mod_.lean + theorems: + - Mod_.assoc_flip + - Mod_.id_hom' + - path: Mathlib/Data/Finset/PImage.lean + theorems: + - Finset.pimage_subset + - Finset.mem_pimage + - Finset.pimage_some + - Part.toFinset_some + - Part.toFinset_none + - Finset.pimage_inter + - Finset.pimage_union + - Finset.pimage_eq_image_filter + - Part.mem_toFinset + - Finset.pimage_empty + - Finset.pimage_congr + - path: Mathlib/Data/Polynomial/Degree/Lemmas.lean + theorems: + - Polynomial.nextCoeff_C_mul_X_add_C + - Polynomial.monic_mul_leadingCoeff_inv + - Polynomial.natDegree_comp_le + - Polynomial.natDegree_C_mul_le + - Polynomial.coeff_sub_eq_left_of_lt + - Polynomial.natDegree_eq_one + - "Polynomial.natDegree_pos_of_eval\u2082_root" + - Polynomial.degree_mul_leadingCoeff_inv + - Polynomial.natDegree_add_coeff_mul + - Polynomial.coeff_pow_eq_ite_of_natDegree_le_of_le + - Polynomial.leadingCoeff_comp + - Polynomial.natDegree_sub + - Polynomial.natDegree_comp + - Polynomial.natDegree_map_eq_iff + - Polynomial.natDegree_mul_C + - Polynomial.degree_mul_C + - Polynomial.degree_mul_leadingCoeff_self_inv + - Polynomial.degree_map_eq_iff + - Polynomial.natDegree_mul_C_le + - Polynomial.natDegree_iterate_comp + - Polynomial.natDegree_add_le_iff_right + - Polynomial.degree_sum_eq_of_disjoint + - Polynomial.natDegree_C_mul + - Polynomial.natDegree_sub_le_iff_left + - Polynomial.coeff_mul_of_natDegree_le + - Polynomial.coeff_add_eq_right_of_lt + - Polynomial.degree_C_mul + - Polynomial.natDegree_C_mul_eq_of_mul_ne_zero + - Polynomial.natDegree_sub_le_iff_right + - Polynomial.natDegree_sum_eq_of_disjoint + - Polynomial.coeff_pow_of_natDegree_le + - Polynomial.natDegree_mul_C_eq_of_mul_ne_zero + - Polynomial.natDegree_add_le_iff_left + - Polynomial.natDegree_pos_of_nextCoeff_ne_zero + - Polynomial.irreducible_mul_leadingCoeff_inv + - Polynomial.degree_pos_of_root + - Polynomial.natDegree_bit1 + - Polynomial.natDegree_mul_C_eq_of_mul_eq_one + - Polynomial.natDegree_C_mul_eq_of_mul_eq_one + - Polynomial.coe_lt_degree + - Polynomial.degree_add_degree_leadingCoeff_inv + - Polynomial.coeff_sub_eq_neg_right_of_lt + - Polynomial.natDegree_le_iff_coeff_eq_zero + - path: .lake/packages/std/Std/Classes/SatisfiesM.lean + theorems: + - SatisfiesM.map + - SatisfiesM.pure + - SatisfiesM_Option_eq + - SatisfiesM_StateRefT_eq + - SatisfiesM_ExceptT_eq + - SatisfiesM.of_true + - SatisfiesM.imp + - SatisfiesM.bind + - SatisfiesM_StateT_eq + - SatisfiesM_ReaderT_eq + - SatisfiesM_Except_eq + - SatisfiesM.seq + - path: Mathlib/Analysis/Analytic/CPolynomial.lean + theorems: + - FormalMultilinearSeries.sum_of_finite + - HasFiniteFPowerSeriesOnBall.eq_const_of_bound_one + - HasFiniteFPowerSeriesOnBall.add + - FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_of_finite + - isOpen_cPolynomialAt + - HasFiniteFPowerSeriesOnBall.cPolynomialAt_of_mem + - CPolynomialOn.continuous + - FormalMultilinearSeries.hasSum_of_finite + - FormalMultilinearSeries.changeOrigin_finite_of_finite + - HasFiniteFPowerSeriesOnBall.changeOrigin + - FormalMultilinearSeries.changeOriginSeries_sum_eq_partialSum_of_finite + - FormalMultilinearSeries.continuousOn_of_finite + - HasFiniteFPowerSeriesOnBall.mk' + - FormalMultilinearSeries.changeOriginSeries_finite_of_finite + - FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_changeOrigin + - CPolynomialAt.sub + - HasFiniteFPowerSeriesOnBall.sub + - ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall + - FormalMultilinearSeries.changeOriginSeriesTerm_bound + - FormalMultilinearSeries.changeOrigin_eval_of_finite + - HasFiniteFPowerSeriesOnBall.eq_partialSum' + - HasFiniteFPowerSeriesOnBall.eq_zero_of_bound_zero + - HasFiniteFPowerSeriesAt.add + - HasFiniteFPowerSeriesOnBall.neg + - HasFiniteFPowerSeriesOnBall.eq_partialSum + - HasFiniteFPowerSeriesAt.sub + - HasFiniteFPowerSeriesOnBall.bound_zero_of_eq_zero + - ContinuousLinearMap.comp_cPolynomialOn + - path: Mathlib/MeasureTheory/Measure/Regular.lean + theorems: + - MeasureTheory.Measure.Regular.smul + - MeasureTheory.Measure.InnerRegular.exists_compact_not_null + - MeasurableSet.exists_isClosed_diff_lt + - MeasureTheory.Measure.Regular.map + - MeasureTheory.Measure.InnerRegularWRT.restrict + - IsCompact.exists_isOpen_lt_of_lt + - MeasureTheory.Measure.InnerRegularWRT.map + - MeasureTheory.Measure.InnerRegularWRT.restrict_of_measure_ne_top + - MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add + - MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen + - MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact + - MeasureTheory.Measure.OuterRegular.of_restrict + - MeasureTheory.Measure.InnerRegular.innerRegularWRT_isClosed_isOpen + - MeasureTheory.Measure.InnerRegularWRT.map' + - MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace + - MeasureTheory.Measure.InnerRegularWRT.isCompact_isClosed + - MeasureTheory.Measure.InnerRegular.map_iff + - MeasureTheory.Measure.Regular.map_iff + - MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top + - MeasureTheory.Measure.InnerRegularWRT.of_restrict + - MeasureTheory.Measure.OuterRegular.smul + - MeasureTheory.Measure.InnerRegularWRT.smul + - MeasureTheory.Measure.OuterRegular.map + - MeasureTheory.Measure.InnerRegularCompactLTTop.map_of_continuous + - MeasurableSet.exists_isCompact_diff_lt + - MeasureTheory.Measure.Regular.exists_compact_not_null + - Set.exists_isOpen_le_add + - MeasureTheory.Measure.WeaklyRegular.smul + - MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite + - MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup + - Set.measure_eq_iInf_isOpen + - IsCompact.measure_eq_iInf_isOpen + - MeasureTheory.Measure.InnerRegularWRT.trans + - Set.exists_isOpen_lt_of_lt + - MeasureTheory.Measure.Regular.restrict_of_measure_ne_top + - MeasureTheory.Measure.InnerRegularWRT.of_sigmaFinite + - MeasurableSet.exists_isOpen_diff_lt + - path: Mathlib/NumberTheory/BernoulliPolynomials.lean + theorems: + - Polynomial.bernoulli_eval_zero + - Polynomial.bernoulli_succ_eval + - Polynomial.derivative_bernoulli_add_one + - Polynomial.bernoulli_generating_function + - Polynomial.bernoulli_eval_one_add + - Polynomial.bernoulli_eq_sub_sum + - Polynomial.derivative_bernoulli + - Polynomial.sum_bernoulli + - Polynomial.bernoulli_eval_one + - Polynomial.bernoulli_def + - Polynomial.bernoulli_zero + - Polynomial.sum_range_pow_eq_bernoulli_sub + - path: Mathlib/LinearAlgebra/Matrix/SchurComplement.lean + theorems: + - "Matrix.isUnit_fromBlocks_zero\u2081\u2082" + - "Matrix.PosSemidef.fromBlocks\u2082\u2082" + - Matrix.det_mul_add_one_comm + - "Matrix.PosSemidef.fromBlocks\u2081\u2081" + - "Matrix.schur_complement_eq\u2082\u2082" + - "Matrix.isUnit_fromBlocks_iff_of_invertible\u2082\u2082" + - "Matrix.fromBlocks_eq_of_invertible\u2082\u2082" + - Matrix.det_one_add_mul_comm + - "Matrix.schur_complement_eq\u2081\u2081" + - "Matrix.inv_fromBlocks_zero\u2082\u2081_of_isUnit_iff" + - "Matrix.inv_fromBlocks_zero\u2081\u2082_of_isUnit_iff" + - Matrix.det_add_mul + - Matrix.det_one_add_col_mul_row + - "Matrix.invOf_fromBlocks_zero\u2081\u2082_eq" + - Matrix.det_add_col_mul_row + - "Matrix.det_fromBlocks\u2082\u2082" + - "Matrix.IsHermitian.fromBlocks\u2081\u2081" + - "Matrix.invOf_fromBlocks_zero\u2082\u2081_eq" + - "Matrix.det_fromBlocks\u2081\u2081" + - "Matrix.det_fromBlocks_one\u2082\u2082" + - "Matrix.fromBlocks_eq_of_invertible\u2081\u2081" + - "Matrix.invOf_fromBlocks\u2081\u2081_eq" + - "Matrix.det_fromBlocks_one\u2081\u2081" + - "Matrix.isUnit_fromBlocks_zero\u2082\u2081" + - "Matrix.IsHermitian.fromBlocks\u2082\u2082" + - "Matrix.isUnit_fromBlocks_iff_of_invertible\u2081\u2081" + - "Matrix.invOf_fromBlocks\u2082\u2082_eq" + - path: Mathlib/LinearAlgebra/JordanChevalley.lean + theorems: + - Module.End.exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow + - Module.End.exists_isNilpotent_isSemisimple + - path: Mathlib/Analysis/BoxIntegral/Partition/Measure.lean + theorems: + - BoxIntegral.Box.measurableSet_coe + - BoxIntegral.Box.volume_apply' + - BoxIntegral.Box.coe_ae_eq_Icc + - BoxIntegral.Prepartition.measure_iUnion_toReal + - BoxIntegral.Box.volume_apply + - BoxIntegral.BoxAdditiveMap.volume_apply + - path: Mathlib/Data/DFinsupp/Basic.lean + theorems: + - DFinsupp.support_erase + - DFinsupp.filter_apply_pos + - DFinsupp.update_eq_single_add_erase + - DFinsupp.filter_pos_add_filter_neg + - DFinsupp.support_subset_iff + - DFinsupp.support_sum + - DFinsupp.comapDomain_zero + - DFinsupp.filter_apply_neg + - DFinsupp.erase_single_same + - DFinsupp.erase_eq_sub_single + - DFinsupp.finite_support + - DFinsupp.mapRange_comp + - DFinsupp.prod_mapRange_index + - DFinsupp.support_filter + - DFinsupp.prod_eq_prod_fintype + - DFinsupp.sigmaUncurry_single + - DFinsupp.extendWith_single_zero + - DFinsupp.mapRange_def + - DFinsupp.single_eq_of_sigma_eq + - DFinsupp.induction + - DFinsupp.mk_injective + - DFinsupp.comapDomain'_add + - DFinsupp.liftAddHom_apply_single + - DFinsupp.single_eq_single_iff + - AddSubmonoid.mem_iSup_iff_exists_dfinsupp' + - DFinsupp.support_mk'_subset + - DFinsupp.sigmaCurry_add + - DFinsupp.eq_mk_support + - DFinsupp.coe_piecewise + - DFinsupp.filter_single_neg + - DFinsupp.support_zipWith + - AddSubmonoid.bsupr_eq_mrange_dfinsupp_sumAddHom + - DFinsupp.mem_support_toFun + - DFinsupp.support_eq_empty + - DFinsupp.comapDomain'_single + - DFinsupp.mapRange_id + - DFinsupp.erase_same + - DFinsupp.comp_liftAddHom + - DFinsupp.comapDomain_add + - DFinsupp.update_eq_sub_add_single + - DFinsupp.sum_single + - DFinsupp.mapRange_add + - DFinsupp.erase_single + - DFinsupp.support_single_ne_zero + - dfinsupp_sumAddHom_mem + - DFinsupp.filter_ne_eq_erase' + - DFinsupp.mapRange.addMonoidHom_comp + - DFinsupp.mk_sub + - DFinsupp.prod_add_index + - DFinsupp.piecewise_single_erase + - DFinsupp.single_add_erase + - DFinsupp.support_mapRange + - DFinsupp.mk_add + - DFinsupp.erase_add + - DFinsupp.single_smul + - DFinsupp.sum_sub_index + - DFinsupp.prod_subtypeDomain_index + - DFinsupp.filter_ne_eq_erase + - DFinsupp.mk_neg + - DFinsupp.sigmaCurry_single + - DFinsupp.support_update + - DFinsupp.subtypeDomain_def + - DFinsupp.update_eq_erase + - DFinsupp.mapRange.addEquiv_trans + - DFinsupp.filter_single_pos + - DFinsupp.liftAddHom_comp_single + - DFinsupp.mk_smul + - DFinsupp.mk_zero + - DFinsupp.comapDomain_single + - DFinsupp.equivFunOnFintype_symm_single + - DFinsupp.single_eq_same + - DFinsupp.addHom_ext + - DFinsupp.sigmaCurry_smul + - DFinsupp.support_subtypeDomain + - DFinsupp.sumAddHom_single + - DFinsupp.erase_single_ne + - DFinsupp.zipWith_def + - AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom + - DFinsupp.comapDomain'_smul + - DFinsupp.erase_ne + - "DFinsupp.induction\u2082" + - DFinsupp.mapRange_single + - DFinsupp.prod_finset_sum_index + - DFinsupp.filter_single + - DFinsupp.mapRange_zero + - DFinsupp.single_apply + - DFinsupp.support_update_ne_zero + - DFinsupp.filter_zero + - DFinsupp.sumAddHom_comm + - DFinsupp.equivFunOnFintype_single + - DFinsupp.single_eq_zero + - DFinsupp.prod_single_index + - DFinsupp.filter_add + - DFinsupp.comapDomain_smul + - DFinsupp.comapDomain'_zero + - DFinsupp.sumAddHom_apply + - DFinsupp.erase_add_single + - DFinsupp.update_self + - DFinsupp.filter_smul + - DFinsupp.zipWith_single_single + - DFinsupp.update_eq_erase_add_single + - path: Mathlib/Data/Polynomial/EraseLead.lean + theorems: + - Polynomial.map_natDegree_eq_sub + - Polynomial.eraseLead_add_C_mul_X_pow + - Polynomial.eraseLead_X_pow + - Polynomial.lt_natDegree_of_mem_eraseLead_support + - Polynomial.eraseLead_add_of_natDegree_lt_left + - Polynomial.eraseLead_coeff_of_ne + - Polynomial.natDegree_pos_of_eraseLead_ne_zero + - Polynomial.nextCoeff_eq_zero_of_eraseLead_eq_zero + - Polynomial.card_support_eq_one + - Polynomial.card_support_eraseLead_add_one + - Polynomial.card_support_eq_two + - Polynomial.eraseLead_monomial + - Polynomial.card_support_eq' + - Polynomial.natDegree_eraseLead_add_one + - Polynomial.eraseLead_support_card_lt + - Polynomial.eraseLead_C_mul_X + - Polynomial.self_sub_C_mul_X_pow + - Polynomial.induction_with_natDegree_le + - Polynomial.eraseLead_coeff_natDegree + - Polynomial.two_le_natDegree_of_nextCoeff_eraseLead + - Polynomial.mono_map_natDegree_eq + - Polynomial.eraseLead_natDegree_lt_or_eraseLead_eq_zero + - Polynomial.card_support_eraseLead + - Polynomial.map_natDegree_eq_natDegree + - Polynomial.eraseLead_coeff + - Polynomial.card_support_eraseLead' + - Polynomial.eraseLead_add_of_natDegree_lt_right + - Polynomial.card_support_le_one_of_eraseLead_eq_zero + - Polynomial.eraseLead_natDegree_le + - Polynomial.leadingCoeff_eraseLead_eq_nextCoeff + - Polynomial.natDegree_eraseLead_le_of_nextCoeff_eq_zero + - Polynomial.natDegree_eraseLead + - Polynomial.eraseLead_support + - Polynomial.card_support_eq + - Polynomial.eraseLead_zero + - Polynomial.card_support_eq_three + - Polynomial.eraseLead_ne_zero + - path: Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean + theorems: + - HasSum.of_nat_of_neg_add_one + - summable_int_iff_summable_nat_and_neg + - HasSum.nat_add_neg + - Summable.tendsto_atTop_zero + - HasSum.nat_add_neg_add_one + - tendsto_sum_nat_add + - HasSum.of_nat_of_neg + - rel_sup_add + - HasSum.zero_add + - "tsum_iSup_decode\u2082" + - HasSum.sum_range_add + - cauchySeq_finset_iff_nat_tsum_vanishing + - Summable.hasSum_iff_tendsto_nat + - summable_int_iff_summable_nat_and_neg_add_one + - hasSum_nat_add_iff + - tsum_eq_zero_add' + - summable_iff_nat_tsum_vanishing + - rel_iSup_tsum + - rel_iSup_sum + - HasSum.even_add_odd + - hasSum_nat_add_iff' + - path: Mathlib/Data/Nat/Bits.lean + theorems: + - Nat.bit0_mod_two + - Nat.bitCasesOn_bit + - Nat.bits_append_bit + - Nat.bit_cases_on_injective + - Nat.bit_eq_zero_iff + - Nat.bit1_mod_two + - Nat.one_bits + - Nat.boddDiv2_eq + - Nat.zero_bits + - Nat.pos_of_bit0_pos + - Nat.bit_ne_zero + - Nat.bit0_eq_zero + - Nat.one_eq_bit1 + - Nat.div2_bits_eq_tail + - Nat.bit1_eq_bit1 + - Nat.bit1_eq_one + - Nat.bodd_eq_bits_head + - Nat.binaryRec_eq' + - Nat.bit0_eq_bit0 + - path: Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean + theorems: + - CategoryTheory.Functor.additive_of_preservesBinaryBiproducts + - CategoryTheory.Functor.additive_of_comp_faithful + - CategoryTheory.Functor.additive_of_iso + - CategoryTheory.Functor.additive_of_full_essSurj_comp + - path: Mathlib/Topology/TietzeExtension.lean + theorems: + - ContinuousMap.exists_extension' + - BoundedContinuousFunction.exists_forall_mem_restrict_eq_of_closed + - BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding + - ContinuousMap.exists_extension_forall_mem_of_closedEmbedding + - TietzeExtension.of_retract + - BoundedContinuousFunction.exists_extension_forall_mem_Icc_of_closedEmbedding + - BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_closedEmbedding + - BoundedContinuousFunction.exists_extension_forall_mem_of_closedEmbedding + - ContinuousMap.exists_extension_forall_mem + - BoundedContinuousFunction.tietze_extension_step + - TietzeExtension.of_homeo + - ContinuousMap.exists_forall_mem_restrict_eq + - ContinuousMap.exists_extension + - BoundedContinuousFunction.exists_extension_norm_eq_of_closedEmbedding' + - path: Mathlib/LinearAlgebra/RootSystem/Basic.lean + theorems: + - RootSystem.ext + - RootPairing.coroot_root_eq_pairing + - RootPairing.coroot_eq_coreflection_of_root_eq_of_span_eq_top' + - RootPairing.coxeterWeight_swap + - RootPairing.reflection_invOn_self + - RootPairing.eq_of_pairing_pairing_eq_two + - RootPairing.ext + - RootPairing.coroot_root_two + - RootPairing.reflection_dualMap_eq_coreflection + - RootPairing.ne_zero + - RootPairing.injOn_dualMap_subtype_span_root_coroot + - RootPairing.IsOrthogonal.symm + - RootPairing.ne_zero' + - RootPairing.IsOrthogonal_comm + - RootPairing.root_ne + - path: Mathlib/Order/Heyting/Boundary.lean + theorems: + - Coheyting.boundary_sup_le + - Coheyting.boundary_idem + - Coheyting.hnot_boundary + - Coheyting.hnot_eq_top_iff_exists_boundary + - Coheyting.boundary_le_boundary_sup_sup_boundary_inf_left + - Coheyting.boundary_le_boundary_sup_sup_boundary_inf_right + - Coheyting.boundary_inf + - Coheyting.boundary_hnot_hnot + - Coheyting.hnot_hnot_sup_boundary + - Coheyting.boundary_top + - path: Mathlib/GroupTheory/Order/Min.lean + theorems: + - Monoid.le_minOrder_iff_forall_subgroup + - Monoid.minOrder_eq_top + - Monoid.le_minOrder + - ZMod.minOrder_of_prime + - ZMod.minOrder + - path: Mathlib/Data/Fintype/Powerset.lean + theorems: + - Fintype.card_finset_len + - Finset.powerset_eq_univ + - Finset.univ_filter_card_eq + - Finset.filter_subset_univ + - Finset.powerset_univ + - path: Mathlib/Topology/Semicontinuous.lean + theorems: + - continuousOn_iff_lower_upperSemicontinuousOn + - lowerSemicontinuousAt_sum + - continuousWithinAt_iff_lower_upperSemicontinuousWithinAt + - continuousAt_iff_lower_upperSemicontinuousAt + - ContinuousAt.comp_lowerSemicontinuousAt + - upperSemicontinuousWithinAt_univ_iff + - upperSemicontinuousOn_univ_iff + - lowerSemicontinuous_iSup + - lowerSemicontinuousWithinAt_ciSup + - IsOpen.lowerSemicontinuous_indicator + - LowerSemicontinuousAt.add' + - lowerSemicontinuousAt_ciSup + - lowerSemicontinuousWithinAt_univ_iff + - LowerSemicontinuousWithinAt.add' + - lowerSemicontinuousOn_iff_le_liminf + - lowerSemicontinuous_iff_isClosed_epigraph + - lowerSemicontinuousOn_iSup + - lowerSemicontinuous_iff_isClosed_preimage + - lowerSemicontinuousAt_iSup + - upperSemicontinuous_iff_isClosed_preimage + - LowerSemicontinuousAt.comp_continuousAt_of_eq + - lowerSemicontinuousWithinAt_sum + - ContinuousAt.comp_lowerSemicontinuousWithinAt + - lowerSemicontinuousWithinAt_iff_le_liminf + - IsClosed.lowerSemicontinuous_indicator + - continuous_iff_lower_upperSemicontinuous + - lowerSemicontinuousAt_iff_le_liminf + - lowerSemicontinuousWithinAt_iSup + - lowerSemicontinuous_iff_le_liminf + - UpperSemicontinuousAt.comp_continuousAt_of_eq + - lowerSemicontinuousOn_univ_iff + - UpperSemicontinuousAt.add' + - path: Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean + theorems: + - Asymptotics.isLittleO_sum_range_of_tendsto_zero + - Asymptotics.IsLittleO.sum_range + - tendsto_pow_div_pow_atTop_zero + - Filter.Tendsto.cesaro_smul + - Asymptotics.IsBigO.trans_tendsto_norm_atTop + - pow_div_pow_eventuallyEq_atBot + - Filter.IsBoundedUnder.isLittleO_sub_self_inv + - tendsto_pow_div_pow_atTop_atTop + - path: Mathlib/Data/Polynomial/Coeff.lean + theorems: + - Polynomial.C_mul' + - Polynomial.coeff_C_mul_X_pow + - Polynomial.coeff_bit0 + - Polynomial.nat_cast_coeff_zero + - Polynomial.coeff_smul + - Polynomial.coeff_X_add_C_pow + - Polynomial.coeff_bit1_mul + - Polynomial.coeff_X_pow_mul + - Polynomial.card_support_trinomial + - Polynomial.coeff_X_add_one_pow + - Polynomial.C_dvd_iff_dvd_coeff + - Polynomial.coeff_one_add_X_pow + - Polynomial.coeff_C_mul_X + - Polynomial.coeff_mul_X + - Polynomial.int_cast_inj + - Polynomial.coeff_X_mul_zero + - Polynomial.int_cast_coeff_zero + - Polynomial.coeff_mul_X_pow' + - Polynomial.coeff_sum + - Polynomial.card_support_mul_le + - Polynomial.support_smul + - Polynomial.coeff_bit0_mul + - Polynomial.coeff_mul_monomial + - Polynomial.coeff_X_pow_mul' + - Polynomial.smul_eq_C_mul + - Polynomial.isRegular_X_pow + - Polynomial.card_support_binomial + - Polynomial.coeff_monomial_mul + - Polynomial.coeff_mul_X_pow + - Polynomial.coeff_X_pow_self + - Polynomial.coeff_X_pow + - Polynomial.update_eq_add_sub_coeff + - Polynomial.coeff_C_mul + - Polynomial.support_trinomial + - Polynomial.nat_cast_inj + - Polynomial.mul_coeff_zero + - Polynomial.coeff_mul_X_zero + - Polynomial.coeff_mul + - Polynomial.coeff_add + - Polynomial.support_binomial + - Polynomial.coeff_X_mul + - path: Mathlib/NumberTheory/Cyclotomic/Rat.lean + theorems: + - IsCyclotomicExtension.Rat.discr_prime_pow_eq_unit_mul_pow' + - IsPrimitiveRoot.subOneIntegralPowerBasis'_gen + - IsPrimitiveRoot.integralPowerBasis'_gen + - IsPrimitiveRoot.zeta_sub_one_prime' + - IsCyclotomicExtension.Rat.absdiscr_prime + - IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime_pow + - IsCyclotomicExtension.Rat.discr_prime_pow_ne_two' + - IsCyclotomicExtension.Rat.discr_odd_prime' + - IsPrimitiveRoot.integralPowerBasis_dim + - IsPrimitiveRoot.integralPowerBasis_gen + - IsPrimitiveRoot.subOneIntegralPowerBasis_gen_prime + - IsCyclotomicExtension.Rat.absdiscr_prime_pow_succ + - IsCyclotomicExtension.Rat.discr_prime_pow' + - IsPrimitiveRoot.subOneIntegralPowerBasis'_gen_prime + - IsCyclotomicExtension.Rat.cyclotomicRing_isIntegralClosure_of_prime + - IsPrimitiveRoot.zeta_sub_one_prime_of_ne_two + - IsPrimitiveRoot.power_basis_int'_dim + - IsCyclotomicExtension.Rat.isIntegralClosure_adjoin_singleton_of_prime + - IsPrimitiveRoot.subOneIntegralPowerBasis_gen + - IsPrimitiveRoot.zeta_sub_one_prime + - IsPrimitiveRoot.zeta_sub_one_prime_of_two_pow + - IsCyclotomicExtension.Rat.absdiscr_prime_pow + - IsCyclotomicExtension.Rat.cyclotomicRing_isIntegralClosure_of_prime_pow + - path: Mathlib/Combinatorics/Schnirelmann.lean + theorems: + - le_schnirelmannDensity_iff + - schnirelmannDensity_eq_one_iff + - schnirelmannDensity_setOf_mod_eq_one + - schnirelmannDensity_congr + - schnirelmannDensity_finset + - schnirelmannDensity_setOf_even + - schnirelmannDensity_lt_iff + - schnirelmannDensity_setOf_modeq_one + - schnirelmannDensity_eq_one_iff_of_zero_mem + - schnirelmannDensity_mul_le_card_filter + - schnirelmannDensity_insert_zero + - schnirelmannDensity_finite + - schnirelmannDensity_univ + - schnirelmannDensity_eq_zero_of_one_not_mem + - schnirelmannDensity_congr' + - schnirelmannDensity_setOf_Odd + - schnirelmannDensity_empty + - schnirelmannDensity_le_one + - schnirelmannDensity_nonneg + - schnirelmannDensity_diff_singleton_zero + - schnirelmannDensity_setOf_prime + - schnirelmannDensity_le_div + - schnirelmannDensity_le_of_subset + - schnirelmannDensity_le_iff_forall + - schnirelmannDensity_le_of_not_mem + - path: Mathlib/Topology/VectorBundle/Basic.lean + theorems: + - Trivialization.mk_coordChangeL + - VectorPrebundle.mk_coordChange + - VectorBundleCore.localTriv_continuousLinearMapAt + - Pretrivialization.coe_linearMapAt + - Trivialization.zeroSection + - "Pretrivialization.symm\u2097_linearMapAt" + - Trivialization.linearMapAt_apply + - "Pretrivialization.linearMapAt_symm\u2097" + - Pretrivialization.linearMapAt_apply + - VectorBundleCore.localTriv_symmL + - Trivialization.coordChangeL_apply' + - Pretrivialization.coe_linearMapAt_of_mem + - VectorBundleCore.localTriv_symm_apply + - VectorPrebundle.toVectorBundle + - Trivialization.coe_linearMapAt_of_mem + - ContinuousLinearMap.inCoordinates_eq + - VectorBundleCore.mem_source_at + - Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm + - Trivialization.comp_continuousLinearEquivAt_eq_coord_change + - VectorBundleCore.inCoordinates_eq + - VectorBundleCore.localTriv_coordChange_eq + - Trivialization.apply_eq_prod_continuousLinearEquivAt + - Trivialization.symm_coordChangeL + - Trivialization.apply_symm_apply_eq_coordChangeL + - path: Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean + theorems: + - jacobiSym.sq_one + - ZMod.isSquare_of_jacobiSym_eq_one + - jacobiSym.eq_neg_one_at_prime_divisor_of_eq_neg_one + - jacobiSym.value_at + - jacobiSym.mod_left + - jacobiSym.even_odd + - fastLegendreSym.eq + - jacobiSym.neg + - jacobiSym.quadratic_reciprocity' + - jacobiSym.div_four_left + - jacobiSym.quadratic_reciprocity_if + - jacobiSym.pow_right + - jacobiSym.quadratic_reciprocity_three_mod_four + - jacobiSym.trichotomy + - jacobiSym.mod_right + - jacobiSym.mul_left + - jacobiSym.list_prod_right + - jacobiSym.list_prod_left + - jacobiSym.ne_zero + - jacobiSym.zero_right + - ZMod.nonsquare_iff_jacobiSym_eq_neg_one + - jacobiSym.quadratic_reciprocity + - jacobiSym.mod_left' + - jacobiSym.prime_dvd_of_eq_neg_one + - jacobiSym.mul_right' + - qrSign.mul_left + - fastJacobiSymAux.eq_jacobiSym + - qrSign.sq_eq_one + - fastJacobiSym.eq + - jacobiSym.quadratic_reciprocity_one_mod_four + - jacobiSym.one_left + - qrSign.symm + - jacobiSym.eq_zero_iff + - jacobiSym.legendreSym.to_jacobiSym + - jacobiSym.sq_one' + - jacobiSym.one_right + - jacobiSym.mod_right' + - qrSign.neg_one_pow + - qrSign.eq_iff_eq + - ZMod.nonsquare_of_jacobiSym_eq_neg_one + - jacobiSym.pow_left + - jacobiSym.zero_left + - path: Mathlib/Topology/ContinuousFunction/ZeroAtInfty.lean + theorems: + - ZeroAtInftyContinuousMap.isClosed_range_toBCF + - ZeroAtInftyContinuousMap.bounded + - ZeroAtInftyContinuousMap.isometry_toBCF + - ZeroAtInftyContinuousMap.zeroAtInftyContinuousMapClass.ofCompact + - ZeroAtInftyContinuousMap.tendsto_iff_tendstoUniformly + - ZeroAtInftyContinuousMap.coe_zsmulRec + - ZeroAtInftyContinuousMap.toBCF_injective + - ZeroAtInftyContinuousMap.coe_nsmulRec + - path: Mathlib/CategoryTheory/Sites/Surjective.lean + theorems: + - CategoryTheory.toSheafify_isLocallySurjective + - CategoryTheory.isLocallySurjective_of_surjective + - CategoryTheory.imageSieve_app + - CategoryTheory.isLocallySurjective_iff_whisker_forget + - CategoryTheory.IsLocallySurjective.comp + - CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top + - CategoryTheory.isLocallySurjective_of_iso + - CategoryTheory.isLocallySurjective_iff_isIso + - CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' + - path: Mathlib/CategoryTheory/Limits/EpiMono.lean + theorems: + - CategoryTheory.mono_iff_fst_eq_snd + - CategoryTheory.mono_iff_isPullback + - CategoryTheory.epi_iff_isPushout + - CategoryTheory.mono_iff_isIso_fst + - CategoryTheory.epi_iff_inl_eq_inr + - CategoryTheory.epi_iff_isIso_inl + - path: Mathlib/Data/W/Cardinal.lean + theorems: + - WType.cardinal_mk_eq_sum' + - WType.cardinal_mk_le_max_aleph0_of_finite + - WType.cardinal_mk_le_max_aleph0_of_finite' + - WType.cardinal_mk_eq_sum + - WType.cardinal_mk_le_of_le' + - WType.cardinal_mk_le_of_le + - path: Mathlib/Data/Fin/OrderHom.lean + theorems: + - Fin.pred_succAbove_pred + - Fin.one_succAbove_zero + - Fin.succAbove_ne_zero_zero + - Fin.predAbove_right_zero + - Fin.predAbove_last_castSucc + - Fin.lt_succAbove_iff_le_castSucc + - Fin.predAbove_zero_succ + - Fin.rev_predAbove + - Fin.succAbove_last + - Fin.succAbove_zero_apply + - Fin.predAbove_last_of_ne_last + - Fin.castPred_succAbove + - Fin.exists_succAbove_eq + - Fin.castPred_succAbove_castPred + - Fin.succAbove_last_apply + - Fin.lt_succAbove_iff_lt_castSucc + - Fin.succAbove_left_injective + - Fin.succAbove_eq_last_iff + - Fin.predAbove_zero + - Fin.succAbove_ne + - Fin.castSucc_succAbove_castSucc + - Fin.succAbove_pred_of_lt + - Fin.succ_succAbove_one + - Fin.succAbove_ne_last_last + - Fin.predAbove_succ_of_le + - Fin.succ_succAbove_succ + - Fin.succ_predAbove_succ + - Fin.predAbove_succ_of_lt + - Fin.pred_succAbove + - Fin.succAbove_lt_iff_succ_le + - Fin.predAbove_right_last + - Fin.one_succAbove_one + - Fin.succAbove_castPred_of_lt + - Fin.predAbove_pred_of_le + - Fin.range_succ + - Fin.castSucc_predAbove_castSucc + - Fin.predAbove_castSucc_of_le + - Fin.exists_succAbove_eq_iff + - Fin.predAbove_castPred_of_le + - Fin.succAbove_rev_left + - Fin.predAbove_right_monotone + - Fin.succ_predAbove_zero + - Fin.predAbove_last_apply + - Fin.predAbove_succAbove + - Fin.succAbove_lt_iff_castSucc_lt + - Fin.zero_succAbove + - Fin.succAbove_rev_right + - Fin.predAbove_left_monotone + - Fin.predAbove_castPred_of_lt + - Fin.predAbove_zero_of_ne_zero + - Fin.predAbove_castSucc_of_lt + - Fin.succAbove_eq_zero_iff + - Fin.succ_succAbove_zero + - Fin.predAbove_pred_of_lt + - Fin.succAbove_predAbove + - Fin.succAbove_pos + - Fin.rev_succAbove + - Fin.predAbove_rev_right + - Fin.one_succAbove_succ + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean + theorems: + - SzemerediRegularity.a_add_one_le_four_pow_parts_card + - SzemerediRegularity.le_stepBound + - SzemerediRegularity.Positivity.eps_pos + - SzemerediRegularity.hundred_le_m + - "SzemerediRegularity.hundred_div_\u03B5_pow_five_le_m" + - "SzemerediRegularity.card_aux\u2081" + - SzemerediRegularity.stepBound_mono + - SzemerediRegularity.Positivity.m_pos + - SzemerediRegularity.hundred_lt_pow_initialBound_mul + - "SzemerediRegularity.card_aux\u2082" + - SzemerediRegularity.stepBound_pos_iff + - SzemerediRegularity.eps_pow_five_pos + - SzemerediRegularity.m_pos + - SzemerediRegularity.initialBound_le_bound + - SzemerediRegularity.coe_m_add_one_pos + - SzemerediRegularity.add_div_le_sum_sq_div_card + - SzemerediRegularity.pow_mul_m_le_card_part + - SzemerediRegularity.coe_stepBound + - path: Mathlib/LinearAlgebra/SymplecticGroup.lean + theorems: + - Matrix.J_squared + - SymplecticGroup.transpose_mem + - SymplecticGroup.mem_iff + - SymplecticGroup.coe_inv' + - SymplecticGroup.transpose_mem_iff + - Matrix.J_transpose + - SymplecticGroup.inv_left_mul_aux + - Matrix.J_inv + - SymplecticGroup.symplectic_det + - SymplecticGroup.mem_iff' + - SymplecticGroup.inv_eq_symplectic_inv + - SymplecticGroup.neg_mem + - Matrix.J_det_mul_J_det + - SymplecticGroup.J_mem + - path: Mathlib/MeasureTheory/Integral/IntegrableOn.lean + theorems: + - MeasureTheory.IntegrableOn.restrict + - integrableOn_Ici_iff_integrableOn_Ioi + - integrableOn_Ici_iff_integrableOn_Ioi' + - MeasureTheory.integrable_indicatorConstLp + - MeasureTheory.IntegrableAtFilter.add + - MeasureTheory.integrableOn_finite_biUnion + - MeasureTheory.IntegrableOn.of_ae_diff_eq_zero + - MeasureTheory.IntegrableAtFilter.smul + - MeasureTheory.integrableOn_map_equiv + - MeasureTheory.integrableOn_const + - MeasurableEmbedding.integrableAtFilter_iff_comap + - integrableOn_Icc_iff_integrableOn_Ico' + - Continuous.integrableAt_nhds + - MeasureTheory.IntegrableOn.restrict_toMeasurable + - MeasureTheory.IntegrableOn.add_measure + - stronglyMeasurableAt_bot + - MeasureTheory.integrableOn_finite_iUnion + - ContinuousOn.aestronglyMeasurable + - MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter + - MeasureTheory.integrableOn_Lp_of_measure_ne_top + - MeasureTheory.integrableOn_univ + - MeasureTheory.IntegrableOn.integrable_of_ae_not_mem_eq_zero + - integrableOn_Ico_iff_integrableOn_Ioo' + - MeasureTheory.integrable_indicator_iff + - MeasureTheory.IntegrableOn.inter_of_restrict + - MeasureTheory.integrableOn_empty + - MeasurableEmbedding.integrableOn_iff_comap + - MeasureTheory.integrableOn_singleton_iff + - MeasureTheory.IntegrableAtFilter.sub + - integrableOn_Icc_iff_integrableOn_Ioo + - integrableOn_Iic_iff_integrableOn_Iio' + - MeasureTheory.IntegrableAtFilter.neg + - MeasureTheory.Integrable.piecewise + - integrableOn_Icc_iff_integrableOn_Ioc' + - integrableOn_Ioc_iff_integrableOn_Ioo + - integrableOn_Icc_iff_integrableOn_Ico + - integrableOn_Ioc_iff_integrableOn_Ioo' + - MeasureTheory.integrable_add_of_disjoint + - integrableOn_Iic_iff_integrableOn_Iio + - ContinuousOn.aestronglyMeasurable_of_isCompact + - MeasurableEmbedding.integrableOn_map_iff + - MeasureTheory.IntegrableAtFilter.inf_ae_iff + - MeasureTheory.integrableAtFilter_top + - integrableOn_Icc_iff_integrableOn_Ioc + - integrableOn_Icc_iff_integrableOn_Ioo' + - ContinuousOn.aestronglyMeasurable_of_isSeparable + - MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter + - MeasureTheory.integrableOn_iff_integrable_of_support_subset + - path: Mathlib/Order/Filter/ENNReal.lean + theorems: + - ENNReal.limsup_const_mul + - ENNReal.limsup_const_mul_of_ne_top + - ENNReal.limsup_liminf_le_liminf_limsup + - ENNReal.limsup_mul_le + - path: Mathlib/Data/Finset/Sigma.lean + theorems: + - Finset.sigma_eq_biUnion + - Finset.not_mem_sigmaLift_of_ne_right + - Finset.mem_sigmaLift + - Finset.sigmaLift_nonempty + - Finset.sup_sigma + - Finset.sigma_eq_empty + - Finset.sigmaLift_eq_empty + - biSup_finsetSigma + - Finset.pairwiseDisjoint_map_sigmaMk + - Finset.not_mem_sigmaLift_of_ne_left + - Finset.mk_mem_sigmaLift + - Finset.sigmaLift_mono + - path: Mathlib/Data/MvPolynomial/Equiv.lean + theorems: + - MvPolynomial.finSuccEquiv_rename_finSuccEquiv + - MvPolynomial.coeff_eval_eq_eval_coeff + - MvPolynomial.eval_eq_eval_mv_eval' + - MvPolynomial.finSuccEquiv_X_succ + - MvPolynomial.finSuccEquiv_X_zero + - MvPolynomial.degree_finSuccEquiv + - MvPolynomial.finSuccEquiv_support + - MvPolynomial.support_coeff_finSuccEquiv + - MvPolynomial.totalDegree_coeff_finSuccEquiv_add_le + - MvPolynomial.mapEquiv_trans + - MvPolynomial.finSuccEquiv_support' + - MvPolynomial.support_finSuccEquiv_nonempty + - MvPolynomial.natDegree_finSuccEquiv + - MvPolynomial.degreeOf_coeff_finSuccEquiv + - MvPolynomial.mapAlgEquiv_trans + - MvPolynomial.finSuccEquiv_comp_C_eq_C + - MvPolynomial.finSuccEquiv_apply + - MvPolynomial.finSuccEquiv_eq + - MvPolynomial.finSuccEquiv_coeff_coeff + - path: Mathlib/Topology/Sheaves/SheafCondition/Sites.lean + theorems: + - TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology + - TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck + - TopCat.Sheaf.extend_hom_app + - TopCat.Presheaf.covering_presieve_eq_self + - TopCat.Sheaf.hom_ext + - TopCat.Opens.coverDense_iff_isBasis + - coverPreserving_opens_map + - IsOpenMap.coverPreserving + - TopCat.Presheaf.isSheaf_of_openEmbedding + - OpenEmbedding.compatiblePreserving + - OpenEmbedding.functor_isContinuous + - path: Mathlib/Data/Real/Archimedean.lean + theorems: + - Real.sInf_le_sSup + - Real.le_sSup_iff + - Real.isLUB_sSup + - Real.sInf_le_iff + - Real.sSup_nonneg + - Real.sInf_nonpos + - Real.iSup_of_isEmpty + - Real.ciSup_const_zero + - Real.sInf_empty + - Real.sSup_le + - Real.cauSeq_converges + - Real.is_glb_sInf + - Real.iInf_of_isEmpty + - Real.isCauSeq_iff_lift + - Real.iInf_Ioi_eq_iInf_rat_gt + - Real.ciInf_const_zero + - Real.sInf_nonneg + - Real.exists_isLUB + - Real.sSup_empty + - path: Mathlib/Algebra/Homology/HomotopyCategory/DegreewiseSplit.lean + theorems: + - "CochainComplex.mappingConeHomOfDegreewiseSplitIso_inv_comp_triangle_mor\u2083" + - CochainComplex.homOfDegreewiseSplit_f + - CochainComplex.mappingCone.cocycleOfDegreewiseSplit_triangleRotateShortComplexSplitting_v + - CochainComplex.shift_f_comp_mappingConeHomOfDegreewiseSplitIso_inv + - path: Mathlib/Analysis/SpecialFunctions/Polynomials.lean + theorems: + - Polynomial.isBigO_of_degree_le + - Polynomial.div_tendsto_zero_of_degree_lt + - Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg + - Polynomial.div_tendsto_atTop_of_degree_gt' + - Polynomial.tendsto_nhds_iff + - Polynomial.isEquivalent_atTop_lead + - Polynomial.abs_div_tendsto_atTop_of_degree_gt + - Polynomial.tendsto_atBot_iff_leadingCoeff_nonpos + - Polynomial.abs_isBoundedUnder_iff + - Polynomial.abs_tendsto_atTop + - Polynomial.isEquivalent_atTop_div + - Polynomial.div_tendsto_zero_iff_degree_lt + - Polynomial.div_tendsto_leadingCoeff_div_of_degree_eq + - Polynomial.div_tendsto_atBot_of_degree_gt' + - path: Mathlib/RingTheory/WittVector/Teichmuller.lean + theorems: + - WittVector.teichmuller_zero + - WittVector.ghostComponent_teichmullerFun + - "WittVector.teichmuller_mul_aux\u2082" + - "WittVector.teichmuller_mul_aux\u2081" + - path: Mathlib/Topology/DiscreteSubset.lean + theorems: + - tendsto_cofinite_cocompact_iff + - Continuous.discrete_of_tendsto_cofinite_cocompact + - isClosed_and_discrete_iff + - tendsto_cofinite_cocompact_of_discrete + - path: Mathlib/SetTheory/Game/Nim.lean + theorems: + - SetTheory.PGame.nim_def + - SetTheory.PGame.moveRight_nim_hEq + - SetTheory.PGame.grundyValue_eq_mex_right + - SetTheory.PGame.nim_fuzzy_zero_of_ne_zero + - SetTheory.PGame.nim_one_moveRight + - SetTheory.PGame.toLeftMovesNim_one_symm + - SetTheory.PGame.nim_add_fuzzy_zero_iff + - SetTheory.PGame.moveRight_nim + - SetTheory.PGame.nim_equiv_iff_eq + - SetTheory.PGame.equiv_nim_grundyValue + - SetTheory.PGame.nim_add_equiv_zero_iff + - SetTheory.PGame.grundyValue_neg + - SetTheory.PGame.leftMoves_nim + - SetTheory.PGame.grundyValue_eq_iff_equiv_nim + - SetTheory.PGame.moveLeft_nim_hEq + - SetTheory.PGame.nim_add_nim_equiv + - SetTheory.PGame.grundyValue_add + - SetTheory.PGame.nim_one_moveLeft + - SetTheory.PGame.grundyValue_iff_equiv_zero + - SetTheory.PGame.grundyValue_nim_add_nim + - SetTheory.PGame.rightMoves_nim + - SetTheory.PGame.toRightMovesNim_one_symm + - SetTheory.PGame.nim_birthday + - SetTheory.PGame.neg_nim + - SetTheory.PGame.grundyValue_eq_mex_left + - SetTheory.PGame.moveLeft_nim + - path: Mathlib/Data/Matrix/Basis.lean + theorems: + - Matrix.StdBasisMatrix.apply_of_row_ne + - Matrix.stdBasisMatrix_zero + - Matrix.matrix_eq_sum_std_basis + - Matrix.StdBasisMatrix.mul_right_apply_same + - Matrix.col_eq_zero_of_commute_stdBasisMatrix + - Matrix.row_eq_zero_of_commute_stdBasisMatrix + - Matrix.StdBasisMatrix.mul_left_apply_of_ne + - Matrix.StdBasisMatrix.apply_of_col_ne + - Matrix.StdBasisMatrix.mul_left_apply_same + - Matrix.smul_stdBasisMatrix + - Matrix.mem_range_scalar_iff_commute_stdBasisMatrix' + - Matrix.std_basis_eq_basis_mul_basis + - Matrix.StdBasisMatrix.mul_same + - Matrix.StdBasisMatrix.trace_zero + - Matrix.induction_on + - Matrix.stdBasisMatrix_add + - Matrix.StdBasisMatrix.diag_same + - Matrix.StdBasisMatrix.trace_eq + - Matrix.mem_range_scalar_of_commute_stdBasisMatrix + - Matrix.diag_eq_of_commute_stdBasisMatrix + - Matrix.StdBasisMatrix.apply_of_ne + - Matrix.induction_on' + - path: Mathlib/Probability/Kernel/CondDistrib.lean + theorems: + - "ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib\u2080" + - ProbabilityTheory.set_lintegral_condDistrib_of_measurableSet + - MeasureTheory.Integrable.norm_integral_condDistrib_map + - MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff + - ProbabilityTheory.condDistrib_ae_eq_of_measure_eq_compProd + - MeasureTheory.Integrable.condDistrib_ae_map + - MeasureTheory.Integrable.comp_snd_map_prod_mk + - ProbabilityTheory.integrable_toReal_condDistrib + - ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib' + - ProbabilityTheory.set_lintegral_preimage_condDistrib + - ProbabilityTheory.condDistrib_ae_eq_condexp + - MeasureTheory.AEStronglyMeasurable.comp_snd_map_prod_mk + - ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib + - MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map + - path: Mathlib/Data/Multiset/Functor.lean + theorems: + - Multiset.traverse_map + - Multiset.id_traverse + - Multiset.map_comp_coe + - Multiset.comp_traverse + - Multiset.map_traverse + - Multiset.naturality + - path: Mathlib/GroupTheory/Congruence.lean + theorems: + - Con.lift_range + - Con.pow + - Con.map_of_mul_left_rel_one + - Con.mapOfSurjective_eq_mapGen + - Con.lift_funext + - Con.ext + - Con.conGen_le + - Con.sSup_eq_conGen + - Con.inv + - Con.coe_sInf + - Con.conGen_of_con + - Con.coe_iInf + - Con.smul + - Con.zpow + - Con.sup_def + - Con.div + - Con.comap_eq + - Con.sInf_toSetoid + - Con.sSup_def + - Con.lift_unique + - Con.conGen_eq + - Con.sup_eq_conGen + - Con.lift_comp_mk' + - Con.induction_on_units + - Con.lift_apply_mk' + - path: Mathlib/Order/Heyting/Basic.lean + theorems: + - himp_inf_distrib + - sup_sdiff_cancel' + - top_himp + - himp_le_himp_himp_himp + - le_compl_comm + - Disjoint.sup_sdiff_cancel_right + - sdiff_le_sdiff_of_sup_le_sup_right + - sup_himp_self_left + - hnot_le_comm + - hnot_hnot_sup_distrib + - lt_compl_self + - sdiff_sup_self + - sdiff_eq_sdiff_iff + - himp_triangle + - le_himp_comm + - sdiff_le_iff' + - Codisjoint.himp_eq_right + - codisjoint_hnot_hnot_left_iff + - compl_compl_inf_distrib + - himp_left_comm + - sdiff_eq_bot_iff + - compl_top + - sdiff_inf_self_left + - himp_inf_self + - le_compl_self + - sdiff_sdiff_self + - sdiff_inf_self_right + - le_himp_iff' + - disjoint_compl_compl_left_iff + - hnot_inf_distrib + - hnot_hnot_sdiff_distrib + - le_compl_iff_disjoint_right + - hnot_sdiff + - compl_compl_himp_distrib + - himp_eq_top_iff + - hnot_bot + - le_sdiff_sup + - compl_sup_distrib + - sdiff_le_iff_left + - sup_sdiff_right_self + - sdiff_triangle + - hnot_top + - disjoint_compl_compl_right_iff + - sdiff_sup_cancel + - himp_eq_himp_iff + - inf_himp + - Codisjoint.himp_inf_cancel_right + - himp_compl + - sup_sdiff_left_self + - hnot_sdiff_comm + - compl_bot + - ne_compl_self + - sdiff_le_comm + - sup_himp_self_right + - sdiff_le_sdiff_of_sup_le_sup_left + - himp_inf_himp_inf_le + - hnot_le_iff_codisjoint_right + - codisjoint_hnot_hnot_right_iff + - sdiff_sdiff + - Disjoint.sdiff_eq_left + - sdiff_sdiff_sdiff_le_sdiff + - himp_compl_comm + - sdiff_bot + - sup_himp_distrib + - Codisjoint.himp_inf_cancel_left + - himp_idem + - le_himp_iff_left + - sdiff_inf_distrib + - inf_himp_le + - sdiff_right_comm + - le_sup_sdiff_sup_sdiff + - sdiff_idem + - himp_himp + - path: Mathlib/Computability/RegularExpressions.lean + theorems: + - RegularExpression.map_id + - RegularExpression.char_rmatch_iff + - RegularExpression.map_pow + - RegularExpression.map_map + - RegularExpression.rmatch_iff_matches' + - RegularExpression.matches'_map + - RegularExpression.one_rmatch_iff + - RegularExpression.mul_rmatch_iff + - RegularExpression.zero_rmatch + - RegularExpression.add_rmatch_iff + - RegularExpression.star_rmatch_iff + - path: Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean + theorems: + - le_iff_exists_mul + - lt_iff_exists_mul + - exists_one_lt_mul_of_lt + - exists_one_lt_mul_of_lt' + - le_of_forall_one_lt_le_mul + - le_iff_exists_mul' + - le_mul_right + - le_mul_self + - le_mul_left + - min_mul_distrib + - min_mul_distrib' + - path: Mathlib/Data/Real/ConjExponents.lean + theorems: + - Real.IsConjExponent.div_conj_eq_sub_one + - NNReal.IsConjExponent.symm + - NNReal.IsConjExponent.div_conj_eq_sub_one + - Real.IsConjExponent.inv_sub_one + - Real.IsConjExponent.mul_eq_add + - Real.isConjExponent_iff_eq_conjExponent + - Real.IsConjExponent.toNNReal + - Real.isConjExponent_one_div + - NNReal.IsConjExponent.inv_add_inv_conj_ennreal + - Real.IsConjExponent.inv_inv + - Real.IsConjExponent.inv_add_inv_conj_ennreal + - NNReal.isConjExponent_coe + - NNReal.IsConjExponent.inv_inv + - NNReal.IsConjExponent.mul_eq_add + - Real.IsConjExponent.conj_eq + - Real.IsConjExponent.symm + - NNReal.isConjExponent_iff_eq_conjExponent + - NNReal.IsConjExponent.conj_eq + - path: Mathlib/ModelTheory/Semantics.lean + theorems: + - FirstOrder.Language.BoundedFormula.realize_iInf + - FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv + - FirstOrder.Language.Formula.realize_iAlls + - FirstOrder.Language.BoundedFormula.realize_foldr_inf + - FirstOrder.Language.Equiv.realize_sentence + - FirstOrder.Language.BoundedFormula.realize_iSup + - FirstOrder.Language.BoundedFormula.realize_imp + - FirstOrder.Language.BoundedFormula.realize_castLE_of_eq + - FirstOrder.Language.BoundedFormula.realize_foldr_sup + - FirstOrder.Language.BoundedFormula.realize_relabel + - FirstOrder.Language.Term.realize_subst + - FirstOrder.Language.Term.realize_relabel + - FirstOrder.Language.Term.realize_constantsToVars + - FirstOrder.Language.model_distinctConstantsTheory + - FirstOrder.Language.BoundedFormula.realize_restrictFreeVar + - FirstOrder.Language.BoundedFormula.realize_toFormula + - FirstOrder.Language.Equiv.realize_boundedFormula + - FirstOrder.Language.BoundedFormula.realize_mapTermRel_add_castLe + - FirstOrder.Language.Formula.realize_graph + - "FirstOrder.Language.Formula.realize_rel\u2082" + - FirstOrder.Language.BoundedFormula.realize_mapTermRel_id + - FirstOrder.Language.LHom.setOf_realize_onFormula + - FirstOrder.Language.elementarilyEquivalent_iff + - FirstOrder.Language.BoundedFormula.realize_alls + - FirstOrder.Language.Term.realize_varsToConstants + - FirstOrder.Language.BoundedFormula.realize_sup + - FirstOrder.Language.model_nonemptyTheory_iff + - FirstOrder.Language.Formula.realize_equivSentence + - FirstOrder.Language.Sentence.realize_cardGe + - FirstOrder.Language.Term.realize_restrictVarLeft + - FirstOrder.Language.Term.realize_constantsVarsEquivLeft + - FirstOrder.Language.Formula.realize_equal + - FirstOrder.Language.BoundedFormula.realize_subst + - FirstOrder.Language.Formula.realize_iExs + - FirstOrder.Language.BoundedFormula.realize_liftAt_one + - FirstOrder.Language.LHom.realize_onTerm + - FirstOrder.Language.Term.realize_restrictVar + - "FirstOrder.Language.BoundedFormula.realize_rel\u2082" + - FirstOrder.Language.Theory.Model.union + - FirstOrder.Language.BoundedFormula.realize_inf + - FirstOrder.Language.BoundedFormula.realize_iExs + - FirstOrder.Language.BoundedFormula.realize_ex + - FirstOrder.Language.BoundedFormula.realize_toPrenexImpRight + - FirstOrder.Language.BoundedFormula.realize_all_liftAt_one_self + - FirstOrder.Language.model_infiniteTheory_iff + - FirstOrder.Language.realize_iff_of_model_completeTheory + - FirstOrder.Language.BoundedFormula.realize_top + - "FirstOrder.Language.Formula.realize_rel\u2081" + - FirstOrder.Language.Hom.realize_term + - "FirstOrder.Language.Term.realize_functions_apply\u2081" + - FirstOrder.Language.Formula.realize_relabel + - FirstOrder.Language.Formula.realize_relabel_sum_inr + - FirstOrder.Language.ElementarilyEquivalent.theory_model_iff + - FirstOrder.Language.BoundedFormula.realize_iAlls + - FirstOrder.Language.LHom.realize_onBoundedFormula + - FirstOrder.Language.Formula.realize_rel + - FirstOrder.Language.Formula.realize_equivSentence_symm_con + - FirstOrder.Language.BoundedFormula.realize_liftAt_one_self + - FirstOrder.Language.BoundedFormula.realize_toPrenexImp + - FirstOrder.Language.LHom.onTheory_model + - FirstOrder.Language.Theory.model_singleton_iff + - "FirstOrder.Language.BoundedFormula.realize_rel\u2081" + - FirstOrder.Language.BoundedFormula.realize_relabelEquiv + - FirstOrder.Language.Equiv.realize_formula + - FirstOrder.Language.BoundedFormula.realize_liftAt + - FirstOrder.Language.BoundedFormula.realize_iff + - "FirstOrder.Language.Term.realize_functions_apply\u2082" + - path: Mathlib/Data/Real/Sign.lean + theorems: + - Real.sign_inv + - Real.sign_int_cast + - Real.sign_eq_zero_iff + - Real.sign_of_neg + - Real.sign_one + - Real.sign_apply_eq + - Real.sign_mul_pos_of_ne_zero + - Real.sign_neg + - Real.sign_of_pos + - Real.inv_sign + - Real.sign_mul_nonneg + - path: Mathlib/FieldTheory/IsAlgClosed/Basic.lean + theorems: + - IsAlgClosed.of_ringEquiv + - IsAlgClosure.equivOfEquiv_symm_algebraMap + - IsAlgClosed.exists_eq_mul_self + - IntermediateField.eq_bot_of_isAlgClosed_of_isAlgebraic + - IsAlgClosed.FractionRing.isAlgebraic + - IsAlgClosed.exists_pow_nat_eq + - IsAlgClosed.of_exists_root + - "IsAlgClosed.exists_eval\u2082_eq_zero_of_injective" + - IsAlgClosed.algebraMap_surjective_of_isIntegral + - IsAlgClosed.splits_codomain + - IsAlgClosed.roots_eq_zero_iff + - path: Mathlib/NumberTheory/RamificationInertia.lean + theorems: + - Ideal.IsDedekindDomain.ramificationIdx_ne_zero + - Ideal.inertiaDeg_algebraMap + - Ideal.quotientToQuotientRangePowQuotSucc_surjective + - Ideal.Factors.inertiaDeg_ne_zero + - Ideal.ramificationIdx_eq_zero + - Ideal.sum_ramification_inertia + - Ideal.le_pow_of_le_ramificationIdx + - Ideal.ramificationIdx_ne_zero + - Ideal.finrank_prime_pow_ramificationIdx + - Ideal.inertiaDeg_of_subsingleton + - Ideal.IsDedekindDomain.ramificationIdx_eq_factors_count + - Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial + - Ideal.IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count + - Ideal.ramificationIdx_of_not_le + - Ideal.ramificationIdx_spec + - Ideal.FinrankQuotientMap.span_eq_top + - Ideal.finrank_quotient_map + - Ideal.ramificationIdx_bot + - Ideal.rank_pow_quot_aux + - Ideal.rank_pow_quot + - Ideal.Factors.finrank_pow_ramificationIdx + - Ideal.quotientToQuotientRangePowQuotSucc_injective + - Ideal.quotientToQuotientRangePowQuotSuccAux_mk + - Ideal.rank_prime_pow_ramificationIdx + - path: Mathlib/Algebra/Homology/ShortComplex/Exact.lean + theorems: + - CategoryTheory.ShortComplex.exact_of_iso + - CategoryTheory.ShortComplex.Exact.isIso_f' + - CategoryTheory.ShortComplex.RightHomologyData.exact_iff_mono_g' + - CategoryTheory.ShortComplex.Exact.liftFromProjective_comp + - CategoryTheory.ShortComplex.Exact.mono_g + - CategoryTheory.ShortComplex.quasiIso_iff_of_zeros' + - CategoryTheory.ShortComplex.Exact.epi_f' + - CategoryTheory.ShortComplex.Exact.mono_g' + - "CategoryTheory.ShortComplex.exact_of_isZero_X\u2082" + - CategoryTheory.ShortComplex.Splitting.s_r + - CategoryTheory.ShortComplex.exact_map_iff_of_faithful + - CategoryTheory.ShortComplex.quasiIso_iff_of_zeros + - CategoryTheory.ShortComplex.Exact.map + - CategoryTheory.ShortComplex.exact_of_f_is_kernel + - CategoryTheory.ShortComplex.exact_iff_mono_cokernel_desc + - CategoryTheory.ShortComplex.HomologyData.exact_iff + - CategoryTheory.ShortComplex.exact_of_g_is_cokernel + - "CategoryTheory.ShortComplex.Exact.isZero_X\u2082_iff" + - CategoryTheory.exact_iff_shortComplex_exact + - CategoryTheory.ShortComplex.Exact.isIso_g' + - CategoryTheory.ShortComplex.exact_iff_homology_iso_zero + - CategoryTheory.ShortComplex.Splitting.epi_g + - CategoryTheory.ShortComplex.Splitting.ext_r + - CategoryTheory.ShortComplex.Exact.map_of_preservesLeftHomologyOf + - CategoryTheory.ShortComplex.HomologyData.exact_iff' + - CategoryTheory.ShortComplex.Splitting.mono_f + - CategoryTheory.ShortComplex.RightHomologyData.exact_iff + - CategoryTheory.ShortComplex.Exact.mono_g_iff + - CategoryTheory.ShortComplex.Exact.epi_f_iff + - CategoryTheory.ShortComplex.Exact.unop + - CategoryTheory.ShortComplex.Splitting.ext_s + - CategoryTheory.ShortComplex.LeftHomologyData.exact_iff_epi_f' + - CategoryTheory.ShortComplex.Splitting.g_s + - CategoryTheory.ShortComplex.exact_and_epi_g_iff_g_is_cokernel + - CategoryTheory.ShortComplex.exact_iff_epi + - CategoryTheory.ShortComplex.Exact.map_of_preservesRightHomologyOf + - "CategoryTheory.ShortComplex.mono_\u03C4\u2082_of_exact_of_mono" + - CategoryTheory.ShortComplex.Exact.desc' + - CategoryTheory.ShortComplex.exact_iff_of_epi_of_isIso_of_mono + - CategoryTheory.ShortComplex.exact_and_mono_f_iff_f_is_kernel + - CategoryTheory.ShortComplex.Splitting.r_f + - CategoryTheory.ShortComplex.Exact.op + - CategoryTheory.ShortComplex.Exact.epi_f + - "CategoryTheory.ShortComplex.Exact.isZero_X\u2082" + - CategoryTheory.ShortComplex.exact_iff_epi_kernel_lift + - CategoryTheory.ShortComplex.exact_and_epi_g_iff_of_iso + - CategoryTheory.ShortComplex.HomologyData.exact_iff_i_p_zero + - "CategoryTheory.ShortComplex.epi_\u03C4\u2082_of_exact_of_epi" + - CategoryTheory.ShortComplex.Exact.comp_descToInjective + - CategoryTheory.ShortComplex.QuasiIso.exact_iff + - CategoryTheory.ShortComplex.Exact.lift' + - CategoryTheory.ShortComplex.LeftHomologyData.exact_iff + - "CategoryTheory.ShortComplex.exact_iff_kernel_\u03B9_comp_cokernel_\u03C0_zero" + - CategoryTheory.ShortComplex.exact_iff_isZero_homology + - CategoryTheory.ShortComplex.exact_iff_mono + - CategoryTheory.ShortComplex.Exact.comp_eq_zero + - CategoryTheory.ShortComplex.exact_and_mono_f_iff_of_iso + - path: Mathlib/FieldTheory/Perfect.lean + theorems: + - not_irreducible_expand + - Polynomial.roots_X_pow_char_pow_sub_C_pow + - Polynomial.roots_expand_pow_map_iterateFrobenius_le + - Polynomial.roots_expand_pow_image_iterateFrobenius_subset + - Polynomial.roots_expand_image_frobenius + - iterateFrobeniusEquiv_zero_apply + - iterateFrobeniusEquiv_symm_add_apply + - Algebra.IsAlgebraic.perfectField + - PerfectRing.toPerfectField + - iterateFrobeniusEquiv_eq_pow + - iterateFrobeniusEquiv_one_apply + - Polynomial.roots_expand + - Polynomial.roots_expand_pow + - iterateFrobeniusEquiv_symm + - Polynomial.roots_expand_image_iterateFrobenius + - Polynomial.roots_X_pow_char_pow_sub_C + - Polynomial.roots_X_pow_char_sub_C_pow + - Polynomial.roots_expand_map_frobenius_le + - Polynomial.roots_X_pow_char_sub_C + - PerfectField.separable_iff_squarefree + - frobenius_comp_frobeniusEquiv_symm + - Polynomial.roots_expand_pow_map_iterateFrobenius + - polynomial_expand_eq + - Polynomial.roots_expand_map_frobenius + - frobeniusEquiv_symm_comp_frobenius + - Polynomial.roots_expand_image_frobenius_subset + - path: Mathlib/Analysis/Calculus/Deriv/AffineMap.lean + theorems: + - AffineMap.hasStrictDerivAt + - AffineMap.hasStrictDerivAt_lineMap + - AffineMap.hasDerivAtFilter + - path: Mathlib/NumberTheory/Padics/PadicNorm.lean + theorems: + - padicNorm.zero + - padicNorm.sum_le + - padicNorm.eq_zpow_of_nonzero + - padicNorm.sum_lt + - padicNorm.neg + - padicNorm.nonarchimedean + - padicNorm.div + - padicNorm.padicNorm_p + - padicNorm.zero_of_padicNorm_eq_zero + - padicNorm.of_int + - padicNorm.dvd_iff_norm_le + - padicNorm.int_lt_one_iff + - padicNorm.nat_lt_one_iff + - padicNorm.one + - padicNorm.sub + - padicNorm.sum_le' + - padicNorm.nonneg + - padicNorm.int_eq_one_iff + - padicNorm.nonarchimedean_aux + - padicNorm.padicNorm_p_lt_one + - padicNorm.nat_eq_one_iff + - padicNorm.add_eq_max_of_ne + - padicNorm.padicNorm_of_prime_of_ne + - padicNorm.mul + - padicNorm.not_int_of_not_padic_int + - padicNorm.values_discrete + - padicNorm.nonzero + - padicNorm.sum_lt' + - path: Mathlib/Topology/Algebra/InfiniteSum/Real.lean + theorems: + - cauchySeq_of_dist_le_of_summable + - "dist_le_tsum_of_dist_le_of_tendsto\u2080" + - cauchySeq_of_edist_le_of_summable + - dist_le_tsum_of_dist_le_of_tendsto + - "dist_le_tsum_dist_of_tendsto\u2080" + - path: Mathlib/Algebra/BigOperators/Ring.lean + theorems: + - Finset.sum_range_succ_mul_sum_range_succ + - Finset.prod_add + - Finset.sum_boole_mul + - Finset.sum_mul_boole + - Finset.sum_pow' + - Finset.prod_range_cast_nat_sub + - Finset.prod_one_sub_ordered + - Finset.dvd_sum + - Finset.prod_sub_ordered + - Finset.sum_div + - Finset.sum_mul_sum + - Finset.sum_prod_piFinset + - Finset.sum_pow_mul_eq_add_pow + - Finset.prod_univ_sum + - Finset.prod_sum + - Finset.prod_add_ordered + - path: Mathlib/MeasureTheory/Function/LocallyIntegrable.lean + theorems: + - MeasureTheory.locallyIntegrable_finset_sum + - MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn + - MeasureTheory.LocallyIntegrableOn.smul_continuousOn + - MeasureTheory.IntegrableOn.mul_continuousOn_of_subset + - MeasureTheory.LocallyIntegrableOn.mono + - MeasureTheory.LocallyIntegrable.integrable_smul_left_of_hasCompactSupport + - MeasureTheory.LocallyIntegrable.exists_nat_integrableOn + - MeasureTheory.LocallyIntegrable.aestronglyMeasurable + - MeasureTheory.LocallyIntegrableOn.exists_countable_integrableOn + - MeasureTheory.locallyIntegrableOn_iff_locallyIntegrable_restrict + - MeasureTheory.LocallyIntegrable.indicator + - "MeasureTheory.Mem\u2112p.locallyIntegrable" + - MonotoneOn.integrableOn_of_measure_ne_top + - MeasureTheory.locallyIntegrable_comap + - MeasureTheory.IntegrableOn.continuousOn_smul + - MeasureTheory.locallyIntegrableOn_iff + - ContinuousOn.integrableOn_compact' + - MonotoneOn.integrableOn_isCompact + - MeasureTheory.locallyIntegrableOn_univ + - Monotone.locallyIntegrable + - MeasureTheory.LocallyIntegrableOn.continuousOn_smul + - MeasureTheory.IntegrableOn.smul_continuousOn + - MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict + - MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport + - MeasureTheory.LocallyIntegrableOn.mul_continuousOn + - MeasureTheory.LocallyIntegrableOn.continuousOn_mul + - MeasureTheory.locallyIntegrable_map_homeomorph + - MeasureTheory.LocallyIntegrable.mono + - MeasureTheory.integrable_iff_integrableAtFilter_cocompact + - MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable + - MeasureTheory.LocallyIntegrable.integrableOn_nhds_isCompact + - path: Mathlib/Data/Nat/GCD/Basic.lean + theorems: + - Nat.gcd_self_add_right + - Nat.coprime_add_self_left + - Nat.coprime_add_self_right + - Nat.dvd_mul + - Nat.gcd_add_mul_right_left + - Nat.coprime_sub_self_left + - Nat.coprime_mul_right_add_right + - Nat.gcd_sub_self_left + - Nat.coprime_iff_isRelPrime + - Nat.coprime_one_right_iff + - Nat.coprime_sub_self_right + - Nat.gcd_mul_right_add_left + - Nat.coprime_mul_left_add_left + - Nat.not_coprime_zero_zero + - Nat.Coprime.lcm_eq_mul + - Nat.gcd_self_add_left + - Nat.coprime_add_mul_right_right + - Nat.gcd_self_sub_right + - Nat.lcm_pos + - Nat.gcd_add_mul_left_left + - Nat.gcd_add_self_left + - Nat.coprime_add_mul_left_left + - Nat.lcm_mul_right + - Nat.gcd_add_mul_right_right + - Nat.coprime_mul_right_add_left + - Nat.coprime_one_left_iff + - Nat.Coprime.mul_add_mul_ne_mul + - Nat.coprime_add_mul_right_left + - Nat.coprime_self_sub_left + - Nat.coprime_pow_right_iff + - Nat.gcd_mul_left_add_right + - Nat.pow_dvd_pow_iff + - Nat.coprime_self_add_left + - Nat.coprime_self_add_right + - Nat.gcd_add_mul_left_right + - Nat.lcm_mul_left + - Nat.gcd_mul_right_add_right + - Nat.gcd_self_sub_left + - Nat.coprime_self_sub_right + - Nat.coprime_add_mul_left_right + - Nat.gcd_sub_self_right + - Nat.coprime_mul_left_add_right + - Nat.gcd_mul_of_coprime_of_dvd + - Nat.gcd_mul_left_add_left + - path: Mathlib/Data/Fintype/Order.lean + theorems: + - Directed.finite_set_le + - Finite.bddAbove_range + - Finite.bddBelow_range + - Directed.finite_le + - path: Mathlib/Data/List/Zip.lean + theorems: + - List.forall_zipWith + - List.revzip_map_snd + - List.unzip_zip_right + - List.get?_zip_with + - List.length_revzip + - List.unzip_right + - List.mem_zip_inits_tails + - List.get?_zip_eq_some + - List.zip_swap + - List.revzip_swap + - List.zipWith_congr + - List.reverse_revzip + - List.unzip_swap + - List.unzip_zip + - List.get?_zip_with_eq_some + - List.get_zipWith + - List.prod_mul_prod_eq_prod_zipWith_of_length_eq + - List.zipWith_distrib_reverse + - List.prod_mul_prod_eq_prod_zipWith_mul_prod_drop + - List.map_uncurry_zip_eq_zipWith + - List.zipWith_comm_of_comm + - List.lt_length_left_of_zipWith + - List.lt_length_right_of_zipWith + - List.map_prod_left_eq_zip + - List.unzip_eq_map + - List.sum_zipWith_distrib_left + - List.zip_of_prod + - List.mem_zip + - List.unzip_left + - List.zip_unzip + - path: Mathlib/Data/List/Sigma.lean + theorems: + - List.sizeOf_dedupKeys + - List.dlookup_kunion_left + - List.lookupAll_eq_nil + - List.kerase_comm + - List.exists_of_kerase + - List.mem_keys_kunion + - List.kerase_of_not_mem_keys + - List.mem_dlookup_kunion + - List.dlookup_kerase_ne + - List.Perm.kreplace + - List.sizeOf_kerase + - List.kunion_kerase + - List.Perm.kunion_right + - List.perm_lookupAll + - List.dlookup_isSome + - List.kreplace_of_forall_not + - List.lookupAll_nodup + - List.kunion_nil + - List.mem_keys_kerase_of_ne + - List.kerase_cons_eq + - List.dlookup_eq_none + - List.Perm.kerase + - List.perm_dlookup + - List.kerase_cons_ne + - List.of_mem_dlookup + - List.keys_kerase + - List.NodupKeys.eq_of_mk_mem + - List.kerase_append_left + - List.map_dlookup_eq_find + - List.dlookup_dedupKeys + - List.not_mem_keys_kerase + - List.lookupAll_sublist + - List.dlookup_kunion_right + - List.nodup_enum_map_fst + - List.nodupKeys_cons + - List.dlookup_kinsert_ne + - List.lookupAll_eq_dlookup + - List.NodupKeys.kunion + - List.kerase_kerase + - List.kreplace_nodupKeys + - List.head?_lookupAll + - List.kreplace_self + - List.kextract_eq_dlookup_kerase + - List.lookupAll_length_le_one + - List.not_eq_key + - List.kerase_append_right + - List.nodupKeys_join + - List.nodupKeys_dedupKeys + - List.mem_dlookup + - List.mem_lookupAll + - List.lookup_ext + - path: Mathlib/Analysis/Analytic/Constructions.lean + theorems: + - HasFPowerSeriesAt.prod + - analyticOn_inv + - analyticAt_inv + - AnalyticAt.pow + - FormalMultilinearSeries.radius_prod_eq_min + - AnalyticAt.div + - Finset.analyticAt_prod + - formalMultilinearSeries_geometric_radius + - HasFPowerSeriesOnBall.prod + - Finset.analyticAt_sum + - hasFPowerSeriesOnBall_inv_one_sub + - AnalyticAt.prod + - path: Mathlib/Data/PNat/Interval.lean + theorems: + - PNat.card_Ioo + - PNat.card_Icc + - PNat.card_fintype_Ioc + - PNat.card_fintype_Ioo + - PNat.card_fintype_Ico + - PNat.card_uIcc + - PNat.card_fintype_Icc + - PNat.card_fintype_uIcc + - PNat.card_Ioc + - path: Mathlib/Topology/Homotopy/Path.lean + theorems: + - Path.Homotopic.hpath_hext + - Path.Homotopy.hcomp_half + - Path.Homotopy.eval_one + - Path.Homotopy.eval_zero + - ContinuousMap.homotopic_const_iff + - Path.Homotopy.hcomp_apply + - path: Mathlib/MeasureTheory/Constructions/Pi.lean + theorems: + - MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic + - MeasureTheory.piPremeasure_pi + - MeasureTheory.measurePreserving_sumPiEquivProdPi_symm + - generateFrom_pi_eq + - MeasureTheory.Measure.pi_ball + - MeasureTheory.piPremeasure_pi_eval + - MeasureTheory.Measure.tprod_tprod + - MeasureTheory.measurePreserving_piFinTwo + - MeasureTheory.Measure.pi_of_empty + - MeasureTheory.Measure.pi_caratheodory + - MeasureTheory.Measure.pi_pi + - MeasureTheory.Measure.pi_empty_univ + - IsPiSystem.pi + - generateFrom_eq_pi + - MeasureTheory.piPremeasure_pi' + - MeasureTheory.Measure.univ_pi_Ioo_ae_eq_Icc + - MeasureTheory.Measure.pi_pi_aux + - MeasureTheory.Measure.pi_eq_generateFrom + - MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc + - MeasureTheory.measurePreserving_piEquivPiSubtypeProd + - MeasureTheory.Measure.pi_eval_preimage_null + - MeasureTheory.measurePreserving_finTwoArrow + - MeasureTheory.measurePreserving_pi_empty + - IsCountablySpanning.pi + - MeasureTheory.Measure.pi_closedBall + - MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc + - MeasureTheory.measurePreserving_piUnique + - MeasureTheory.OuterMeasure.pi_pi_le + - MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici + - MeasureTheory.Measure.pi_univ + - MeasureTheory.Measure.pi'_pi + - MeasureTheory.measurePreserving_piCongrLeft + - MeasureTheory.OuterMeasure.le_pi + - MeasureTheory.measurePreserving_piFinSuccAbove + - path: Mathlib/LinearAlgebra/Vandermonde.lean + theorems: + - Matrix.vandermonde_cons + - Matrix.det_eval_matrixOfPolynomials_eq_det_vandermonde + - Matrix.det_vandermonde_add + - Matrix.eq_zero_of_forall_index_sum_mul_pow_eq_zero + - Matrix.vandermonde_succ + - Matrix.eval_matrixOfPolynomials_eq_vandermonde_mul_matrixOfPolynomials + - Matrix.det_vandermonde_sub + - Matrix.vandermonde_transpose_mul_vandermonde + - Matrix.det_vandermonde_ne_zero_iff + - Matrix.det_vandermonde_eq_zero_iff + - Matrix.vandermonde_mul_vandermonde_transpose + - path: Mathlib/Order/GameAdd.lean + theorems: + - Prod.gameAdd_swap_swap + - Sym2.GameAdd.fix_eq + - Sym2.GameAdd.fst_snd + - Sym2.gameAdd_iff + - Acc.prod_gameAdd + - Prod.rprod_le_transGen_gameAdd + - Prod.gameAdd_iff + - Acc.sym2_gameAdd + - Sym2.GameAdd.snd_fst + - path: Mathlib/Topology/Connected/Basic.lean + theorems: + - connectedComponentIn_mono + - IsPreconnected.image + - IsPreconnected.subset_left_of_subset_union + - isPreconnected_iff_preconnectedSpace + - Sigma.isConnected_iff + - subsingleton_of_disjoint_isClopen + - IsPreconnected.biUnion_of_reflTransGen + - Continuous.exists_lift_sigma + - isPreconnected_closed_iff + - Subtype.preconnectedSpace + - mem_connectedComponentIn + - nonempty_inter + - connectedSpace_iff_connectedComponent + - IsPreconnected.subset_of_closure_inter_subset + - Sum.isConnected_iff + - isConnected_univ_pi + - "IsPreconnected.induction\u2082'" + - IsPreconnected.biUnion_of_chain + - isPreconnected_iff_subset_of_fully_disjoint_closed + - nonempty_frontier_iff + - connectedComponentIn_subset + - subsingleton_of_disjoint_isOpen_iUnion_eq_univ + - IsPreconnected.sUnion_directed + - isPreconnected_of_forall_pair + - Sum.isPreconnected_iff + - "PreconnectedSpace.induction\u2082" + - preimage_connectedComponent_connected + - IsConnected.iUnion_of_chain + - IsPreconnected.union + - IsPreconnected.iUnion_of_chain + - isClopen_iff + - isPreconnected_connectedComponentIn + - IsPreconnected.subset_isClopen + - IsPreconnected.iUnion_of_reflTransGen + - Inducing.isPreconnected_image + - IsPreconnected.subset_or_subset + - connectedComponents_preimage_singleton + - connectedComponents_preimage_image + - Sigma.isPreconnected_iff + - isConnected_iff_sUnion_disjoint_open + - connectedComponentIn_nonempty_iff + - subsingleton_of_disjoint_isClosed_iUnion_eq_univ + - preconnectedSpace_iff_connectedComponent + - isConnected_connectedComponentIn_iff + - IsPreconnected.subset_connectedComponentIn + - isPreconnected_iff_subset_of_disjoint + - IsPreconnected.preimage_of_isClosedMap + - "IsPreconnected.induction\u2082" + - Function.Surjective.connectedSpace + - isPreconnected_of_forall + - isPreconnected_iff_subset_of_disjoint_closed + - IsPreconnected.preimage_of_isOpenMap + - isPreconnected_sUnion + - isPreconnected_univ_pi + - path: Mathlib/LinearAlgebra/Prod.lean + theorems: + - Submodule.fst_map_snd + - LinearMap.prod_eq_sup_map + - Submodule.fst_map_fst + - LinearMap.coprod_map_prod + - LinearMap.range_prod_eq + - LinearMap.tailings_zero + - Submodule.range_snd + - LinearEquiv.snd_comp_prodComm + - Submodule.ker_inr + - LinearMap.coprod_inl_inr + - LinearEquiv.fst_comp_prodComm + - LinearMap.tailings_disjoint_tunnel + - LinearMap.tailing_le_tunnel + - LinearMap.range_coprod + - LinearMap.inr_injective + - LinearMap.graph_eq_range_prod + - LinearMap.coprod_inl + - Submodule.prod_map_snd + - Submodule.ker_inl + - LinearMap.tailing_disjoint_tunnel_succ + - Submodule.sup_eq_range + - Submodule.snd_map_snd + - Submodule.le_prod_iff + - Submodule.range_fst + - LinearMap.snd_eq_coprod + - LinearMap.map_coprod_prod + - Submodule.map_inr + - LinearMap.range_prod_le + - LinearMap.fst_eq_coprod + - LinearMap.coprod_inr + - Submodule.prod_le_iff + - LinearMap.tailings_succ + - LinearMap.inr_map_mul + - LinearMap.ker_prodMap + - LinearMap.ker_coprod_of_disjoint_range + - Submodule.fst_inf_snd + - LinearMap.isCompl_range_inl_inr + - LinearMap.span_inl_union_inr + - LinearMap.inl_map_mul + - Submodule.prod_eq_top_iff + - Submodule.comap_fst + - Submodule.snd_map_fst + - LinearMap.range_inr + - Submodule.prod_comap_inr + - LinearMap.ker_prod_ker_le_ker_coprod + - Submodule.map_inl + - Submodule.comap_snd + - LinearMap.range_inl + - Submodule.fst_sup_snd + - LinearMap.inl_injective + - LinearMap.ker_prod + - Submodule.prod_eq_bot_iff + - LinearMap.disjoint_inl_inr + - Submodule.prod_map_fst + - LinearMap.graph_eq_ker_coprod + - path: Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean + theorems: + - MvPolynomial.isWeightedHomogeneous_monomial + - MvPolynomial.isWeightedHomogeneous_of_total_degree_zero + - MvPolynomial.coeff_weightedHomogeneousComponent + - MvPolynomial.IsWeightedHomogeneous.inj_right + - MvPolynomial.IsWeightedHomogeneous.prod + - MvPolynomial.IsWeightedHomogeneous.weighted_total_degree + - MvPolynomial.weightedTotalDegree'_zero + - MvPolynomial.weightedTotalDegree_zero + - MvPolynomial.weightedHomogeneousComponent_zero + - MvPolynomial.sum_weightedHomogeneousComponent + - MvPolynomial.weightedHomogeneousComponent_isWeightedHomogeneous + - MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported + - MvPolynomial.isWeightedHomogeneous_X + - MvPolynomial.IsWeightedHomogeneous.coeff_eq_zero + - MvPolynomial.weightedHomogeneousComponent_eq_zero + - MvPolynomial.weightedHomogeneousSubmodule_mul + - MvPolynomial.weightedHomogeneousComponent_apply + - MvPolynomial.weightedTotalDegree_coe + - MvPolynomial.weightedHomogeneousComponent_C_mul + - MvPolynomial.weightedHomogeneousComponent_finsupp + - MvPolynomial.weightedHomogeneousComponent_weighted_homogeneous_polynomial + - MvPolynomial.weightedHomogeneousComponent_eq_zero' + - MvPolynomial.weightedTotalDegree'_eq_bot_iff + - path: Mathlib/Init/Data/Bool/Lemmas.lean + theorems: + - Bool.coe_sort_false + - Bool.coe_and_iff + - Bool.coe_true + - Bool.bool_iff_false + - Bool.ite_eq_false_distrib + - Bool.decide_congr + - Bool.not_eq_true_eq_eq_false + - Bool.cond_self + - Bool.or_eq_true_eq_eq_true_or_eq_true + - Bool.eq_true_eq_not_eq_false + - Bool.true_eq_false_eq_False + - Bool.ite_eq_true_distrib + - Bool.coe_false + - Bool.coe_sort_true + - Bool.decide_iff + - Bool.false_eq_true_eq_False + - Bool.not_eq_false_eq_eq_true + - Bool.coe_xor_iff + - Bool.eq_false_eq_not_eq_true + - Bool.or_eq_false_eq_eq_false_and_eq_false + - Bool.coe_or_iff + - Bool.and_eq_true_eq_eq_true_and_eq_true + - path: Mathlib/CategoryTheory/MorphismProperty.lean + theorems: + - CategoryTheory.MorphismProperty.RespectsIso.of_respects_arrow_iso + - CategoryTheory.MorphismProperty.map_isoClosure + - CategoryTheory.MorphismProperty.map_subset_iff + - CategoryTheory.MorphismProperty.StableUnderBaseChange.respectsIso + - CategoryTheory.MorphismProperty.StableUnderCobaseChange.mk + - CategoryTheory.MorphismProperty.StableUnderComposition.epimorphisms + - CategoryTheory.MorphismProperty.inverseImage_map_eq_of_isEquivalence + - CategoryTheory.MorphismProperty.monotone_map + - CategoryTheory.MorphismProperty.StableUnderBaseChange.baseChange_map + - CategoryTheory.MorphismProperty.isoClosure_subset_iff + - CategoryTheory.MorphismProperty.RespectsIso.arrow_iso_iff + - CategoryTheory.MorphismProperty.isomorphisms.iff + - CategoryTheory.MorphismProperty.StableUnderBaseChange.mk + - CategoryTheory.MorphismProperty.StableUnderComposition.diagonal + - CategoryTheory.MorphismProperty.IsInvertedBy.iff_map_subset_isomorphisms + - CategoryTheory.MorphismProperty.IsInvertedBy.pi + - CategoryTheory.MorphismProperty.IsInvertedBy.of_comp + - CategoryTheory.MorphismProperty.IsInvertedBy.leftOp + - CategoryTheory.MorphismProperty.naturalityProperty.stableUnderComposition + - CategoryTheory.MorphismProperty.map_eq_of_iso + - CategoryTheory.MorphismProperty.isoClosure_respectsIso + - CategoryTheory.MorphismProperty.IsInvertedBy.iff_of_iso + - CategoryTheory.MorphismProperty.inverseImage_equivalence_inverse_eq_map_functor + - CategoryTheory.MorphismProperty.RespectsIso.cancel_right_isIso + - CategoryTheory.MorphismProperty.RespectsIso.isomorphisms + - CategoryTheory.MorphismProperty.StableUnderComposition.universally + - CategoryTheory.MorphismProperty.StableUnderBaseChange.diagonal + - CategoryTheory.MorphismProperty.FunctorsInverting.ext + - CategoryTheory.MorphismProperty.StableUnderComposition.isomorphisms + - CategoryTheory.MorphismProperty.IsInvertedBy.unop + - CategoryTheory.MorphismProperty.map_id_eq_isoClosure + - CategoryTheory.MorphismProperty.map_map + - CategoryTheory.MorphismProperty.IsInvertedBy.rightOp + - CategoryTheory.MorphismProperty.StableUnderComposition.inverseImage + - CategoryTheory.MorphismProperty.RespectsIso.cancel_left_isIso + - CategoryTheory.MorphismProperty.RespectsIso.inverseImage + - CategoryTheory.MorphismProperty.RespectsIso.diagonal + - CategoryTheory.MorphismProperty.epimorphisms.iff + - CategoryTheory.MorphismProperty.monomorphisms.iff + - CategoryTheory.MorphismProperty.map_id + - CategoryTheory.MorphismProperty.top_apply + - CategoryTheory.MorphismProperty.IsInvertedBy.op + - CategoryTheory.MorphismProperty.universally_respectsIso + - CategoryTheory.MorphismProperty.StableUnderBaseChange.pullback_map + - CategoryTheory.NatTrans.isIso_app_iff_of_iso + - CategoryTheory.MorphismProperty.IsInvertedBy.prod + - CategoryTheory.MorphismProperty.IsInvertedBy.iff_comp + - CategoryTheory.MorphismProperty.IsInvertedBy.isoClosure_iff + - CategoryTheory.MorphismProperty.StableUnderComposition.monomorphisms + - CategoryTheory.MorphismProperty.RespectsIso.monomorphisms + - CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk + - CategoryTheory.MorphismProperty.IsInvertedBy.map_iff + - CategoryTheory.MorphismProperty.monotone_isoClosure + - CategoryTheory.MorphismProperty.map_respectsIso + - CategoryTheory.MorphismProperty.bijective_stableUnderComposition + - CategoryTheory.MorphismProperty.surjective_stableUnderComposition + - CategoryTheory.MorphismProperty.RespectsIso.epimorphisms + - CategoryTheory.MorphismProperty.injective_stableUnderComposition + - CategoryTheory.MorphismProperty.naturalityProperty.stableUnderInverse + - CategoryTheory.MorphismProperty.ext + - CategoryTheory.MorphismProperty.map_inverseImage_eq_of_isEquivalence + - CategoryTheory.MorphismProperty.universally_le + - path: Mathlib/RingTheory/IsTensorProduct.lean + theorems: + - Algebra.IsPushout.symm + - Algebra.pushoutDesc_apply + - IsTensorProduct.lift_eq + - TensorProduct.isTensorProduct + - TensorProduct.isBaseChange + - IsBaseChange.of_lift_unique + - IsTensorProduct.equiv_symm_apply + - IsBaseChange.equiv_symm_apply + - IsBaseChange.comp + - IsBaseChange.iff_lift_unique + - IsTensorProduct.inductionOn + - IsBaseChange.lift_eq + - Algebra.pushoutDesc_left + - Algebra.pushoutDesc_right + - IsBaseChange.ofEquiv + - Algebra.IsPushout.algHom_ext + - IsBaseChange.algHom_ext + - IsTensorProduct.map_eq + - path: Mathlib/Data/Real/GoldenRatio.lean + theorems: + - one_sub_goldConj + - gold_lt_two + - gold_mul_goldConj + - Real.coe_fib_eq + - goldConj_mul_gold + - gold_irrational + - fib_golden_conj_exp + - gold_sub_goldConj + - one_sub_gold + - fib_isSol_fibRec + - gold_add_goldConj + - Real.coe_fib_eq' + - inv_goldConj + - goldConj_sq + - neg_one_lt_goldConj + - fib_golden_exp' + - goldConj_irrational + - inv_gold + - geom_goldConj_isSol_fibRec + - gold_pos + - one_lt_gold + - gold_pow_sub_gold_pow + - gold_sq + - fibRec_charPoly_eq + - path: Mathlib/Data/Nat/Pairing.lean + theorems: + - Nat.unpair_lt + - Nat.pair_unpair' + - Nat.pair_lt_max_add_one_sq + - Nat.pair_unpair + - Nat.unpair_pair + - Nat.left_le_pair + - Nat.max_sq_add_min_le_pair + - Set.iUnion_unpair_prod + - Nat.add_le_pair + - Nat.pair_lt_pair_right + - Nat.pair_lt_pair_left + - Nat.unpair_zero + - Nat.right_le_pair + - Nat.unpair_right_le + - iSup_unpair + - path: Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean + theorems: + - Set.ordConnectedProj_eq + - Set.ordConnectedComponent_eq_empty + - Set.dual_ordSeparatingSet + - Set.dual_ordConnectedComponent + - Set.eq_of_mem_ordConnectedSection_of_uIcc_subset + - Set.disjoint_ordT5Nhd + - Set.ordConnectedComponent_inter + - Set.ordConnectedComponent_univ + - Set.dual_ordConnectedSection + - Set.mem_ordConnectedComponent_comm + - path: Mathlib/Data/UnionFind.lean + theorems: + - UnionFind.model' + - UFModel.Models.setParent + - UFModel.Models.rank_eq + - UFModel.Agrees.set + - UnionFind.rank_lt + - UnionFind.parent_lt + - UFModel.Agrees.size_eq + - UnionFind.lt_rankMax + - UFModel.Agrees.get_eq + - UnionFind.rank_eq + - UFModel.Agrees.mk' + - UFModel.Models.push + - UFModel.Agrees.push + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean + theorems: + - WeierstrassCurve.Affine.degree_polynomial + - WeierstrassCurve.Affine.baseChange_addX + - WeierstrassCurve.Affine.natDegree_polynomial + - WeierstrassCurve.Affine.map_slope + - WeierstrassCurve.Affine.nonsingular_zero + - WeierstrassCurve.Affine.irreducible_polynomial + - WeierstrassCurve.Affine.slope_of_Yne + - WeierstrassCurve.Affine.nonsingular_iff' + - WeierstrassCurve.Affine.eval_polynomial_zero + - WeierstrassCurve.Affine.eval_polynomial + - WeierstrassCurve.Affine.baseChange_slope + - WeierstrassCurve.Affine.equation_iff + - WeierstrassCurve.Affine.slope_of_Yne_eq_eval + - WeierstrassCurve.Affine.Point.some_add_some_of_Yne + - WeierstrassCurve.Affine.equation_iff_variableChange + - WeierstrassCurve.Affine.equation_zero + - WeierstrassCurve.Affine.monic_polynomial + - WeierstrassCurve.Affine.slope_of_Xne + - WeierstrassCurve.Affine.eval_polynomialY + - WeierstrassCurve.Affine.polynomial_ne_zero + - WeierstrassCurve.Affine.nonsingular_neg_iff + - WeierstrassCurve.Affine.Point.some_add_some_of_Xne + - WeierstrassCurve.Affine.eval_polynomialX_zero + - WeierstrassCurve.Affine.eval_negPolynomial + - WeierstrassCurve.Affine.map_addY' + - "WeierstrassCurve.Affine.nonsingular_of_\u0394_ne_zero" + - WeierstrassCurve.Affine.baseChange_nonsingular + - WeierstrassCurve.Affine.eval_polynomialX + - WeierstrassCurve.Affine.baseChange_negY + - WeierstrassCurve.Affine.Point.map_map + - WeierstrassCurve.Affine.slope_of_Yeq + - WeierstrassCurve.Affine.equation_add' + - "WeierstrassCurve.Affine.nonsingular_zero_of_\u0394_ne_zero" + - WeierstrassCurve.Affine.baseChange_equation + - WeierstrassCurve.Affine.nonsingular_add' + - WeierstrassCurve.Affine.negY_negY + - WeierstrassCurve.Affine.map_addY + - WeierstrassCurve.Affine.addPolynomial_eq + - WeierstrassCurve.Affine.map_negY + - WeierstrassCurve.Affine.equation_neg_iff + - WeierstrassCurve.Affine.baseChange_addY' + - WeierstrassCurve.Affine.Point.map_id + - WeierstrassCurve.Affine.equation_add_iff + - WeierstrassCurve.Affine.addPolynomial_slope + - WeierstrassCurve.Affine.equation_iff' + - WeierstrassCurve.Affine.Yeq_of_Xeq + - WeierstrassCurve.Affine.Point.map_baseChange + - WeierstrassCurve.Affine.nonsingular_add_of_eval_derivative_ne_zero + - WeierstrassCurve.Affine.map_addX + - WeierstrassCurve.Affine.Point.map_injective + - WeierstrassCurve.Affine.derivative_addPolynomial_slope + - WeierstrassCurve.Affine.map_nonsingular + - WeierstrassCurve.Affine.nonsingular_iff + - WeierstrassCurve.Affine.baseChange_addY + - WeierstrassCurve.Affine.nonsingular_iff_variableChange + - WeierstrassCurve.Affine.eval_polynomialY_zero + - WeierstrassCurve.Affine.C_addPolynomial + - WeierstrassCurve.Affine.Point.some_add_some_of_Yeq + - WeierstrassCurve.Affine.polynomial_eq + - WeierstrassCurve.Affine.map_equation + - path: Mathlib/Topology/ProperMap.lean + theorems: + - isProperMap_iff_isCompact_preimage + - isProperMap_iff_isClosedMap_and_tendsto_cofinite + - IsProperMap.isCompact_preimage + - isProperMap_iff_isClosedMap_ultrafilter + - IsProperMap.isClosedMap + - isProperMap_iff_isClosedMap_and_compact_fibers + - isProperMap_iff_tendsto_cocompact + - isProperMap_iff_universally_closed + - isProperMap_iff_ultrafilter + - Continuous.isProperMap + - IsProperMap.prod_map + - IsProperMap.pi_map + - Homeomorph.isProperMap + - isProperMap_iff_isClosedMap_filter + - path: Mathlib/Algebra/Lie/Nilpotent.lean + theorems: + - LieModule.lowerCentralSeriesLast_le_of_not_isTrivial + - LieSubmodule.ucs_le_of_normalizer_eq_self + - LieModule.nilpotencyLength_eq_zero_iff + - LieModule.coe_lcs_range_toEndomorphism_eq + - LieModule.disjoint_lowerCentralSeries_maxTrivSubmodule_iff + - LieAlgebra.nilpotent_of_nilpotent_quotient + - LieSubmodule.lowerCentralSeries_tensor_eq_baseChange + - LieModule.maxGenEigenSpace_toEndomorphism_eq_top + - LieIdeal.coe_lcs_eq + - LieModule.derivedSeries_le_lowerCentralSeries + - Function.Surjective.lieAlgebra_isNilpotent + - LieSubmodule.ucs_bot_one + - "LieModule.isNilpotent_toEndomorphism_of_isNilpotent\u2082" + - LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot + - LieModule.coe_lowerCentralSeries_ideal_le + - LieModule.isNilpotent_quotient_iff + - LieModule.map_lowerCentralSeries_le + - LieSubmodule.ucs_eq_self_of_normalizer_eq_self + - LieSubmodule.lowerCentralSeries_map_eq_lcs + - LieModule.isNilpotent_range_toEndomorphism_iff + - LieIdeal.lowerCentralSeries_map_eq + - LieIdeal.map_lowerCentralSeries_le + - LieModule.nilpotencyLength_eq_succ_iff + - LieModule.eventually_iInf_lowerCentralSeries_eq + - LieModule.map_lowerCentralSeries_eq + - LieSubmodule.isNilpotent_iff_exists_self_le_ucs + - LieModule.isNilpotent_toEndomorphism_of_isNilpotent + - Function.Surjective.lieModule_lcs_map_eq + - LieModule.nilpotencyLength_eq_one_iff + - LieEquiv.nilpotent_iff_equiv_nilpotent + - LieModule.trivial_iff_lower_central_eq_bot + - Function.Injective.lieAlgebra_isNilpotent + - LieModule.exists_forall_pow_toEndomorphism_eq_zero + - LieSubmodule.lcs_add_le_iff + - LieModule.iterate_toEndomorphism_mem_lowerCentralSeries + - LieSubmodule.ucs_eq_top_iff + - "LieModule.iterate_toEndomorphism_mem_lowerCentralSeries\u2082" + - LieModule.nilpotentOfNilpotentQuotient + - LieSubmodule.lcs_sup + - LieSubmodule.lcs_le_iff + - LieAlgebra.ad_nilpotent_of_nilpotent + - Function.Surjective.lieModuleIsNilpotent + - LieSubmodule.ucs_comap_incl + - LieModule.iInf_lcs_le_of_isNilpotent_quot + - LieSubalgebra.isNilpotent_ad_of_isNilpotent_ad + - coe_lowerCentralSeries_ideal_quot_eq + - LieSubmodule.lcs_le_self + - LieModule.nontrivial_lowerCentralSeriesLast + - LieSubmodule.ucs_mono + - Equiv.lieModule_isNilpotent_iff + - LieModule.lowerCentralSeriesLast_le_max_triv + - LieModule.antitone_lowerCentralSeries + - LieModule.iInf_lowerCentralSeries_eq_bot_of_isNilpotent + - LieAlgebra.isNilpotent_range_ad_iff + - path: Mathlib/Topology/Order.lean + theorems: + - continuous_generateFrom_iff + - nhds_false + - continuous_inf_rng + - mem_nhds_induced + - TopologicalSpace.isOpen_top_iff + - discreteTopology_iff_nhds_ne + - le_induced_generateFrom + - tendsto_nhds_Prop + - discreteTopology_iff_nhds + - continuous_discrete_rng + - continuous_sSup_dom + - isClosed_induced_iff' + - isOpen_iSup_iff + - le_nhdsAdjoint_iff + - dense_discrete + - continuous_sup_dom + - TopologicalSpace.nhds_mkOfNhds_single + - map_nhds_induced_of_surjective + - gc_nhds + - denseRange_discrete + - continuous_iSup_dom + - closure_induced + - induced_iff_nhds_eq + - le_of_nhds_le_nhds + - continuous_le_rng + - le_nhdsAdjoint_iff' + - continuous_iInf_rng + - TopologicalSpace.nhds_mkOfNhds_of_hasBasis + - Equiv.induced_symm + - TopologicalSpace.nhds_generateFrom + - map_nhds_induced_of_mem + - mem_nhds_discrete + - isClosed_iSup_iff + - continuous_le_dom + - isClosed_induced_iff + - tendsto_nhds_true + - nhds_nhdsAdjoint_same + - continuous_coinduced_dom + - TopologicalSpace.tendsto_nhds_generateFrom_iff + - map_nhds_induced_eq + - continuous_Prop + - preimage_nhds_coinduced + - continuous_induced_rng + - discreteTopology_iff_singleton_mem_nhds + - continuous_sInf_rng + - nhds_induced + - path: Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean + theorems: + - SimplicialObject.Splitting.comp_PInfty_eq_zero_iff + - "SimplicialObject.Splitting.\u03B9Summand_comp_d_comp_\u03C0Summand_eq_zero" + - SimplicialObject.Splitting.decomposition_id + - SimplicialObject.Splitting.cofan_inj_comp_PInfty_eq_zero + - "SimplicialObject.Splitting.\u03C3_comp_\u03C0Summand_id_eq_zero" + - "SimplicialObject.Splitting.\u03C0Summand_comp_cofan_inj_id_comp_PInfty_eq_PInfty" + - "SimplicialObject.Splitting.PInfty_comp_\u03C0Summand_id" + - "SimplicialObject.Splitting.cofan_inj_\u03C0Summand_eq_zero" + - path: Mathlib/MeasureTheory/Function/SimpleFunc.lean + theorems: + - MeasureTheory.SimpleFunc.restrict_preimage + - MeasureTheory.SimpleFunc.le_sup_lintegral + - MeasureTheory.SimpleFunc.range_const + - MeasureTheory.SimpleFunc.lintegral_eq_of_subset + - MeasureTheory.SimpleFunc.map_lintegral + - MeasureTheory.SimpleFunc.bind_const + - MeasureTheory.SimpleFunc.lintegral_eq_of_measure_preimage + - Measurable.ennreal_induction + - MeasureTheory.SimpleFunc.range_one + - MeasureTheory.SimpleFunc.const_lintegral + - MeasureTheory.SimpleFunc.piecewise_compl + - MeasureTheory.SimpleFunc.ennrealRatEmbed_encode + - MeasureTheory.SimpleFunc.forall_mem_range + - MeasureTheory.SimpleFunc.coe_injective + - MeasureTheory.SimpleFunc.sum_eapproxDiff + - Measurable.simpleFunc_add + - MeasureTheory.SimpleFunc.restrict_empty + - MeasureTheory.SimpleFunc.range_map + - MeasureTheory.SimpleFunc.restrict_apply + - MeasureTheory.SimpleFunc.range_indicator + - MeasureTheory.SimpleFunc.support_eq + - MeasureTheory.SimpleFunc.tsum_eapproxDiff + - MeasureTheory.SimpleFunc.restrict_const_lintegral + - MeasureTheory.SimpleFunc.finset_sup_apply + - MeasureTheory.SimpleFunc.FinMeasSupp.mul + - MeasureTheory.SimpleFunc.approx_comp + - MeasureTheory.SimpleFunc.induction + - MeasureTheory.SimpleFunc.measurableSet_cut + - MeasureTheory.SimpleFunc.mem_restrict_range + - MeasureTheory.SimpleFunc.FinMeasSupp.of_lintegral_ne_top + - MeasureTheory.SimpleFunc.lintegral_sum + - MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty + - MeasureTheory.SimpleFunc.range_const_subset + - MeasureTheory.SimpleFunc.coe_restrict + - MeasureTheory.SimpleFunc.piecewise_univ + - MeasureTheory.SimpleFunc.iSup_approx_apply + - MeasureTheory.SimpleFunc.measurableSet_support + - MeasureTheory.SimpleFunc.lintegral_map' + - MeasureTheory.SimpleFunc.iSup_eapprox_apply + - MeasureTheory.SimpleFunc.const_lintegral_restrict + - MeasureTheory.SimpleFunc.restrict_mono + - MeasureTheory.SimpleFunc.pair_preimage_singleton + - MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict + - MeasureTheory.SimpleFunc.finMeasSupp_iff + - MeasureTheory.SimpleFunc.lintegral_congr + - MeasureTheory.SimpleFunc.lintegral_restrict + - MeasureTheory.SimpleFunc.eapprox_lt_top + - MeasureTheory.SimpleFunc.exists_range_iff + - MeasureTheory.SimpleFunc.restrict_lintegral + - MeasureTheory.SimpleFunc.map_preimage + - MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_top + - MeasureTheory.SimpleFunc.map_restrict_of_zero + - MeasureTheory.SimpleFunc.range_comp_subset_range + - MeasureTheory.SimpleFunc.simpleFunc_bot + - MeasureTheory.SimpleFunc.lintegral_mono + - MeasureTheory.SimpleFunc.sum_range_measure_preimage_singleton + - MeasureTheory.SimpleFunc.restrict_univ + - MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict + - MeasureTheory.SimpleFunc.add_lintegral + - MeasureTheory.SimpleFunc.piecewise_empty + - Measurable.add_simpleFunc + - MeasureTheory.SimpleFunc.const_mul_lintegral + - path: Mathlib/Analysis/InnerProductSpace/Calculus.lean + theorems: + - HasDerivAt.norm_sq + - HasDerivWithinAt.inner + - differentiableWithinAt_euclidean + - contDiffAt_euclidean + - differentiable_euclidean + - hasStrictFDerivAt_norm_sq + - ContDiffAt.dist + - contDiff_norm_sq + - PartialHomeomorph.contDiffOn_univUnitBall_symm + - fderiv_inner_apply + - DifferentiableAt.dist + - PartialHomeomorph.contDiffOn_univBall_symm + - PartialHomeomorph.contDiff_univUnitBall + - hasStrictFDerivAt_euclidean + - hasFDerivWithinAt_euclidean + - ContDiffWithinAt.dist + - contDiffOn_euclidean + - DifferentiableWithinAt.dist + - contDiffWithinAt_euclidean + - differentiableOn_euclidean + - contDiffAt_norm + - PartialHomeomorph.contDiff_univBall + - HasDerivAt.inner + - contDiff_euclidean + - differentiableAt_euclidean + - path: Mathlib/Analysis/Convex/Side.lean + theorems: + - AffineSubspace.sOppSide_pointReflection + - AffineSubspace.isPreconnected_setOf_sOppSide + - Function.Injective.sSameSide_map_iff + - AffineSubspace.wOppSide_comm + - AffineSubspace.wOppSide_self_iff + - "Wbtw.wOppSide\u2081\u2083" + - AffineSubspace.wOppSide_vadd_left_iff + - AffineSubspace.WSameSide.not_sOppSide + - AffineSubspace.WSameSide.trans_wOppSide + - AffineSubspace.sSameSide_smul_vsub_vadd_left + - AffineSubspace.isPreconnected_setOf_wOppSide + - Function.Injective.wOppSide_map_iff + - AffineSubspace.SOppSide.exists_sbtw + - AffineSubspace.wSameSide_iff_exists_right + - AffineSubspace.isConnected_setOf_sSameSide + - AffineSubspace.isPreconnected_setOf_sSameSide + - AffineSubspace.wOppSide_of_left_mem + - AffineSubspace.WOppSide.map + - AffineSubspace.sOppSide_smul_vsub_vadd_left + - AffineSubspace.setOf_wSameSide_eq_image2 + - AffineSubspace.sOppSide_vadd_right_iff + - AffineSubspace.WSameSide.trans + - AffineSubspace.setOf_wOppSide_eq_image2 + - AffineSubspace.sSameSide_comm + - AffineSubspace.wOppSide_smul_vsub_vadd_left + - AffineSubspace.WSameSide.map + - AffineSubspace.WOppSide.trans + - AffineSubspace.setOf_sSameSide_eq_image2 + - AffineSubspace.wSameSide_vadd_right_iff + - AffineSubspace.wSameSide_iff_exists_left + - AffineSubspace.wSameSide_of_left_mem + - Function.Injective.wSameSide_map_iff + - AffineSubspace.isConnected_setOf_wSameSide + - AffineSubspace.wSameSide_smul_vsub_vadd_left + - AffineSubspace.sOppSide_iff_exists_right + - AffineSubspace.sSameSide_iff_exists_right + - AffineSubspace.wSameSide_and_wOppSide_iff + - AffineSubspace.isConnected_setOf_wOppSide + - AffineSubspace.not_sOppSide_self + - AffineSubspace.sSameSide_vadd_left_iff + - AffineSubspace.wOppSide_iff_exists_wbtw + - AffineSubspace.sOppSide_comm + - AffineSubspace.isConnected_setOf_sOppSide + - Sbtw.sOppSide_of_not_mem_of_mem + - AffineSubspace.setOf_sOppSide_eq_image2 + - Function.Injective.sOppSide_map_iff + - AffineSubspace.SSameSide.not_wOppSide + - AffineSubspace.sOppSide_iff_exists_left + - AffineSubspace.wSameSide_vadd_left_iff + - AffineSubspace.wOppSide_vadd_right_iff + - AffineSubspace.isPreconnected_setOf_wSameSide + - AffineSubspace.sOppSide_vadd_left_iff + - "Wbtw.wSameSide\u2082\u2083" + - AffineSubspace.wOppSide_iff_exists_left + - path: Mathlib/FieldTheory/PurelyInseparable.lean + theorems: + - IntermediateField.adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable + - IsPurelyInseparable.injective_comp_algebraMap + - map_mem_perfectClosure_iff + - IsPurelyInseparable.tower_bot + - perfectField_of_perfectClosure_eq_bot + - separableClosure.adjoin_eq_of_isAlgebraic + - Polynomial.Separable.map_irreducible_of_isPurelyInseparable + - isPurelyInseparable_iff_minpoly_eq_X_pow_sub_C + - isPurelyInseparable_iff_pow_mem + - le_perfectClosure + - AlgEquiv.isPurelyInseparable + - separableClosure_le + - IntermediateField.isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one + - IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable + - minpoly.map_eq_of_separable_of_isPurelyInseparable + - Field.sepDegree_eq_of_isPurelyInseparable_of_isSeparable + - Field.finSepDegree_eq + - IsPurelyInseparable.tower_top + - Field.sepDegree_mul_sepDegree_of_isAlgebraic + - mem_perfectClosure_iff_pow_mem + - separableClosure.eq_bot_iff + - separableClosure.isPurelyInseparable + - isPurelyInseparable_iff_fd_isPurelyInseparable + - LinearIndependent.map_pow_expChar_pow_of_isSeparable + - Field.span_map_pow_expChar_pow_eq_top_of_isSeparable + - separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable + - LinearIndependent.map_of_isPurelyInseparable_of_separable + - LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable + - IntermediateField.isPurelyInseparable_adjoin_iff_pow_mem + - IsPurelyInseparable.trans + - eq_separableClosure_iff + - Field.finSepDegree_mul_finInsepDegree + - isPurelyInseparable_iff_perfectClosure_eq_top + - IntermediateField.isPurelyInseparable_adjoin_simple_iff_pow_mem + - isSepClosed_iff_isPurelyInseparable_algebraicClosure + - Field.lift_rank_mul_lift_sepDegree_of_isSeparable + - IsPurelyInseparable.insepDegree_eq + - mem_perfectClosure_iff_natSepDegree_eq_one + - le_perfectClosure_iff + - IntermediateField.sepDegree_adjoin_eq_of_isAlgebraic_of_isPurelyInseparable' + - IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparable + - LinearIndependent.map_pow_expChar_pow_of_separable + - isPurelyInseparable_iff_minpoly_eq_X_sub_C_pow + - Field.sepDegree_eq_of_isPurelyInseparable + - isPurelyInseparable_iff_natSepDegree_eq_one + - Field.lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic + - IsPurelyInseparable.sepDegree_eq_one + - Field.rank_mul_sepDegree_of_isSeparable + - separableClosure_le_iff + - perfectClosure.comap_eq_of_algHom + - path: Mathlib/Algebra/Lie/Solvable.lean + theorems: + - LieAlgebra.derivedSeriesOfIdeal_add + - LieAlgebra.derivedSeriesOfIdeal_add_le_add + - LieAlgebra.abelian_iff_derived_succ_eq_bot + - Function.Surjective.lieAlgebra_isSolvable + - LieAlgebra.solvable_iff_equiv_solvable + - LieAlgebra.abelian_iff_derived_one_eq_bot + - LieAlgebra.derivedLength_zero + - LieAlgebra.derivedSeries_of_derivedLength_succ + - LieAlgebra.radical_eq_top_of_isSolvable + - LieAlgebra.abelian_of_solvable_ideal_eq_bot_iff + - LieIdeal.derivedSeries_eq_bot_iff + - LieIdeal.derivedSeries_eq_derivedSeriesOfIdeal_comap + - Function.Injective.lieAlgebra_isSolvable + - LieIdeal.derivedSeries_add_eq_bot + - LieAlgebra.derivedSeries_of_bot_eq_bot + - LieAlgebra.derivedLength_eq_derivedLengthOfIdeal + - LieIdeal.derivedSeries_eq_derivedSeriesOfIdeal_map + - LieIdeal.derivedSeries_map_eq + - LieAlgebra.abelian_derivedAbelianOfIdeal + - LieIdeal.derivedSeries_map_le + - path: Mathlib/RingTheory/RootsOfUnity/Basic.lean + theorems: + - IsPrimitiveRoot.zero + - IsPrimitiveRoot.pow_of_dvd + - IsPrimitiveRoot.mk_of_lt + - IsPrimitiveRoot.primitiveRoots_one + - IsPrimitiveRoot.injOn_pow_mul + - isPrimitiveRoot_of_mem_primitiveRoots + - IsPrimitiveRoot.zpow_eq_one_iff_dvd + - IsPrimitiveRoot.pow_iff_coprime + - IsPrimitiveRoot.eq_pow_of_mem_rootsOfUnity + - IsPrimitiveRoot.card_primitiveRoots + - IsPrimitiveRoot.zpow_of_gcd_eq_one + - IsPrimitiveRoot.card_nthRootsFinset + - IsPrimitiveRoot.coe_submonoidClass_iff + - rootsOfUnity_one + - IsPrimitiveRoot.nthRoots_one_eq_biUnion_primitiveRoots + - IsPrimitiveRoot.isPrimitiveRoot_iff' + - IsPrimitiveRoot.inv + - IsPrimitiveRoot.card_nthRoots_one + - IsPrimitiveRoot.pow_eq_one_iff_dvd + - IsPrimitiveRoot.iff + - IsPrimitiveRoot.nthRoots_eq + - IsPrimitiveRoot.isPrimitiveRoot_iff + - IsPrimitiveRoot.one_right_iff + - IsPrimitiveRoot.inv_iff + - IsPrimitiveRoot.pow + - IsPrimitiveRoot.zmodEquivZPowers_apply_coe_nat + - IsPrimitiveRoot.zpowers_eq + - IsPrimitiveRoot.zpow_eq_one + - IsPrimitiveRoot.pow_of_coprime + - mem_rootsOfUnity' + - IsPrimitiveRoot.map_rootsOfUnity + - IsPrimitiveRoot.card_rootsOfUnity' + - IsPrimitiveRoot.nthRoots_nodup + - mem_rootsOfUnity_prime_pow_mul_iff' + - IsPrimitiveRoot.card_rootsOfUnity + - IsPrimitiveRoot.zmodEquivZPowers_symm_apply_zpow + - IsPrimitiveRoot.pow_sub_one_eq + - IsPrimitiveRoot.injOn_pow + - rootsOfUnity_le_of_dvd + - rootsOfUnityEquivOfPrimitiveRoots_symm_apply + - IsPrimitiveRoot.neZero' + - IsPrimitiveRoot.eq_pow_of_pow_eq_one + - IsPrimitiveRoot.map_of_injective + - IsPrimitiveRoot.geom_sum_eq_zero + - primitiveRoots_zero + - IsPrimitiveRoot.autToPow_spec + - map_rootsOfUnity_eq_pow_self + - mem_rootsOfUnity_iff_mem_nthRoots + - map_rootsOfUnity + - IsPrimitiveRoot.zmodEquivZPowers_apply_coe_int + - mem_rootsOfUnity_prime_pow_mul_iff + - IsPrimitiveRoot.card_nthRoots + - IsPrimitiveRoot.eq_neg_one_of_two_right + - mem_primitiveRoots + - IsPrimitiveRoot.zmodEquivZPowers_symm_apply_pow + - IsPrimitiveRoot.isUnit + - IsPrimitiveRoot.pow_inj + - IsPrimitiveRoot.coe_units_iff + - IsPrimitiveRoot.neg_one + - IsPrimitiveRoot.nthRoots_one_eq_biUnion_primitiveRoots' + - rootsOfUnity.coe_pow + - path: Mathlib/RingTheory/Valuation/Integers.lean + theorems: + - Valuation.Integers.dvd_of_le + - Valuation.Integers.one_of_isUnit + - Valuation.Integers.le_of_dvd + - Valuation.Integers.isUnit_of_one + - Valuation.mem_integer_iff + - path: Mathlib/CategoryTheory/Bicategory/Adjunction.lean + theorems: + - CategoryTheory.Bicategory.adjointifyCounit_left_triangle + - CategoryTheory.Bicategory.rightZigzagIso_inv + - CategoryTheory.Bicategory.Adjunction.comp_right_triangle_aux + - CategoryTheory.Bicategory.leftZigzagIso_inv + - CategoryTheory.Bicategory.Adjunction.comp_left_triangle_aux + - CategoryTheory.Bicategory.rightZigzag_idempotent_of_left_triangle + - path: Mathlib/Topology/Algebra/Constructions.lean + theorems: + - Units.embedding_val_mk' + - path: Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean + theorems: + - PartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq + - map_extChartAt_nhdsWithin_eq_image' + - ModelWithCorners.symm_map_nhdsWithin_image + - ModelWithCorners.symm_map_nhdsWithin_range + - PartialHomeomorph.extend_target_subset_range + - writtenInExtChartAt_extChartAt_symm + - PartialHomeomorph.continuousAt_extend_symm + - extChartAt_target_subset_range + - PartialHomeomorph.extend_source_mem_nhds + - extChartAt_preimage_mem_nhdsWithin' + - mem_analyticGroupoid_of_boundaryless + - contDiffGroupoid_le + - contDiffGroupoid_zero_eq + - PartialHomeomorph.continuousOn_extend + - ModelWithCorners.left_inv + - ModelWithCorners.image_eq + - extChartAt_image_nhd_mem_nhds_of_boundaryless + - extChartAt_model_space_eq_id + - ModelWithCorners.locallyCompactSpace + - PartialHomeomorph.extend_preimage_mem_nhdsWithin + - writtenInExtChartAt_extChartAt + - PartialHomeomorph.map_extend_nhds_of_boundaryless + - PartialHomeomorph.extend_coord_change_source_mem_nhdsWithin + - writtenInExtChartAt_chartAt_symm_comp + - modelWithCornersSelf_prod + - PartialHomeomorph.map_extend_symm_nhdsWithin_range + - extChartAt_target_mem_nhds + - writtenInExtChartAt_chartAt_symm + - PartialHomeomorph.continuousOn_writtenInExtend_iff + - isOpen_extChartAt_target + - PartialHomeomorph.extend_target_mem_nhdsWithin + - isOpen_extChartAt_preimage + - extChartAt_preimage_mem_nhds' + - PartialHomeomorph.extend_image_source_inter + - OpenEmbedding.singleton_smoothManifoldWithCorners + - PartialHomeomorph.interior_extend_target_subset_interior_range + - PartialHomeomorph.extend_target' + - PartialHomeomorph.extend_left_inv + - PartialHomeomorph.isOpen_extend_source + - PartialHomeomorph.extend_source + - map_extChartAt_symm_nhdsWithin' + - ModelWithCorners.range_prod + - ModelWithCorners.symm_continuousWithinAt_comp_right_iff + - PartialHomeomorph.contDiffWithinAt_extend_coord_change + - PartialHomeomorph.extend_symm_continuousWithinAt_comp_right_iff + - extChartAt_preimage_mem_nhds + - extChartAt_source_mem_nhds' + - PartialHomeomorph.mem_interior_extend_target + - contDiffGroupoid_prod + - PartialHomeomorph.map_extend_symm_nhdsWithin + - ofSet_mem_contDiffGroupoid + - PartialHomeomorph.tendsto_extend_comp_iff + - extChartAt_prod + - PartialHomeomorph.extend_preimage_inter_eq + - mem_extChartAt_source + - PartialHomeomorph.extend_preimage_mem_nhds + - PartialHomeomorph.map_extend_nhdsWithin + - PartialHomeomorph.map_extend_nhdsWithin_eq_image + - extChartAt_target_mem_nhdsWithin' + - PartialHomeomorph.map_extend_nhdsWithin_eq_image_of_subset + - PartialHomeomorph.extend_coord_change_source_mem_nhdsWithin' + - writtenInExtChartAt_chartAt_comp + - Manifold.locallyCompact_of_finiteDimensional + - PartialHomeomorph.map_extend_nhds + - PartialHomeomorph.isOpen_extend_target + - PartialHomeomorph.continuousWithinAt_writtenInExtend_iff + - writtenInExtChartAt_chartAt + - map_extChartAt_nhds_of_boundaryless + - PartialHomeomorph.extend_coord_change_source + - PartialHomeomorph.mapsTo_extend + - continuousAt_extChartAt' + - PartialHomeomorph.extend_target + - map_extChartAt_symm_nhdsWithin_range' + - smoothManifoldWithCorners_of_contDiffOn + - extChartAt_preimage_inter_eq + - nhdsWithin_extChartAt_target_eq' + - ModelWithCorners.target_eq + - PartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq_aux + - map_extChartAt_nhdsWithin' + - uniqueDiffOn_extChartAt_target + - PartialHomeomorph.contDiffWithinAt_extend_coord_change' + - ModelWithCorners.unique_diff_preimage + - PartialHomeomorph.isOpen_extend_preimage + - ofSet_mem_analyticGroupoid + - ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range + - map_extChartAt_nhds' + - path: Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean + theorems: + - SimpleGraph.extend_finset_to_connected + - SimpleGraph.Subgraph.Connected.nonempty + - SimpleGraph.Subgraph.induce_union_connected + - SimpleGraph.Subgraph.Connected.adj_union + - SimpleGraph.induce_connected_adj_union + - SimpleGraph.Subgraph.Connected.mono' + - SimpleGraph.induce_union_connected + - SimpleGraph.Walk.connected_induce_support + - SimpleGraph.Subgraph.Connected.preconnected + - SimpleGraph.induce_connected_of_patches + - SimpleGraph.Walk.toSubgraph_connected + - SimpleGraph.Subgraph.connected_iff + - SimpleGraph.induce_sUnion_connected_of_pairwise_not_disjoint + - SimpleGraph.Subgraph.connected_iff_forall_exists_walk_subgraph + - SimpleGraph.Subgraph.Connected.mono + - SimpleGraph.Subgraph.Connected.induce_verts + - SimpleGraph.induce_pair_connected_of_adj + - SimpleGraph.Subgraph.top_induce_pair_connected_of_adj + - SimpleGraph.Subgraph.Connected.sup + - SimpleGraph.Subgraph.subgraphOfAdj_connected + - SimpleGraph.Subgraph.singletonSubgraph_connected + - SimpleGraph.connected_induce_iff + - SimpleGraph.Subgraph.preconnected_iff_forall_exists_walk_subgraph + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean + theorems: + - Real.Angle.toReal_neg_iff_sign_neg + - Real.Angle.two_zsmul_toReal_eq_two_mul_sub_two_pi + - Real.Angle.toReal_eq_zero_iff + - Real.Angle.sin_ne_zero_iff + - Real.Angle.coe_abs_toReal_of_sign_nonneg + - Real.Angle.cos_pos_iff_abs_toReal_lt_pi_div_two + - Real.Angle.toReal_neg_pi_div_two + - Real.Angle.sin_toReal + - Real.Angle.cos_pi_div_two_sub + - Real.Angle.toReal_eq_neg_pi_div_two_iff + - Real.Angle.tan_coe + - Real.Angle.sign_coe_pi + - Real.Angle.toReal_coe_eq_self_iff_mem_Ioc + - Real.Angle.neg_coe_pi + - Real.Angle.two_zsmul_eq_iff + - Real.Angle.eq_iff_abs_toReal_eq_of_sign_eq + - Real.Angle.abs_sin_eq_of_two_nsmul_eq + - Real.Angle.sin_coe_pi + - Real.Angle.cos_nonneg_iff_abs_toReal_le_pi_div_two + - Real.Angle.two_nsmul_toReal_eq_two_mul_add_two_pi + - Real.Angle.toReal_zero + - Real.Angle.sin_eq_real_sin_iff_eq_or_add_eq_pi + - Real.Angle.cos_coe_pi + - Real.Angle.tan_eq_of_two_nsmul_eq + - Real.Angle.coe_int_mul_eq_zsmul + - Real.Angle.cos_eq_real_cos_iff_eq_or_eq_neg + - Real.Angle.tan_eq_inv_of_two_zsmul_add_two_zsmul_eq_pi + - Real.Angle.sign_ne_zero_iff + - Real.Angle.toReal_injective + - Real.Angle.two_zsmul_neg_pi_div_two + - Real.Angle.two_nsmul_neg_pi_div_two + - Real.Angle.cos_neg + - Real.Angle.coe_nat_mul_eq_nsmul + - Real.Angle.cos_eq_iff_eq_or_eq_neg + - Real.Angle.tan_toReal + - Real.Angle.sign_two_zsmul_eq_sign_iff + - Real.Angle.eq_neg_self_iff + - Real.Angle.cos_add + - Real.Angle.two_nsmul_ne_zero_iff + - Real.Angle.two_zsmul_toReal_eq_two_mul_add_two_pi + - Real.Angle.two_zsmul_coe_div_two + - Real.Angle.sign_antiperiodic + - Real.Angle.cos_neg_iff_pi_div_two_lt_abs_toReal + - Real.Angle.toReal_pi + - Real.Angle.toReal_le_pi + - Real.Angle.coe_toReal + - Real.Angle.sign_neg + - Real.Angle.cos_sin_inj + - Real.Angle.abs_cos_eq_of_two_nsmul_eq + - Real.Angle.toReal_coe_eq_self_sub_two_pi_iff + - Real.Angle.toReal_coe_eq_self_iff + - Real.Angle.neg_ne_self_iff + - Real.Angle.tan_periodic + - Real.Angle.two_nsmul_coe_div_two + - Real.Angle.sin_eq_iff_eq_or_add_eq_pi + - Real.Angle.pi_div_two_ne_zero + - Real.Angle.sign_coe_pi_div_two + - Real.Angle.abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi + - Real.Angle.toIocMod_toReal + - Real.Angle.cos_sub_pi_div_two + - Real.Angle.sin_pi_div_two_sub + - Real.Angle.sin_eq_iff_coe_eq_or_add_eq_pi + - Real.Angle.two_zsmul_ne_zero_iff + - Real.Angle.sign_toReal + - Real.Angle.sign_neg_coe_nonpos_of_nonneg_of_le_pi + - Real.Angle.toReal_pi_div_two + - Real.Angle.coe_toIcoMod + - Real.Angle.two_zsmul_eq_zero_iff + - Real.Angle.tan_eq_of_two_zsmul_eq + - Real.Angle.toReal_nonneg_iff_sign_nonneg + - Real.Angle.sin_zero + - Real.Angle.toReal_eq_pi_div_two_iff + - Real.Angle.abs_sin_eq_of_two_zsmul_eq + - Real.Angle.neg_pi_lt_toReal + - Real.Angle.nsmul_toReal_eq_mul + - Real.Angle.two_nsmul_eq_iff + - Real.Angle.cos_eq_iff_coe_eq_or_eq_neg + - Real.Angle.two_zsmul_coe_pi + - Real.Angle.tan_zero + - Real.Angle.coe_pi_add_coe_pi + - Real.Angle.toReal_eq_pi_iff + - Real.Angle.sin_neg + - Real.Angle.sin_antiperiodic + - Real.Angle.sign_coe_neg_pi_div_two + - Real.Angle.cos_add_pi_div_two + - Real.Angle.abs_cos_eq_abs_sin_of_two_nsmul_add_two_nsmul_eq_pi + - Real.Angle.two_zsmul_toReal_eq_two_mul + - Real.Angle.sin_add_pi_div_two + - Real.Angle.tan_coe_pi + - Real.Angle.cos_sq_add_sin_sq + - Real.Angle.angle_eq_iff_two_pi_dvd_sub + - Real.Angle.sin_sub_pi_div_two + - Real.Angle.cos_eq_zero_iff + - Real.Angle.abs_toReal_eq_pi_div_two_iff + - Real.Angle.cos_zero + - Real.Angle.two_nsmul_eq_pi_iff + - Real.Angle.two_nsmul_eq_zero_iff + - Real.Angle.two_zsmul_eq_pi_iff + - Real.Angle.sign_pi_add + - Real.Angle.tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi + - Real.Angle.pi_ne_zero + - Real.Angle.sign_pi_sub + - Real.Angle.toReal_coe_eq_self_add_two_pi_iff + - Real.Angle.abs_toReal_neg_coe_eq_self_iff + - Real.Angle.coe_toIocMod + - Real.Angle.two_nsmul_coe_pi + - Real.Angle.sin_add + - Real.Angle.sign_zero + - Real.Angle.neg_pi_div_two_ne_zero + - Real.Angle.sign_eq_zero_iff + - ContinuousOn.angle_sign_comp + - Real.Angle.coe_two_pi + - Real.Angle.sin_eq_zero_iff + - Real.Angle.neg_eq_self_iff + - Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi + - Real.Angle.toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff + - Real.Angle.sub_coe_pi_eq_add_coe_pi + - Real.Angle.eq_iff_sign_eq_and_abs_toReal_eq + - Real.Angle.ne_neg_self_iff + - Real.Angle.sign_two_nsmul_eq_sign_iff + - Real.Angle.abs_cos_eq_of_two_zsmul_eq + - Real.Angle.cos_antiperiodic + - Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi + - path: Mathlib/Logic/Relation.lean + theorems: + - Relation.TransGen.trans_right + - Relation.church_rosser + - Relation.TransGen.lift + - Relation.reflTransGen_minimal + - Relation.map_apply_apply + - Relation.ReflTransGen.lift' + - Relation.ReflTransGen.head_induction_on + - Relation.ReflGen.mono + - Relation.ReflTransGen.swap + - Relation.reflGen_eq_self + - Relation.comp_iff + - Relation.reflTransGen_eq_self + - Reflexive.rel_of_ne_imp + - Relation.transGen_eq_self + - Relation.reflGen_transGen + - Relation.reflTransGen_reflGen + - Relation.map_id_id + - Relation.reflTransGen_transGen + - Relation.reflTransGen_of_transitive_reflexive + - Acc.of_fibration + - Relation.ReflTransGen.trans + - Relation.TransGen.trans_left + - Relation.iff_comp + - Relation.TransGen.head'_iff + - Relation.ReflTransGen.total_of_right_unique + - Relation.flip_comp + - Equivalence.eqvGen_iff + - Relation.reflTransGen_iff_eq_or_transGen + - Relation.ReflTransGen.cases_head_iff + - Relation.ReflTransGen.head + - Relation.reflTransGen_eq_transGen + - Relation.ReflTransGen.cases_head + - Relation.TransGen.tail'_iff + - Relation.reflTransGen_iff_eq + - Relation.transGen_minimal + - Relation.TransGen.swap + - Acc.TransGen + - Relation.TransGen.trans_induction_on + - Relation.reflGen_minimal + - Relation.TransGen.to_reflTransGen + - Relation.transGen_reflGen + - EqvGen.mono + - Relation.TransGen.lift' + - Relation.ReflGen.to_reflTransGen + - Relation.TransGen.tail' + - Relation.ReflTransGen.symmetric + - path: Mathlib/Algebra/Order/Hom/Monoid.lean + theorems: + - OrderMonoidWithZeroHom.cancel_left + - OrderMonoidHom.toMonoidHom_injective + - strictMono_iff_map_pos + - map_nonpos + - OrderMonoidHom.mk_coe + - OrderMonoidWithZeroHom.cancel_right + - OrderMonoidHom.toOrderHom_injective + - OrderMonoidHom.cancel_right + - OrderMonoidWithZeroHom.toMonoidWithZeroHom_eq_coe + - map_nonneg + - monotone_iff_map_nonneg + - OrderMonoidWithZeroHom.toMonoidWithZeroHom_injective + - OrderMonoidHom.cancel_left + - path: Mathlib/Topology/Bases.lean + theorems: + - TopologicalSpace.separableSpace_iff_countable + - TopologicalSpace.IsTopologicalBasis.isOpen_iff + - TopologicalSpace.secondCountableTopology_induced + - TopologicalSpace.IsTopologicalBasis.eq_of_forall_subset_iff + - TopologicalSpace.IsTopologicalBasis.inf + - TopologicalSpace.IsTopologicalBasis.inducing + - TopologicalSpace.IsTopologicalBasis.quotientMap + - TopologicalSpace.isOpen_biUnion_countable + - TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds + - TopologicalSpace.exists_dense_seq + - TopologicalSpace.IsSeparable.prod + - Dense.isSeparable_iff + - TopologicalSpace.IsTopologicalBasis.sigma + - IsTopologicalBasis.iInf_induced + - isTopologicalBasis_pi + - TopologicalSpace.isTopologicalBasis_of_subbasis + - TopologicalSpace.countable_cover_nhds + - TopologicalSpace.IsTopologicalBasis.of_hasBasis_nhds + - TopologicalSpace.isSeparable_union + - TopologicalSpace.IsTopologicalBasis.open_eq_iUnion + - TopologicalSpace.exists_countable_basis + - TopologicalSpace.IsTopologicalBasis.nhds_hasBasis + - TopologicalSpace.IsSeparable.mono + - TopologicalSpace.IsTopologicalBasis.diff_empty + - TopologicalSpace.IsTopologicalBasis.mem_nhds_iff + - QuotientMap.secondCountableTopology + - TopologicalSpace.IsTopologicalBasis.inf_induced + - TopologicalSpace.IsSeparable.image + - TopologicalSpace.isTopologicalBasis_opens + - exists_countable_dense_bot_top + - TopologicalSpace.IsTopologicalBasis.insert_empty + - TopologicalSpace.isSeparable_closure + - TopologicalSpace.isSeparable_pi + - TopologicalSpace.isTopologicalBasis_of_cover + - TopologicalSpace.IsTopologicalBasis.dense_iff + - TopologicalSpace.IsSeparable.univ_pi + - TopologicalSpace.IsTopologicalBasis.isOpenMap_iff + - TopologicalSpace.IsTopologicalBasis.isOpen + - IsTopologicalBasis.iInf + - TopologicalSpace.IsTopologicalBasis.subset_of_forall_subset + - TopologicalSpace.isOpen_sUnion_countable + - TopologicalSpace.countable_cover_nhdsWithin + - TopologicalSpace.IsTopologicalBasis.continuous_iff + - TopologicalSpace.isOpen_iUnion_countable + - TopologicalSpace.secondCountableTopology_iInf + - Dense.exists_countable_dense_subset_bot_top + - TopologicalSpace.IsTopologicalBasis.sum + - Inducing.secondCountableTopology + - TopologicalSpace.IsSeparable.iUnion + - TopologicalSpace.IsTopologicalBasis.mem_closure_iff + - path: Mathlib/LinearAlgebra/Matrix/IsDiag.lean + theorems: + - Matrix.isDiag_conjTranspose_iff + - Matrix.IsDiag.add + - Matrix.IsDiag.kronecker + - Matrix.IsDiag.smul + - Matrix.isDiag_fromBlocks_iff + - Matrix.IsDiag.neg + - Matrix.IsDiag.fromBlocks + - Matrix.IsDiag.fromBlocks_of_isSymm + - Matrix.IsDiag.sub + - Matrix.IsDiag.diagonal_diag + - Matrix.IsDiag.map + - Matrix.IsDiag.isSymm + - path: Mathlib/NumberTheory/ZetaFunction.lean + theorems: + - riemannZeta_zero + - summable_exp_neg_pi_mul_nat_sq + - riemannZeta_residue_one + - completed_zeta_eq_mellin_of_one_lt_re + - completed_zeta_eq_tsum_of_one_lt_re + - "mellin_zetaKernel\u2082_eq_of_lt_re" + - "locally_integrable_zetaKernel\u2082" + - riemannZeta_neg_two_mul_nat_add_one + - "zetaKernel\u2081_eq_jacobiTheta" + - "continuousAt_zetaKernel\u2081" + - riemannCompletedZeta_one_sub + - riemannCompletedZeta_residue_one + - "riemannCompletedZeta\u2080_one_sub" + - riemannZeta_neg_nat_eq_bernoulli + - "mellin_zetaKernel\u2081_eq_tsum" + - "isBigO_zero_zetaKernel\u2082" + - "isBigO_zero_zetaKernel\u2081" + - zeta_nat_eq_tsum_of_gt_one + - zeta_eq_tsum_one_div_nat_add_one_cpow + - "zetaKernel\u2082_one_div" + - differentiableAt_completed_zeta + - integral_cpow_mul_exp_neg_pi_mul_sq + - riemannZeta_two_mul_nat + - "isBigO_atTop_zetaKernel\u2081" + - "isBigO_zero_zetaKernel\u2082_rpow" + - differentiableAt_riemannZeta + - hasMellin_one_div_sqrt_sub_one_div_two_Ioc + - "isBigO_atTop_zetaKernel\u2082" + - riemannZeta_four + - zeta_eq_tsum_one_div_nat_cpow + - riemannZeta_two + - hasMellin_one_div_sqrt_Ioc + - path: Mathlib/Topology/Germ.lean + theorems: + - forall_restrictGermPredicate_iff + - restrictGermPredicate_congr + - Filter.Eventually.germ_congr_set + - eq_of_germ_isConstant + - IsLocallyConstant.of_germ_isConstant + - eq_of_germ_isConstant_on + - path: Mathlib/GroupTheory/Abelianization.lean + theorems: + - commutator_eq_normalClosure + - commutator_eq_closure + - card_commutator_closureCommutatorRepresentatives + - Abelianization.commutator_subset_ker + - rank_closureCommutatorRepresentatives_le + - card_commutatorSet_closureCommutatorRepresentatives + - commutator_centralizer_commutator_le_center + - rank_commutator_le_card + - image_commutatorSet_closureCommutatorRepresentatives + - path: Mathlib/LinearAlgebra/AffineSpace/Independent.lean + theorems: + - affineIndependent_iff_indicator_eq_of_affineCombination_eq + - AffineIndependent.comp_embedding + - affineCombination_mem_affineSpan_pair + - affineIndependent_iff_of_fintype + - Affine.Simplex.centroid_eq_iff + - Affine.Simplex.reindex_reindex_symm + - Affine.Simplex.face_eq_mkOfPoint + - affineIndependent_of_ne_of_mem_of_not_mem_of_mem + - affineIndependent_of_ne + - AffineIndependent.eq_zero_of_sum_eq_zero_subtype + - sign_eq_of_affineCombination_mem_affineSpan_pair + - sign_eq_of_affineCombination_mem_affineSpan_single_lineMap + - affineIndependent_set_iff_linearIndependent_vsub + - AffineIndependent.range + - AffineIndependent.injective + - AffineIndependent.mem_affineSpan_iff + - affineIndependent_of_ne_of_mem_of_mem_of_not_mem + - affineIndependent_iff_linearIndependent_vsub + - Affine.Simplex.reindex_symm_reindex + - Affine.Simplex.ext + - AffineEquiv.affineIndependent_set_of_eq_iff + - Affine.Simplex.reindex_range_points + - AffineIndependent.affineIndependent_of_not_mem_span + - Affine.Simplex.face_centroid_eq_iff + - exists_subset_affineIndependent_affineSpan_eq_top + - AffineIndependent.units_lineMap + - Affine.Simplex.range_face_points + - AffineIndependent.of_comp + - affineIndependent_of_ne_of_not_mem_of_mem_of_mem + - AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan + - affineIndependent_equiv + - affineIndependent_iff + - linearIndependent_set_iff_affineIndependent_vadd_union_singleton + - exists_nontrivial_relation_sum_zero_of_not_affine_ind + - AffineIndependent.affineSpan_disjoint_of_disjoint + - weightedVSub_mem_vectorSpan_pair + - AffineIndependent.not_mem_affineSpan_diff + - Affine.Simplex.face_centroid_eq_centroid + - exists_affineIndependent + - AffineIndependent.map' + - AffineIndependent.eq_of_sum_eq_sum + - Affine.Simplex.centroid_eq_of_range_eq + - affineIndependent_iff_eq_of_fintype_affineCombination_eq + - AffineIndependent.eq_of_sum_eq_sum_subtype + - path: Mathlib/Data/Nat/Log.lean + theorems: + - Nat.clog_of_two_le + - Nat.clog_of_right_le_one + - Nat.log_eq_zero_iff + - Nat.clog_pos + - Nat.log_div_mul_self + - Nat.pow_log_le_add_one + - Nat.log_anti_left + - Nat.clog_anti_left + - Nat.log_pos_iff + - Nat.le_log_of_pow_le + - Nat.add_pred_div_lt + - Nat.log_monotone + - Nat.log_of_one_lt_of_le + - Nat.log_div_base + - Nat.clog_of_left_le_one + - Nat.log_eq_iff + - Nat.le_pow_iff_clog_le + - Nat.log_eq_one_iff' + - Nat.pow_le_of_le_log + - Nat.pow_pred_clog_lt_self + - Nat.log_eq_of_pow_le_of_lt_pow + - Nat.log_mul_base + - Nat.pow_le_iff_le_log + - Nat.clog_pow + - Nat.clog_mono_right + - path: Mathlib/NumberTheory/LSeries/Basic.lean + theorems: + - LSeriesSummable_of_le_const_mul_rpow + - LSeriesHasSum_congr + - LSeriesSummable.congr' + - LSeries.norm_term_le_of_re_le_re + - LSeriesSummable.le_const_mul_rpow + - LSeriesSummable_of_bounded_of_one_lt_re + - LSeriesSummable.isBigO_rpow + - LSeries.norm_term_eq + - LSeriesSummable_of_isBigO_rpow + - LSeries.norm_term_le + - LSeriesSummable.of_re_le_re + - zeta_LSeriesSummable_iff_one_lt_re + - LSeries.term_congr + - LSeriesSummable.one_iff_one_lt_re + - LSeriesSummable_zero + - LSeriesSummable_of_bounded_of_one_lt_real + - path: Mathlib/Order/SymmDiff.lean + theorems: + - inf_sup_symmDiff + - bihimp_of_ge + - symmDiff_eq_top + - symmDiff_symmDiff_right' + - symmDiff_assoc + - symmDiff_self + - bihimp_eq_inf_himp_inf + - bihimp_right_comm + - bihimp_bihimp_cancel_right + - IsCompl.symmDiff_eq_top + - symmDiff_eq_right + - IsCompl.bihimp_eq_bot + - sdiff_symmDiff_right + - le_symmDiff_iff_left + - symmDiff_eq_sup + - top_bihimp + - symmDiff_eq_sup_sdiff_inf + - disjoint_symmDiff_inf + - symmDiff_eq_left + - symmDiff_top' + - Codisjoint.bihimp_eq_inf + - symmDiff_symmDiff_cancel_right + - bot_symmDiff + - bihimp_left_comm + - bihimp_self + - symmDiff_symmDiff_self' + - Disjoint.le_symmDiff_sup_symmDiff_right + - himp_bihimp + - symmDiff_bot + - symmDiff_symmDiff_cancel_left + - Disjoint.le_symmDiff_sup_symmDiff_left + - bihimp_bot + - symmDiff_sup_inf + - symmDiff_eq_bot + - symmDiff_triangle + - symmDiff_right_comm + - bihimp_top + - symmDiff_sdiff_left + - symmDiff_hnot_self + - sup_himp_bihimp + - symmDiff_le_iff + - top_symmDiff' + - hnot_symmDiff_self + - Disjoint.symmDiff_left + - inf_symmDiff_distrib_left + - bihimp_bihimp_self + - symmDiff_sdiff + - bihimp_bihimp_bihimp_comm + - sdiff_symmDiff_left + - compl_symmDiff + - symmDiff_of_le + - symmDiff_symmDiff_left + - symmDiff_sdiff_right + - symmDiff_symmDiff_inf + - le_symmDiff_iff_right + - symmDiff_of_ge + - symmDiff_symmDiff_right + - symmDiff_comm + - bot_bihimp + - Bool.symmDiff_eq_xor + - compl_symmDiff_compl + - symmDiff_sdiff_eq_sup + - symmDiff_sdiff_inf + - le_bihimp_iff + - sdiff_symmDiff_eq_sup + - sup_sdiff_symmDiff + - inf_symmDiff_symmDiff + - sdiff_symmDiff' + - bihimp_bihimp_cancel_left + - bihimp_eq_bot + - symmDiff_eq_iff_sdiff_eq + - inf_symmDiff_distrib_right + - symmDiff_symmDiff_symmDiff_comm + - symmDiff_eq' + - path: Mathlib/Algebra/Star/SelfAdjoint.lean + theorems: + - selfAdjoint.mem_iff + - skewAdjoint.mem_iff + - IsSelfAdjoint.mul_star_self + - IsSelfAdjoint.conjugate' + - skewAdjoint.smul_mem + - IsSelfAdjoint.neg + - IsSelfAdjoint.conjugate + - isSelfAdjoint_star_add_self + - IsSelfAdjoint.zpow + - IsSelfAdjoint.pow + - IsSelfAdjoint.add + - IsSelfAdjoint.star_iff + - IsSelfAdjoint.div + - skewAdjoint.bit0_mem + - IsSelfAdjoint.isStarNormal + - IsSelfAdjoint.bit1 + - IsSelfAdjoint.inv + - IsSelfAdjoint.mul + - skewAdjoint.conjugate' + - IsSelfAdjoint.sub + - IsSelfAdjoint.commute_iff + - IsSelfAdjoint.smul + - IsSelfAdjoint.bit0 + - IsSelfAdjoint.star_mul_self + - skewAdjoint.isStarNormal_of_mem + - skewAdjoint.conjugate + - isSelfAdjoint_add_star_self + - path: Mathlib/LinearAlgebra/Projection.lean + theorems: + - Submodule.linear_proj_add_linearProjOfIsCompl_eq_self + - LinearMap.IsProj.eq_conj_prod_map' + - LinearMap.ofIsCompl_smul + - Submodule.linearProjOfIsCompl_apply_eq_zero_iff + - LinearMap.IsProj.isCompl + - Submodule.prodEquivOfIsCompl_symm_apply_left + - LinearMap.ofIsCompl_add + - LinearMap.ofIsCompl_eq + - LinearMap.IsProj.codRestrict_apply_cod + - LinearMap.ofIsCompl_right_apply + - Submodule.linearProjOfIsCompl_apply_left + - Submodule.prodEquivOfIsCompl_symm_apply_right + - LinearMap.IsProj.eq_conj_prodMap + - LinearMap.ker_id_sub_eq_of_proj + - LinearMap.isProj_iff_idempotent + - Submodule.prodEquivOfIsCompl_symm_apply_fst_eq_zero + - Submodule.prodEquivOfIsCompl_symm_apply_snd_eq_zero + - LinearMap.linearProjOfIsCompl_of_proj + - LinearMap.isCompl_of_proj + - LinearMap.ofIsCompl_left_apply + - path: Mathlib/NumberTheory/ZetaValues.lean + theorems: + - hasSum_one_div_nat_pow_mul_fourier + - bernoulliFun_eval_one + - bernoulliFourierCoeff_recurrence + - hasDerivAt_bernoulliFun + - bernoulliFun_endpoints_eq_of_ne_one + - antideriv_bernoulliFun + - hasSum_L_function_mod_four_eval_three + - hasSum_one_div_nat_pow_mul_cos + - hasSum_zeta_four + - hasSum_one_div_pow_mul_fourier_mul_bernoulliFun + - Polynomial.bernoulli_three_eval_one_quarter + - hasSum_one_div_nat_pow_mul_sin + - fourierCoeff_bernoulli_eq + - hasSum_zeta_two + - integral_bernoulliFun_eq_zero + - bernoulliFourierCoeff_eq + - summable_bernoulli_fourier + - bernoulli_zero_fourier_coeff + - bernoulliFourierCoeff_zero + - bernoulliFun_eval_zero + - hasSum_zeta_nat + - path: Mathlib/Algebra/Lie/Weights/Chain.lean + theorems: + - LieModule.lie_mem_weightSpaceChain_of_weightSpace_eq_bot_right + - LieModule.exists_forall_mem_rootSpaceProductNegSelf_smul_add_eq_zero + - LieModule.lie_mem_weightSpaceChain_of_weightSpace_eq_bot_left + - LieModule.eventually_weightSpace_smul_add_eq_bot + - LieModule.trace_toEndomorphism_weightSpaceChain_eq_zero + - "LieModule.exists\u2082_weightSpace_smul_add_eq_bot" + - LieModule.weightSpaceChain_def' + - path: Mathlib/CategoryTheory/Bicategory/Coherence.lean + theorems: + - CategoryTheory.FreeBicategory.normalize_naturality + - CategoryTheory.FreeBicategory.normalizeAux_congr + - "CategoryTheory.FreeBicategory.preinclusion_map\u2082" + - CategoryTheory.FreeBicategory.normalizeAux_nil_comp + - path: Mathlib/Analysis/Convex/Mul.lean + theorems: + - ConvexOn.smul' + - Even.convexOn_pow + - ConcaveOn.smul_convexOn + - ConvexOn.smul_concaveOn + - ConcaveOn.smul'' + - ConcaveOn.smul' + - convexOn_zpow + - ConvexOn.pow + - ConvexOn.smul'' + - ConcaveOn.smul_convexOn' + - ConvexOn.smul_concaveOn' + - path: Mathlib/Analysis/Calculus/Rademacher.lean + theorems: + - LipschitzWith.ae_differentiableAt + - LipschitzWith.hasFderivAt_of_hasLineDerivAt_of_closure + - LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul' + - LipschitzOnWith.ae_differentiableWithinAt_of_mem_pi + - LipschitzOnWith.ae_differentiableWithinAt_of_mem_of_real + - LipschitzWith.ae_lineDeriv_sum_eq + - LipschitzOnWith.ae_differentiableWithinAt_of_mem + - LipschitzWith.ae_differentiableAt_of_real + - LipschitzWith.ae_exists_fderiv_of_countable + - LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul + - LipschitzWith.integral_lineDeriv_mul_eq + - LipschitzWith.ae_lineDifferentiableAt + - LipschitzOnWith.ae_differentiableWithinAt + - path: Mathlib/Data/Part.lean + theorems: + - Part.eq_none_iff + - Part.bind_le + - Part.ext' + - Part.div_mem_div + - Part.bind_toOption + - Part.map_none + - Part.bind_some_right + - Part.Dom.bind + - Part.union_mem_union + - Part.mul_mem_mul + - Part.some_div_some + - Part.some_append_some + - Part.le_total_of_le_of_le + - Part.assert_neg + - Part.mem_toOption + - Part.mod_mem_mod + - Part.ne_none_iff + - Part.ofOption_eq_get + - Part.assert_pos + - Part.elim_toOption + - Part.inter_get_eq + - Part.bind_assoc + - Part.get_eq_get_of_eq + - Part.bind_some + - Part.map_map + - Part.append_mem_append + - Part.some_sdiff_some + - Part.inv_mem_inv + - Part.union_get_eq + - Part.bind_map + - Part.map_id' + - Part.some_inter_some + - Part.bind_of_mem + - Part.some_mod_some + - Part.to_ofOption + - Part.ofOption_dom + - Part.get_eq_iff_eq_some + - Part.sdiff_mem_sdiff + - Part.toOption_isSome + - Part.some_ne_none + - Part.some_union_some + - Part.map_bind + - Part.inter_mem_inter + - Part.div_get_eq + - Part.mod_get_eq + - Part.toOption_eq_some_iff + - Part.append_get_eq + - Part.some_mul_some + - path: Mathlib/Algebra/BigOperators/Fin.lean + theorems: + - Fin.prod_univ_add + - Fin.prod_univ_six + - List.prod_ofFn + - Fin.prod_ofFn + - Fin.prod_univ_three + - finFunctionFinEquiv_single + - Fin.prod_const + - List.prod_take_ofFn + - finPiFinEquiv_single + - Fin.prod_univ_castSucc + - Fin.prod_congr' + - Fin.partialProd_zero + - Fin.prod_univ_seven + - Fin.prod_univ_one + - Fin.prod_trunc + - Fin.prod_univ_succAbove + - Fin.prod_univ_four + - Fin.partialProd_right_inv + - Fin.partialProd_succ + - Fin.prod_univ_two + - Fin.prod_univ_eight + - Fin.prod_Ioi_zero + - Fin.inv_partialProd_mul_eq_contractNth + - Fin.sum_pow_mul_eq_add_pow + - Fin.partialProd_left_inv + - Fin.prod_Ioi_succ + - Fin.sum_const + - List.alternatingProd_eq_finset_prod + - Fin.prod_univ_def + - Fin.prod_univ_five + - Fin.prod_cons + - path: Mathlib/RingTheory/FractionalIdeal/Norm.lean + theorems: + - FractionalIdeal.absNorm_eq' + - FractionalIdeal.abs_det_basis_change + - FractionalIdeal.absNorm_span_singleton + - FractionalIdeal.absNorm_eq_zero_iff + - FractionalIdeal.coeIdeal_absNorm + - FractionalIdeal.absNorm_nonneg + - FractionalIdeal.absNorm_div_norm_eq_absNorm_div_norm + - FractionalIdeal.absNorm_one + - path: Mathlib/Analysis/NormedSpace/PiLp.lean + theorems: + - PiLp.edist_eq_of_L2 + - PiLp.nndist_eq_iSup + - PiLp.nndist_eq_sum + - PiLp.basis_toMatrix_basisFun_mul + - PiLp.norm_eq_ciSup + - PiLp.norm_eq_of_nat + - PiLp.nnnorm_equiv_symm_single + - PiLp.dist_eq_of_L2 + - PiLp.norm_eq_of_L2 + - LinearIsometryEquiv.piLpCongrLeft_symm + - PiLp.nnnorm_eq_sum + - PiLp.edist_self + - PiLp.antilipschitzWith_equiv_aux + - PiLp.nnnorm_eq_ciSup + - PiLp.dist_eq_iSup + - PiLp.norm_sq_eq_of_L2 + - PiLp.edist_eq_iSup + - PiLp.infty_equiv_isometry + - PiLp.norm_equiv_symm_const + - PiLp.lipschitzWith_equiv_aux + - PiLp.nndist_equiv_symm_single_same + - PiLp.edist_comm + - PiLp.nndist_eq_of_L2 + - PiLp.edist_equiv_symm_single_same + - PiLp.nnnorm_equiv_symm_const + - PiLp.nnnorm_equiv + - PiLp.nnnorm_equiv_symm_const' + - PiLp.nnnorm_eq_of_L2 + - LinearIsometryEquiv.piLpCongrLeft_single + - PiLp.iSup_edist_ne_top_aux + - PiLp.basisFun_apply + - PiLp.aux_uniformity_eq + - path: Mathlib/Analysis/MeanInequalitiesPow.lean + theorems: + - ENNReal.rpow_add_rpow_le + - Real.arith_mean_le_rpow_mean + - NNReal.add_rpow_le_one_of_add_le_one + - ENNReal.rpow_add_le_mul_rpow_add_rpow + - ENNReal.add_rpow_le_rpow_add + - ENNReal.rpow_arith_mean_le_arith_mean2_rpow + - NNReal.rpow_add_le_add_rpow + - NNReal.rpow_add_rpow_le_add + - NNReal.rpow_add_rpow_le + - ENNReal.rpow_add_le_add_rpow + - NNReal.pow_sum_div_card_le_sum_pow + - Real.pow_sum_div_card_le_sum_pow + - ENNReal.rpow_arith_mean_le_arith_mean_rpow + - NNReal.add_rpow_le_rpow_add + - NNReal.rpow_add_le_mul_rpow_add_rpow + - NNReal.rpow_arith_mean_le_arith_mean2_rpow + - path: Mathlib/Control/Basic.lean + theorems: + - map_bind + - guard_true + - guard_false + - CommApplicative.commutative_map + - seq_bind_eq + - map_seq + - seq_map_assoc + - path: Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean + theorems: + - MeasureTheory.SimpleFunc.integrable_pair + - MeasureTheory.SimpleFunc.measure_support_lt_top + - MeasureTheory.Lp.simpleFunc.norm_toSimpleFunc + - MeasureTheory.SimpleFunc.integrable_approxOn_range + - "MeasureTheory.SimpleFunc.norm_approxOn_y\u2080_le" + - MeasureTheory.SimpleFunc.integrable_approxOn + - MeasureTheory.SimpleFunc.snorm'_eq + - MeasureTheory.SimpleFunc.tendsto_approxOn_L1_nnnorm + - MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg + - MeasureTheory.Lp.simpleFunc.toSimpleFunc_toLp + - MeasureTheory.SimpleFunc.norm_approxOn_zero_le + - MeasureTheory.Lp.simpleFunc.exists_simpleFunc_nonneg_ae_eq + - MeasureTheory.Lp.simpleFunc.coeFn_le + - MeasureTheory.Lp.simpleFunc.smul_toSimpleFunc + - "MeasureTheory.SimpleFunc.mem\u2112p_approxOn" + - MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_snorm + - MeasureTheory.Lp.simpleFunc.zero_toSimpleFunc + - MeasureTheory.L1.SimpleFunc.integrable + - MeasureTheory.Lp.simpleFunc.toSimpleFunc_eq_toFun + - MeasureTheory.Lp.simpleFunc.add_toSimpleFunc + - MeasureTheory.SimpleFunc.nnnorm_approxOn_le + - MeasureTheory.Lp.simpleFunc.neg_toSimpleFunc + - "MeasureTheory.SimpleFunc.measure_preimage_lt_top_of_mem\u2112p" + - "MeasureTheory.Mem\u2112p.exists_simpleFunc_snorm_sub_lt" + - "MeasureTheory.SimpleFunc.mem\u2112p_approxOn_range" + - MeasureTheory.Lp.simpleFunc.denseEmbedding + - MeasureTheory.SimpleFunc.tendsto_approxOn_range_L1_nnnorm + - "MeasureTheory.SimpleFunc.mem\u2112p_of_finite_measure_preimage" + - "MeasureTheory.Mem\u2112p.induction" + - MeasureTheory.Lp.simpleFunc.sub_toSimpleFunc + - MeasureTheory.Lp.simpleFunc.coeFn_nonneg + - MeasureTheory.Lp.induction + - MeasureTheory.SimpleFunc.tendsto_approxOn_range_Lp_snorm + - MeasureTheory.Lp.simpleFunc.induction + - MeasureTheory.Integrable.induction + - "MeasureTheory.Mem\u2112p.induction_dense" + - MeasureTheory.Lp.simpleFunc.toLp_sub + - path: Mathlib/Analysis/InnerProductSpace/Dual.lean + theorems: + - InnerProductSpace.ext_inner_right_basis + - InnerProductSpace.unique_continuousLinearMapOfBilin + - InnerProductSpace.continuousLinearMapOfBilin_apply + - InnerProductSpace.toDual_symm_apply + - InnerProductSpace.ext_inner_left_basis + - path: Mathlib/NumberTheory/Liouville/LiouvilleWith.lean + theorems: + - LiouvilleWith.sub_rat_iff + - LiouvilleWith.int_sub_iff + - LiouvilleWith.sub_int_iff + - LiouvilleWith.neg + - LiouvilleWith.nat_mul + - LiouvilleWith.mono + - LiouvilleWith.mul_rat_iff + - Liouville.liouvilleWith + - LiouvilleWith.rat_add_iff + - Liouville.frequently_exists_num + - LiouvilleWith.frequently_lt_rpow_neg + - LiouvilleWith.exists_pos + - liouvilleWith_one + - LiouvilleWith.rat_sub_iff + - LiouvilleWith.nat_mul_iff + - LiouvilleWith.nat_sub_iff + - LiouvilleWith.nat_add_iff + - LiouvilleWith.rat_mul_iff + - LiouvilleWith.int_mul_iff + - LiouvilleWith.add_int_iff + - LiouvilleWith.mul_nat_iff + - LiouvilleWith.ne_cast_int + - LiouvilleWith.mul_rat + - LiouvilleWith.mul_int_iff + - forall_liouvilleWith_iff + - LiouvilleWith.sub_nat_iff + - LiouvilleWith.add_rat + - LiouvilleWith.add_nat_iff + - LiouvilleWith.add_rat_iff + - LiouvilleWith.irrational + - path: Mathlib/Combinatorics/Quiver/Covering.lean + theorems: + - Prefunctor.bijective_costar_iff_bijective_star + - Prefunctor.symmetrifyCostar + - Prefunctor.costar_conj_star + - Prefunctor.pathStar_surjective + - Prefunctor.IsCovering.symmetrify + - Prefunctor.pathStar_injective + - Prefunctor.IsCovering.of_comp_left + - Prefunctor.symmetrifyStar + - Prefunctor.IsCovering.map_injective + - path: Mathlib/CategoryTheory/Preadditive/Biproducts.lean + theorems: + - CategoryTheory.Limits.biproduct.lift_desc + - CategoryTheory.Biprod.ofComponents_fst + - CategoryTheory.Limits.biproduct.matrix_map + - CategoryTheory.Limits.biprod.total + - CategoryTheory.Limits.biprod.map_eq + - CategoryTheory.Limits.inr_of_isLimit + - CategoryTheory.Limits.HasBiproduct.of_hasProduct + - CategoryTheory.Limits.HasBiproduct.of_hasCoproduct + - CategoryTheory.Limits.IsBilimit.total + - CategoryTheory.Limits.biprod.add_eq_lift_id_desc + - CategoryTheory.Limits.biproduct.matrix_desc + - CategoryTheory.Limits.biprod.desc_eq + - CategoryTheory.Biprod.inr_ofComponents + - CategoryTheory.Biproduct.column_nonzero_of_iso' + - CategoryTheory.Limits.snd_of_isColimit + - CategoryTheory.Limits.biproduct.lift_eq + - CategoryTheory.Biprod.ofComponents_snd + - CategoryTheory.Limits.biprod.lift_eq + - CategoryTheory.Limits.inl_of_isLimit + - CategoryTheory.Limits.biprod.add_eq_lift_desc_id + - CategoryTheory.Biprod.inl_ofComponents + - CategoryTheory.Limits.biproduct.lift_matrix + - CategoryTheory.Limits.biproduct.desc_eq + - CategoryTheory.Biprod.ofComponents_comp + - CategoryTheory.Limits.IsBilimit.binary_total + - CategoryTheory.Limits.fst_of_isColimit + - CategoryTheory.Limits.biproduct.map_matrix + - CategoryTheory.Limits.biprod.lift_desc + - CategoryTheory.Limits.biproduct.map_eq + - CategoryTheory.Biprod.column_nonzero_of_iso + - CategoryTheory.Biprod.ofComponents_eq + - path: Mathlib/GroupTheory/SpecificGroups/Alternating.lean + theorems: + - alternatingGroup.normalClosure_swap_mul_swap_five + - alternatingGroup.nontrivial_of_three_le_card + - Equiv.Perm.finRotate_bit1_mem_alternatingGroup + - Equiv.Perm.IsThreeCycle.alternating_normalClosure + - alternatingGroup.isConj_of + - two_mul_card_alternatingGroup + - alternatingGroup.isThreeCycle_isConj + - Equiv.Perm.prod_list_swap_mem_alternatingGroup_iff_even_length + - alternatingGroup.normalClosure_finRotate_five + - alternatingGroup.isConj_swap_mul_swap_of_cycleType_two + - Equiv.Perm.closure_three_cycles_eq_alternating + - Equiv.Perm.isThreeCycle_sq_of_three_mem_cycleType_five + - path: Mathlib/LinearAlgebra/Span.lean + theorems: + - Submodule.span_nat_eq_addSubmonoid_closure + - Submodule.mem_span_singleton + - Submodule.span_smul_le + - Submodule.mem_sSup_of_directed + - Submodule.span_singleton_le_iff_mem + - Submodule.iSup_toAddSubmonoid + - Submodule.span_smul_eq_of_isUnit + - Submodule.wcovBy_span_singleton_sup + - Submodule.span_zero_singleton + - Submodule.span_preimage_le + - Submodule.coe_sup + - LinearMap.span_singleton_sup_ker_eq_top + - Submodule.mem_span_pair + - LinearEquiv.toSpanNonzeroSingleton_apply + - Submodule.covBy_span_singleton_sup + - Submodule.sup_toAddSubgroup + - LinearEquiv.coord_self + - Submodule.sup_toAddSubmonoid + - Submodule.le_span_singleton_iff + - Submodule.span_singleton_smul_eq + - LinearMap.toSpanSingleton_zero + - LinearEquiv.toSpanNonzeroSingleton_one + - Submodule.map_subtype_span_singleton + - Submodule.biSup_comap_subtype_eq_top + - Submodule.span_attach_biUnion + - Submodule.span_insert_zero + - Submodule.biSup_comap_eq_top_of_range_eq_biSup + - Submodule.prod_sup_prod + - Submodule.apply_mem_span_image_iff_mem_span + - Submodule.closure_induction + - Submodule.mem_span_singleton_trans + - Submodule.span_singleton_smul_le + - LinearMap.map_eq_top_iff + - Submodule.finset_span_isCompactElement + - Submodule.span_sup + - Submodule.disjoint_span_singleton + - LinearMap.span_singleton_eq_range + - Submodule.singleton_span_isCompactElement + - Submodule.mem_iSup_of_directed + - Submodule.comap_map_eq + - LinearMap.range_domRestrict_eq_range_iff + - Submodule.span_insert + - Submodule.iSup_induction + - Submodule.span_eq_iSup_of_singleton_spans + - Submodule.mem_sup' + - Submodule.biSup_comap_eq_top_of_surjective + - LinearMap.isIdempotentElem_apply_one_iff + - Submodule.prod_bot + - Submodule.iSup_induction' + - LinearMap.map_le_map_iff' + - Submodule.span_induction + - Submodule.prod_top + - Submodule.submodule_eq_sSup_le_nonzero_spans + - Submodule.closure_induction' + - Submodule.span_eq_closure + - Submodule.lt_sup_iff_not_mem + - Submodule.span_range_eq_iSup + - Submodule.span_singleton_eq_bot + - Submodule.exists_add_eq_of_codisjoint + - Submodule.mem_span_insert' + - Submodule.span_int_eq + - Submodule.span_singleton_eq_span_singleton + - Submodule.mem_span_finite_of_mem_span + - Submodule.span_singleton_eq_top_iff + - Submodule.comap_map_eq_self + - Submodule.nontrivial_span_singleton + - Submodule.mem_iSup + - Submodule.span_singleton_group_smul_eq + - Submodule.span_span_coe_preimage + - LinearMap.eqOn_span_iff + - Submodule.mem_span_insert + - Submodule.span_nat_eq + - Submodule.mem_sup + - LinearMap.toSpanSingleton_isIdempotentElem_iff + - Submodule.span_neg + - Submodule.span_induction' + - Submodule.span_zero + - LinearMap.map_le_map_iff + - Submodule.apply_mem_span_image_of_mem_span + - LinearMap.surjective_domRestrict_iff + - path: Mathlib/NumberTheory/Divisors.lean + theorems: + - Nat.image_snd_divisorsAntidiagonal + - Nat.prod_properDivisors_prime_pow + - Nat.swap_mem_divisorsAntidiagonal_aux + - Nat.prime_divisors_eq_to_filter_divisors_prime + - Nat.prod_div_divisors + - Nat.image_fst_divisorsAntidiagonal + - Nat.sum_properDivisors_dvd + - Nat.snd_mem_divisors_of_mem_antidiagonal + - Nat.sum_properDivisors_eq_one_iff_prime + - Nat.dvd_of_mem_divisors + - Nat.cons_self_properDivisors + - Nat.properDivisors_prime_pow + - Nat.properDivisors_one + - Nat.map_swap_divisorsAntidiagonal + - Nat.mem_divisorsAntidiagonal + - Nat.Prime.prod_properDivisors + - Nat.Prime.prod_divisors + - Nat.eq_properDivisors_of_subset_of_sum_eq_sum + - Nat.properDivisors.not_self_mem + - Nat.divisors_prime_pow + - Nat.divisors_one + - Nat.one_mem_properDivisors_iff_one_lt + - Nat.filter_dvd_eq_divisors + - Nat.Prime.divisors + - Nat.nonempty_divisors + - Nat.insert_self_properDivisors + - Nat.mem_properDivisors_iff_exists + - Nat.prod_divisorsAntidiagonal + - Nat.prod_divisors_prime_pow + - Nat.one_mem_divisors + - Nat.divisorsAntidiagonal_zero + - Nat.prime_divisors_filter_dvd_of_dvd + - Nat.fst_mem_divisors_of_mem_antidiagonal + - Nat.divisorsAntidiagonal_one + - Nat.sum_divisors_eq_sum_properDivisors_add_self + - Nat.map_div_right_divisors + - Nat.Prime.properDivisors + - Nat.one_lt_div_of_mem_properDivisors + - Nat.mem_properDivisors_prime_pow + - Nat.mem_properDivisors + - Nat.map_div_left_divisors + - Nat.mem_divisors_prime_pow + - Nat.pos_of_mem_divisors + - Nat.mem_divisors + - Nat.divisor_le + - Nat.divisors_subset_properDivisors + - Nat.divisors_filter_dvd_of_dvd + - Nat.perfect_iff_sum_divisors_eq_two_mul + - Nat.filter_dvd_eq_properDivisors + - Nat.prod_divisorsAntidiagonal' + - Nat.properDivisors_zero + - Nat.divisors_zero + - Nat.properDivisors_eq_singleton_one_iff_prime + - path: Mathlib/MeasureTheory/Measure/NullMeasurable.lean + theorems: + - "MeasureTheory.measure_union_add_inter\u2080" + - "MeasureTheory.measure_union\u2080" + - "MeasureTheory.measure_union_add_inter\u2080'" + - MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq + - "MeasureTheory.measure_union\u2080_aux" + - Finset.nullMeasurableSet + - MeasureTheory.measure_iUnion + - "MeasureTheory.measure_iUnion\u2080" + - "MeasureTheory.measure_add_measure_compl\u2080" + - MeasureTheory.NullMeasurableSet.exists_measurable_superset_ae_eq + - "MeasureTheory.measure_inter_add_diff\u2080" + - MeasureTheory.exists_subordinate_pairwise_disjoint + - MeasureTheory.NullMeasurable.congr + - MeasureTheory.NullMeasurableSet.sUnion + - "MeasureTheory.measure_union\u2080'" + - path: Mathlib/NumberTheory/Padics/RingHoms.lean + theorems: + - PadicInt.denseRange_nat_cast + - PadicInt.nthHomSeq_mul + - PadicInt.nthHom_zero + - PadicInt.lift_spec + - PadicInt.dvd_appr_sub_appr + - PadicInt.lift_self + - PadicInt.limNthHom_zero + - PadicInt.appr_spec + - PadicInt.modPart_lt_p + - PadicInt.nthHomSeq_add + - PadicInt.isUnit_den + - PadicInt.exists_mem_range + - PadicInt.appr_lt + - PadicInt.appr_mono + - PadicInt.norm_sub_modPart_aux + - PadicInt.cast_toZModPow + - PadicInt.isCauSeq_nthHom + - PadicInt.norm_sub_modPart + - PadicInt.pow_dvd_nthHom_sub + - PadicInt.lift_sub_val_mem_span + - PadicInt.zmod_congr_of_sub_mem_max_ideal + - PadicInt.zmod_congr_of_sub_mem_span_aux + - PadicInt.zmod_congr_of_sub_mem_span + - PadicInt.limNthHom_spec + - PadicInt.zmod_cast_comp_toZModPow + - PadicInt.lift_unique + - PadicInt.ker_toZModPow + - PadicInt.nthHomSeq_one + - PadicInt.ker_toZMod + - PadicInt.toZModPow_eq_iff_ext + - PadicInt.toZMod_spec + - path: Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean + theorems: + - "WeakFEPair.\u039B_residue_zero" + - WeakFEPair.h_feq' + - WeakFEPair.f_modif_aux1 + - "WeakFEPair.differentiableAt_\u039B" + - WeakFEPair.hasMellin + - StrongFEPair.hf_zero' + - StrongFEPair.functional_equation + - StrongFEPair.hf_top' + - WeakFEPair.hf_zero + - WeakFEPair.f_modif_aux2 + - WeakFEPair.hf_modif_FE + - WeakFEPair.hf_zero' + - "WeakFEPair.symm_\u039B\u2080_eq" + - "WeakFEPair.\u039B\u2080_eq" + - WeakFEPair.functional_equation + - WeakFEPair.hf_modif_int + - "WeakFEPair.\u039B_residue_k" + - path: Mathlib/Analysis/SpecialFunctions/Exponential.lean + theorems: + - hasStrictDerivAt_exp_smul_const_of_mem_ball + - hasStrictFDerivAt_exp_zero_of_radius_pos + - hasStrictDerivAt_exp_smul_const_of_mem_ball' + - hasStrictDerivAt_exp_of_mem_ball + - "Complex.exp_eq_exp_\u2102" + - hasFDerivAt_exp_smul_const_of_mem_ball' + - "Real.exp_eq_exp_\u211D" + - hasStrictFDerivAt_exp_smul_const_of_mem_ball' + - hasFDerivAt_exp_of_mem_ball + - hasFDerivAt_exp_smul_const_of_mem_ball + - path: Mathlib/Tactic/NormNum/Inv.lean + theorems: + - Mathlib.Meta.NormNum.isRat_ratCast + - Mathlib.Meta.NormNum.isInt_ratCast + - Mathlib.Meta.NormNum.isRat_inv_neg + - Mathlib.Meta.NormNum.isRat_inv_zero + - Mathlib.Meta.NormNum.isRat_inv_neg_one + - Mathlib.Meta.NormNum.isRat_mkRat + - Mathlib.Meta.NormNum.isRat_inv_pos + - Mathlib.Meta.NormNum.isNat_ratCast + - Mathlib.Meta.NormNum.isRat_inv_one + - path: Mathlib/Topology/UniformSpace/Equiv.lean + theorems: + - UniformEquiv.comap_eq + - UniformEquiv.toEquiv_injective + - UniformEquiv.uniformInducing + - path: Mathlib/Analysis/NormedSpace/BanachSteinhaus.lean + theorems: + - banach_steinhaus + - banach_steinhaus_iSup_nnnorm + - path: Mathlib/Analysis/Calculus/ContDiff/Defs.lean + theorems: + - contDiff_iff_contDiffAt + - norm_fderivWithin_iteratedFDerivWithin + - ContDiffWithinAt.eventually + - ContDiffWithinAt.differentiable_within_at' + - contDiffAt_one_iff + - contDiff_iff_forall_nat_le + - ContDiffAt.eventually + - iteratedFDerivWithin_one_apply + - norm_iteratedFDerivWithin_fderivWithin + - hasFTaylorSeriesUpToOn_zero_iff + - contDiffOn_succ_iff_hasFDerivWithin + - tsupport_iteratedFDeriv_subset + - contDiffAt_succ_iff_hasFDerivAt + - contDiffOn_succ_iff_fderivWithin + - ContDiffOn.ftaylorSeriesWithin + - ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin + - norm_iteratedFDeriv_zero + - contDiff_succ_iff_hasFDerivAt + - HasFTaylorSeriesUpToOn.shift_of_succ + - contDiff_zero + - contDiffWithinAt_succ_iff_hasFDerivWithinAt + - Filter.EventuallyEq.iteratedFDerivWithin' + - fderivWithin_iteratedFDerivWithin + - iteratedFDerivWithin_eventually_congr_set' + - ContDiffAt.congr_of_eventuallyEq + - contDiff_iff_continuous_differentiable + - contDiffWithinAt_insert + - contDiffOn_top_iff_fderivWithin + - Filter.EventuallyEq.iteratedFDerivWithin_eq + - ContDiffWithinAt.contDiffAt + - contDiffAt_zero + - hasFTaylorSeriesUpTo_succ_iff_right + - hasFTaylorSeriesUpTo_zero_iff + - iteratedFDeriv_succ_apply_right + - hasFTaylorSeriesUpToOn_succ_iff_right + - contDiffOn_succ_iff_hasFDerivWithinAt + - contDiff_iff_ftaylorSeries + - contDiff_succ_iff_fderiv + - ContDiffAt.continuousAt + - fderiv_iteratedFDeriv + - contDiffOn_all_iff_nat + - contDiffWithinAt_insert_self + - HasFTaylorSeriesUpToOn.contDiffOn + - contDiff_top_iff_fderiv + - ContDiffWithinAt.contDiffOn' + - hasFTaylorSeriesUpTo_top_iff + - contDiff_top + - contDiffAt_top + - HasFTaylorSeriesUpTo.hasFDerivAt + - hasFTaylorSeriesUpToOn_univ_iff + - contDiffWithinAt_zero + - iteratedFDerivWithin_of_isOpen + - contDiffOn_of_locally_contDiffOn + - contDiffOn_of_continuousOn_differentiableOn + - HasFTaylorSeriesUpToOn.zero_eq' + - hasFTaylorSeriesUpToOn_succ_iff_left + - HasFTaylorSeriesUpToOn.hasFDerivWithinAt + - iteratedFDerivWithin_univ + - ContDiffOn.fderivWithin + - contDiffOn_top_iff_fderiv_of_isOpen + - contDiff_all_iff_nat + - HasFTaylorSeriesUpToOn.congr + - contDiffOn_succ_of_fderivWithin + - iteratedFDeriv_succ_eq_comp_right + - norm_iteratedFDeriv_fderiv + - HasFTaylorSeriesUpTo.ofLe + - contDiffOn_univ + - iteratedFDeriv_one_apply + - contDiffOn_succ_iff_fderiv_of_isOpen + - ContDiffAt.contDiffOn + - norm_fderiv_iteratedFDeriv + - iteratedFDerivWithin_succ_eq_comp_right + - norm_iteratedFDerivWithin_zero + - HasFTaylorSeriesUpToOn.continuousOn + - hasFTaylorSeriesUpTo_top_iff' + - contDiffWithinAt_top + - HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn + - ContDiffAt.differentiableAt + - HasFTaylorSeriesUpTo.zero_eq' + - contDiffOn_top + - contDiffOn_zero + - ContDiffWithinAt.mono_of_mem + - path: Mathlib/Probability/Distributions/Geometric.lean + theorems: + - ProbabilityTheory.geometricPMFReal_pos + - ProbabilityTheory.geometricPMFReal_nonneg + - ProbabilityTheory.measurable_geometricPMFReal + - path: Mathlib/Algebra/Lie/Normalizer.lean + theorems: + - LieSubmodule.le_normalizer + - LieIdeal.idealizer_eq_normalizer + - LieSubalgebra.mem_normalizer_iff + - LieSubalgebra.ideal_in_normalizer + - LieSubalgebra.lie_mem_sup_of_mem_normalizer + - LieSubalgebra.exists_nested_lieIdeal_ofLe_normalizer + - LieSubmodule.comap_normalizer + - LieSubalgebra.normalizer_eq_self_iff + - LieSubmodule.monotone_normalizer + - LieSubmodule.normalizer_inf + - LieSubalgebra.mem_normalizer_iff' + - LieSubmodule.top_lie_le_iff_le_normalizer + - path: Mathlib/Probability/Distributions/Gamma.lean + theorems: + - ProbabilityTheory.lintegral_gammaPDF_eq_one + - ProbabilityTheory.isProbabilityMeasureGamma + - lintegral_Iic_eq_lintegral_Iio_add_Icc + - ProbabilityTheory.gammaPDFReal_pos + - ProbabilityTheory.gammaPDF_of_neg + - ProbabilityTheory.gammaPDF_of_nonneg + - ProbabilityTheory.gammaCDFReal_eq_lintegral + - ProbabilityTheory.gammaPDFReal_nonneg + - ProbabilityTheory.lintegral_gammaPDF_of_nonpos + - ProbabilityTheory.gammaCDFReal_eq_integral + - path: Mathlib/Topology/MetricSpace/IsometricSMul.lean + theorems: + - EMetric.preimage_smul_closedBall + - EMetric.preimage_smul_ball + - Metric.preimage_mul_right_closedBall + - dist_div_right + - EMetric.preimage_mul_right_closedBall + - edist_div_right + - Metric.preimage_smul_closedBall + - edist_inv_inv + - Metric.preimage_mul_right_ball + - Metric.preimage_smul_ball + - edist_inv + - dist_div_left + - Metric.preimage_smul_sphere + - EMetric.preimage_mul_right_ball + - edist_div_left + - nndist_div_right + - path: Mathlib/NumberTheory/Cyclotomic/Basic.lean + theorems: + - IsCyclotomicExtension.finite_of_singleton + - IsCyclotomicExtension.numberField + - IsCyclotomicExtension.neZero + - IsCyclotomicExtension.finite + - IsCyclotomicExtension.union_right + - IsPrimitiveRoot.adjoin_isCyclotomicExtension + - IsCyclotomicExtension.empty + - IsCyclotomicExtension.isSplittingField_X_pow_sub_one + - IsCyclotomicExtension.trans + - IsCyclotomicExtension.equiv + - IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots + - CyclotomicRing.eq_adjoin_primitive_root + - IsCyclotomicExtension.neZero' + - IsCyclotomicExtension.union_left + - IsCyclotomicExtension.iff_union_singleton_one + - IsCyclotomicExtension.splits_X_pow_sub_one + - IsCyclotomicExtension.singleton_zero_of_bot_eq_top + - IsCyclotomicExtension.splitting_field_cyclotomic + - IsCyclotomicExtension.subsingleton_iff + - IsCyclotomicExtension.singleton_one_of_bot_eq_top + - IsCyclotomicExtension.adjoin_primitive_root_eq_top + - IsAlgClosed.isCyclotomicExtension + - IsCyclotomicExtension.singleton_one + - IsCyclotomicExtension.splits_cyclotomic + - IsCyclotomicExtension.iff_singleton + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean + theorems: + - Real.contDiff_arctan + - Real.hasStrictDerivAt_tan + - Real.hasDerivAt_tan + - Real.continuousAt_tan + - Real.deriv_tan + - Real.tendsto_abs_tan_of_cos_eq_zero + - Real.hasStrictDerivAt_arctan + - path: Mathlib/MeasureTheory/Integral/CircleIntegral.lean + theorems: + - hasSum_two_pi_I_cauchyPowerSeries_integral + - circleIntegral.norm_integral_le_of_norm_le_const + - periodic_circleMap + - range_circleMap + - circleIntegrable_zero_radius + - circleIntegrable_sub_inv_iff + - image_circleMap_Ioc + - circleIntegral.integral_smul + - abs_circleMap_zero + - norm_cauchyPowerSeries_le + - hasFPowerSeriesOn_cauchy_integral + - circleIntegral.integral_smul_const + - circleMap_mem_sphere + - circleIntegral.norm_two_pi_i_inv_smul_integral_le_of_norm_le_const + - circleIntegral.integral_sub + - circleIntegrable_sub_zpow_iff + - le_radius_cauchyPowerSeries + - circleMap_mem_sphere' + - circleIntegrable_iff + - circleMap_not_mem_ball + - circleIntegral.integral_congr + - circleIntegral.integral_sub_center_inv + - circleMap_eq_center_iff + - circleMap_sub_center + - circleIntegral.integral_eq_zero_of_hasDerivWithinAt' + - cauchyPowerSeries_apply + - hasSum_cauchyPowerSeries_integral + - lipschitzWith_circleMap + - circleIntegral.integral_radius_zero + - circleIntegral.integral_sub_zpow_of_ne + - circleIntegral_def_Icc + - circleIntegral.integral_sub_zpow_of_undef + - circleIntegral.integral_sub_inv_of_mem_ball + - hasDerivAt_circleMap + - deriv_circleMap_eq_zero_iff + - continuous_circleMap_inv + - circleIntegral.integral_sub_inv_smul_sub_smul + - CircleIntegrable.out + - circleIntegral.norm_integral_lt_of_norm_le_const_of_lt + - circleIntegral.norm_integral_le_of_norm_le_const' + - path: Mathlib/Algebra/FreeAlgebra.lean + theorems: + - FreeAlgebra.algebraMap_leftInverse + - "FreeAlgebra.quot_mk_eq_\u03B9" + - FreeAlgebra.lift_unique + - "FreeAlgebra.\u03B9_comp_lift" + - "FreeAlgebra.lift_comp_\u03B9" + - FreeAlgebra.liftAux_eq + - "FreeAlgebra.adjoin_range_\u03B9" + - "FreeAlgebra.\u03B9_injective" + - FreeAlgebra.hom_ext + - Algebra.adjoin_eq_range_freeAlgebra_lift + - "FreeAlgebra.lift_\u03B9_apply" + - FreeAlgebra.lift_symm_apply + - "FreeAlgebra.\u03B9_ne_algebraMap" + - path: Mathlib/Analysis/NormedSpace/Extend.lean + theorems: + - "ContinuousLinearMap.norm_extendTo\U0001D55C'_bound" + - "LinearMap.norm_extendTo\U0001D55C'_apply_sq" + - "LinearMap.extendTo\U0001D55C'_apply_re" + - path: Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean + theorems: + - UpperHalfPlane.normSq_pos + - UpperHalfPlane.mul_smul' + - UpperHalfPlane.linear_ne_zero + - UpperHalfPlane.c_mul_im_sq_le_normSq_denom + - UpperHalfPlane.ModularGroup.det_coe' + - UpperHalfPlane.exists_SL2_smul_eq_of_apply_zero_one_eq_zero + - UpperHalfPlane.exists_SL2_smul_eq_of_apply_zero_one_ne_zero + - UpperHalfPlane.ModularGroup.im_smul_eq_div_normSq + - UpperHalfPlane.ModularGroup.SL_neg_smul + - UpperHalfPlane.denom_cocycle + - UpperHalfPlane.denom_ne_zero + - UpperHalfPlane.ModularGroup.denom_apply + - UpperHalfPlane.modular_T_zpow_smul + - UpperHalfPlane.modular_S_smul + - UpperHalfPlane.neg_smul + - UpperHalfPlane.smulAux'_im + - UpperHalfPlane.modular_T_smul + - path: Mathlib/Data/Matrix/Rank.lean + theorems: + - Matrix.rank_eq_finrank_span_cols + - Matrix.rank_transpose_mul_self + - Matrix.rank_zero + - Matrix.ker_mulVecLin_transpose_mul_self + - Matrix.rank_mul_eq_left_of_isUnit_det + - Matrix.rank_submatrix + - Matrix.rank_self_mul_transpose + - Matrix.rank_self_mul_conjTranspose + - Matrix.rank_mul_eq_right_of_isUnit_det + - Matrix.rank_submatrix_le + - Matrix.rank_mul_le_right + - Matrix.rank_one + - Matrix.rank_reindex + - Matrix.rank_mul_le_left + - Matrix.rank_eq_finrank_span_row + - Matrix.rank_le_card_height + - Matrix.rank_of_isUnit + - Matrix.rank_le_card_width + - Matrix.rank_unit + - Matrix.ker_mulVecLin_conjTranspose_mul_self + - Matrix.rank_eq_finrank_range_toLin + - Matrix.rank_conjTranspose_mul_self + - path: Mathlib/Data/Nat/WithBot.lean + theorems: + - Nat.WithBot.add_eq_one_iff + - Nat.WithBot.coe_nonneg + - Nat.WithBot.add_eq_two_iff + - Nat.WithBot.one_le_iff_zero_lt + - Nat.WithBot.add_eq_zero_iff + - Nat.WithBot.add_one_le_of_lt + - Nat.WithBot.lt_one_iff_le_zero + - Nat.WithBot.add_eq_three_iff + - path: Mathlib/Algebra/QuadraticDiscriminant.lean + theorems: + - quadratic_eq_zero_iff_discrim_eq_sq + - discrim_eq_sq_of_quadratic_eq_zero + - quadratic_eq_zero_iff_of_discrim_eq_zero + - discrim_le_zero_of_nonpos + - quadratic_eq_zero_iff + - discrim_neg + - exists_quadratic_eq_zero + - discrim_lt_zero + - discrim_le_zero + - discrim_lt_zero_of_neg + - path: Mathlib/NumberTheory/NumberField/Units.lean + theorems: + - NumberField.Units.fun_eq_repr + - NumberField.Units.dirichletUnitTheorem.seq_decreasing + - NumberField.Units.rank_modTorsion + - NumberField.Units.dirichletUnitTheorem.exists_unit + - NumberField.isUnit_iff_norm + - NumberField.Units.rootsOfUnity_eq_torsion + - NumberField.Units.dirichletUnitTheorem.unitLattice_span_eq_top + - Rat.RingOfIntegers.isUnit_iff + - NumberField.Units.dirichletUnitTheorem.logEmbedding_component_le + - NumberField.Units.rootsOfUnity_eq_one + - NumberField.Units.coe_pow + - NumberField.Units.finrank_eq_rank + - NumberField.Units.dirichletUnitTheorem.seq_norm_le + - NumberField.Units.dirichletUnitTheorem.seq_ne_zero + - NumberField.Units.dirichletUnitTheorem.unitLattice_inter_ball_finite + - NumberField.Units.unitLattice_rank + - NumberField.Units.dirichletUnitTheorem.logEmbedding_eq_zero_iff + - NumberField.Units.coe_zpow + - NumberField.Units.dirichletUnitTheorem.mult_log_place_eq_zero + - NumberField.Units.dirichletUnitTheorem.seq_next + - NumberField.Units.mem_torsion + - NumberField.Units.dirichletUnitTheorem.log_le_of_logEmbedding_le + - NumberField.Units.coe_injective + - NumberField.Units.dirichletUnitTheorem.sum_logEmbedding_component + - NumberField.Units.exist_unique_eq_mul_prod + - path: Mathlib/Order/Hom/Lattice.lean + theorems: + - InfTopHom.cancel_left + - LatticeHom.cancel_left + - Disjoint.map + - Codisjoint.map + - BoundedLatticeHom.cancel_left + - LatticeHom.withTopWithBot_comp + - map_sdiff' + - map_symmDiff' + - SupHom.cancel_left + - SupBotHom.cancel_left + - InfHom.cancel_left + - path: Mathlib/MeasureTheory/Covering/Differentiation.lean + theorems: + - VitaliFamily.measure_le_mul_of_subset_limRatioMeas_lt + - VitaliFamily.ae_tendsto_div + - VitaliFamily.ae_tendsto_rnDeriv + - VitaliFamily.le_mul_withDensity + - VitaliFamily.ae_tendsto_limRatio + - VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable + - VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div_of_integrable + - VitaliFamily.ae_tendsto_average_norm_sub + - VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet + - VitaliFamily.measure_limRatioMeas_top + - VitaliFamily.ae_tendsto_average + - VitaliFamily.ae_tendsto_lintegral_div + - VitaliFamily.aemeasurable_limRatio + - VitaliFamily.ae_tendsto_lintegral_div' + - VitaliFamily.withDensity_le_mul + - VitaliFamily.exists_measurable_supersets_limRatio + - VitaliFamily.ae_eventually_measure_pos + - VitaliFamily.mul_measure_le_of_subset_lt_limRatioMeas + - VitaliFamily.ae_eventually_measure_zero_of_singular + - VitaliFamily.ae_tendsto_rnDeriv_of_absolutelyContinuous + - VitaliFamily.eventually_filterAt_integrableOn + - VitaliFamily.ae_tendsto_lintegral_nnnorm_sub_div + - VitaliFamily.measure_limRatioMeas_zero + - VitaliFamily.null_of_frequently_le_of_frequently_ge + - VitaliFamily.ae_tendsto_limRatioMeas + - VitaliFamily.ae_tendsto_measure_inter_div + - VitaliFamily.withDensity_limRatioMeas_eq + - VitaliFamily.measure_le_of_frequently_le + - VitaliFamily.eventually_measure_lt_top + - path: Mathlib/Data/Nat/Bitwise.lean + theorems: + - Nat.testBit_two_pow_self + - Nat.lt_of_testBit + - Nat.bit_mod_two + - Nat.and_two_pow + - Nat.xor_left_injective + - Nat.xor_eq_zero + - Nat.bitwise_of_ne_zero + - Nat.xor_trichotomy + - Nat.xor_right_injective + - Nat.bitwise_zero + - Nat.bitwise_bit' + - Nat.bitwise_swap + - Nat.bit_mod_two_eq_zero_iff + - Nat.xor_zero + - Nat.bit_mod_two_eq_one_iff + - Nat.lor_assoc + - Nat.bitwise_lt + - Nat.exists_most_significant_bit + - Nat.bitwise_eq_binaryRec + - Nat.testBit_eq_inth + - Nat.two_pow_and + - Nat.bit_ne_zero_iff + - Nat.bitwise_zero_right + - Nat.append_lt + - Nat.xor_cancel_right + - Nat.land_assoc + - Nat.shiftLeft_lt + - Nat.testBit_eq_false_of_lt + - Nat.binaryRec_of_ne_zero + - Nat.zero_xor + - Nat.bitwise_bit + - Nat.xor_cancel_left + - Nat.bitwise_comm + - Nat.bit_eq_zero + - Nat.bit_lt_two_pow_succ_iff + - Nat.zero_of_testBit_eq_false + - Nat.testBit_two_pow_of_ne + - Nat.testBit_two_pow + - Nat.xor_assoc + - path: Mathlib/MeasureTheory/Function/EssSup.lean + theorems: + - meas_lt_essInf + - OrderIso.essSup_apply + - essSup_eq_sInf + - essSup_mono_measure + - essSup_map_measure_of_measurable + - meas_essSup_lt + - ENNReal.essSup_piecewise + - ENNReal.coe_essSup + - essSup_comp_le_essSup_map_measure + - essInf_eq_sSup + - essSup_le_of_ae_le + - ENNReal.essSup_indicator_eq_essSup_restrict + - essSup_map_measure + - MeasurableEmbedding.essSup_map_measure + - essSup_measure_zero + - essInf_antitone_measure + - ENNReal.essSup_liminf_le + - essSup_smul_measure + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean + theorems: + - InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero + - InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero + - EuclideanGeometry.dist_div_tan_angle_of_angle_eq_pi_div_two + - InnerProductGeometry.sin_angle_add_of_inner_eq_zero + - InnerProductGeometry.norm_div_tan_angle_sub_of_inner_eq_zero + - EuclideanGeometry.sin_angle_of_angle_eq_pi_div_two + - InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero + - EuclideanGeometry.sin_angle_mul_dist_of_angle_eq_pi_div_two + - InnerProductGeometry.angle_sub_pos_of_inner_eq_zero + - InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero + - InnerProductGeometry.cos_angle_sub_mul_norm_of_inner_eq_zero + - InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero + - InnerProductGeometry.cos_angle_add_of_inner_eq_zero + - InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero + - InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero + - EuclideanGeometry.angle_eq_arccos_of_angle_eq_pi_div_two + - InnerProductGeometry.angle_add_lt_pi_div_two_of_inner_eq_zero + - InnerProductGeometry.tan_angle_add_of_inner_eq_zero + - InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero + - EuclideanGeometry.angle_pos_of_angle_eq_pi_div_two + - EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two + - InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero + - InnerProductGeometry.angle_sub_le_pi_div_two_of_inner_eq_zero + - InnerProductGeometry.norm_add_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two + - EuclideanGeometry.dist_div_sin_angle_of_angle_eq_pi_div_two + - InnerProductGeometry.tan_angle_sub_of_inner_eq_zero + - EuclideanGeometry.angle_eq_arctan_of_angle_eq_pi_div_two + - EuclideanGeometry.cos_angle_of_angle_eq_pi_div_two + - InnerProductGeometry.angle_add_le_pi_div_two_of_inner_eq_zero + - InnerProductGeometry.angle_sub_lt_pi_div_two_of_inner_eq_zero + - EuclideanGeometry.tan_angle_mul_dist_of_angle_eq_pi_div_two + - EuclideanGeometry.dist_div_cos_angle_of_angle_eq_pi_div_two + - EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two + - InnerProductGeometry.sin_angle_sub_of_inner_eq_zero + - InnerProductGeometry.tan_angle_sub_mul_norm_of_inner_eq_zero + - InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two + - EuclideanGeometry.angle_le_pi_div_two_of_angle_eq_pi_div_two + - InnerProductGeometry.cos_angle_sub_of_inner_eq_zero + - EuclideanGeometry.tan_angle_of_angle_eq_pi_div_two + - InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero + - EuclideanGeometry.cos_angle_mul_dist_of_angle_eq_pi_div_two + - InnerProductGeometry.angle_add_pos_of_inner_eq_zero + - InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero + - path: Mathlib/Algebra/Module/Equiv.lean + theorems: + - LinearEquiv.coe_toLinearMap_mul + - LinearEquiv.eq_comp_toLinearMap_symm + - LinearEquiv.ofSubsingleton_self + - LinearEquiv.symm_trans_self + - AddEquiv.toIntLinearEquiv_toAddEquiv + - AddEquiv.toNatLinearEquiv_toAddEquiv + - LinearEquiv.comp_toLinearMap_symm_eq + - LinearEquiv.self_trans_symm + - LinearEquiv.toLinearMap_symm_comp_eq + - LinearEquiv.symm_symm + - LinearEquiv.eq_toLinearMap_symm_comp + - path: Mathlib/RingTheory/HahnSeries/Basic.lean + theorems: + - HahnSeries.support_nonempty_iff + - HahnSeries.embDomain_single + - HahnSeries.support_embDomain_subset + - HahnSeries.embDomain_coeff + - HahnSeries.embDomain_injective + - HahnSeries.coeff_eq_zero_of_lt_order + - HahnSeries.single_coeff + - HahnSeries.forallLTEqZero_supp_BddBelow + - HahnSeries.BddBelow_zero + - HahnSeries.single_injective + - HahnSeries.order_ofForallLtEqZero + - HahnSeries.coeff_order_ne_zero + - HahnSeries.embDomain_zero + - path: Mathlib/Data/Int/Interval.lean + theorems: + - Int.card_fintype_Icc + - Int.card_Ico_of_le + - Int.card_fintype_Ioc_of_le + - Int.card_fintype_Ioo + - Int.card_fintype_Icc_of_le + - Int.card_fintype_Ioc + - Int.card_fintype_uIcc + - Int.card_Ioo_of_lt + - Int.card_Icc_of_le + - Int.card_fintype_Ico_of_le + - Int.image_Ico_emod + - Int.card_Ioc_of_le + - Int.card_uIcc + - Int.card_fintype_Ico + - Int.card_fintype_Ioo_of_lt + - path: Mathlib/Data/Real/Basic.lean + theorems: + - Real.ofCauchy_inf + - Real.cauchy_neg + - Real.ofCauchy_sup + - Real.cauchy_zero + - Real.cauchy_one + - Real.le_def' + - Real.cauchy_inv + - Real.cauchy_mul + - Real.mk_add + - Real.ind_mk + - Real.mk_le_of_forall_le + - Real.mul_pos + - Real.mk_pos + - Real.ofCauchy_inv + - Real.ext_cauchy_iff + - Real.cauchy_add + - Real.mk_mul + - Real.mk_one + - Real.add_lt_add_iff_left + - Real.mk_zero + - Real.mk_le + - Real.le_mk_of_forall_le + - Real.ratCast_lt + - Real.mk_neg + - Real.zero_lt_one + - Real.lt_cauchy + - Real.ofCauchy_sub + - path: Mathlib/GroupTheory/CommutingProbability.lean + theorems: + - Subgroup.commProb_subgroup_le + - commProb_eq_one_iff + - commProb_pi + - DihedralGroup.commProb_reciprocal + - commProb_le_one + - Subgroup.commProb_quotient_le + - commProb_function + - commProb_def' + - DihedralGroup.commProb_odd + - commProb_prod + - DihedralGroup.reciprocalFactors_odd + - DihedralGroup.commProb_cons + - DihedralGroup.commProb_nil + - DihedralGroup.div_four_lt + - DihedralGroup.reciprocalFactors_even + - path: Mathlib/Algebra/CharP/ExpChar.lean + theorems: + - add_pow_expChar_pow_of_commute + - char_zero_of_expChar_one + - charZero_of_expChar_one' + - expChar_one_iff_char_zero + - ExpChar.eq + - expChar_pos + - iterateFrobenius_zero_apply + - ExpChar.neg_one_pow_expChar_pow + - expChar_of_injective_ringHom + - add_pow_expChar + - RingHom.expChar + - expChar_one_of_char_zero + - ExpChar.neg_one_pow_expChar + - ringExpChar.eq + - expChar_is_prime_or_one + - sub_pow_expChar_pow_of_commute + - ExpChar.exists + - sub_pow_expChar_of_commute + - iterateFrobenius_one_apply + - char_prime_of_ne_zero + - iterateFrobenius_mul_apply + - char_eq_expChar_iff + - sub_pow_expChar + - add_pow_expChar_of_commute + - add_pow_expChar_pow + - ringExpChar.eq_one + - iterateFrobenius_add_apply + - sub_pow_expChar_pow + - path: Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean + theorems: + - BoundedContinuousFunction.apply_le_nndist_zero + - BoundedContinuousFunction.toReal_lintegral_coe_eq_integral + - BoundedContinuousFunction.tendsto_integral_of_forall_limsup_integral_le_integral + - BoundedContinuousFunction.lintegral_lt_top_of_nnreal + - BoundedContinuousFunction.isBounded_range_integral + - BoundedContinuousFunction.tendsto_integral_of_forall_integral_le_liminf_integral + - BoundedContinuousFunction.integrable_of_nnreal + - BoundedContinuousFunction.lintegral_nnnorm_le + - BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub + - BoundedContinuousFunction.norm_integral_le_mul_norm + - BoundedContinuousFunction.integrable + - BoundedContinuousFunction.integral_const_sub + - BoundedContinuousFunction.norm_integral_le_norm + - BoundedContinuousFunction.integral_add_const + - BoundedContinuousFunction.lintegral_le_edist_mul + - path: Mathlib/CategoryTheory/Adjunction/Mates.lean + theorems: + - CategoryTheory.transferNatTransSelf_symm_id + - CategoryTheory.unit_transferNatTransSelf + - CategoryTheory.transferNatTransSelf_id + - CategoryTheory.transferNatTransSelf_comp + - CategoryTheory.transferNatTransSelf_symm_comm + - CategoryTheory.transferNatTrans_counit + - CategoryTheory.transferNatTransSelf_of_iso + - CategoryTheory.transferNatTransSelf_comm + - CategoryTheory.transferNatTransSelf_counit + - CategoryTheory.unit_transferNatTrans + - CategoryTheory.transferNatTransSelf_symm_comp + - CategoryTheory.transferNatTransSelf_symm_of_iso + - CategoryTheory.transferNatTransSelf_adjunction_id_symm + - CategoryTheory.transferNatTransSelf_adjunction_id + - path: Mathlib/Data/List/InsertNth.lean + theorems: + - List.mem_insertNth + - List.nthLe_insertNth_self + - List.insertNth_length_self + - List.removeNth_insertNth + - List.insertNth_removeNth_of_ge + - List.get_insertNth_add_succ + - List.insertNth_injective + - List.insertNth_comm + - List.get_insertNth_of_lt + - List.insertNth_of_length_lt + - List.length_le_length_insertNth + - List.length_insertNth_le_succ + - List.get_insertNth_self + - List.nthLe_insertNth_add_succ + - path: Mathlib/Data/List/Lemmas.lean + theorems: + - List.mapAccumr_eq_foldr + - List.foldl_range_subset_of_range_subset + - "List.mapAccumr\u2082_eq_foldr" + - List.foldr_range_subset_of_range_subset + - List.injOn_insertNth_index_of_not_mem + - path: Mathlib/MeasureTheory/Measure/GiryMonad.lean + theorems: + - MeasureTheory.Measure.join_eq_bind + - MeasureTheory.Measure.measurable_map + - MeasureTheory.Measure.measurable_join + - MeasureTheory.Measure.bind_dirac + - MeasureTheory.Measure.lintegral_join + - MeasureTheory.Measure.measurable_lintegral + - MeasureTheory.Measure.dirac_bind + - MeasureTheory.Measure.bind_bind + - MeasureTheory.Measure.join_map_map + - MeasureTheory.Measure.join_zero + - MeasureTheory.Measure.bind_zero_left + - MeasureTheory.Measure.measurable_dirac + - MeasureTheory.Measure.bind_apply + - MeasureTheory.Measure.bind_zero_right + - path: Mathlib/RingTheory/IntegralDomain.lean + theorems: + - sum_hom_units + - isCyclic_of_subgroup_isDomain + - Finset.exists_eq_pow_of_mul_eq_pow_of_coprime + - Finite.isField_of_domain + - card_nthRoots_subgroup_units + - card_fiber_eq_of_mem_range + - Polynomial.div_eq_quo_add_rem_div + - exists_eq_pow_of_mul_eq_pow_of_coprime + - path: Mathlib/Algebra/GCDMonoid/Basic.lean + theorems: + - gcd_mul_left + - CommGroupWithZero.coe_normUnit + - gcd_eq_of_dvd_sub_left + - exists_associated_pow_of_mul_eq_pow + - lcm_eq_zero_iff + - isUnit_gcd_of_eq_mul_gcd + - gcd_mul_right + - CommGroupWithZero.normalize_eq_one + - Associates.out_dvd_iff + - gcd_mul_left' + - Irreducible.gcd_eq_one_iff + - normalize_idem + - gcd_eq_right_iff + - lcm_mul_left + - gcd_eq_of_dvd_sub_right + - dvd_mul_gcd_of_dvd_mul + - gcd_pow_right_dvd_pow_gcd + - normalize_eq_zero + - lcm_dvd_iff + - gcd_mul_right' + - lcm_eq_right_iff + - normalize_eq_normalize + - gcd_eq_zero_iff + - lcm_eq_one_iff + - normUnit_mul_normUnit + - Irreducible.isUnit_gcd_iff + - extract_gcd + - pow_dvd_of_mul_eq_pow + - Associates.out_mul + - path: Mathlib/Algebra/IsPrimePow.lean + theorems: + - isPrimePow_nat_iff_bounded + - Prime.isPrimePow + - IsPrimePow.pow + - Nat.disjoint_divisors_filter_isPrimePow + - isPrimePow_iff_pow_succ + - IsPrimePow.dvd + - not_isPrimePow_zero + - path: Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean + theorems: + - CategoryTheory.BraidedCategory.yang_baxter + - "CategoryTheory.tensor_\u03BC_natural_right" + - CategoryTheory.braiding_rightUnitor + - CategoryTheory.braiding_tensorUnit_left + - CategoryTheory.braiding_tensorUnit_right + - CategoryTheory.BraidedCategory.braiding_inv_tensor_left + - "CategoryTheory.tensor_\u03BC_natural" + - CategoryTheory.braiding_leftUnitor + - "CategoryTheory.tensor_\u03BC_natural_left" + - CategoryTheory.rightUnitor_inv_braiding + - CategoryTheory.BraidedCategory.hexagon_reverse_inv + - CategoryTheory.BraidedCategory.braiding_naturality + - CategoryTheory.tensor_associativity + - CategoryTheory.SymmetricCategory.reverseBraiding_eq + - CategoryTheory.BraidedCategory.yang_baxter' + - CategoryTheory.rightUnitor_monoidal + - CategoryTheory.leftUnitor_inv_braiding + - CategoryTheory.tensor_right_unitality + - CategoryTheory.BraidedCategory.braiding_tensor_right + - CategoryTheory.BraidedCategory.braiding_tensor_left + - CategoryTheory.associator_monoidal + - "CategoryTheory.braiding_leftUnitor_aux\u2082" + - "CategoryTheory.braiding_rightUnitor_aux\u2082" + - CategoryTheory.BraidedCategory.hexagon_forward_inv + - "CategoryTheory.braiding_rightUnitor_aux\u2081" + - CategoryTheory.tensor_left_unitality + - CategoryTheory.BraidedCategory.braiding_inv_tensor_right + - CategoryTheory.leftUnitor_monoidal + - "CategoryTheory.braiding_leftUnitor_aux\u2081" + - path: Mathlib/Data/Int/Order/Units.lean + theorems: + - Int.units_inv_eq_self + - Int.isUnit_iff_abs_eq + - Int.units_sq + - Int.isUnit_sq + - Int.units_div_eq_mul + - Int.neg_one_pow_ne_zero + - Int.units_coe_mul_self + - Int.units_pow_eq_pow_mod_two + - Int.units_mul_self + - path: Mathlib/Algebra/Group/Prod.lean + theorems: + - MonoidHom.coprod_comp_inr + - MonoidHom.coprod_comp_inl + - MonoidHom.coprod_unique + - Prod.one_mk_mul_one_mk + - MulHom.comp_coprod + - MonoidHom.prod_unique + - Prod.mk_one_mul_mk_one + - MonoidHom.comp_coprod + - MulHom.prod_unique + - path: .lake/packages/std/Std/Data/RBMap/WF.lean + theorems: + - Std.RBNode.DelProp.redred + - Std.RBNode.Ordered.insert + - Std.RBNode.Ordered.append + - Std.RBNode.Ordered.del + - Std.RBNode.balance2_All + - Std.RBNode.All.ins + - Std.RBNode.Balanced.balRight + - Std.RBNode.All.append + - Std.RBNode.All.del + - Std.RBNode.All.balRight + - Std.RBNode.Ordered.setRed + - Std.RBNode.All.balLeft + - Std.RBNode.Ordered.ins + - Std.RBNode.Ordered.balance1 + - Std.RBNode.Ordered.setBlack + - Std.RBNode.Ordered.balance2 + - Std.RBNode.setBlack_idem + - Std.RBNode.Balanced.balLeft + - Std.RBNode.Ordered.balLeft + - Std.RBNode.Balanced.ins + - Std.RBNode.Ordered.map + - Std.RBNode.insert_setBlack + - Std.RBNode.All.setRed + - Std.RBNode.Balanced.insert + - Std.RBNode.balance1_eq + - Std.RBNode.Balanced.del + - Std.RBNode.All_and + - Std.RBNode.WF.out + - Std.RBNode.Ordered.balRight + - Std.RBNode.balance1_All + - Std.RBNode.RedRed.balance1 + - Std.RBNode.Balanced.append + - path: Mathlib/Algebra/Ring/Units.lean + theorems: + - Units.divp_add_divp_same + - Units.add_divp + - Units.divp_sub_divp_same + - Units.sub_divp + - Units.neg_divp + - Units.divp_sub_divp + - Units.map_neg_one + - Units.add_eq_mul_one_add_div + - Units.divp_add + - Units.map_neg + - Units.divp_sub + - Units.divp_add_divp + - path: Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean + theorems: + - ExistsContDiffBumpBase.y_neg + - ExistsContDiffBumpBase.u_exists + - "ExistsContDiffBumpBase.w_mul_\u03C6_nonneg" + - ExistsContDiffBumpBase.y_pos_of_mem_ball + - ExistsContDiffBumpBase.u_compact_support + - ExistsContDiffBumpBase.w_integral + - ExistsContDiffBumpBase.y_eq_one_of_mem_closedBall + - IsOpen.exists_smooth_support_eq + - ExistsContDiffBumpBase.y_le_one + - ExistsContDiffBumpBase.w_nonneg + - ExistsContDiffBumpBase.w_compact_support + - ExistsContDiffBumpBase.y_smooth + - exists_smooth_tsupport_subset + - ExistsContDiffBumpBase.w_support + - ExistsContDiffBumpBase.u_int_pos + - ExistsContDiffBumpBase.w_def + - ExistsContDiffBumpBase.y_eq_zero_of_not_mem_ball + - path: Mathlib/Geometry/Manifold/Instances/Sphere.lean + theorems: + - range_mfderiv_coe_sphere + - stereographic'_source + - stereo_left_inv + - stereoInvFunAux_mem + - stereographic'_symm_apply + - stereographic'_target + - sphere_ext_iff + - ContMDiff.codRestrict_sphere + - hasFDerivAt_stereoInvFunAux + - stereoInvFun_ne_north_pole + - hasFDerivAt_stereoInvFunAux_comp_coe + - stereographic_neg_apply + - mfderiv_coe_sphere_injective + - contMDiff_neg_sphere + - contDiffOn_stereoToFun + - stereo_right_inv + - contDiff_stereoInvFunAux + - stereographic_source + - path: Mathlib/LinearAlgebra/Matrix/Trace.lean + theorems: + - Matrix.trace_col_mul_row + - Matrix.trace_blockDiagonal' + - Matrix.trace_eq_zero_of_isEmpty + - Matrix.trace_blockDiagonal + - Matrix.trace_submatrix_succ + - Matrix.trace_one + - Matrix.trace_mul_cycle' + - Matrix.trace_diagonal + - Matrix.trace_mul_cycle + - Matrix.trace_fin_three + - Matrix.trace_mul_comm + - path: Mathlib/Data/MvPolynomial/Monad.lean + theorems: + - "MvPolynomial.join\u2082_map" + - "MvPolynomial.eval\u2082Hom_comp_bind\u2082" + - "MvPolynomial.vars_bind\u2081" + - "MvPolynomial.eval\u2082Hom_comp_C" + - "MvPolynomial.bind\u2082_monomial" + - "MvPolynomial.rename_comp_bind\u2081" + - "MvPolynomial.bind\u2081_comp_rename" + - "MvPolynomial.bind\u2082_C_left" + - "MvPolynomial.map_bind\u2082" + - "MvPolynomial.bind\u2082_monomial_one" + - "MvPolynomial.hom_bind\u2081" + - "MvPolynomial.mem_vars_bind\u2081" + - "MvPolynomial.map_bind\u2081" + - "MvPolynomial.bind\u2081_monomial" + - "MvPolynomial.eval\u2082Hom_bind\u2081" + - MvPolynomial.map_comp_C + - "MvPolynomial.bind\u2082_comp_bind\u2082" + - "MvPolynomial.bind\u2081_X_left" + - "MvPolynomial.bind\u2082_map" + - "MvPolynomial.bind\u2081_comp_bind\u2081" + - "MvPolynomial.bind\u2081_bind\u2081" + - MvPolynomial.aeval_id_rename + - "MvPolynomial.aeval_comp_bind\u2081" + - path: Mathlib/NumberTheory/EllipticDivisibilitySequence.lean + theorems: + - normEDS'_one + - normEDS_two + - normEDS_odd + - normEDS_three + - normEDS_even + - normEDS'_odd + - IsEllSequence_mul + - normEDS'_three + - normEDS_neg + - normEDS_one + - IsEllSequence_id + - normEDS'_zero + - normEDS'_two + - normEDS'_even + - normEDS_zero + - normEDS'_four + - normEDS_four + - path: Mathlib/MeasureTheory/PiSystem.lean + theorems: + - isPiSystem_iUnion_of_directed_le + - MeasurableSpace.DynkinSystem.has_diff + - isPiSystem_Icc_mem + - isPiSystem_Ixx_mem + - generateFrom_piiUnionInter_singleton_left + - mem_generatePiSystem_iUnion_elim' + - generatePiSystem_mono + - generateFrom_generatePiSystem_eq + - IsPiSystem.comap + - MeasurableSpace.DynkinSystem.has_compl_iff + - MeasurableSpace.DynkinSystem.generateHas_compl + - MeasurableSpace.DynkinSystem.generate_inter + - generatePiSystem_measurableSet + - isPiSystem_Ixx + - IsPiSystem.insert_univ + - isPiSystem_Icc + - measurableSet_iSup_of_mem_piiUnionInter + - isPiSystem_image_Iio + - MeasurableSpace.DynkinSystem.has_iUnion + - mem_generatePiSystem_iUnion_elim + - MeasurableSpace.induction_on_inter + - MeasurableSpace.DynkinSystem.generateFrom_eq + - piiUnionInter_singleton + - generatePiSystem_subset_self + - isPiSystem_piiUnionInter + - generateFrom_piiUnionInter_measurableSet + - subset_piiUnionInter + - piiUnionInter_singleton_left + - IsPiSystem.insert_empty + - MeasurableSpace.DynkinSystem.has_union + - MeasurableSpace.DynkinSystem.has_univ + - IsPiSystem.singleton + - isPiSystem_image_Iic + - path: Mathlib/Data/Holor.lean + theorems: + - Holor.cast_type + - HolorIndex.drop_drop + - Holor.cprankMax_nil + - HolorIndex.take_take + - Holor.slice_eq + - Holor.cprankMax_1 + - Holor.cprankMax_sum + - HolorIndex.cast_type + - Holor.slice_sum + - Holor.slice_unitVec_mul + - Holor.holor_index_cons_decomp + - Holor.mul_assoc0 + - Holor.mul_scalar_mul + - Holor.cprankMax_add + - Holor.cprankMax_mul + - Holor.sum_unitVec_mul_slice + - Holor.mul_assoc + - HolorIndex.drop_take + - path: Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean + theorems: + - ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def' + - ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atTop + - ProbabilityTheory.toRatCDF_unit_prod + - ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_unit_prod + - ProbabilityTheory.tendsto_defaultRatCDF_atTop + - ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_eq + - ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_univ + - ProbabilityTheory.IsMeasurableRatCDF.measurable_stieltjesFunction + - ProbabilityTheory.IsMeasurableRatCDF.measure_stieltjesFunction_Iic + - ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_le_one + - ProbabilityTheory.tendsto_defaultRatCDF_atBot + - ProbabilityTheory.IsRatStieltjesPoint.ite + - ProbabilityTheory.isMeasurableRatCDF_toRatCDF + - ProbabilityTheory.toRatCDF_of_isRatStieltjesPoint + - ProbabilityTheory.IsMeasurableRatCDF.measurable_measure_stieltjesFunction + - ProbabilityTheory.IsMeasurableRatCDF.tendsto_stieltjesFunction_atBot + - ProbabilityTheory.IsMeasurableRatCDF.monotone_stieltjesFunctionAux + - ProbabilityTheory.isRatStieltjesPoint_unit_prod_iff + - ProbabilityTheory.stieltjesOfMeasurableRat_unit_prod + - ProbabilityTheory.iInf_rat_gt_defaultRatCDF + - ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_nonneg + - ProbabilityTheory.measurableSet_isRatStieltjesPoint + - ProbabilityTheory.defaultRatCDF_nonneg + - ProbabilityTheory.monotone_defaultRatCDF + - ProbabilityTheory.defaultRatCDF_le_one + - ProbabilityTheory.IsMeasurableRatCDF.continuousWithinAt_stieltjesFunctionAux_Ici + - path: Mathlib/FieldTheory/IsAlgClosed/Classification.lean + theorems: + - IsAlgClosed.ringEquivOfCardinalEqOfCharP + - IsAlgClosed.ringEquivOfCardinalEqOfCharEq + - Algebra.IsAlgebraic.cardinal_mk_le_sigma_polynomial + - IsAlgClosed.ringEquivOfCardinalEqOfCharZero + - IsAlgClosed.isAlgClosure_of_transcendence_basis + - IsAlgClosed.cardinal_le_max_transcendence_basis + - IsAlgClosed.cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt + - Algebra.IsAlgebraic.cardinal_mk_le_max + - path: Mathlib/Topology/Category/Profinite/Nobeling.lean + theorems: + - Profinite.NobelingProof.Products.eval_eq + - Profinite.NobelingProof.proj_eq_of_subset + - Profinite.NobelingProof.Products.max_eq_eval + - Profinite.NobelingProof.GoodProducts.linearIndependent_comp_of_eval + - Profinite.NobelingProof.Products.evalCons + - Profinite.NobelingProof.factors_prod_eq_basis_of_eq + - Profinite.NobelingProof.fin_comap_jointlySurjective + - Profinite.NobelingProof.GoodProducts.linearIndependent_iff_smaller + - Profinite.NobelingProof.GoodProducts.smaller_factorization + - Profinite.NobelingProof.Products.lt_nil_empty + - Profinite.NobelingProof.GoodProducts.smaller_mono + - Profinite.NobelingProof.ord_term + - Profinite.NobelingProof.succ_exact + - Profinite.NobelingProof.GoodProducts.span_iff_products + - Profinite.NobelingProof.GoodProducts.sum_to_range + - Profinite.NobelingProof.proj_eq_self + - Profinite.NobelingProof.Products.max_eq_o_cons_tail' + - Profinite.NobelingProof.projRestricts_comp_projRestrict + - Profinite.NobelingProof.Products.span_nil_eq_top + - Profinite.NobelingProof.CC_exact + - Profinite.NobelingProof.GoodProducts.injective_sum_to + - Profinite.NobelingProof.Products.lt_iff_lex_lt + - Profinite.NobelingProof.GoodProducts.maxTail_isGood + - Profinite.NobelingProof.CC_comp_zero + - Profinite.NobelingProof.C1_projOrd + - Profinite.NobelingProof.term_ord_aux + - Profinite.NobelingProof.spanFinBasis.span + - Profinite.NobelingProof.GoodProducts.maxToGood_injective + - Profinite.NobelingProof.projRestricts_eq_id + - Profinite.NobelingProof.ord_term_aux + - Profinite.NobelingProof.continuous_proj + - Profinite.NobelingProof.swapTrue_mem_C1 + - Profinite.NobelingProof.GoodProducts.linearIndependentSingleton + - Profinite.NobelingProof.one_sub_e_mem_of_false + - Profinite.NobelingProof.succ_mono + - Profinite.NobelingProof.GoodProducts.P0 + - Profinite.NobelingProof.GoodProducts.good_lt_maxProducts + - Profinite.NobelingProof.GoodProducts.linearIndependentAux + - Profinite.NobelingProof.union_C0C1_eq + - Profinite.NobelingProof.isClosed_C0 + - Profinite.NobelingProof.GoodProducts.max_eq_eval_unapply + - Profinite.NobelingProof.Products.isGood_nil + - Profinite.NobelingProof.contained_proj + - Profinite.NobelingProof.GoodProducts.head!_eq_o_of_maxProducts + - Profinite.NobelingProof.Products.max_eq_o_cons_tail + - Profinite.NobelingProof.GoodProducts.square_commutes + - Profinite.NobelingProof.Products.isGood_mono + - Profinite.NobelingProof.GoodProducts.injective + - Profinite.NobelingProof.e_mem_of_eq_true + - "Profinite.NobelingProof.Products.eval_\u03C0s_image" + - Profinite.NobelingProof.GoodProducts.linearIndependent_iff_union_smaller + - "Profinite.NobelingProof.Products.eval_\u03C0s_image'" + - Profinite.NobelingProof.GoodProducts.linearIndependent_iff_range + - Profinite.NobelingProof.factors_prod_eq_basis + - Profinite.NobelingProof.proj_comp_of_subset + - Profinite.NobelingProof.projRestricts_eq_comp + - Profinite.NobelingProof.swapTrue_eq_true + - Profinite.Nobeling.embedding + - Profinite.NobelingProof.GoodProducts.union + - Profinite.NobelingProof.Products.limitOrdinal + - "Profinite.NobelingProof.coe_\u03C0s" + - Profinite.NobelingProof.proj_prop_eq_self + - Profinite.NobelingProof.GoodProducts.range_equiv_smaller_toFun_bijective + - Profinite.NobelingProof.Products.evalFacProp + - Profinite.NobelingProof.list_prod_apply + - Profinite.NobelingProof.GoodProducts.sum_equiv_comp_eval_eq_elim + - "Profinite.NobelingProof.eval_eq_\u03C0J" + - Profinite.NobelingProof.GoodProducts.Plimit + - Profinite.NobelingProof.Products.evalFacProps + - Profinite.NobelingProof.Products.prop_of_isGood + - Profinite.NobelingProof.iso_map_bijective + - Profinite.NobelingProof.GoodProducts.finsupp_sum_mem_span_eval + - Profinite.NobelingProof.mem_C'_eq_false + - Profinite.NobelingProof.GoodProducts.union_succ + - Profinite.NobelingProof.GoodProducts.linearIndependent_iff_sum + - Profinite.NobelingProof.Products.prop_of_isGood_of_contained + - Profinite.NobelingProof.isClosed_C1 + - "Profinite.NobelingProof.Products.eval_\u03C0s'" + - Profinite.NobelingProof.GoodProducts.span_sum + - Profinite.NobelingProof.continuous_swapTrue + - Profinite.NobelingProof.GoodProducts.chain'_cons_of_lt + - Profinite.NobelingProof.C0_projOrd + - Profinite.NobelingProof.GoodProducts.span + - Profinite.NobelingProof.factors_prod_eq_basis_of_ne + - Profinite.NobelingProof.GoodProducts.spanFin + - "Profinite.NobelingProof.coe_\u03C0s'" + - path: Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean + theorems: + - NonUnitalSubsemiring.coe_closure_eq + - NonUnitalSubsemiring.mem_center_iff + - NonUnitalRingHom.srange_eq_map + - NonUnitalSubsemiring.mem_map_equiv + - NonUnitalSubsemiring.coe_iSup_of_directed + - NonUnitalRingHom.srange_top_iff_surjective + - NonUnitalSubsemiring.mem_sSup_of_directedOn + - NonUnitalRingHom.map_srange + - NonUnitalSubsemiring.top_prod + - NonUnitalSubsemiring.coe_sSup_of_directedOn + - NonUnitalSubsemiring.mem_map + - NonUnitalSubsemiring.closure_addSubmonoid_closure + - Subsemigroup.nonUnitalSubsemiringClosure_eq_closure + - Set.mem_center_iff_addMonoidHom + - NonUnitalSubsemiring.mem_iSup_of_directed + - NonUnitalSubsemiring.prod_top + - path: Mathlib/Algebra/Homology/Exact.lean + theorems: + - CategoryTheory.epi_iff_exact_zero_right + - CategoryTheory.kernelSubobject_arrow_eq_zero_of_exact_zero_left + - CategoryTheory.exact_epi_comp + - CategoryTheory.exact_comp_hom_inv_comp_iff + - CategoryTheory.comp_eq_zero_of_exact + - CategoryTheory.Preadditive.exact_iff_exact_of_iso + - CategoryTheory.exact_iso_comp + - CategoryTheory.exact_kernelSubobject_arrow + - CategoryTheory.Exact.epi_kernel_lift + - CategoryTheory.Preadditive.exact_of_iso_of_exact + - CategoryTheory.exact_of_zero + - CategoryTheory.exact_comp_mono + - CategoryTheory.exact_comp_mono_iff + - "CategoryTheory.kernel_\u03B9_eq_zero_of_exact_zero_left" + - "CategoryTheory.exact_kernel_\u03B9" + - CategoryTheory.Preadditive.exact_iff_homology'_zero + - CategoryTheory.exact_comp_hom_inv_comp + - CategoryTheory.comp_eq_zero_of_image_eq_kernel + - CategoryTheory.imageToKernel_isIso_of_image_eq_kernel + - CategoryTheory.mono_iff_exact_zero_left + - path: Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean + theorems: + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_\u03B4\u2080'" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.map_on_summand\u2080" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.map_on_summand" + - "AlgebraicTopology.DoldKan.Is\u03B4\u2080.iff" + - "AlgebraicTopology.DoldKan.HigherFacesVanish.on_\u0393\u2080_summand_id" + - "AlgebraicTopology.DoldKan.PInfty_on_\u0393\u2080_splitting_summand_eq_self" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_id" + - "AlgebraicTopology.DoldKan.Is\u03B4\u2080.eq_\u03B4\u2080" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_naturality" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_\u03B4\u2080" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_eq_zero" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.map_on_summand'" + - "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_comp" + - path: Mathlib/MeasureTheory/Function/Jacobian.lean + theorems: + - MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul + - MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux1 + - MeasureTheory.aemeasurable_toNNReal_abs_det_fderivWithin + - MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar + - MeasureTheory.addHaar_image_le_mul_of_det_lt + - MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul' + - MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux2 + - MeasureTheory.det_one_smulRight + - MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar + - MeasureTheory.aemeasurable_ofReal_abs_det_fderivWithin + - MeasureTheory.mul_le_addHaar_image_of_lt_det + - MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul + - MeasureTheory.addHaar_image_eq_zero_of_differentiableOn_of_addHaar_eq_zero + - MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv_aux2 + - MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul' + - MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul + - exists_partition_approximatesLinearOn_of_hasFDerivWithinAt + - MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul + - exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt + - MeasureTheory.aemeasurable_fderivWithin + - MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul + - MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul + - MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1 + - MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul + - MeasureTheory.lintegral_image_eq_lintegral_abs_det_fderiv_mul + - ApproximatesLinearOn.norm_fderiv_sub_le + - MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero_aux + - MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero + - MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv + - MeasureTheory.integrableOn_image_iff_integrableOn_abs_deriv_smul + - MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image + - MeasureTheory.integral_image_eq_integral_abs_deriv_smul + - path: Mathlib/Data/List/Permutation.lean + theorems: + - List.permutations_nil + - List.permutationsAux_cons + - List.permutations'Aux_eq_permutationsAux2 + - List.map_permutationsAux + - List.permutationsAux_nil + - List.permutationsAux2_append + - List.permutationsAux2_comp_append + - List.map_permutations' + - List.permutationsAux_append + - List.length_foldr_permutationsAux2' + - List.permutations_append + - List.foldr_permutationsAux2 + - List.map_permutationsAux2' + - List.permutationsAux2_snd_eq + - List.length_permutationsAux2 + - List.map_map_permutations'Aux + - List.permutationsAux2_fst + - List.mem_foldr_permutationsAux2 + - List.map_permutationsAux2 + - List.mem_permutationsAux2 + - List.map_permutations + - List.permutationsAux2_snd_cons + - List.mem_permutationsAux2' + - path: Mathlib/Data/Matrix/RowCol.lean + theorems: + - Matrix.updateRow_subsingleton + - Matrix.updateRow_apply + - Matrix.updateColumn_submatrix_equiv + - Matrix.transpose_row + - Matrix.updateRow_submatrix_equiv + - Matrix.row_add + - Matrix.map_updateRow + - Matrix.updateColumn_transpose + - Matrix.diagonal_updateRow_single + - Matrix.submatrix_updateColumn_equiv + - Matrix.col_vecMul + - Matrix.updateColumn_subsingleton + - Matrix.submatrix_updateRow_equiv + - Matrix.updateRow_transpose + - Matrix.col_mulVec + - Matrix.updateColumn_conjTranspose + - Matrix.transpose_col + - Matrix.diagonal_updateColumn_single + - Matrix.row_vecMul + - Matrix.col_smul + - Matrix.vecMulVec_eq + - Matrix.updateRow_conjTranspose + - Matrix.row_smul + - Matrix.conjTranspose_col + - Matrix.col_add + - Matrix.diag_col_mul_row + - Matrix.row_mulVec + - Matrix.updateColumn_apply + - Matrix.conjTranspose_row + - Matrix.map_updateColumn + - path: Mathlib/MeasureTheory/Measure/Content.lean + theorems: + - MeasureTheory.Content.le_outerMeasure_compacts + - MeasureTheory.Content.mono + - MeasureTheory.Content.outerMeasure_exists_open + - MeasureTheory.Content.innerContent_iUnion_nat + - MeasureTheory.Content.measure_eq_content_of_regular + - MeasureTheory.Content.outerMeasure_lt_top_of_isCompact + - MeasureTheory.Content.innerContent_iSup_nat + - MeasureTheory.Content.outerMeasure_preimage + - MeasureTheory.Content.is_mul_left_invariant_innerContent + - MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant + - MeasureTheory.Content.contentRegular_exists_compact + - MeasureTheory.Content.sup_le + - MeasureTheory.Content.outerMeasure_caratheodory + - MeasureTheory.Content.outerMeasure_exists_compact + - MeasureTheory.Content.empty + - MeasureTheory.Content.sup_disjoint + - MeasureTheory.Content.innerContent_exists_compact + - MeasureTheory.Content.innerContent_bot + - MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant + - MeasureTheory.Content.innerContent_comap + - MeasureTheory.Content.borel_le_caratheodory + - MeasureTheory.Content.is_mul_left_invariant_outerMeasure + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean + theorems: + - InnerProductGeometry.norm_add_eq_norm_sub_iff_angle_eq_pi_div_two + - InnerProductGeometry.cos_eq_one_iff_angle_eq_zero + - LinearIsometry.angle_map + - InnerProductGeometry.norm_sub_eq_abs_sub_norm_iff_angle_eq_zero + - InnerProductGeometry.continuousAt_angle + - InnerProductGeometry.angle_neg_neg + - InnerProductGeometry.norm_sub_eq_abs_sub_norm_of_angle_eq_zero + - InnerProductGeometry.angle_zero_left + - InnerProductGeometry.angle_smul_right_of_neg + - InnerProductGeometry.angle_smul_right_of_pos + - InnerProductGeometry.angle_zero_right + - InnerProductGeometry.angle_smul_left_of_neg + - InnerProductGeometry.inner_eq_neg_mul_norm_iff_angle_eq_pi + - InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two + - InnerProductGeometry.inner_eq_mul_norm_iff_angle_eq_zero + - InnerProductGeometry.angle_eq_pi_iff + - InnerProductGeometry.inner_eq_mul_norm_of_angle_eq_zero + - InnerProductGeometry.angle_self + - InnerProductGeometry.cos_eq_zero_iff_angle_eq_pi_div_two + - InnerProductGeometry.angle_neg_right + - InnerProductGeometry.angle_smul_smul + - InnerProductGeometry.angle_neg_left + - InnerProductGeometry.sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi + - InnerProductGeometry.angle_eq_zero_iff + - InnerProductGeometry.norm_sub_eq_add_norm_iff_angle_eq_pi + - InnerProductGeometry.sin_eq_one_iff_angle_eq_pi_div_two + - InnerProductGeometry.angle_self_neg_of_nonzero + - InnerProductGeometry.sin_angle_mul_norm_mul_norm + - InnerProductGeometry.inner_eq_neg_mul_norm_of_angle_eq_pi + - InnerProductGeometry.cos_eq_neg_one_iff_angle_eq_pi + - InnerProductGeometry.angle_add_angle_eq_pi_of_angle_eq_pi + - InnerProductGeometry.norm_sub_eq_add_norm_of_angle_eq_pi + - InnerProductGeometry.angle_smul_left_of_pos + - InnerProductGeometry.norm_add_eq_add_norm_of_angle_eq_zero + - InnerProductGeometry.norm_add_eq_add_norm_iff_angle_eq_zero + - InnerProductGeometry.angle_comm + - InnerProductGeometry.angle_neg_self_of_nonzero + - path: Mathlib/RingTheory/Polynomial/Nilpotent.lean + theorems: + - Polynomial.isNilpotent_C_mul_pow_X_of_isNilpotent + - Polynomial.isUnit_iff' + - Polynomial.isNilpotent_pow_X_mul_C_of_isNilpotent + - Polynomial.isNilpotent_aeval_sub_of_isNilpotent_sub + - Polynomial.isNilpotent_reflect_iff + - Polynomial.isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub + - Polynomial.isUnit_of_coeff_isUnit_isNilpotent + - Polynomial.isNilpotent_iff + - Polynomial.coeff_isUnit_isNilpotent_of_isUnit + - Polynomial.isUnit_C_add_X_mul_iff + - Polynomial.isNilpotent_C_iff + - Polynomial.isNilpotent_mul_X_iff + - Polynomial.not_isUnit_of_natDegree_pos_of_isReduced + - Polynomial.isNilpotent_monomial_iff + - path: Mathlib/Algebra/Homology/HomotopyCategory/Shift.lean + theorems: + - CochainComplex.shiftFunctorAdd_inv_app_f + - CochainComplex.shiftFunctorZero_inv_app_f + - CochainComplex.shiftFunctorComm_hom_app_f + - CochainComplex.shiftFunctorAdd'_inv_app_f' + - CochainComplex.shiftFunctorAdd_hom_app_f + - CochainComplex.shiftFunctorAdd_eq + - CochainComplex.XIsoOfEq_shift + - CochainComplex.shiftFunctorAdd'_hom_app_f' + - CochainComplex.shiftFunctorZero_hom_app_f + - CochainComplex.shiftFunctorZero_eq + - CochainComplex.shiftFunctorAdd'_eq + - path: Mathlib/Algebra/Module/Submodule/Ker.lean + theorems: + - LinearMap.le_ker_iff_map + - LinearMap.ker_restrict + - LinearMap.disjoint_ker + - LinearMap.ker_eq_bot' + - LinearMap.ker_le_comap + - LinearMap.ker_comp_of_ker_eq_bot + - LinearMap.ker_zero + - LinearMap.injective_domRestrict_iff + - LinearMap.injective_restrict_iff_disjoint + - LinearMap.ker_le_ker_comp + - LinearMap.sub_mem_ker_iff + - LinearMap.disjoint_ker' + - LinearMapClass.ker_eq_bot + - Submodule.ker_inclusion + - LinearMap.ker_codRestrict + - LinearMap.ker_eq_bot_of_inverse + - LinearMap.ker_sup_ker_le_ker_comp_of_commute + - LinearMap.ker_eq_bot_of_injective + - path: Mathlib/Order/Disjointed.lean + theorems: + - disjoint_disjointed + - partialSups_disjointed + - disjointed_eq_inf_compl + - disjointed_le_id + - Monotone.disjointed_succ_sup + - preimage_find_eq_disjointed + - Monotone.disjointed_succ + - disjointed_unique + - path: Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean + theorems: + - HomologicalComplex.biprodXIso_hom_fst + - HomologicalComplex.inr_biprodXIso_inv + - HomologicalComplex.biprod_inl_snd_f + - HomologicalComplex.biprod_inl_fst_f + - HomologicalComplex.biprod_lift_fst_f + - HomologicalComplex.inl_biprodXIso_inv + - HomologicalComplex.biprod_lift_snd_f + - HomologicalComplex.biprodXIso_hom_snd + - HomologicalComplex.biprod_inr_desc_f + - HomologicalComplex.biprod_inl_desc_f + - HomologicalComplex.biprod_inr_snd_f + - HomologicalComplex.biprod_inr_fst_f + - path: Mathlib/Data/Polynomial/Laurent.lean + theorems: + - LaurentPolynomial.mul_T_assoc + - LaurentPolynomial.degree_C_mul_T_ite + - LaurentPolynomial.C_apply + - Polynomial.toLaurent_C_mul_X_pow + - Polynomial.toLaurent_C_mul_eq + - Polynomial.toLaurent_C + - LaurentPolynomial.degree_C_ite + - LaurentPolynomial.isLocalization + - LaurentPolynomial.degree_C_mul_T_le + - LaurentPolynomial.leftInverse_trunc_toLaurent + - Polynomial.toLaurent_X + - LaurentPolynomial.degree_C + - LaurentPolynomial.support_C_mul_T_of_ne_zero + - LaurentPolynomial.degree_C_le + - LaurentPolynomial.toLaurent_reverse + - LaurentPolynomial.commute_T + - LaurentPolynomial.T_add + - LaurentPolynomial.trunc_C_mul_T + - LaurentPolynomial.degree_C_mul_T + - LaurentPolynomial.involutive_invert + - LaurentPolynomial.invert_comp_C + - LaurentPolynomial.support_C_mul_T + - LaurentPolynomial.degree_T + - Polynomial.toLaurent_X_pow + - LaurentPolynomial.reduce_to_polynomial_of_mul_T + - LaurentPolynomial.induction_on' + - LaurentPolynomial.induction_on_mul_T + - LaurentPolynomial.T_pow + - LaurentPolynomial.single_eq_C_mul_T + - LaurentPolynomial.induction_on + - LaurentPolynomial.degree_eq_bot_iff + - LaurentPolynomial.degree_T_le + - Polynomial.toLaurent_C_mul_T + - LaurentPolynomial.toLaurent_support + - LaurentPolynomial.exists_T_pow + - path: Mathlib/Algebra/Module/Submodule/LinearMap.lean + theorems: + - LinearMap.submodule_pow_eq_zero_of_pow_eq_zero + - LinearMap.restrict_commute + - LinearMap.pow_apply_mem_of_forall_mem + - LinearMap.pow_restrict + - path: Mathlib/Data/List/Destutter.lean + theorems: + - List.destutter'_cons_neg + - List.destutter_eq_self_iff + - List.destutter'_sublist + - List.destutter'_is_chain + - List.mem_destutter' + - List.destutter'_cons_pos + - List.destutter'_singleton + - List.destutter'_is_chain' + - List.destutter'_of_chain + - List.destutter'_eq_self_iff + - path: Mathlib/CategoryTheory/Conj.lean + theorems: + - CategoryTheory.Iso.conjAut_apply + - CategoryTheory.Iso.homCongr_apply + - CategoryTheory.Iso.symm_self_conj + - CategoryTheory.Iso.homCongr_refl + - CategoryTheory.Iso.trans_conjAut + - CategoryTheory.Iso.homCongr_trans + - CategoryTheory.Iso.refl_conj + - CategoryTheory.Functor.map_homCongr + - CategoryTheory.Functor.map_conjAut + - CategoryTheory.Iso.homCongr_comp + - path: Mathlib/CategoryTheory/Sites/CoversTop.lean + theorems: + - CategoryTheory.GrothendieckTopology.CoversTop.sections_ext + - CategoryTheory.GrothendieckTopology.CoversTop.ext + - CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.exists_unique_section + - CategoryTheory.GrothendieckTopology.coversTop_iff_of_isTerminal + - CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.familyOfElements_apply + - CategoryTheory.Presheaf.FamilyOfElementsOnObjects.IsCompatible.familyOfElements_isCompatible + - path: Mathlib/CategoryTheory/GlueData.lean + theorems: + - "CategoryTheory.GlueData.\u03B9_jointly_surjective" + - "CategoryTheory.GlueData.types_\u03B9_jointly_surjective" + - "CategoryTheory.GlueData.\u03B9_gluedIso_hom" + - CategoryTheory.GlueData.t'_iji + - CategoryTheory.GlueData.t'_iij + - "CategoryTheory.GlueData.\u03B9_gluedIso_inv" + - CategoryTheory.GlueData.t_inv + - CategoryTheory.GlueData.t'_inv + - CategoryTheory.GlueData.t'_comp_eq_pullbackSymmetry + - CategoryTheory.GlueData.t'_jii + - path: Mathlib/Logic/Equiv/Option.lean + theorems: + - Equiv.removeNone_aux_inv + - Equiv.optionSubtype_apply_symm_apply + - Equiv.optionSubtypeNe_symm_self + - Equiv.optionSubtypeNe_symm_of_ne + - Equiv.removeNone_aux_some + - Equiv.some_removeNone_iff + - Equiv.removeNone_aux_none + - Equiv.option_symm_apply_none_iff + - Equiv.removeNone_optionCongr + - Equiv.optionSubtype_symm_apply_symm_apply + - path: Mathlib/CategoryTheory/Sites/Spaces.lean + theorems: + - Opens.pretopology_toGrothendieck + - Opens.pretopology_ofGrothendieck + - path: Mathlib/Data/ZMod/Quotient.lean + theorems: + - finite_zpowers + - MulAction.orbitZPowersEquiv_symm_apply' + - Nat.card_zpowers + - AddAction.orbitZMultiplesEquiv_symm_apply' + - MulAction.minimalPeriod_eq_card + - path: Mathlib/RingTheory/Congruence.lean + theorems: + - RingCon.ringConGen_of_ringCon + - RingCon.sup_def + - RingCon.coe_iInf + - RingCon.sSup_def + - RingCon.sSup_eq_ringConGen + - RingCon.sup_eq_ringConGen + - RingCon.coe_sInf + - RingCon.ext + - RingCon.ringConGen_le + - RingCon.sInf_toSetoid + - path: Mathlib/MeasureTheory/Integral/MeanInequalities.lean + theorems: + - ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero + - ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top + - ENNReal.funMulInvSnorm_rpow + - ENNReal.lintegral_mul_prod_norm_pow_le + - ENNReal.ae_eq_zero_of_lintegral_rpow_eq_zero + - ENNReal.lintegral_rpow_funMulInvSnorm_eq_one + - ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top + - ENNReal.lintegral_Lp_add_le_of_le_one + - ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one + - ENNReal.lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top + - ENNReal.lintegral_mul_norm_pow_le + - ENNReal.lintegral_prod_norm_pow_le + - ENNReal.fun_eq_funMulInvSnorm_mul_snorm + - ENNReal.lintegral_Lp_add_le_aux + - NNReal.lintegral_mul_le_Lp_mul_Lq + - ENNReal.lintegral_Lp_add_le + - ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr + - ENNReal.lintegral_mul_le_Lp_mul_Lq + - ENNReal.lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow + - ENNReal.lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add + - path: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean + theorems: + - StarSubalgebra.isUnit_coe_inv_mem + - spectrum_star_mul_self_of_isStarNormal + - StarSubalgebra.coe_isUnit + - elementalStarAlgebra.bijective_characterSpaceToSpectrum + - path: Mathlib/Logic/Equiv/Basic.lean + theorems: + - Equiv.Perm.eq_of_prodExtendRight_ne + - Equiv.Perm.subtypeCongr.symm + - Equiv.subtypeEquiv_refl + - Equiv.piCongrLeft_sum_inl + - Equiv.swap_apply_self + - Equiv.prodCongr_refl_left + - Equiv.Perm.extendDomain_apply_not_subtype + - Equiv.prodCongrRight_trans_prodComm + - Function.update_comp_equiv + - Equiv.swap_apply_right + - Equiv.Perm.subtypeCongr.right_apply + - Equiv.Perm.subtypeCongr.refl + - Equiv.setValue_eq + - Equiv.Perm.sumCongr_swap_refl + - Equiv.subtypeEquiv_symm + - Equiv.swapCore_swapCore + - "Equiv.semiconj\u2082_conj" + - Equiv.Perm.sumCongr_refl_swap + - Equiv.swap_apply_eq_iff + - Equiv.swapCore_self + - Equiv.Perm.extendDomain_trans + - Equiv.piCongrLeft'_symm + - Equiv.swapCore_comm + - Equiv.swap_apply_of_ne_of_ne + - Equiv.apply_swap_eq_self + - Equiv.prodCongr_refl_right + - Equiv.Perm.extendDomain_refl + - Equiv.swap_eq_refl_iff + - Equiv.semiconj_conj + - Equiv.symm_trans_swap_trans + - Equiv.Perm.subtypeCongr.left_apply + - Equiv.sigmaEquivProd_sigmaCongrRight + - Equiv.piCongr_apply_apply + - Equiv.swap_comp_apply + - Function.Injective.map_swap + - Equiv.piCongr'_symm_apply_symm_apply + - Equiv.Perm.extendDomain_apply_image + - Equiv.piCongrLeft_sum_inr + - Equiv.swap_eq_update + - Equiv.Perm.fst_prodExtendRight + - Function.piCongrLeft'_symm_update + - Equiv.Perm.subtypeCongr.apply + - Equiv.swap_apply_ne_self_iff + - Function.piCongrLeft'_update + - Equiv.Perm.extendDomain_apply_subtype + - Function.Injective.swap_apply + - Equiv.coe_subtypeEquivCodomain_symm + - Equiv.sigmaCongrRight_sigmaEquivProd + - Equiv.comp_swap_eq_update + - Equiv.prodCongrLeft_trans_prodComm + - Equiv.Perm.subtypeCongr.trans + - Equiv.sumCongr_trans + - Equiv.piCongrLeft'_symm_apply_apply + - path: Mathlib/Data/Sum/Basic.lean + theorems: + - Sum.map_surjective + - Sum.LiftRel.isLeft_congr + - Sum.elim_update_left + - Sum.LiftRel.isRight_congr + - Sum.LiftRel.isRight_right + - Sum.isSome_getLeft?_iff_isLeft + - Sum.exists_sum + - Sum.getRight_eq_getRight? + - Sum.update_elim_inl + - Sum.update_inl_apply_inl + - Sum.LiftRel.isLeft_right + - Sum.sum_rec_congr + - Sum.eq_right_iff_getRight_eq + - Sum.update_elim_inr + - Sum.getLeft_eq_getLeft? + - Sum.update_inr_apply_inr + - Sum.isSome_getRight?_iff_isRight + - Sum.elim_update_right + - Sum.LiftRel.isRight_left + - Sum.LiftRel.exists_of_isLeft_left + - Sum.LiftRel.exists_of_isRight_left + - Sum.LiftRel.isLeft_left + - Sum.eq_left_iff_getLeft_eq + - path: Mathlib/Algebra/Order/Sub/Defs.lean + theorems: + - tsub_right_comm + - AddLECancellable.lt_add_of_tsub_lt_left + - tsub_le_iff_left + - add_tsub_add_eq_tsub_right + - tsub_le_tsub_add_tsub + - AddLECancellable.le_add_tsub + - add_le_add_add_tsub + - add_tsub_le_assoc + - AddLECancellable.lt_tsub_of_add_lt_left + - tsub_tsub_tsub_le_tsub + - le_tsub_add_add + - tsub_nonpos + - tsub_add_eq_tsub_tsub_swap + - tsub_le_iff_tsub_le + - add_tsub_add_le_tsub_add_tsub + - add_tsub_le_tsub_add + - AddLECancellable.tsub_eq_of_eq_add + - add_tsub_add_le_tsub_right + - AddLECancellable.lt_tsub_of_add_lt_right + - add_tsub_add_eq_tsub_left + - AddLECancellable.add_tsub_cancel_right + - tsub_tsub + - add_tsub_add_le_tsub_left + - AddLECancellable.le_tsub_of_add_le_right + - path: Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean + theorems: + - UniformOnFun.continuous_rng_iff + - UniformOnFun.iInf_eq + - UniformOnFun.uniformContinuous_restrict + - UniformOnFun.postcomp_uniformContinuous + - UniformOnFun.hasBasis_uniformity_of_covering_of_basis + - UniformOnFun.precomp_uniformContinuous + - UniformFun.gc + - UniformOnFun.comap_eq + - UniformOnFun.isClosed_setOf_continuous_of_le + - UniformFun.inf_eq + - "UniformOnFun.hasBasis_uniformity_of_basis_aux\u2081" + - UniformFun.uniformContinuous_eval + - UniformFun.comap_eq + - UniformOnFun.hasAntitoneBasis_uniformity + - UniformFun.iInf_eq + - UniformFun.isClosed_setOf_continuous + - UniformOnFun.uniformContinuous_toFun + - UniformOnFun.inf_eq + - UniformFun.tendsto_iff_tendstoUniformly + - UniformOnFun.postcomp_uniformInducing + - UniformFun.postcomp_uniformContinuous + - UniformFun.uniformContinuous_toFun + - UniformOnFun.t2Space_of_covering + - UniformFun.precomp_uniformContinuous + - UniformOnFun.gen_eq_preimage_restrict + - UniformOnFun.hasBasis_uniformity_of_basis + - path: .lake/packages/std/Std/Data/PairingHeap.lean + theorems: + - Std.PairingHeapImp.Heap.size_merge_node + - Std.PairingHeapImp.Heap.noSibling_tail + - Std.PairingHeapImp.Heap.noSibling_deleteMin + - Std.PairingHeapImp.Heap.size_combine + - Std.PairingHeapImp.Heap.WF.tail + - Std.PairingHeapImp.Heap.size_deleteMin + - Std.PairingHeapImp.Heap.noSibling_combine + - Std.PairingHeapImp.Heap.size_tail + - Std.PairingHeapImp.Heap.WF.tail? + - Std.PairingHeapImp.Heap.WF.deleteMin + - Std.PairingHeapImp.Heap.WF.merge_node + - Std.PairingHeapImp.Heap.size_tail?_lt + - Std.PairingHeapImp.Heap.size_merge + - Std.PairingHeapImp.Heap.size_deleteMin_lt + - Std.PairingHeapImp.Heap.noSibling_tail? + - Std.PairingHeapImp.Heap.size_tail? + - path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean + theorems: + - AlgebraicGeometry.ProjIsoSpecTopComponent.fromSpec_toSpec + - AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_fromSpec + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff' + - AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_ne_top + - AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.carrier_eq_span + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.homogeneous + - AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.MemCarrier.clear_denominator' + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.denom_not_mem + - AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.add_mem + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.zero_mem + - AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.smul_mem + - AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.disjoint + - AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_injective + - path: Mathlib/Combinatorics/Additive/Energy.lean + theorems: + - Finset.multiplicativeEnergy_univ_right + - Finset.multiplicativeEnergy_empty_right + - Finset.multiplicativeEnergy_comm + - Finset.multiplicativeEnergy_univ_left + - Finset.multiplicativeEnergy_empty_left + - Finset.multiplicativeEnergy_pos_iff + - Finset.multiplicativeEnergy_eq_zero_iff + - Finset.le_multiplicativeEnergy + - path: Mathlib/RingTheory/Int/Basic.lean + theorems: + - Int.gcd_eq_one_iff_coprime + - Int.Prime.dvd_pow + - Int.eq_pow_of_mul_eq_pow_bit1_right + - Nat.factors_multiset_prod_of_irreducible + - Int.associated_iff + - induction_on_primes + - Int.normalize_of_nonpos + - multiplicity.finite_int_iff + - Int.exists_unit_of_abs + - Nat.factors_eq + - Int.nonneg_of_normalize_eq_self + - Int.sq_of_gcd_eq_one + - Int.natAbs_euclideanDomain_gcd + - Int.eq_pow_of_mul_eq_pow_bit1_left + - Int.Prime.dvd_mul' + - Int.gcd_ne_one_iff_gcd_mul_right_ne_one + - multiplicity.finite_int_iff_natAbs_finite + - Int.normalize_of_nonneg + - Int.abs_eq_normalize + - Int.Prime.dvd_pow' + - Int.span_natAbs + - Int.associated_iff_natAbs + - Int.exists_prime_and_dvd + - prime_two_or_dvd_of_dvd_two_mul_pow_self_two + - Int.Prime.dvd_mul + - Int.coprime_iff_nat_coprime + - path: Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean + theorems: + - "AlgebraicGeometry.\u0393Spec.adjunction_homEquiv_apply" + - "AlgebraicGeometry.Spec\u0393Identity_hom_app_presheaf_obj" + - "AlgebraicGeometry.\u0393Spec.adjunction_homEquiv_symm_apply" + - "AlgebraicGeometry.LocallyRingedSpace.to\u0393SpecCApp_iff" + - "AlgebraicGeometry.LocallyRingedSpace.toStalk_stalkMap_to\u0393Spec" + - "AlgebraicGeometry.\u0393Spec.adjunction_counit_app" + - "AlgebraicGeometry.\u0393Spec.adjunction_unit_app_app_top" + - AlgebraicGeometry.LocallyRingedSpace.not_mem_prime_iff_unit_in_stalk + - "AlgebraicGeometry.LocallyRingedSpace.to\u0393Spec_preim_basicOpen_eq" + - "AlgebraicGeometry.LocallyRingedSpace.\u0393_Spec_left_triangle" + - "AlgebraicGeometry.\u0393Spec.adjunction_unit_app" + - "AlgebraicGeometry.LocallyRingedSpace.isUnit_res_to\u0393SpecMapBasicOpen" + - AlgebraicGeometry.LocallyRingedSpace.comp_ring_hom_ext + - "AlgebraicGeometry.LocallyRingedSpace.to\u0393Spec_continuous" + - "AlgebraicGeometry.LocallyRingedSpace.to\u0393SpecSheafedSpace_app_eq" + - path: Mathlib/LinearAlgebra/Projectivization/Subspace.lean + theorems: + - Projectivization.Subspace.sup_span + - Projectivization.Subspace.span_univ + - Projectivization.Subspace.span_eq_sInf + - Projectivization.Subspace.span_sup + - Projectivization.Subspace.mem_span + - path: Mathlib/GroupTheory/FreeGroup/Basic.lean + theorems: + - FreeGroup.of_injective + - FreeGroup.invRev_length + - FreeGroup.Red.not_step_nil + - FreeGroup.Red.Step.cons_cons_iff + - FreeGroup.Red.Step.invRev + - FreeGroup.toWord_injective + - FreeGroup.Red.Step.append_left + - FreeGroup.map_eq_lift + - FreeGroup.mk_toWord + - FreeGroup.reduce_invRev + - FreeGroup.Red.step_invRev_iff + - FreeGroup.map.comp + - FreeGroup.reduce.min + - FreeGroup.map.id + - FreeGroup.Red.church_rosser + - FreeGroup.Red.Step.diamond_aux + - FreeGroup.Red.not_step_singleton + - FreeGroup.map.unique + - FreeGroup.closure_range_of + - FreeGroup.Red.Step.append_right + - FreeGroup.Red.inv_of_red_of_ne + - FreeGroup.reduce.red + - FreeGroup.Red.length + - FreeGroup.lift.range_le + - FreeGroup.norm_inv_eq + - FreeGroup.equivalence_join_red + - FreeGroup.lift_eq_prod_map + - FreeGroup.Red.Step.length + - FreeGroup.Red.Step.cons_left_iff + - FreeGroup.Red.cons_nil_iff_singleton + - FreeGroup.Red.append_append_left_iff + - FreeGroup.toWord_inv + - FreeGroup.Red.to_append_iff + - FreeGroup.Red.red_iff_irreducible + - FreeGroup.red_invRev_iff + - FreeGroup.reduce.not + - FreeGroup.lift.range_eq_closure + - FreeGroup.Red.sizeof_of_step + - FreeGroup.Red.Step.lift + - FreeGroup.Red.Step.not_rev + - FreeGroup.Red.cons_cons_iff + - FreeGroup.norm_eq_zero + - FreeGroup.Red.Step.sublist + - FreeGroup.norm_mul_le + - FreeGroup.Red.Step.append_left_iff + - FreeGroup.reduce_toWord + - path: Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean + theorems: + - "ExteriorAlgebra.\u03B9_eq_algebraMap_iff" + - "ExteriorAlgebra.\u03B9_range_disjoint_one" + - "ExteriorAlgebra.comp_\u03B9_sq_zero" + - "ExteriorAlgebra.\u03B9_eq_zero_iff" + - "ExteriorAlgebra.map_apply_\u03B9Multi" + - "ExteriorAlgebra.\u03B9Inv_comp_map" + - ExteriorAlgebra.algebraMap_leftInverse + - "ExteriorAlgebra.\u03B9_ne_one" + - "ExteriorAlgebra.\u03B9Multi_succ_curryLeft" + - "TensorAlgebra.toExterior_\u03B9" + - ExteriorAlgebra.map_surjective_iff + - "ExteriorAlgebra.\u03B9_leftInverse" + - ExteriorAlgebra.leftInverse_map_iff + - ExteriorAlgebra.toTrivSqZeroExt_comp_map + - "ExteriorAlgebra.\u03B9Multi_range" + - "ExteriorAlgebra.\u03B9_mul_prod_list" + - "ExteriorAlgebra.map_comp_\u03B9Multi" + - path: Mathlib/Analysis/SpecialFunctions/Log/Base.lean + theorems: + - Real.logb_le_iff_le_rpow_of_base_lt_one + - Real.rpow_logb_of_neg + - Real.logb_nonpos_iff + - Real.logb_self_eq_one_iff + - Real.mul_logb + - Real.b_ne_one' + - Real.logb_pos + - Real.logb_one + - Real.strictAntiOn_logb + - Real.logb_nonneg_iff_of_base_lt_one + - Real.logb_lt_logb_of_base_lt_one + - Real.logb_neg_eq_logb + - Real.logb_lt_logb_iff + - Real.logb_div + - Real.log_b_ne_zero + - Real.inv_logb_div_base + - Real.b_pos + - Real.logb_le_iff_le_rpow + - Real.logb_zero + - Real.logb_neg_iff + - Real.logb_nonneg_of_base_lt_one + - Real.rpow_logb + - Real.logb_rpow + - Real.logb_lt_iff_lt_rpow + - Real.logb_nonpos_iff' + - Real.logb_le_logb_of_le + - Real.logb_eq_zero + - Real.logb_div_base + - Real.logb_nonneg_iff + - Real.logb_lt_logb_iff_of_base_lt_one + - Real.logb_pos_of_base_lt_one + - Real.inv_logb_mul_base + - Real.tendsto_logb_atTop_of_base_lt_one + - Real.logb_mul_base + - Real.logb_le_logb_of_base_lt_one + - Real.logb_prod + - Real.b_ne_one + - Real.logb_mul + - Real.surjOn_logb' + - Real.strictMonoOn_logb_of_base_lt_one + - Real.ceil_logb_nat_cast + - Real.induction_Ico_mul + - Real.inv_logb + - Real.le_logb_iff_rpow_le + - Real.logb_lt_logb + - Real.logb_pos_iff + - Real.logb_pos_iff_of_base_lt_one + - Real.le_logb_iff_rpow_le_of_base_lt_one + - Real.floor_logb_nat_cast + - Real.logb_nonpos_iff_of_base_lt_one + - Real.logb_lt_iff_lt_rpow_of_base_lt_one + - Real.logb_le_logb + - Real.logb_neg_iff_of_base_lt_one + - Real.lt_logb_iff_rpow_lt + - Real.rpow_logb_eq_abs + - Real.lt_logb_iff_rpow_lt_of_base_lt_one + - Real.logb_abs + - path: Mathlib/LinearAlgebra/SesquilinearForm.lean + theorems: + - LinearMap.IsRefl.nondegenerate_of_separatingLeft + - "LinearMap.linearIndependent_of_isOrtho\u1D62" + - LinearMap.isAdjointPair_zero + - LinearMap.isOrtho_flip + - LinearMap.SeparatingLeft.congr + - LinearMap.isPairSelfAdjoint_equiv + - "LinearMap.isAdjointPair_iff_comp_eq_compl\u2082" + - LinearMap.IsRefl.ker_flip_eq_bot + - "LinearMap.IsOrtho\u1D62.separatingLeft_of_not_isOrtho_basis_self" + - LinearMap.separatingLeft_iff_linear_nontrivial + - LinearMap.isOrtho_zero_left + - LinearMap.IsSymm.domRestrict + - LinearMap.IsRefl.ker_eq_bot_iff_ker_flip_eq_bot + - "LinearMap.isOrtho\u1D62_flip" + - LinearMap.span_singleton_sup_orthogonal_eq_top + - LinearMap.ortho_smul_left + - LinearMap.ortho_smul_right + - LinearMap.flip_separatingLeft + - LinearMap.orthogonal_span_singleton_eq_to_lin_ker + - LinearMap.separatingRight_iff_flip_ker_eq_bot + - LinearMap.IsAlt.neg + - LinearMap.isSymm_iff_eq_flip + - LinearMap.IsAlt.isRefl + - LinearMap.isAlt_iff_eq_neg_flip + - LinearMap.IsAdjointPair.smul + - LinearMap.span_singleton_inf_orthogonal_eq_bot + - "LinearMap.IsOrtho\u1D62.not_isOrtho_basis_self_of_separatingRight" + - LinearMap.IsSymm.isRefl + - "LinearMap.IsOrtho\u1D62.not_isOrtho_basis_self_of_separatingLeft" + - LinearMap.IsAdjointPair.add + - "LinearMap.IsOrtho\u1D62.separatingRight_iff_not_isOrtho_basis_self" + - LinearMap.isSkewAdjoint_iff_neg_self_adjoint + - LinearMap.mem_skewAdjointSubmodule + - LinearMap.IsAdjointPair.comp + - LinearMap.separatingLeft_congr_iff + - LinearMap.separatingRight_iff_linear_flip_nontrivial + - LinearMap.nondegenerateRestrictOfDisjointOrthogonal + - path: Mathlib/Order/Filter/CountableSeparatingOn.lean + theorems: + - Filter.exists_mem_singleton_mem_of_mem_of_nonempty_of_forall_separating + - Filter.exists_subset_subsingleton_mem_of_forall_separating + - exists_seq_separating + - Filter.EventuallyEq.of_eventually_mem_of_forall_separating_mem_iff + - Filter.exists_singleton_mem_of_mem_of_forall_separating + - HasCountableSeparatingOn.of_subtype + - path: Mathlib/Probability/Distributions/Gaussian.lean + theorems: + - ProbabilityTheory.gaussianReal_const_add + - ProbabilityTheory.gaussianPDFReal_sub + - ProbabilityTheory.gaussianReal_apply + - ProbabilityTheory.gaussianPDFReal_nonneg + - ProbabilityTheory.gaussianReal_map_add_const + - ProbabilityTheory.gaussianPDFReal_zero_var + - ProbabilityTheory.gaussianPDF_pos + - ProbabilityTheory.gaussianReal_mul_const + - ProbabilityTheory.gaussianPDF_zero_var + - ProbabilityTheory.lintegral_gaussianPDFReal_eq_one + - ProbabilityTheory.gaussianReal_map_const_add + - MeasurableEmbedding.gaussianReal_comap_apply + - ProbabilityTheory.gaussianReal_absolutelyContinuous + - ProbabilityTheory.gaussianPDFReal_add + - ProbabilityTheory.gaussianPDFReal_inv_mul + - MeasurableEquiv.gaussianReal_map_symm_apply + - ProbabilityTheory.rnDeriv_gaussianReal + - ProbabilityTheory.gaussianReal_map_mul_const + - ProbabilityTheory.integral_gaussianPDFReal_eq_one + - ProbabilityTheory.integrable_gaussianPDFReal + - ProbabilityTheory.gaussianPDFReal_mul + - ProbabilityTheory.gaussianReal_map_const_mul + - ProbabilityTheory.gaussianPDFReal_pos + - ProbabilityTheory.gaussianReal_const_mul + - ProbabilityTheory.gaussianReal_apply_eq_integral + - ProbabilityTheory.gaussianReal_add_const + - ProbabilityTheory.gaussianReal_absolutelyContinuous' + - path: Mathlib/Order/Filter/Pi.lean + theorems: + - Filter.pi_neBot + - Filter.pi_inf_principal_pi_eq_bot + - "Filter.map_pi_map_coprod\u1D62_le" + - Filter.hasBasis_pi + - Filter.pi_inf_principal_pi_neBot + - Filter.pi_inf_principal_univ_pi_neBot + - "Filter.coprod\u1D62_eq_bot_iff" + - "Filter.mem_coprod\u1D62_iff" + - Filter.mem_of_pi_mem_pi + - Filter.map_eval_pi + - "Filter.coprod\u1D62_neBot_iff'" + - Filter.mem_pi + - Filter.tendsto_pi + - Filter.pi_inf_principal_univ_pi_eq_bot + - Filter.pi_eq_bot + - "Filter.coprod\u1D62_eq_bot_iff'" + - Filter.pi_inj + - "Filter.compl_mem_coprod\u1D62" + - "Filter.coprod\u1D62_neBot_iff" + - Filter.pi_pure + - Filter.pi_principal + - path: Mathlib/Analysis/MellinInversion.lean + theorems: + - mellin_inversion + - mellinInv_eq_fourierIntegralInv + - mellin_eq_fourierIntegral + - rexp_cexp_aux + - path: Mathlib/Algebra/Order/Sub/Canonical.lean + theorems: + - AddLECancellable.tsub_add_tsub_comm + - tsub_add_eq_max + - AddLECancellable.eq_tsub_iff_add_eq_of_le + - AddLECancellable.lt_tsub_iff_left_of_le + - add_tsub_cancel_of_le + - AddLECancellable.tsub_lt_tsub_right_of_le + - AddLECancellable.le_tsub_iff_right + - add_tsub_eq_max + - AddLECancellable.lt_of_tsub_lt_tsub_left_of_le + - AddLECancellable.add_tsub_assoc_of_le + - AddLECancellable.tsub_tsub_assoc + - tsub_pos_iff_not_le + - tsub_min + - AddLECancellable.tsub_lt_iff_tsub_lt + - AddLECancellable.tsub_lt_tsub_iff_right + - AddLECancellable.tsub_lt_self + - AddLECancellable.tsub_right_inj + - AddLECancellable.le_tsub_iff_le_tsub + - tsub_add_min + - tsub_tsub_tsub_cancel_right + - AddLECancellable.lt_tsub_iff_right_of_le + - AddLECancellable.tsub_lt_self_iff + - AddLECancellable.tsub_inj_right + - AddLECancellable.tsub_tsub_tsub_cancel_left + - tsub_add_tsub_cancel + - AddLECancellable.tsub_lt_iff_right + - tsub_le_tsub_iff_right + - tsub_add_cancel_of_le + - tsub_eq_tsub_min + - AddLECancellable.tsub_le_tsub_iff_left + - AddLECancellable.add_tsub_tsub_cancel + - tsub_add_cancel_iff_le + - AddLECancellable.tsub_add_eq_add_tsub + - AddLECancellable.tsub_eq_iff_eq_add_of_le + - tsub_eq_zero_iff_le + - tsub_left_inj + - tsub_pos_iff_lt + - lt_of_tsub_lt_tsub_right_of_le + - path: Mathlib/Algebra/Lie/OfAssociative.lean + theorems: + - LieModule.toEndomorphism_pow_lie + - AlgHom.toLieHom_injective + - LieSubmodule.coe_map_toEndomorphism_le + - LieSubmodule.toEndomorphism_restrict_eq_toEndomorphism + - LieSubalgebra.ad_comp_incl_eq + - LieSubmodule.coe_toEndomorphism_pow + - LieSubmodule.toEndomorphism_comp_subtype_mem + - LieModule.toEndomorphism_lie + - LieModule.toEndomorphism_module_end + - LieAlgebra.ad_eq_lmul_left_sub_lmul_right + - LieModule.toEndomorphism_pow_comp_lieHom + - LieSubmodule.mapsTo_pow_toEndomorphism_sub_algebraMap + - path: Mathlib/Tactic/NormNum/Prime.lean + theorems: + - Mathlib.Meta.NormNum.isNat_minFac_3 + - Mathlib.Meta.NormNum.isNat_minFac_2 + - Mathlib.Meta.NormNum.minFacHelper_2 + - Mathlib.Meta.NormNum.minFacHelper_3 + - Mathlib.Meta.NormNum.minFacHelper_0 + - Mathlib.Meta.NormNum.isNat_minFac_4 + - Mathlib.Meta.NormNum.minFacHelper_1 + - Mathlib.Meta.NormNum.MinFacHelper.one_lt + - path: Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean + theorems: + - "AlgebraicTopology.DoldKan.N\u2081\u0393\u2080_hom_app_f_f" + - "AlgebraicTopology.DoldKan.N\u2081\u0393\u2080_hom_app" + - "AlgebraicTopology.DoldKan.N\u2082\u0393\u2082ToKaroubiIso_hom_app" + - "AlgebraicTopology.DoldKan.N\u2081\u0393\u2080_inv_app_f_f" + - "AlgebraicTopology.DoldKan.N\u2081\u0393\u2080_app" + - "AlgebraicTopology.DoldKan.N\u2082\u0393\u2082_inv_app_f_f" + - "AlgebraicTopology.DoldKan.whiskerLeft_toKaroubi_N\u2082\u0393\u2082_hom" + - "AlgebraicTopology.DoldKan.N\u2081\u0393\u2080_inv_app" + - "AlgebraicTopology.DoldKan.N\u2082\u0393\u2082ToKaroubiIso_inv_app" + - path: Mathlib/Order/ConditionallyCompleteLattice/Basic.lean + theorems: + - exists_lt_of_lt_csSup + - ciSup_eq_ite + - exists_lt_of_lt_ciSup' + - Set.Iic_ciInf + - exists_lt_of_lt_csSup' + - WithTop.coe_sSup' + - WithTop.coe_sSup + - ciSup_le + - ciSup_neg + - Monotone.le_csInf_image + - csSup_image2_eq_csSup_csSup + - GaloisConnection.l_ciSup_set + - WithTop.isLUB_sSup' + - Monotone.csInf_image_le + - isLUB_csSup' + - WithTop.iSup_coe_eq_top + - WithTop.coe_sInf' + - ciSup_const + - ciSup_of_empty + - le_ciSup_iff' + - csSup_eq_csSup_of_forall_exists_le + - GaloisConnection.l_csSup' + - isLUB_ciSup_set + - WithTop.coe_sInf + - WithTop.isGLB_sInf + - csSup_Iio + - sSup_iUnion_Iic + - WithTop.iInf_coe_lt_top + - cbiSup_eq_of_not_forall + - WithTop.isGLB_sInf' + - GaloisConnection.l_ciSup + - csInf_Ioi + - ciSup_mono + - isLeast_csInf + - WithTop.isLUB_sSup + - ciInf_eq_of_forall_ge_of_forall_gt_exists_lt + - csSup_eq_univ_of_not_bddAbove + - cbiSup_eq_of_forall + - Monotone.ciSup_mem_iInter_Icc_of_antitone + - WithTop.iInf_coe_eq_top + - WithTop.coe_iInf + - path: Mathlib/Data/Int/Order/Basic.lean + theorems: + - Int.abs_le_one_iff + - Int.ediv_dvd_of_dvd + - Int.sign_mul_abs + - bit0_mul + - Int.sign_eq_ediv_abs + - zpow_abs_eq_one + - Int.natAbs_sq + - Int.toNat_lt_toNat + - Int.natAbs_sub_pos_iff + - Int.le_add_one_iff + - Int.natAbs_le_self_sq + - Int.exists_lt_and_lt_iff_not_dvd + - Int.add_emod_eq_add_mod_right + - Int.cast_mul_eq_zsmul_cast + - Int.toNat_pred_coe_of_pos + - Int.coe_nat_ne_zero + - Int.abs_lt_one_iff + - Int.le_induction + - bit1_mul + - Int.emod_two_eq_zero_or_one + - Int.neg_emod_two + - Nat.cast_natAbs + - Int.le_induction_down + - Int.natAbs_abs + - Int.toNat_le + - Int.ediv_dvd_ediv + - mul_bit0 + - Int.abs_ediv_le_abs + - Int.toNat_le_toNat + - Int.abs_sign_of_nonzero + - Int.ediv_eq_zero_of_lt_abs + - mul_bit1 + - Int.sign_add_eq_of_sign_eq + - Int.emod_lt + - path: .lake/packages/std/Std/Data/Fin/Lemmas.lean + theorems: + - Fin.foldr_eq_foldr_list + - Fin.list_succ + - Fin.foldl_loop_eq + - Fin.get_list + - Fin.foldl_loop + - Fin.foldl_loop_lt + - Fin.foldr_loop + - Fin.foldl_eq_foldl_list + - Fin.length_list + - path: Mathlib/RingTheory/WittVector/Basic.lean + theorems: + - WittVector.mapFun.surjective + - WittVector.mapFun.sub + - WittVector.mapFun.injective + - WittVector.ghostFun_add + - WittVector.mapFun.zsmul + - WittVector.mapFun.zero + - WittVector.mapFun.int_cast + - WittVector.ghostFun_zsmul + - WittVector.mapFun.nsmul + - WittVector.ghostFun_pow + - WittVector.mapFun.one + - WittVector.ghostFun_nat_cast + - WittVector.matrix_vecEmpty_coeff + - WittVector.ghostFun_one + - WittVector.ghostFun_zero + - WittVector.ghostFun_int_cast + - WittVector.ghostFun_mul + - WittVector.ghostFun_neg + - WittVector.mapFun.pow + - WittVector.ghostFun_nsmul + - WittVector.mapFun.add + - WittVector.mapFun.mul + - WittVector.mapFun.neg + - WittVector.mapFun.nat_cast + - path: Mathlib/MeasureTheory/Measure/WithDensity.lean + theorems: + - MeasureTheory.lintegral_withDensity_eq_lintegral_mul + - MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul + - MeasureTheory.withDensity_smul' + - MeasureTheory.withDensity_indicator_one + - "MeasureTheory.lintegral_withDensity_eq_lintegral_mul\u2080'" + - MeasureTheory.aemeasurable_withDensity_ennreal_iff + - MeasureTheory.withDensity_const + - "MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable\u2080" + - MeasureTheory.withDensity_smul + - MeasureTheory.withDensity_ofReal_mutuallySingular + - "MeasureTheory.withDensity_mul\u2080" + - MeasureTheory.withDensity_sum + - MeasureTheory.withDensity_indicator + - "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080" + - MeasureTheory.IsLocallyFiniteMeasure.withDensity_coe + - MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable + - MeasureTheory.withDensity_mono + - MeasureTheory.SigmaFinite.withDensity_of_ne_top + - "MeasureTheory.withDensity_inv_same\u2080" + - MeasureTheory.withDensity_apply_le + - MeasureTheory.withDensity_tsum + - MeasureTheory.SigmaFinite.withDensity_ofReal + - MeasureTheory.SigmaFinite.withDensity_of_ne_top' + - MeasureTheory.SigmaFinite.withDensity + - MeasureTheory.withDensity_apply' + - MeasureTheory.ae_withDensity_iff + - MeasureTheory.isFiniteMeasure_withDensity + - "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable\u2080\ + '" + - MeasureTheory.withDensity_congr_ae + - MeasureTheory.withDensity_zero + - MeasureTheory.withDensity_zero_left + - MeasureTheory.withDensity_inv_same_le + - MeasureTheory.withDensity_apply_eq_zero + - MeasureTheory.withDensity_smul_measure + - MeasureTheory.withDensity_absolutelyContinuous' + - MeasureTheory.withDensity_add_left + - MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable + - MeasureTheory.withDensity_absolutelyContinuous + - MeasureTheory.exists_absolutelyContinuous_isFiniteMeasure + - "MeasureTheory.set_lintegral_withDensity_eq_lintegral_mul\u2080'" + - MeasureTheory.lintegral_withDensity_le_lintegral_mul + - MeasureTheory.ae_withDensity_iff_ae_restrict + - MeasureTheory.restrict_withDensity' + - MeasureTheory.withDensity_one + - MeasureTheory.withDensity_add_right + - path: Mathlib/RingTheory/DedekindDomain/Ideal.lean + theorems: + - Ideal.mem_normalizedFactors_iff + - irreducible_pow_sup + - Ideal.coprime_of_no_prime_ge + - FractionalIdeal.mul_right_le_iff + - IsDedekindDomainInv.dimensionLEOne + - FractionalIdeal.coe_ideal_span_singleton_mul_inv + - FractionalIdeal.inv_le_comm + - FractionalIdeal.div_eq_mul_inv + - Ideal.eq_prime_pow_of_succ_lt_of_le + - idealFactorsEquivOfQuotEquiv_mem_normalizedFactors_of_mem_normalizedFactors + - FractionalIdeal.coe_ideal_mul_inv + - Ideal.le_mul_of_no_prime_factors + - IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot + - FractionalIdeal.adjoinIntegral_eq_one_of_isUnit + - IsDedekindDomain.inf_prime_pow_eq_prod + - idealFactorsFunOfQuotHom_id + - FractionalIdeal.spanSingleton_div_spanSingleton + - Ideal.inf_eq_mul_of_coprime + - Ideal.factors_span_eq + - FractionalIdeal.right_inverse_eq + - Ideal.pow_lt_self + - Ideal.prod_le_prime + - normalizedFactorsEquivOfQuotEquiv_multiplicity_eq_multiplicity + - Ideal.squarefree_span_singleton + - IsDedekindDomainInv.integrallyClosed + - FractionalIdeal.mul_inv_cancel_iff + - FractionalIdeal.mul_left_le_iff + - FractionalIdeal.inv_anti_mono + - Ideal.prime_of_isPrime + - FractionalIdeal.coe_inv_of_nonzero + - multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity + - FractionalIdeal.isPrincipal_inv + - Ideal.IsPrime.mul_mem_pow + - IsDedekindDomainInv.isNoetherianRing + - FractionalIdeal.spanSingleton_inv_mul + - FractionalIdeal.mul_inv_cancel_of_le_one + - FractionalIdeal.invertible_iff_generator_nonzero + - singleton_span_mem_normalizedFactors_of_mem_normalizedFactors + - idealFactorsFunOfQuotHom_comp + - Ideal.le_of_pow_le_prime + - Ideal.dvdNotUnit_iff_lt + - IsDedekindDomain.exists_representative_mod_finset + - idealFactorsEquivOfQuotEquiv_is_dvd_iso + - exists_multiset_prod_cons_le_and_prod_not_le + - Ideal.isCoprime_iff_gcd + - multiplicity_eq_multiplicity_span + - FractionalIdeal.le_inv_comm + - Ideal.isPrime_of_prime + - FractionalIdeal.map_inv + - FractionalIdeal.not_inv_le_one_of_ne_bot + - irreducible_pow_sup_of_ge + - Ideal.dvd_iff_le + - Ideal.sup_mul_inf + - FractionalIdeal.spanSingleton_mul_inv + - FractionalIdeal.one_mem_inv_coe_ideal + - sup_eq_prod_inf_factors + - Associates.le_singleton_iff + - FractionalIdeal.coe_ideal_span_singleton_div_self + - isDedekindDomainInv_iff + - FractionalIdeal.mul_generator_self_inv + - FractionalIdeal.spanSingleton_div_self + - IsDedekindDomain.exists_forall_sub_mem_ideal + - Ideal.exist_integer_multiples_not_mem + - path: Mathlib/RingTheory/WittVector/WittPolynomial.lean + theorems: + - wittPolynomial_zmod_self + - xInTermsOfW_aux + - xInTermsOfW_vars_aux + - constantCoeff_wittPolynomial + - xInTermsOfW_zero + - wittPolynomial_vars + - "bind\u2081_xInTermsOfW_wittPolynomial" + - map_wittPolynomial + - wittPolynomial_one + - wittPolynomial_zero + - "bind\u2081_wittPolynomial_xInTermsOfW" + - xInTermsOfW_eq + - aeval_wittPolynomial + - wittPolynomial_eq_sum_C_mul_X_pow + - constantCoeff_xInTermsOfW + - wittPolynomial_vars_subset + - path: Mathlib/Data/MvPolynomial/Supported.lean + theorems: + - MvPolynomial.supported_eq_range_rename + - MvPolynomial.supported_empty + - MvPolynomial.mem_supported_vars + - MvPolynomial.exists_restrict_to_vars + - MvPolynomial.supportedEquivMvPolynomial_symm_C + - MvPolynomial.mem_supported + - MvPolynomial.X_mem_supported + - MvPolynomial.supported_le_supported_iff + - MvPolynomial.supported_univ + - MvPolynomial.supportedEquivMvPolynomial_symm_X + - path: Mathlib/GroupTheory/SpecificGroups/Cyclic.lean + theorems: + - ZMod.exponent + - IsSimpleGroup.prime_card + - powers_eq_top_of_prime_card + - CommGroup.is_simple_iff_isCyclic_and_prime_card + - isCyclic_of_card_pow_eq_one_le + - IsCyclic.image_range_orderOf + - MonoidHom.map_cyclic + - IsCyclic.exists_monoid_generator + - orderOf_eq_card_of_forall_mem_zpowers + - exists_pow_ne_one_of_isCyclic + - "card_orderOf_eq_totient_aux\u2082" + - zpowers_eq_top_of_prime_card + - IsCyclic.card_orderOf_eq_totient + - mem_powers_of_prime_card + - isCyclic_of_orderOf_eq_card + - Infinite.orderOf_eq_zero_of_forall_mem_zpowers + - isSimpleGroup_of_prime_card + - mem_zpowers_of_prime_card + - Nontrivial.of_not_isCyclic + - card_pow_eq_one_eq_orderOf_aux + - zmultiplesHom_ker_eq + - IsCyclic.image_range_card + - isCyclic_of_surjective + - isCyclic_of_prime_card + - IsCyclic.exponent_eq_card + - IsCyclic.card_pow_eq_one_le + - commutative_of_cyclic_center_quotient + - "card_orderOf_eq_totient_aux\u2081" + - IsCyclic.unique_zpow_zmod + - not_isCyclic_iff_exponent_eq_prime + - path: Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean + theorems: + - Matrix.SpecialLinearGroup.scalar_eq_self_of_mem_center + - Matrix.SpecialLinearGroup.SL2_inv_expl_det + - Matrix.SpecialLinearGroup.isCoprime_col + - ModularGroup.T_inv_mul_apply_one + - ModularGroup.coe_T_zpow + - Matrix.SpecialLinearGroup.center_eq_bot_of_subsingleton + - ModularGroup.T_mul_apply_one + - IsCoprime.exists_SL2_row + - Matrix.SpecialLinearGroup.mem_center_iff + - Matrix.SpecialLinearGroup.SL2_inv_expl + - Matrix.SpecialLinearGroup.det_ne_zero + - IsCoprime.exists_SL2_col + - IsCoprime.mulVecSL + - Matrix.SpecialLinearGroup.row_ne_zero + - ModularGroup.coe_T_inv + - IsCoprime.vecMulSL + - Matrix.SpecialLinearGroup.isCoprime_row + - ModularGroup.T_pow_mul_apply_one + - Matrix.SpecialLinearGroup.fin_two_exists_eq_mk_of_apply_zero_one_eq_zero + - Matrix.SpecialLinearGroup.fin_two_induction + - path: Mathlib/Algebra/GroupWithZero/Basic.lean + theorems: + - mul_self_mul_inv + - pow_eq_zero_iff' + - "mul_left_eq_self\u2080" + - inv_eq_zero + - pow_eq_zero_iff + - "inv_mul_cancel_left\u2080" + - one_div_ne_zero + - div_zero + - pow_eq_zero_of_le + - inv_eq_of_mul + - ne_zero_pow + - div_div_self + - GroupWithZero.mul_right_injective + - GroupWithZero.mul_left_injective + - "mul_eq_left\u2080" + - div_self_mul_self' + - "mul_left_surjective\u2080" + - mul_self_div_self + - "inv_mul_cancel_right\u2080" + - ne_zero_of_one_div_ne_zero + - div_self_mul_self + - mul_eq_mul_right_iff + - mul_eq_zero_of_ne_zero_imp_eq_zero + - "mul_right_surjective\u2080" + - zero_div + - "div_mul_eq_mul_div\u2080" + - zero_pow + - inv_mul_mul_self + - zero_pow_eq + - "right_eq_mul\u2080" + - "mul_right_eq_self\u2080" + - eq_zero_of_zero_eq_one + - pow_eq_zero + - mul_eq_mul_left_iff + - mul_inv_mul_self + - zero_pow_eq_zero + - "left_eq_mul\u2080" + - path: Mathlib/Data/List/Chain.lean + theorems: + - List.Chain'.cons_of_le + - List.chain'_pair + - List.chain'_iff_get + - List.chain'_map + - List.Chain'.rel_head? + - List.chain_iff_get + - List.chain'_reverse + - List.chain_append_cons_cons + - List.Chain.iff_mem + - List.chain_pmap_of_chain + - List.Chain.sublist + - List.chain'_append + - List.Chain'.sublist + - List.Chain.pairwise + - List.chain'_append_cons_cons + - List.chain_singleton + - List.chain_iff_nthLe + - Acc.list_chain' + - List.chain'_join + - "List.chain_iff_forall\u2082" + - List.Chain'.append_overlap + - "List.chain_append_singleton_iff_forall\u2082" + - List.chain_map + - List.chain_split + - List.chain'_isInfix + - List.Chain.induction + - List.Chain'.imp + - List.Chain.rel + - List.chain_of_chain_pmap + - List.exists_chain_of_relationReflTransGen + - List.chain'_iff_nthLe + - path: Mathlib/Algebra/Invertible/Defs.lean + theorems: + - mul_invOf_mul_self_cancel + - invertible_unique + - invOf_mul_self_assoc + - mul_invOf_eq_iff_eq_mul_right + - invOf_mul_self_assoc' + - mul_left_eq_iff_eq_invOf_mul + - Invertible.congr + - mul_invOf_self_assoc' + - invOf_mul + - mul_right_eq_iff_eq_mul_invOf + - mul_mul_invOf_self_cancel + - invOf_mul_eq_iff_eq_mul_left + - mul_invOf_mul_self_cancel' + - mul_invOf_self_assoc + - mul_mul_invOf_self_cancel' + - path: Mathlib/RingTheory/HahnSeries/Addition.lean + theorems: + - HahnSeries.order_neg + - HahnSeries.support_neg + - HahnSeries.sub_coeff + - HahnSeries.embDomain_add + - HahnSeries.min_order_le_order_add + - HahnSeries.support_add_subset + - HahnSeries.sub_coeff' + - HahnSeries.order_smul_not_lt + - HahnSeries.embDomain_smul + - path: Mathlib/CategoryTheory/Sites/IsSheafFor.lean + theorems: + - CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPullback + - CategoryTheory.Presieve.compatible_iff_sieveCompatible + - CategoryTheory.Presieve.restrict_extend + - CategoryTheory.Presieve.FamilyOfElements.Compatible.sieveExtend + - CategoryTheory.Presieve.extension_iff_amalgamation + - CategoryTheory.Presieve.restrict_inj + - CategoryTheory.Presieve.FamilyOfElements.Compatible.compPresheafMap + - CategoryTheory.Presieve.Arrows.Compatible.exists_familyOfElements + - CategoryTheory.Presieve.isSheafFor_singleton_iso + - CategoryTheory.Presieve.pullbackCompatible_iff + - CategoryTheory.Presieve.isSeparatedFor_iff_generate + - CategoryTheory.Presieve.isSeparatedFor_top + - CategoryTheory.Presieve.isSheafFor_subsieve_aux + - CategoryTheory.Presieve.extend_restrict + - CategoryTheory.Presieve.isAmalgamation_sieveExtend + - CategoryTheory.Presieve.Arrows.pullbackCompatible_iff + - CategoryTheory.Presieve.FamilyOfElements.comp_of_compatible + - CategoryTheory.Presieve.IsSeparatedFor.isSheafFor + - CategoryTheory.Presieve.isSheafFor_iff_yonedaSheafCondition + - CategoryTheory.Presieve.isSheafFor_subsieve + - CategoryTheory.Presieve.Arrows.Compatible.familyOfElements_compatible + - CategoryTheory.Presieve.isSheafFor_iso + - CategoryTheory.Presieve.FamilyOfElements.IsAmalgamation.compPresheafMap + - CategoryTheory.Presieve.is_compatible_of_exists_amalgamation + - CategoryTheory.Presieve.isSheafFor_top_sieve + - CategoryTheory.Presieve.isSheafFor_arrows_iff + - CategoryTheory.Presieve.extend_agrees + - CategoryTheory.Presieve.isSheafFor_iff_generate + - CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor + - path: Mathlib/Data/Polynomial/Induction.lean + theorems: + - Polynomial.mem_span_C_coeff + - Polynomial.induction_on + - Polynomial.induction_on' + - Polynomial.span_le_of_C_coeff_mem + - path: Mathlib/NumberTheory/Pell.lean + theorems: + - Pell.existsUnique_pos_generator + - "Pell.Solution\u2081.y_mul_pos" + - Pell.IsFundamental.exists_of_not_isSquare + - "Pell.Solution\u2081.x_ne_zero" + - Pell.IsFundamental.subsingleton + - Pell.pos_generator_iff_fundamental + - "Pell.Solution\u2081.prop_y" + - "Pell.Solution\u2081.eq_zero_of_d_neg" + - Pell.IsFundamental.zpow_y_lt_iff_lt + - "Pell.Solution\u2081.x_zpow_pos" + - "Pell.Solution\u2081.eq_one_of_x_eq_one" + - "Pell.Solution\u2081.d_nonsquare_of_one_lt_x" + - "Pell.Solution\u2081.d_pos_of_one_lt_x" + - "Pell.Solution\u2081.sign_y_zpow_eq_sign_of_x_pos_of_y_pos" + - Pell.IsFundamental.mul_inv_y_nonneg + - Pell.IsFundamental.mul_inv_x_lt_x + - Pell.IsFundamental.eq_pow_of_nonneg + - Pell.IsFundamental.zpow_ne_neg_zpow + - Pell.IsFundamental.eq_zpow_or_neg_zpow + - "Pell.Solution\u2081.y_zpow_pos" + - "Pell.Solution\u2081.prop_x" + - "Pell.Solution\u2081.x_pow_pos" + - Pell.IsFundamental.y_strictMono + - "Pell.Solution\u2081.x_mul" + - "Pell.Solution\u2081.exists_nontrivial_of_not_isSquare" + - "Pell.Solution\u2081.exists_pos_variant" + - "Pell.Solution\u2081.y_ne_zero_of_one_lt_x" + - "Pell.Solution\u2081.exists_pos_of_not_isSquare" + - Pell.IsFundamental.zpow_eq_one_iff + - Pell.IsFundamental.mul_inv_x_pos + - "Pell.Solution\u2081.y_pow_succ_pos" + - Pell.is_pell_solution_iff_mem_unitary + - "Pell.Solution\u2081.eq_one_or_neg_one_iff_y_eq_zero" + - Pell.exists_iff_not_isSquare + - Pell.IsFundamental.y_le_y + - Pell.IsFundamental.x_mul_y_le_y_mul_x + - "Pell.Solution\u2081.x_mul_pos" + - path: Mathlib/RingTheory/RootsOfUnity/Minpoly.lean + theorems: + - IsPrimitiveRoot.totient_le_degree_minpoly + - IsPrimitiveRoot.isIntegral + - IsPrimitiveRoot.separable_minpoly_mod + - IsPrimitiveRoot.pow_isRoot_minpoly + - IsPrimitiveRoot.minpoly_eq_pow + - IsPrimitiveRoot.minpoly_dvd_x_pow_sub_one + - IsPrimitiveRoot.is_roots_of_minpoly + - IsPrimitiveRoot.minpoly_dvd_pow_mod + - path: Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean + theorems: + - "TopCat.Sheaf.eq_of_locally_eq\u2082" + - TopCat.Sheaf.existsUnique_gluing' + - TopCat.Sheaf.eq_of_locally_eq' + - TopCat.Sheaf.eq_of_locally_eq + - TopCat.Presheaf.isGluing_iff_pairwise + - TopCat.Presheaf.isSheaf_iff_isSheafUniqueGluing_types + - TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff + - path: Mathlib/Algebra/Group/Semiconj/Basic.lean + theorems: + - SemiconjBy.inv_right_iff + - SemiconjBy.inv_inv_symm_iff + - SemiconjBy.inv_symm_left_iff + - path: Mathlib/Data/Finset/Fold.lean + theorems: + - Finset.fold_const + - Finset.fold_min_le + - Finset.fold_min_lt + - Finset.fold_max_lt + - Finset.fold_image_idem + - Finset.fold_max_le + - Finset.fold_map + - Finset.fold_hom + - Finset.fold_op_distrib + - Finset.fold_insert + - Finset.fold_op_rel_iff_and + - Finset.fold_cons + - Finset.fold_op_rel_iff_or + - Finset.fold_insert_idem + - Finset.fold_max_add + - Finset.fold_congr + - Finset.fold_ite' + - Finset.fold_union_empty_singleton + - path: Mathlib/Algebra/Category/ModuleCat/Presheaf.lean + theorems: + - PresheafOfModules.Hom.ext + - PresheafOfModules.map_id + - PresheafOfModules.map_comp + - path: Mathlib/MeasureTheory/Integral/CircleTransform.lean + theorems: + - Complex.circleTransformDeriv_eq + - Complex.continuous_circleTransformDeriv + - Complex.circleTransformDeriv_bound + - Complex.circleTransformDeriv_periodic + - Complex.continuous_circleTransform + - Complex.continuousOn_abs_circleTransformBoundingFunction + - Complex.abs_circleTransformBoundingFunction_le + - Complex.continuousOn_prod_circle_transform_function + - Complex.integral_circleTransform + - path: Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean + theorems: + - separatingDual_iff_injective + - SeparatingDual.t2Space + - SeparatingDual.exists_eq_one_ne_zero_of_ne_zero_pair + - SeparatingDual.exists_eq_one + - SeparatingDual.exists_separating_of_ne + - SeparatingDual.exists_continuousLinearEquiv_apply_eq + - SeparatingDual.dualMap_surjective_iff + - SeparatingDual.t1Space + - path: Mathlib/Data/Rat/Lemmas.lean + theorems: + - Rat.add_den_dvd + - Rat.den_mk + - Rat.mul_self_num + - Rat.coe_int_div_self + - Rat.mul_num_den' + - Rat.mul_self_den + - Rat.inv_coe_nat_den_of_pos + - Rat.inv_coe_int_den_of_pos + - Rat.inv_def'' + - Rat.num_div_eq_of_coprime + - Rat.den_dvd + - Rat.exists + - Rat.inv_coe_int_num_of_pos + - Rat.coe_int_div + - Rat.mul_den_dvd + - Rat.inv_neg + - Rat.den_div_cast_eq_one_iff + - Rat.num_mk + - Rat.mul_num + - Rat.forall + - Rat.inv_coe_int_num + - Rat.den_div_eq_of_coprime + - Rat.num_den_mk + - Rat.coe_nat_div + - Rat.mul_den + - Rat.add_num_den + - Rat.inv_coe_int_den + - Rat.mul_den_eq_num + - Rat.exists_eq_mul_div_num_and_eq_mul_div_den + - Rat.substr_num_den' + - Rat.add_num_den' + - Rat.inv_coe_nat_den + - Rat.num_dvd + - Rat.div_int_inj + - path: Mathlib/Order/Height.lean + theorems: + - Set.exists_chain_of_le_chainHeight + - Set.le_chainHeight_TFAE + - Set.wellFoundedGT_of_chainHeight_ne_top + - Set.chainHeight_eq_iSup_Iic + - Set.chainHeight_eq_iSup_Ici + - Set.chainHeight_dual + - Set.one_le_chainHeight_iff + - Set.chainHeight_insert_of_forall_gt + - Set.chainHeight_le_chainHeight_TFAE + - Set.wellFoundedLT_of_chainHeight_ne_top + - Set.chainHeight_union_le + - Set.cons_mem_subchain_iff + - Set.singleton_mem_subchain_iff + - Set.chainHeight_eq_top_iff + - Set.chainHeight_insert_of_forall_lt + - Set.le_chainHeight_add_nat_iff + - Set.chainHeight_add_le_chainHeight_add + - Set.chainHeight_union_eq + - Set.chainHeight_image + - Set.chainHeight_eq_zero_iff + - path: Mathlib/Algebra/Homology/ModuleCat.lean + theorems: + - ModuleCat.cycles'_ext + - ModuleCat.cycles'Map_toCycles' + - ModuleCat.homology'_ext + - path: Mathlib/MeasureTheory/Group/AddCircle.lean + theorems: + - AddCircle.closedBall_ae_eq_ball + - AddCircle.volume_of_add_preimage_eq + - AddCircle.isAddFundamentalDomain_of_ae_ball + - path: Mathlib/Geometry/Manifold/LocalInvariantProperties.lean + theorems: + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_of_liftPropAt_of_mem_nhds + - StructureGroupoid.LocalInvariantProp.congr_set + - StructureGroupoid.LocalInvariantProp.liftPropAt_congr_iff_of_eventuallyEq + - StructureGroupoid.LocalInvariantProp.liftPropOn_symm_of_mem_maximalAtlas + - StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_maximalAtlas + - StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_subtype_val + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart' + - StructureGroupoid.LocalInvariantProp.liftPropOn_of_liftProp + - StructureGroupoid.LocalInvariantProp.right_invariance + - StructureGroupoid.isLocalStructomorphWithinAt_localInvariantProp + - PartialHomeomorph.isLocalStructomorphWithinAt_source_iff + - StructureGroupoid.LocalInvariantProp.liftPropAt_of_mem_maximalAtlas + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux + - PartialHomeomorph.isLocalStructomorphWithinAt_iff + - StructureGroupoid.LocalInvariantProp.congr_iff_nhdsWithin + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_of_liftPropAt + - StructureGroupoid.HasGroupoid.comp + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_inter' + - StructureGroupoid.LocalInvariantProp.liftProp_subtype_val + - ChartedSpace.liftPropAt_iff + - StructureGroupoid.LocalInvariantProp.liftProp_id + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_mono_of_mem + - StructureGroupoid.LocalInvariantProp.liftPropAt_symm_of_mem_maximalAtlas + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_aux + - PartialHomeomorph.isLocalStructomorphWithinAt_iff' + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux2 + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_mono + - ChartedSpace.liftProp_iff + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_iff + - StructureGroupoid.LocalInvariantProp.liftPropOn_indep_chart + - StructureGroupoid.LocalInvariantProp.liftPropAt_chart_symm + - StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_inclusion + - StructureGroupoid.LocalInvariantProp.liftProp_inclusion + - StructureGroupoid.LocalInvariantProp.liftPropAt_of_liftPropWithinAt + - StructureGroupoid.LocalInvariantProp.liftProp_of_locally_liftPropOn + - StructureGroupoid.LocalInvariantProp.left_invariance + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux + - StructureGroupoid.liftPropOn_univ + - StructureGroupoid.LocalInvariantProp.liftPropOn_of_locally_liftPropOn + - StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_of_eventuallyEq + - path: Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean + theorems: + - strictConcaveOn_log_Iio + - one_add_mul_self_lt_rpow_one_add + - one_add_mul_self_le_rpow_one_add + - strictConvexOn_exp + - strictConvexOn_rpow + - Real.exp_mul_le_cosh_add_mul_sinh + - convexOn_rpow + - strictConcaveOn_log_Ioi + - path: Mathlib/Logic/Small/Group.lean + theorems: + - equivShrink_div + - equivShrink_symm_sub + - equivShrink_symm_mul + - equivShrink_sub + - equivShrink_symm_div + - equivShrink_symm_neg + - equivShrink_add + - equivShrink_symm_inv + - equivShrink_mul + - equivShrink_symm_add + - equivShrink_inv + - equivShrink_neg + - path: Mathlib/Topology/MetricSpace/Thickening.lean + theorems: + - Metric.eventually_not_mem_thickening_of_infEdist_pos + - Metric.thickening_cthickening_subset + - Metric.ball_subset_thickening + - Metric.closure_subset_thickening + - Metric.cthickening_max_zero + - Metric.cthickening_cthickening_subset + - IsCompact.cthickening_eq_biUnion_closedBall + - Metric.diam_cthickening_le + - Metric.frontier_cthickening_disjoint + - Metric.cthickening_closure + - Metric.thickening_thickening_subset + - Metric.cthickening_eq_iInter_thickening + - Metric.thickening_iUnion + - Metric.closure_eq_iInter_thickening + - Metric.closedBall_subset_cthickening_singleton + - Disjoint.exists_cthickenings + - Bornology.IsBounded.thickening + - Metric.thickening_empty + - Metric.cthickening_eq_iInter_cthickening' + - Metric.thickening_union + - Metric.cthickening_thickening_subset + - Metric.closure_subset_cthickening + - Bornology.IsBounded.cthickening + - Metric.cthickening_union + - Metric.frontier_thickening_disjoint + - Disjoint.exists_thickenings + - Metric.closure_eq_iInter_cthickening' + - Metric.cthickening_singleton + - Metric.closedBall_subset_cthickening + - Metric.cthickening_eq_iInter_thickening'' + - Metric.thickening_eq_biUnion_ball + - IsCompact.exists_isCompact_cthickening + - Metric.cthickening_eq_iInter_thickening' + - Metric.cthickening_subset_iUnion_closedBall_of_lt + - IsClosed.cthickening_eq_biUnion_closedBall + - Metric.mem_cthickening_of_dist_le + - Metric.cthickening_empty + - Metric.closure_eq_iInter_thickening' + - Metric.cthickening_of_nonpos + - Metric.closure_eq_iInter_cthickening + - Metric.thickening_closure + - Metric.thickening_subset_cthickening + - Metric.thickening_singleton + - Metric.infEdist_le_infEdist_cthickening_add + - Metric.diam_thickening_le + - Metric.eventually_not_mem_cthickening_of_infEdist_pos + - Metric.cthickening_eq_iInter_cthickening + - path: Mathlib/Algebra/Star/Subalgebra.lean + theorems: + - StarSubalgebra.top_toSubalgebra + - StarSubalgebra.eq_top_iff + - StarSubalgebra.inf_toSubalgebra + - StarSubalgebra.toSubalgebra_injective + - StarSubalgebra.comap_comap + - Subalgebra.star_adjoin_comm + - StarSubalgebra.coe_iInf + - StarSubalgebra.gc + - StarSubalgebra.mem_centralizer_iff + - StarSubalgebra.iInf_toSubalgebra + - StarSubalgebra.mem_sInf + - Subalgebra.star_mem_star_iff + - StarSubalgebra.adjoin_induction' + - StarAlgHom.range_eq_map_top + - "StarSubalgebra.adjoin_induction\u2082" + - StarSubalgebra.mem_iInf + - StarAlgHom.ext_adjoin + - path: Mathlib/Analysis/Analytic/Basic.lean + theorems: + - HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal + - HasFPowerSeriesAt.congr + - FormalMultilinearSeries.radius_eq_top_of_eventually_eq_zero + - HasFPowerSeriesOnBall.hasSum_sub + - AnalyticOn.continuous + - isOpen_analyticAt + - FormalMultilinearSeries.radius_neg + - AnalyticAt.sub + - HasFPowerSeriesAt.sub + - HasFPowerSeriesAt.isBigO_sub_partialSum_pow + - HasFPowerSeriesOnBall.eventually_hasSum_sub + - HasFPowerSeriesAt.add + - FormalMultilinearSeries.summable_norm_apply + - HasFPowerSeriesOnBall.congr + - FormalMultilinearSeries.le_radius_of_eventually_le + - FormalMultilinearSeries.continuousOn + - FormalMultilinearSeries.constFormalMultilinearSeries_radius + - hasFPowerSeriesOnBall_const + - hasFPowerSeriesAt_iff' + - FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm + - HasFPowerSeriesOnBall.sub + - FormalMultilinearSeries.hasFPowerSeriesOnBall + - "FormalMultilinearSeries.changeOriginSeries_summable_aux\u2082" + - "FormalMultilinearSeries.changeOriginSeries_summable_aux\u2081" + - FormalMultilinearSeries.norm_changeOriginSeriesTerm + - HasFPowerSeriesAt.apply_eq_zero + - HasFPowerSeriesOnBall.tendstoUniformlyOn + - FormalMultilinearSeries.radius_eq_top_iff_summable_norm + - FormalMultilinearSeries.le_radius_of_summable + - FormalMultilinearSeries.summable_nnnorm_mul_pow + - HasFPowerSeriesOnBall.analyticAt_of_mem + - HasFPowerSeriesOnBall.neg + - FormalMultilinearSeries.le_mul_pow_of_radius_pos + - Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero + - FormalMultilinearSeries.changeOrigin_radius + - FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le + - HasFPowerSeriesAt.eq_formalMultilinearSeries + - HasFPowerSeriesOnBall.eventually_hasSum + - HasFPowerSeriesOnBall.coeff_zero + - FormalMultilinearSeries.lt_radius_of_isBigO + - FormalMultilinearSeries.nnnorm_changeOriginSeries_apply_le_tsum + - HasFPowerSeriesAt.eq_zero + - HasFPowerSeriesOnBall.image_sub_sub_deriv_le + - ContinuousLinearMap.comp_analyticOn + - HasFPowerSeriesOnBall.uniform_geometric_approx + - FormalMultilinearSeries.changeOriginSeriesTerm_changeOriginIndexEquiv_symm + - FormalMultilinearSeries.summable_norm_mul_pow + - HasFPowerSeriesOnBall.uniform_geometric_approx' + - HasFPowerSeriesOnBall.changeOrigin + - HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub + - FormalMultilinearSeries.nnnorm_changeOrigin_le + - HasFPowerSeriesOnBall.tendstoUniformlyOn' + - "FormalMultilinearSeries.changeOriginSeries_summable_aux\u2083" + - hasFPowerSeriesAt_iff + - HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn + - FormalMultilinearSeries.changeOrigin_eval + - FormalMultilinearSeries.isLittleO_of_lt_radius + - FormalMultilinearSeries.min_radius_le_radius_add + - FormalMultilinearSeries.norm_mul_pow_le_mul_pow_of_lt_radius + - HasFPowerSeriesOnBall.comp_sub + - ContinuousLinearMap.comp_hasFPowerSeriesOnBall + - FormalMultilinearSeries.partialSum_continuous + - FormalMultilinearSeries.radius_le_radius_continuousLinearMap_comp + - FormalMultilinearSeries.nnnorm_changeOriginSeries_le_tsum + - path: Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean + theorems: + - SimpleGraph.FarFromTriangleFree.mono + - SimpleGraph.FarFromTriangleFree.nonpos + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean + theorems: + - "MeasureTheory.condexp_stronglyMeasurable_mul\u2080" + - MeasureTheory.condexp_stronglyMeasurable_mul + - MeasureTheory.rnDeriv_ae_eq_condexp + - MeasureTheory.set_integral_abs_condexp_le + - "MeasureTheory.condexp_stronglyMeasurable_mul_of_bound\u2080" + - MeasureTheory.Integrable.uniformIntegrable_condexp + - MeasureTheory.integral_abs_condexp_le + - MeasureTheory.condexp_stronglyMeasurable_mul_of_bound + - MeasureTheory.ae_bdd_condexp_of_ae_bdd + - MeasureTheory.condexp_stronglyMeasurable_simpleFunc_mul + - path: Mathlib/Analysis/SumIntegralComparisons.lean + theorems: + - AntitoneOn.integral_le_sum + - MonotoneOn.sum_le_integral + - AntitoneOn.integral_le_sum_Ico + - AntitoneOn.sum_le_integral + - MonotoneOn.integral_le_sum_Ico + - MonotoneOn.sum_le_integral_Ico + - AntitoneOn.sum_le_integral_Ico + - MonotoneOn.integral_le_sum + - path: Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean + theorems: + - CategoryTheory.Limits.eq_zero_of_epi_kernel + - "CategoryTheory.Limits.CokernelCofork.\u03C0_eq_zero" + - CategoryTheory.Limits.cokernelZeroIsoTarget_hom + - CategoryTheory.Limits.kernelComparison_comp_kernel_map + - "CategoryTheory.Limits.\u03C0_comp_cokernelIsoOfEq_inv" + - CategoryTheory.Limits.cokernel_map_comp_cokernelComparison + - CategoryTheory.Limits.cokernelComparison_map_desc + - CategoryTheory.Limits.kernel.lift_zero + - "CategoryTheory.Limits.MonoFactorisation.kernel_\u03B9_comp" + - CategoryTheory.Limits.map_lift_kernelComparison + - CategoryTheory.Limits.lift_comp_kernelIsoOfEq_inv + - "CategoryTheory.Limits.kernelIsoOfEq_inv_comp_\u03B9" + - "CategoryTheory.Limits.colimit_\u03B9_zero_cokernel_desc" + - CategoryTheory.Limits.isKernelCompMono_lift + - CategoryTheory.Limits.kernelIsoOfEq_trans + - CategoryTheory.Limits.kernel.lift_map + - CategoryTheory.Limits.CokernelCofork.condition + - CategoryTheory.Limits.eq_zero_of_mono_cokernel + - CategoryTheory.Limits.cokernelIsoOfEq_inv_comp_desc + - CategoryTheory.Limits.isCokernelEpiComp_desc + - "CategoryTheory.Limits.KernelFork.IsLimit.isIso_\u03B9" + - CategoryTheory.Limits.kernelZeroIsoSource_inv + - CategoryTheory.Limits.cokernelIsoOfEq_refl + - CategoryTheory.Limits.cokernel.desc_zero + - CategoryTheory.Limits.kernelIsoOfEq_refl + - CategoryTheory.Limits.cokernelIsoOfEq_hom_comp_desc + - "CategoryTheory.Limits.\u03C0_comp_cokernelIsoOfEq_hom" + - "CategoryTheory.Limits.kernelIsoOfEq_hom_comp_\u03B9" + - CategoryTheory.Limits.lift_comp_kernelIsoOfEq_hom + - CategoryTheory.Limits.KernelFork.condition + - CategoryTheory.Limits.cokernelIsoOfEq_trans + - path: Mathlib/RingTheory/WittVector/Truncated.lean + theorems: + - TruncatedWittVector.ext_iff + - WittVector.mem_ker_truncate + - WittVector.out_truncateFun + - WittVector.truncate_comp_lift + - WittVector.truncateFun_add + - TruncatedWittVector.out_injective + - TruncatedWittVector.mk_coeff + - WittVector.coeff_truncateFun + - WittVector.truncateFun_neg + - TruncatedWittVector.truncate_comp + - TruncatedWittVector.coeff_out + - TruncatedWittVector.coeff_truncate + - TruncatedWittVector.truncateFun_out + - WittVector.truncateFun_sub + - WittVector.truncateFun_nsmul + - TruncatedWittVector.iInf_ker_truncate + - WittVector.truncate_mk' + - TruncatedWittVector.truncate_truncate + - WittVector.truncateFun_mul + - WittVector.truncateFun_pow + - TruncatedWittVector.coeff_zero + - WittVector.truncate_liftFun + - TruncatedWittVector.truncate_surjective + - WittVector.lift_unique + - WittVector.truncateFun_zsmul + - path: Mathlib/LinearAlgebra/Matrix/Transvection.lean + theorems: + - Matrix.Pivot.mul_listTransvecRow_last_col_take + - Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_col + - Matrix.Pivot.mul_listTransvecRow_last_col + - Matrix.TransvectionStruct.prod_mul_reverse_inv_prod + - Matrix.mem_range_scalar_iff_commute_transvectionStruct + - Matrix.TransvectionStruct.sumInl_toMatrix_prod_mul + - Matrix.TransvectionStruct.det_toMatrix_prod + - Matrix.diagonal_transvection_induction_of_det_ne_zero + - Matrix.TransvectionStruct.mul_inv + - Matrix.transvection_zero + - Matrix.diagonal_transvection_induction + - Matrix.updateRow_eq_transvection + - Matrix.mul_transvection_apply_of_ne + - Matrix.Pivot.listTransvecCol_mul_last_row_drop + - Matrix.mul_transvection_apply_same + - Matrix.TransvectionStruct.reverse_inv_prod_mul_prod + - Matrix.Pivot.exists_list_transvec_mul_diagonal_mul_list_transvec + - Matrix.Pivot.exists_isTwoBlockDiagonal_of_ne_zero + - Matrix.TransvectionStruct.toMatrix_sumInl + - Matrix.Pivot.isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow + - Matrix.transvection_mul_apply_same + - Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction + - Matrix.Pivot.listTransvecCol_mul_last_col + - Matrix.TransvectionStruct.toMatrix_reindexEquiv + - Matrix.transvection_mul_transvection_same + - Matrix.Pivot.exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec + - Matrix.TransvectionStruct.inv_mul + - Matrix.TransvectionStruct.toMatrix_reindexEquiv_prod + - Matrix.Pivot.reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal + - Matrix.Pivot.mul_listTransvecRow_last_row + - Matrix.TransvectionStruct.mul_sumInl_toMatrix_prod + - Matrix.Pivot.listTransvecCol_mul_last_row + - Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_row + - Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux + - Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal + - Matrix.mem_range_scalar_of_commute_transvectionStruct + - path: Mathlib/RingTheory/Ideal/Prod.lean + theorems: + - Ideal.map_fst_prod + - Ideal.ideal_prod_prime_aux + - Ideal.ideal_prod_prime + - Ideal.prod.ext_iff + - Ideal.map_prodComm_prod + - Ideal.isPrime_of_isPrime_prod_top + - Ideal.map_snd_prod + - Ideal.isPrime_ideal_prod_top' + - Ideal.prod_top_top + - Ideal.ideal_prod_eq + - Ideal.isPrime_ideal_prod_top + - Ideal.isPrime_of_isPrime_prod_top' + - path: Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean + theorems: + - TrivSqZeroExt.norm_inl + - TrivSqZeroExt.hasSum_expSeries_of_smul_comm + - TrivSqZeroExt.exp_def_of_smul_comm + - TrivSqZeroExt.norm_def + - TrivSqZeroExt.fst_exp + - TrivSqZeroExt.snd_exp + - TrivSqZeroExt.snd_expSeries_of_smul_comm + - TrivSqZeroExt.nnnorm_inr + - TrivSqZeroExt.fst_expSeries + - TrivSqZeroExt.norm_inr + - TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm + - TrivSqZeroExt.nnnorm_inl + - TrivSqZeroExt.eq_smul_exp_of_invertible + - TrivSqZeroExt.exp_inr + - TrivSqZeroExt.nnnorm_def + - TrivSqZeroExt.exp_inl + - path: Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean + theorems: + - CategoryTheory.Limits.Concrete.multiequalizer_ext + - CategoryTheory.Limits.Concrete.widePushout_exists_rep + - CategoryTheory.Limits.Concrete.prodEquiv_symm_apply_fst + - CategoryTheory.Limits.Concrete.cokernel_funext + - CategoryTheory.Limits.Concrete.Pi.map_ext + - CategoryTheory.Limits.Concrete.initial_iff_empty_of_preserves_of_reflects + - CategoryTheory.Limits.Concrete.widePullback_ext' + - CategoryTheory.Limits.Concrete.widePullback_ext + - CategoryTheory.Limits.Concrete.empty_of_initial_of_preserves + - "CategoryTheory.Limits.Concrete.productEquiv_symm_apply_\u03C0" + - CategoryTheory.Limits.Concrete.prodEquiv_symm_apply_snd + - CategoryTheory.Limits.Concrete.widePushout_exists_rep' + - path: Mathlib/CategoryTheory/Abelian/Pseudoelements.lean + theorems: + - CategoryTheory.Abelian.Pseudoelement.pseudoApply_aux + - CategoryTheory.Abelian.Pseudoelement.apply_zero + - CategoryTheory.Abelian.Pseudoelement.epi_of_pseudo_surjective + - CategoryTheory.Abelian.Pseudoelement.zero_morphism_ext + - CategoryTheory.Abelian.Pseudoelement.pseudo_pullback + - CategoryTheory.Abelian.Pseudoelement.ModuleCat.eq_range_of_pseudoequal + - CategoryTheory.Abelian.Pseudoelement.pseudo_surjective_of_epi + - CategoryTheory.Abelian.Pseudoelement.pseudoZero_iff + - CategoryTheory.Abelian.Pseudoelement.zero_of_map_zero + - CategoryTheory.Abelian.Pseudoelement.comp_apply + - CategoryTheory.Abelian.pseudoEqual_trans + - CategoryTheory.Abelian.Pseudoelement.apply_eq_zero_of_comp_eq_zero + - CategoryTheory.Abelian.pseudoEqual_refl + - CategoryTheory.Abelian.Pseudoelement.zero_apply + - CategoryTheory.Abelian.Pseudoelement.pseudoZero_aux + - CategoryTheory.Abelian.Pseudoelement.pseudo_exact_of_exact + - CategoryTheory.Abelian.Pseudoelement.pseudo_injective_of_mono + - CategoryTheory.Abelian.Pseudoelement.exact_of_pseudo_exact + - CategoryTheory.Abelian.Pseudoelement.sub_of_eq_image + - CategoryTheory.Abelian.Pseudoelement.eq_zero_iff + - path: Mathlib/Topology/MetricSpace/Cauchy.lean + theorems: + - cauchySeq_iff_le_tendsto_0 + - Metric.uniformCauchySeqOn_iff + - Metric.exists_subseq_bounded_of_cauchySeq + - cauchySeq_bdd + - path: Mathlib/LinearAlgebra/BilinearForm/Properties.lean + theorems: + - BilinForm.IsAdjointPair.comp + - BilinForm.isSymm_iff_flip + - BilinForm.IsAdjointPair.mul + - BilinForm.IsAdjointPair.smul + - BilinForm.dualBasis_repr_apply + - BilinForm.dualBasis_dualBasis_flip + - BilinForm.isAdjointPair_iff_compLeft_eq_compRight + - BilinForm.nondegenerate_congr_iff + - BilinForm.IsAlt.isRefl + - BilinForm.IsAdjointPair.add + - BilinForm.apply_dualBasis_left + - BilinForm.isPairSelfAdjoint_equiv + - BilinForm.symmCompOfNondegenerate_left_apply + - BilinForm.nondegenerate_iff_ker_eq_bot + - BilinForm.comp_symmCompOfNondegenerate_apply + - BilinForm.IsAlt.neg_eq + - BilinForm.Nondegenerate.flip + - BilinForm.isSkewAdjoint_iff_neg_self_adjoint + - BilinForm.apply_dualBasis_right + - BilinForm.compLeft_injective + - BilinForm.isAdjointPair_zero + - BilinForm.IsAdjointPair.sub + - BilinForm.isAdjointPair_unique_of_nondegenerate + - BilinForm.mem_skewAdjointSubmodule + - path: Mathlib/Topology/ContinuousFunction/Compact.lean + theorems: + - ContinuousMap.dist_le_iff_of_nonempty + - ContinuousMap.dist_le + - ContinuousMap.dist_lt_iff_of_nonempty + - ContinuousLinearMap.toLinear_compLeftContinuousCompact + - ContinuousMap.summable_of_locally_summable_norm + - ContinuousMap.dist_apply_le_dist + - ContinuousMap.dist_lt_iff + - ContinuousMap.uniformInducing_equivBoundedOfCompact + - path: Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean + theorems: + - CategoryTheory.ShortComplex.RightHomologyMapData.rightHomologyMap_eq + - CategoryTheory.ShortComplex.RightHomologyData.ofIsLimitKernelFork_g' + - CategoryTheory.ShortComplex.RightHomologyData.opcyclesIso_inv_comp_descOpcycles + - CategoryTheory.ShortComplex.RightHomologyMapData.opcyclesMap_comm + - CategoryTheory.ShortComplex.RightHomologyData.pOpcycles_comp_opcyclesIso_hom + - CategoryTheory.ShortComplex.RightHomologyData.p_comp_opcyclesIso_inv + - "CategoryTheory.ShortComplex.RightHomologyData.isIso_\u03B9" + - CategoryTheory.ShortComplex.opcyclesMap'_g' + - "CategoryTheory.ShortComplex.RightHomologyMapData.congr_\u03C6Q" + - "CategoryTheory.ShortComplex.rightHomology\u03B9_descOpcycles_\u03C0_eq_zero_of_boundary" + - "CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_hom_comp_\u03B9" + - CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono_p + - "CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_inv_comp_rightHomology\u03B9" + - CategoryTheory.ShortComplex.isIso_opcyclesMap'_of_isIso_of_epi + - CategoryTheory.ShortComplex.RightHomologyData.isIso_p + - CategoryTheory.ShortComplex.rightHomologyMap'_comp + - CategoryTheory.ShortComplex.leftHomologyMap_op + - CategoryTheory.ShortComplex.rightHomologyMap'_op + - CategoryTheory.ShortComplex.opcycles_ext + - CategoryTheory.ShortComplex.RightHomologyData.opcyclesIso_hom_comp_descQ + - CategoryTheory.ShortComplex.rightHomology_ext + - CategoryTheory.ShortComplex.leftHomologyMap'_op + - CategoryTheory.ShortComplex.RightHomologyMapData.rightHomologyMap_comm + - CategoryTheory.ShortComplex.opcyclesMap_comp_descOpcycles + - CategoryTheory.ShortComplex.RightHomologyData.ofZeros_g' + - CategoryTheory.ShortComplex.descOpcycles_comp + - CategoryTheory.ShortComplex.opcycles_ext_iff + - CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono'_p + - CategoryTheory.ShortComplex.rightHomologyMap_op + - "CategoryTheory.ShortComplex.RightHomologyData.\u03B9_descQ_eq_zero_of_boundary" + - "CategoryTheory.ShortComplex.rightHomology\u03B9_comp_fromOpcycles" + - CategoryTheory.ShortComplex.rightHomology_ext_iff + - "CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono'_g'_\u03C4\ + \u2083" + - CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono_g' + - "CategoryTheory.ShortComplex.RightHomologyMapData.congr_\u03C6H" + - CategoryTheory.ShortComplex.RightHomologyData.ofIsColimitCokernelCofork_g' + - path: Mathlib/FieldTheory/Normal.lean + theorems: + - IntermediateField.splits_of_mem_adjoin + - Normal.of_algEquiv + - AlgHom.restrictNormal_comp + - Normal.minpoly_eq_iff_mem_orbit + - Normal.exists_isSplittingField + - AlgEquiv.restrictNormal_trans + - Normal.of_isSplittingField + - AlgHom.fieldRange_of_normal + - isSolvable_of_isScalarTower + - Normal.tower_top_of_normal + - path: Mathlib/Probability/Moments.lean + theorems: + - ProbabilityTheory.mgf_zero_measure + - ProbabilityTheory.IndepFun.mgf_add + - ProbabilityTheory.iIndepFun.mgf_sum + - ProbabilityTheory.measure_le_le_exp_cgf + - ProbabilityTheory.iIndepFun.integrable_exp_mul_sum + - ProbabilityTheory.mgf_const + - ProbabilityTheory.mgf_pos' + - ProbabilityTheory.measure_ge_le_exp_mul_mgf + - ProbabilityTheory.aestronglyMeasurable_exp_mul_sum + - ProbabilityTheory.mgf_undef + - ProbabilityTheory.centralMoment_one + - ProbabilityTheory.IndepFun.mgf_add' + - ProbabilityTheory.moment_zero + - ProbabilityTheory.cgf_zero' + - ProbabilityTheory.centralMoment_two_eq_variance + - ProbabilityTheory.mgf_nonneg + - ProbabilityTheory.cgf_zero_fun + - ProbabilityTheory.measure_le_le_exp_mul_mgf + - ProbabilityTheory.cgf_const + - ProbabilityTheory.measure_ge_le_exp_cgf + - ProbabilityTheory.mgf_zero_fun + - ProbabilityTheory.centralMoment_zero + - ProbabilityTheory.cgf_const' + - ProbabilityTheory.cgf_undef + - ProbabilityTheory.IndepFun.exp_mul + - ProbabilityTheory.centralMoment_one' + - ProbabilityTheory.mgf_const' + - ProbabilityTheory.IndepFun.integrable_exp_mul_add + - ProbabilityTheory.cgf_zero + - ProbabilityTheory.mgf_neg + - ProbabilityTheory.mgf_zero + - ProbabilityTheory.mgf_zero' + - ProbabilityTheory.aestronglyMeasurable_exp_mul_add + - ProbabilityTheory.IndepFun.cgf_add + - ProbabilityTheory.iIndepFun.cgf_sum + - ProbabilityTheory.cgf_neg + - ProbabilityTheory.cgf_zero_measure + - path: Mathlib/Data/Set/Intervals/WithBotTop.lean + theorems: + - WithTop.image_coe_Ico + - WithTop.preimage_coe_Ioo + - WithTop.image_coe_Ioi + - WithTop.image_coe_Iio + - WithTop.image_coe_Icc + - WithBot.preimage_coe_Ioi_bot + - WithBot.image_coe_Iio + - WithTop.preimage_coe_Ico + - WithTop.image_coe_Ioc + - WithBot.image_coe_Ici + - WithBot.preimage_coe_Ioo_bot + - WithTop.image_coe_Iic + - WithBot.preimage_coe_Ioo + - WithBot.image_coe_Ioo + - WithBot.preimage_coe_Icc + - WithTop.range_coe + - WithBot.image_coe_Icc + - WithBot.image_coe_Ico + - WithBot.preimage_coe_Ico + - WithBot.preimage_coe_Ioc_bot + - WithBot.preimage_coe_Ioc + - WithTop.preimage_coe_Ico_top + - WithBot.image_coe_Ioi + - WithTop.preimage_coe_Ioo_top + - WithTop.image_coe_Ioo + - WithBot.image_coe_Ioc + - WithTop.image_coe_Ici + - WithTop.preimage_coe_Icc + - WithBot.image_coe_Iic + - path: Mathlib/RingTheory/Localization/Integral.lean + theorems: + - IsFractionRing.isAlgebraic_iff + - isAlgebraic_of_isLocalization + - IsFractionRing.ideal_span_singleton_map_subset + - "IsLocalization.integerNormalization_eval\u2082_eq_zero" + - IsLocalization.integerNormalization_aeval_eq_zero + - IsIntegral.exists_multiple_integral_of_isLocalization + - RingHom.isIntegralElem_localization_at_leadingCoeff + - IsLocalization.coeffIntegerNormalization_of_not_mem_support + - IsLocalization.integerNormalization_map_to_map + - IsIntegralClosure.isFractionRing_of_algebraic + - isAlgebraic_of_isFractionRing + - IsLocalization.coeffIntegerNormalization_mem_support + - isIntegral_localization + - IsFractionRing.isAlgebraic_iff' + - IsFractionRing.integerNormalization_eq_zero_iff + - IsLocalization.integerNormalization_spec + - IsLocalization.integerNormalization_coeff + - IsLocalization.scaleRoots_commonDenom_mem_lifts + - path: Mathlib/RingTheory/FiniteType.lean + theorems: + - AddMonoidAlgebra.finiteType_iff_group_fg + - AddMonoidAlgebra.mem_adjoin_support + - Algebra.FiniteType.of_restrictScalars_finiteType + - MonoidAlgebra.mvPolynomial_aeval_of_surjective_of_closure + - AddMonoidAlgebra.support_gen_of_gen + - MonoidAlgebra.exists_finset_adjoin_eq_top + - MonoidAlgebra.finiteType_iff_group_fg + - AddMonoidAlgebra.mem_closure_of_mem_span_closure + - Algebra.FiniteType.iff_quotient_freeAlgebra + - RingHom.FiniteType.comp + - Algebra.FiniteType.iff_quotient_mvPolynomial'' + - RingHom.FiniteType.of_surjective + - AddMonoidAlgebra.exists_finset_adjoin_eq_top + - Module.Finite.injective_of_surjective_endomorphism + - MonoidAlgebra.mem_closure_of_mem_span_closure + - MonoidAlgebra.freeAlgebra_lift_of_surjective_of_closure + - AddMonoidAlgebra.support_gen_of_gen' + - RingHom.FiniteType.of_comp_finiteType + - MonoidAlgebra.of_mem_span_of_iff + - AddMonoidAlgebra.mvPolynomial_aeval_of_surjective_of_closure + - AddMonoidAlgebra.finiteType_iff_fg + - Algebra.FiniteType.iff_quotient_freeAlgebra' + - Algebra.FiniteType.iff_quotient_mvPolynomial' + - Algebra.FiniteType.freeAlgebra + - AddMonoidAlgebra.of'_mem_span + - MonoidAlgebra.mem_adjoin_support + - MonoidAlgebra.support_gen_of_gen + - RingHom.FiniteType.comp_surjective + - Algebra.FiniteType.isNoetherianRing + - Algebra.FiniteType.iff_quotient_mvPolynomial + - MonoidAlgebra.support_gen_of_gen' + - Algebra.FiniteType.of_surjective + - AddMonoidAlgebra.freeAlgebra_lift_of_surjective_of_closure + - Algebra.FiniteType.mvPolynomial + - Algebra.FiniteType.polynomial + - path: Mathlib/Analysis/NormedSpace/Star/Multiplier.lean + theorems: + - DoubleCentralizer.ext + - DoubleCentralizer.norm_snd + - DoubleCentralizer.range_toProdMulOpposite + - DoubleCentralizer.range_toProd + - DoubleCentralizer.coe_eq_algebraMap + - path: Mathlib/CategoryTheory/Sites/Types.lean + theorems: + - CategoryTheory.isSheaf_yoneda' + - CategoryTheory.eval_map + - CategoryTheory.typesGrothendieckTopology_eq_canonical + - CategoryTheory.typesGlue_eval + - CategoryTheory.eval_typesGlue + - path: Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean + theorems: + - EuclideanGeometry.right_ne_of_oangle_sign_eq_one + - EuclideanGeometry.two_zsmul_oangle_of_vectorSpan_eq + - EuclideanGeometry.oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq + - "EuclideanGeometry.oangle_swap\u2081\u2083_sign" + - EuclideanGeometry.dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint + - EuclideanGeometry.oangle_rotate_sign + - Collinear.oangle_sign_of_sameRay_vsub + - EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two + - Wbtw.oangle_eq_right + - Wbtw.oangle_sign_eq_of_ne_right + - EuclideanGeometry.oangle_self_right + - EuclideanGeometry.left_ne_right_of_oangle_sign_eq_one + - EuclideanGeometry.abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq + - EuclideanGeometry.oangle_midpoint_right + - EuclideanGeometry.angle_eq_pi_div_two_of_oangle_eq_pi_div_two + - EuclideanGeometry.left_ne_of_oangle_sign_eq_one + - EuclideanGeometry.collinear_iff_of_two_zsmul_oangle_eq + - Collinear.two_zsmul_oangle_eq_left + - AffineSubspace.SOppSide.oangle_sign_eq_neg + - EuclideanGeometry.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero + - Sbtw.oangle_eq_left_right + - EuclideanGeometry.oangle_midpoint_left + - Sbtw.oangle_eq_add_pi_left + - "Wbtw.oangle\u2082\u2081\u2083_eq_zero" + - EuclideanGeometry.oangle_eq_pi_iff_sbtw + - Wbtw.oangle_sign_eq_of_ne_left + - EuclideanGeometry.affineIndependent_iff_of_two_zsmul_oangle_eq + - EuclideanGeometry.oangle_self_left + - Wbtw.oangle_eq_left + - "Sbtw.oangle\u2083\u2082\u2081_eq_pi" + - EuclideanGeometry.right_ne_of_oangle_sign_eq_neg_one + - Sbtw.oangle_sign_eq + - EuclideanGeometry.left_ne_right_of_oangle_ne_zero + - AffineSubspace.SSameSide.oangle_sign_eq + - EuclideanGeometry.continuousAt_oangle + - EuclideanGeometry.oangle_ne_zero_and_ne_pi_iff_affineIndependent + - EuclideanGeometry.left_ne_of_oangle_sign_eq_neg_one + - EuclideanGeometry.oangle_eq_zero_iff_wbtw + - EuclideanGeometry.oangle_eq_oangle_of_dist_eq + - Sbtw.oangle_eq_add_pi_right + - "EuclideanGeometry.oangle_swap\u2082\u2083_sign" + - EuclideanGeometry.two_zsmul_oangle_of_parallel + - EuclideanGeometry.oangle_midpoint_rev_left + - EuclideanGeometry.left_ne_right_of_oangle_sign_eq_neg_one + - EuclideanGeometry.angle_rev_eq_pi_div_two_of_oangle_eq_neg_pi_div_two + - EuclideanGeometry.angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two + - EuclideanGeometry.left_ne_of_oangle_ne_zero + - EuclideanGeometry.oangle_midpoint_rev_right + - EuclideanGeometry.oangle_eq_zero_or_eq_pi_iff_collinear + - EuclideanGeometry.right_ne_of_oangle_ne_zero + - path: Mathlib/RingTheory/AlgebraicIndependent.lean + theorems: + - algebraicIndependent_iUnion_of_directed + - algebraicIndependent_finset_map_embedding_subtype + - AlgebraicIndependent.algebraMap_injective + - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some + - AlgebraicIndependent.option_iff + - AlgebraicIndependent.to_subtype_range + - algebraicIndependent_comp_subtype + - AlgebraicIndependent.map + - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C + - algebraicIndependent_empty_type + - AlgebraicIndependent.image + - AlgebraicIndependent.of_comp + - AlgebraicIndependent.linearIndependent + - AlgebraicIndependent.image_of_comp + - AlgebraicIndependent.restrictScalars + - algebraicIndependent_empty_type_iff + - AlgebraicIndependent.mono + - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none + - IsTranscendenceBasis.isAlgebraic + - AlgebraicIndependent.coe_range + - AlgebraicIndependent.comp + - AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin + - algebraicIndependent_sUnion_of_directed + - algebraicIndependent_subtype + - algebraicIndependent_empty_iff + - AlgebraicIndependent.isTranscendenceBasis_iff + - exists_isTranscendenceBasis + - algebraicIndependent_of_finite + - algebraicIndependent_bounded_of_finset_algebraicIndependent_bounded + - exists_maximal_algebraicIndependent + - path: Mathlib/GroupTheory/Transfer.lean + theorems: + - MonoidHom.transfer_eq_pow + - MonoidHom.transfer_eq_pow_aux + - Subgroup.leftTransversals.diff_mul_diff + - Subgroup.leftTransversals.smul_diff_smul + - MonoidHom.ker_transferSylow_isComplement' + - MonoidHom.transfer_center_eq_pow + - MonoidHom.transfer_def + - MonoidHom.transfer_eq_prod_quotient_orbitRel_zpowers_quot + - path: Mathlib/CategoryTheory/Sites/DenseSubsite.lean + theorems: + - CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_apply + - CategoryTheory.Functor.IsCoverDense.sheafHom_restrict_eq + - CategoryTheory.Functor.IsCoverDense.iso_of_restrict_iso + - CategoryTheory.Presieve.in_coverByImage + - CategoryTheory.Functor.IsCoverDense.functorPullback_pushforward_covering + - CategoryTheory.Functor.IsCoverDense.ext + - CategoryTheory.Functor.IsCoverDense.sheafHom_eq + - CategoryTheory.Functor.is_cover_of_isCoverDense + - CategoryTheory.Functor.IsCoverDense.compatiblePreserving + - CategoryTheory.Functor.IsCoverDense.Types.appHom_valid_glue + - CategoryTheory.Functor.IsCoverDense.Types.pushforwardFamily_compatible + - path: Mathlib/Topology/PartialHomeomorph.lean + theorems: + - PartialHomeomorph.eventually_nhds' + - PartialHomeomorph.transHomeomorph_transHomeomorph + - PartialHomeomorph.ext_iff + - PartialHomeomorph.restr_toPartialEquiv' + - PartialHomeomorph.restr_source_inter + - PartialHomeomorph.isOpen_image_of_subset_source + - PartialHomeomorph.continuousAt_iff_continuousAt_comp_right + - PartialHomeomorph.isOpen_image_iff_of_subset_source + - PartialHomeomorph.symm_map_nhds_eq + - PartialHomeomorph.replaceEquiv_eq_self + - Homeomorph.trans_transPartialHomeomorph + - PartialHomeomorph.restr_source' + - OpenEmbedding.toPartialHomeomorph_right_inv + - PartialHomeomorph.nhds_eq_comap_inf_principal + - PartialHomeomorph.Set.EqOn.restr_eqOn_source + - PartialHomeomorph.map_subtype_source + - PartialHomeomorph.eventually_nhdsWithin' + - PartialHomeomorph.trans_ofSet + - PartialHomeomorph.eventually_nhds + - PartialHomeomorph.isOpen_symm_image_iff_of_subset_target + - PartialHomeomorph.IsImage.iff_symm_preimage_eq' + - PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left + - PartialHomeomorph.tendsto_symm + - PartialHomeomorph.IsImage.map_nhdsWithin_eq + - PartialHomeomorph.trans_transHomeomorph + - PartialHomeomorph.continuous_iff_continuous_comp_left + - PartialHomeomorph.continuousAt_iff_continuousAt_comp_left + - PartialHomeomorph.eventually_nhdsWithin + - OpenEmbedding.toPartialHomeomorph_left_inv + - PartialHomeomorph.trans_of_set' + - PartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr + - PartialHomeomorph.ofSet_univ_eq_refl + - PartialHomeomorph.ofSet_trans + - PartialHomeomorph.openEmbedding_restrict + - PartialHomeomorph.IsImage.closure + - PartialHomeomorph.prod_eq_prod_of_nonempty' + - PartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter + - PartialHomeomorph.ofSet_trans_ofSet + - PartialHomeomorph.subtypeRestr_symm_eqOn_of_le + - Homeomorph.transPartialHomeomorph_trans + - PartialHomeomorph.subtypeRestr_source + - PartialHomeomorph.eventually_ne_nhdsWithin + - PartialHomeomorph.continuousOn_iff_continuousOn_comp_right + - PartialHomeomorph.map_nhdsWithin_preimage_eq + - TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_target + - path: Mathlib/Order/JordanHolder.lean + theorems: + - CompositionSeries.jordan_holder + - CompositionSeries.toList_injective + - CompositionSeries.Equivalent.snoc + - CompositionSeries.length_toList + - CompositionSeries.exists_top_eq_snoc_equivalant + - CompositionSeries.ext + - CompositionSeries.length_pos_of_bot_eq_bot_of_top_eq_top_of_length_pos + - CompositionSeries.eraseTop_top_le + - CompositionSeries.isMaximal_eraseTop_top + - CompositionSeries.append_natAdd_aux + - CompositionSeries.append_succ_natAdd + - CompositionSeries.append_succ_natAdd_aux + - CompositionSeries.append_castAdd + - CompositionSeries.chain'_toList + - CompositionSeries.ext_fun + - CompositionSeries.Equivalent.append + - CompositionSeries.toList_ofList + - JordanHolderLattice.isMaximal_of_eq_inf + - CompositionSeries.toList_ne_nil + - CompositionSeries.eq_snoc_eraseTop + - CompositionSeries.toList_sorted + - CompositionSeries.bot_snoc + - CompositionSeries.append_castAdd_aux + - CompositionSeries.mem_toList + - CompositionSeries.total + - CompositionSeries.append_succ_castAdd + - CompositionSeries.Equivalent.snoc_snoc_swap + - CompositionSeries.mem_snoc + - CompositionSeries.top_eraseTop + - CompositionSeries.append_succ_castAdd_aux + - CompositionSeries.Equivalent.length_eq + - CompositionSeries.length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero + - JordanHolderLattice.isMaximal_inf_right_of_isMaximal_sup + - CompositionSeries.ofList_toList + - CompositionSeries.snoc_eraseTop_top + - CompositionSeries.append_natAdd + - CompositionSeries.eq_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero + - CompositionSeries.mem_eraseTop_of_ne_of_mem + - path: Mathlib/Analysis/SpecialFunctions/Complex/Log.lean + theorems: + - Complex.range_exp + - Complex.tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero + - Complex.countable_preimage_exp + - Complex.log_zero + - Complex.two_pi_I_ne_zero + - continuousAt_clog + - Complex.log_conj_eq_ite + - Complex.log_ofReal_mul + - Complex.map_exp_comap_re_atTop + - Complex.tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero + - Complex.log_ofReal_re + - Complex.log_re + - Complex.exp_eq_exp_iff_exp_sub_eq_one + - Complex.log_inv_eq_ite + - Complex.log_exp + - Complex.log_im + - Complex.log_neg_one + - Complex.log_neg_I + - Complex.ofReal_log + - Complex.log_im_le_pi + - Complex.log_mul_eq_add_log_iff + - Complex.continuousWithinAt_log_of_re_neg_of_im_zero + - Complex.neg_pi_lt_log_im + - Complex.exp_eq_one_iff + - Complex.log_inv + - Complex.exp_inj_of_neg_pi_lt_of_le_pi + - Complex.log_one + - Complex.log_conj + - Complex.exp_eq_exp_iff_exists_int + - Complex.log_mul_ofReal + - Complex.map_exp_comap_re_atBot + - Complex.log_I + - Complex.exp_log + - path: Mathlib/Data/Real/Cardinality.lean + theorems: + - Cardinal.mk_Ioi_real + - Cardinal.cantorFunctionAux_true + - Cardinal.cantorFunctionAux_succ + - Cardinal.cantorFunctionAux_eq + - Cardinal.cantorFunction_injective + - Cardinal.cantorFunction_succ + - Cardinal.mk_Iio_real + - Cardinal.mk_Ioo_real + - Cardinal.mk_real + - Cardinal.cantorFunctionAux_false + - Cardinal.not_countable_real + - Cardinal.mk_univ_real + - Cardinal.increasing_cantorFunction + - Cardinal.cantorFunction_le + - Cardinal.cantorFunctionAux_nonneg + - Cardinal.cantorFunctionAux_zero + - Cardinal.summable_cantor_function + - path: Mathlib/GroupTheory/GroupAction/Period.lean + theorems: + - MulAction.period_bounded_of_exponent_pos + - MulAction.period_dvd_orderOf + - MulAction.period_inv + - MulAction.period_dvd_exponent + - MulAction.period_eq_one_iff + - path: Mathlib/Analysis/Complex/RealDeriv.lean + theorems: + - HasDerivAt.complexToReal_fderiv + - HasDerivWithinAt.complexToReal_fderiv + - HasStrictDerivAt.real_of_complex + - ContDiffAt.real_of_complex + - DifferentiableAt.conformalAt + - HasDerivAt.comp_ofReal + - HasDerivAt.complexToReal_fderiv' + - conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj + - HasStrictDerivAt.complexToReal_fderiv' + - HasDerivWithinAt.complexToReal_fderiv' + - HasDerivAt.real_of_complex + - HasDerivAt.ofReal_comp + - HasStrictDerivAt.complexToReal_fderiv + - path: Mathlib/Topology/MetricSpace/MetricSeparated.lean + theorems: + - IsMetricSeparated.finite_iUnion_left_iff + - IsMetricSeparated.finite_iUnion_right_iff + - IsMetricSeparated.union_left + - IsMetricSeparated.disjoint + - path: Mathlib/Data/Nat/Cast/Basic.lean + theorems: + - nsmul_eq_mul + - nsmul_one + - MonoidHom.ext_mnat + - map_natCast' + - AddMonoidHom.apply_nat + - MonoidHom.apply_mnat + - eq_natCast' + - ext_nat' + - NeZero.nat_of_neZero + - nsmul_eq_mul' + - ext_nat'' + - Nat.cast_pow + - ext_nat + - Nat.cast_mul + - path: Mathlib/MeasureTheory/Group/Pointwise.lean + theorems: + - "MeasurableSet.const_smul\u2080" + - MeasurableSet.const_smul_of_ne_zero + - MeasurableSet.const_smul + - path: Mathlib/Topology/Perfect.lean + theorems: + - perfectSpace_iff_forall_not_isolated + - AccPt.nhds_inter + - Preperfect.perfect_closure + - exists_countable_union_perfect_of_isClosed + - preperfect_iff_nhds + - exists_perfect_nonempty_of_isClosed_of_not_countable + - Preperfect.open_inter + - preperfect_iff_perfect_closure + - Perfect.closure_nhds_inter + - Perfect.splitting + - path: Mathlib/GroupTheory/Submonoid/Operations.lean + theorems: + - MonoidHom.mrange_top_iff_surjective + - Nat.addSubmonoid_closure_one + - Submonoid.bot_or_nontrivial + - MonoidHom.mker_one + - Submonoid.prod_eq_bot_iff + - Submonoid.prod_top + - Submonoid.mrange_inl + - Submonoid.top_prod + - MonoidHom.mker_prod_map + - Submonoid.closure_closure_coe_preimage + - MonoidHom.mker_inl + - MonoidHom.submonoidMap_surjective + - Submonoid.mem_map + - Submonoid.bot_prod_bot + - MonoidHom.map_mrange + - Submonoid.mrange_inr + - Submonoid.mk_eq_one + - Submonoid.nontrivial_iff_exists_ne_one + - Submonoid.le_prod_iff + - Submonoid.prod_eq_top_iff + - Submonoid.eq_bot_iff_forall + - MonoidHom.mrangeRestrict_mker + - Submonoid.prod_le_iff + - Submonoid.comap_id + - MonoidHom.restrict_mrange + - MonoidHom.mker_inr + - path: Mathlib/LinearAlgebra/Matrix/ZPow.lean + theorems: + - Matrix.zpow_neg_one + - Matrix.zpow_mul' + - Matrix.zpow_add_one_of_ne_neg_one + - Matrix.zpow_bit0 + - Matrix.zpow_neg_mul_zpow_self + - Matrix.one_div_pow + - Matrix.inv_zpow + - Matrix.zero_zpow + - Matrix.one_zpow + - Matrix.zpow_add_one + - Matrix.zpow_add + - Matrix.Commute.mul_zpow + - Matrix.zpow_bit1' + - Matrix.isUnit_det_zpow_iff + - Matrix.SemiconjBy.zpow_right + - Matrix.zpow_sub + - Matrix.Commute.zpow_right + - Matrix.zpow_mul + - Matrix.zpow_one_add + - Matrix.zero_zpow_eq + - Matrix.IsSymm.zpow + - Matrix.inv_pow' + - Matrix.zpow_add_of_nonpos + - Matrix.pow_sub' + - Matrix.zpow_neg + - Matrix.zpow_add_of_nonneg + - Matrix.zpow_neg_coe_nat + - Matrix.transpose_zpow + - Matrix.inv_zpow' + - IsUnit.det_zpow + - Matrix.one_div_zpow + - Matrix.zpow_bit1 + - Matrix.zpow_ne_zero_of_isUnit_det + - Matrix.zpow_sub_one + - Matrix.conjTranspose_zpow + - Matrix.coe_units_zpow + - path: Mathlib/Analysis/BoundedVariation.lean + theorems: + - variationOnFromTo.eq_neg_swap + - eVariationOn.Icc_add_Icc + - variationOnFromTo.self + - variationOnFromTo.eq_zero_iff_of_le + - variationOnFromTo.antitoneOn + - eVariationOn.sum_le_of_monotoneOn_Icc + - eVariationOn.union + - variationOnFromTo.eq_zero_iff + - eVariationOn.comp_le_of_antitoneOn + - eVariationOn.comp_eq_of_monotoneOn + - eVariationOn.add_le_union + - eVariationOn.edist_le + - eVariationOn.add_point + - eVariationOn.comp_ofDual + - variationOnFromTo.monotoneOn + - eVariationOn.comp_eq_of_antitoneOn + - LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_pi + - eVariationOn.eq_zero_iff + - eVariationOn.lowerSemicontinuous_aux + - LocallyBoundedVariationOn.ae_differentiableAt + - eVariationOn.lowerSemicontinuous + - eVariationOn.lowerSemicontinuous_uniformOn + - eVariationOn.eq_of_edist_zero_on + - variationOnFromTo.add + - eVariationOn.comp_inter_Icc_eq_of_monotoneOn + - eVariationOn.mono + - variationOnFromTo.nonpos_of_ge + - eVariationOn.eq_of_eqOn + - BoundedVariationOn.sub_le + - MonotoneOn.eVariationOn_le + - eVariationOn.sum_le_of_monotoneOn_Iic + - variationOnFromTo.sub_self_monotoneOn + - LipschitzOnWith.comp_eVariationOn_le + - BoundedVariationOn.dist_le + - eVariationOn.constant_on + - variationOnFromTo.comp_eq_of_monotoneOn + - variationOnFromTo.nonneg_of_le + - variationOnFromTo.eq_zero_iff_of_ge + - eVariationOn.nonempty_monotone_mem + - LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn + - variationOnFromTo.eq_of_ge + - LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem + - variationOnFromTo.edist_zero_of_eq_zero + - path: Mathlib/SetTheory/Ordinal/Basic.lean + theorems: + - Ordinal.lift_inj + - Ordinal.le_add_left + - Cardinal.lift_lt_univ + - Ordinal.type_def + - Ordinal.one_lt_card + - Ordinal.lift_down + - Ordinal.lt_one_iff_zero + - Cardinal.card_le_iff + - Ordinal.relIso_enum' + - Ordinal.nat_le_card + - Cardinal.ord_nat + - Ordinal.enum_zero_le' + - Cardinal.lift_lt_univ' + - Ordinal.type_fintype + - Ordinal.lift_type_lt + - nonempty_embedding_to_cardinal + - Ordinal.typein_top + - Ordinal.typein_apply + - Cardinal.lt_univ + - Cardinal.lift_ord + - Ordinal.type_fin + - Ordinal.le_one_iff + - Ordinal.lt_wf + - Ordinal.card_nat + - Ordinal.card_eq_nat + - Ordinal.relIso_enum + - Cardinal.ord_le + - Ordinal.card_le_one + - Cardinal.ord_injective + - Ordinal.enum_zero_eq_bot + - Ordinal.lift_lt + - Ordinal.typein_lt_typein + - Ordinal.typein_le_typein' + - Cardinal.mk_ordinal_out + - Ordinal.enum_le_enum + - Ordinal.card_eq_zero + - Ordinal.one_le_card + - Ordinal.le_add_right + - Ordinal.typein_one_out + - Ordinal.succ_le_iff' + - Ordinal.card_succ + - Ordinal.enum_lt_enum + - Ordinal.typein_le_typein + - Ordinal.typein_lt_self + - Ordinal.type_def' + - Ordinal.type_uLift + - Cardinal.ord_univ + - Cardinal.mk_ord_out + - Ordinal.one_out_eq + - WellOrder.eta + - Ordinal.type_out + - List.Sorted.lt_ord_of_lt + - Cardinal.ord_one + - Ordinal.le_enum_succ + - Ordinal.enum_zero_le + - Ordinal.succ_one + - Cardinal.card_ord + - Cardinal.card_typein_lt + - Ordinal.type_preimage_aux + - Ordinal.card_eq_one + - Cardinal.card_typein_out_lt + - Ordinal.zero_lt_card + - Ordinal.nat_lt_card + - Cardinal.small_iff_lift_mk_lt_univ + - Cardinal.card_le_of_le_ord + - Ordinal.lift.principalSeg_top' + - Ordinal.one_le_iff_pos + - Cardinal.lt_univ' + - Ordinal.one_le_iff_ne_zero + - Ordinal.type_ne_zero_iff_nonempty + - Ordinal.lift_le + - path: Mathlib/Topology/Connected/TotallyDisconnected.lean + theorems: + - Continuous.connectedComponentsLift_continuous + - totallyDisconnectedSpace_subtype_iff + - IsPreconnected.eqOn_const_of_mapsTo + - IsPreconnected.constant_of_mapsTo + - Embedding.isTotallyDisconnected_image + - exists_isClopen_of_totally_separated + - totallyDisconnectedSpace_iff_connectedComponent_subsingleton + - totallyDisconnectedSpace_iff_connectedComponent_singleton + - Embedding.isTotallyDisconnected_range + - isTotallyDisconnected_of_isClopen_set + - isTotallyDisconnected_of_isTotallySeparated + - path: Mathlib/RingTheory/MvPolynomial/Basic.lean + theorems: + - MvPolynomial.mem_restrictDegree + - MvPolynomial.mem_restrictDegree_iff_sup + - MvPolynomial.mapRange_eq_map + - MvPolynomial.mem_restrictTotalDegree + - path: Mathlib/SetTheory/Cardinal/Cofinality.lean + theorems: + - Ordinal.exists_lsub_cof + - Ordinal.lsub_lt_ord_lift + - Cardinal.sum_lt_of_isRegular + - Cardinal.lsub_lt_ord_of_isRegular + - Cardinal.isRegular_aleph_one + - Ordinal.IsFundamentalSequence.cof_eq + - Cardinal.le_range_of_union_finset_eq_top + - Ordinal.le_cof_iff_blsub + - Ordinal.cof_add + - Ordinal.IsFundamentalSequence.ord_cof + - Ordinal.bsup_lt_ord + - Cardinal.lt_power_cof + - Ordinal.le_cof_iff_lsub + - Ordinal.cof_sup_le_lift + - Ordinal.cof_sup_le + - Cardinal.nfpFamily_lt_ord_lift_of_isRegular + - Cardinal.nfp_lt_ord_of_isRegular + - Ordinal.lsub_lt_ord + - Ordinal.iSup_lt_lift + - Cardinal.lsub_lt_ord_lift_of_isRegular + - Ordinal.infinite_pigeonhole + - Ordinal.cof_univ + - Ordinal.blsub_lt_ord_lift + - Ordinal.IsFundamentalSequence.trans + - Ordinal.iSup_lt + - Cardinal.derivBFamily_lt_ord + - Ordinal.lt_cof_type + - Cardinal.IsRegular.ord_pos + - Ordinal.IsNormal.cof_le + - Cardinal.exists_infinite_fiber + - Cardinal.nfpBFamily_lt_ord_lift_of_isRegular + - Cardinal.isRegular_aleph0 + - Ordinal.IsNormal.isFundamentalSequence + - Cardinal.isRegular_aleph_succ + - Cardinal.derivBFamily_lt_ord_lift + - Ordinal.blsub_lt_ord + - Ordinal.cof_blsub_le_lift + - Ordinal.cof_lsub_le_lift + - Cardinal.isStrongLimit_aleph0 + - Ordinal.cof_succ + - Cardinal.univ_inaccessible + - Ordinal.cof_eq' + - Ordinal.ord_cof_eq + - Cardinal.nfpFamily_lt_ord_of_isRegular + - Ordinal.unbounded_of_unbounded_sUnion + - Cardinal.iSup_lt_of_isRegular + - Cardinal.derivFamily_lt_ord_lift + - Ordinal.cof_le_card + - Ordinal.IsFundamentalSequence.succ + - Cardinal.isRegular_succ + - Cardinal.sup_lt_ord_of_isRegular + - Ordinal.sup_lt_ord + - Ordinal.nfpBFamily_lt_ord_lift + - Ordinal.cof_bsup_le + - Ordinal.nfpFamily_lt_ord + - Cardinal.sup_lt_ord_lift_of_isRegular + - Ordinal.cof_eq_sInf_lsub + - Cardinal.blsub_lt_ord_lift_of_isRegular + - Ordinal.le_cof_type + - Ordinal.cof_blsub_le + - Ordinal.nfpBFamily_lt_ord + - Cardinal.isStrongLimit_beth + - Ordinal.sup_sequence_lt_omega1 + - Ordinal.exists_fundamental_sequence + - Ordinal.cof_omega + - Cardinal.mk_bounded_subset + - Cardinal.iSup_lt_lift_of_isRegular + - Ordinal.nfp_lt_ord + - Ordinal.unbounded_of_unbounded_iUnion + - RelIso.cof_le_lift + - Cardinal.mk_subset_mk_lt_cof + - Ordinal.lift_cof + - Ordinal.cof_ord_le + - Ordinal.infinite_pigeonhole_card + - Ordinal.IsFundamentalSequence.monotone + - Ordinal.IsFundamentalSequence.zero + - Cardinal.bsup_lt_ord_of_isRegular + - Cardinal.derivFamily_lt_ord + - Cardinal.isRegular_aleph'_succ + - Ordinal.cof_bsup_le_lift + - Cardinal.blsub_lt_ord_of_isRegular + - Ordinal.cof_cof + - Order.le_cof + - Ordinal.exists_blsub_cof + - Ordinal.infinite_pigeonhole_set + - Cardinal.deriv_lt_ord + - Cardinal.lt_cof_power + - Ordinal.cof_lsub_le + - Ordinal.cof_zero + - Cardinal.infinite_pigeonhole_card_lt + - Cardinal.bsup_lt_ord_lift_of_isRegular + - Ordinal.nfpFamily_lt_ord_lift + - Ordinal.cof_eq_zero + - path: Mathlib/Data/Int/ModEq.lean + theorems: + - Int.add_modEq_right + - Int.ModEq.of_mul_left + - Int.ModEq.add_left_cancel + - Int.ModEq.add_right_cancel + - Int.ModEq.add + - Int.ModEq.cancel_left_div_gcd + - Int.ModEq.sub + - Int.ModEq.pow + - Int.modEq_iff_dvd + - Int.gcd_a_modEq + - Int.modEq_neg + - Int.coe_nat_modEq_iff + - Int.modEq_iff_add_fac + - Int.ModEq.mul_left' + - Int.modEq_zero_iff + - Int.ModEq.of_div + - Int.ModEq.mul_right' + - Int.modEq_sub_fac + - Int.modEq_add_fac + - Int.mod_coprime + - Int.modEq_and_modEq_iff_modEq_mul + - Int.neg_modEq_neg + - Int.modEq_zero_iff_dvd + - Int.ModEq.neg + - Int.exists_unique_equiv + - Int.ModEq.cancel_right_div_gcd + - Int.add_modEq_left + - Int.exists_unique_equiv_nat + - path: Mathlib/Geometry/Euclidean/Circumcenter.lean + theorems: + - EuclideanGeometry.circumsphere_eq_of_cospherical + - EuclideanGeometry.existsUnique_dist_eq_of_insert + - Affine.Simplex.circumradius_pos + - EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq + - Affine.Simplex.sum_centroidWeightsWithCircumcenter + - Affine.Simplex.circumcenter_eq_affineCombination_of_pointsWithCircumcenter + - Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq + - Affine.Simplex.circumcenter_eq_of_range_eq + - Affine.Simplex.circumcenter_eq_centroid + - Affine.Simplex.eq_circumcenter_of_dist_eq + - EuclideanGeometry.circumcenter_eq_of_cospherical_subset + - EuclideanGeometry.exists_circumsphere_eq_of_cospherical + - EuclideanGeometry.cospherical_iff_exists_mem_of_complete + - Affine.Simplex.circumcenter_reindex + - Affine.Simplex.eq_circumradius_of_dist_eq + - EuclideanGeometry.eq_or_eq_reflection_of_dist_eq + - Affine.Simplex.sum_circumcenterWeightsWithCircumcenter + - Affine.Simplex.centroid_eq_affineCombination_of_pointsWithCircumcenter + - Affine.Simplex.sum_reflectionCircumcenterWeightsWithCircumcenter + - EuclideanGeometry.exists_circumradius_eq_of_cospherical + - Affine.Simplex.sum_pointWeightsWithCircumcenter + - EuclideanGeometry.exists_circumsphere_eq_of_cospherical_subset + - EuclideanGeometry.circumsphere_eq_of_cospherical_subset + - EuclideanGeometry.circumradius_eq_of_cospherical_subset + - Affine.Simplex.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter + - EuclideanGeometry.circumradius_eq_of_cospherical + - Affine.Simplex.circumsphere_reindex + - EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq + - EuclideanGeometry.exists_circumcenter_eq_of_cospherical_subset + - Affine.Simplex.dist_circumcenter_eq_circumradius' + - EuclideanGeometry.exists_circumradius_eq_of_cospherical_subset + - Affine.Simplex.circumcenter_eq_point + - EuclideanGeometry.exists_circumcenter_eq_of_cospherical + - Affine.Simplex.orthogonalProjection_circumcenter + - EuclideanGeometry.circumcenter_eq_of_cospherical + - Affine.Simplex.sum_pointsWithCircumcenter + - AffineIndependent.existsUnique_dist_eq + - Affine.Simplex.circumradius_reindex + - Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq + - Affine.Simplex.point_eq_affineCombination_of_pointsWithCircumcenter + - path: Mathlib/GroupTheory/Subsemigroup/Membership.lean + theorems: + - Subsemigroup.mem_iSup_of_mem + - Subsemigroup.iSup_induction' + - Subsemigroup.coe_iSup_of_directed + - Subsemigroup.mem_sup_left + - Subsemigroup.coe_sSup_of_directed_on + - Subsemigroup.mem_sSup_of_directed_on + - Subsemigroup.iSup_induction + - Subsemigroup.mem_sup_right + - Subsemigroup.mem_iSup_of_directed + - path: Mathlib/Data/Bool/Basic.lean + theorems: + - Bool.and_intro + - Bool.injective_iff + - Bool.right_le_or + - Bool.decide_coe + - Bool.forall_bool' + - Bool.and_elim_left + - Bool.not_not_eq + - Bool.eq_iff_eq_true_iff + - Bool.decide_and + - Bool.self_ne_not + - Bool.or_le + - Bool.coe_iff_coe + - Bool.eq_not_iff + - Bool.cond_not + - Bool.or_inr + - Bool.beq_comm + - Bool.eq_false_of_not_eq_true' + - Bool.toNat_bne_one + - Bool.decide_or + - Bool.toNat_le_toNat + - Bool.xor_iff_ne + - Bool.cond_eq_ite + - Bool.le_and + - Bool.beq_eq_decide_eq + - Bool.eq_or_eq_not + - Bool.cond_decide + - Bool.or_inl + - Bool.not_eq_iff + - Bool.toNat_beq_one + - Bool.decide_not + - Bool.toNat_bne_zero + - Bool.eq_false_of_ne_true + - Bool.le_iff_imp + - Bool.apply_apply_apply + - Bool.not_ne_self + - Bool.ofNat_le_ofNat + - Bool.and_le_right + - Bool.lt_iff + - Bool.ofNat_toNat + - Bool.not_eq_not + - Bool.left_le_or + - Bool.eq_true_of_not_eq_false' + - Bool.and_elim_right + - Bool.exists_bool' + - Bool.bne_eq_xor + - Bool.eq_true_of_ne_false + - Bool.not_iff_not + - Bool.dichotomy + - Bool.and_le_left + - Bool.toNat_beq_zero + - path: Mathlib/Geometry/Manifold/ContMDiff/Product.lean + theorems: + - ContMDiff.prod_map + - ContMDiffWithinAt.prod_mk_space + - contMDiff_prod_iff + - ContMDiffAt.prod_map + - ContMDiffAt.prod_map' + - contMDiffWithinAt_pi_space + - contMDiffAt_prod_iff + - ContMDiffWithinAt.prod_mk + - contMDiffWithinAt_fst + - contMDiffWithinAt_snd + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean + theorems: + - CategoryTheory.Limits.biproduct.mapBiproduct_hom_desc + - CategoryTheory.Functor.biproductComparison'_comp_biproductComparison + - CategoryTheory.Functor.biprodComparison'_comp_biprodComparison + - CategoryTheory.Limits.biprod.mapBiprod_inv_map_desc + - CategoryTheory.Limits.biprod.map_lift_mapBiprod + - CategoryTheory.Limits.biproduct.map_lift_mapBiprod + - CategoryTheory.Limits.biprod.lift_mapBiprod + - CategoryTheory.Limits.biproduct.mapBiproduct_inv_map_desc + - path: Mathlib/Topology/Algebra/InfiniteSum/Defs.lean + theorems: + - Summable.hasSum + - Function.Injective.hasSum_iff + - HasSum.unique + - tsum_eq_zero_of_not_summable + - hasSum_subtype_iff_of_support_subset + - Finset.hasSum + - path: Mathlib/Analysis/MeanInequalities.lean + theorems: + - Real.geom_mean_le_arith_mean2_weighted + - NNReal.geom_mean_le_arith_mean4_weighted + - Real.geom_mean_weighted_of_constant + - ENNReal.young_inequality + - Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg + - Real.Lp_add_le_tsum_of_nonneg + - NNReal.Lp_add_le + - Real.rpow_sum_le_const_mul_sum_rpow + - NNReal.isGreatest_Lp + - NNReal.inner_le_Lp_mul_Lp_of_norm_le_one + - NNReal.rpow_sum_le_const_mul_sum_rpow + - NNReal.geom_mean_le_arith_mean3_weighted + - Real.rpow_sum_le_const_mul_sum_rpow_of_nonneg + - NNReal.young_inequality_real + - Real.young_inequality_of_nonneg + - ENNReal.Lp_add_le + - ENNReal.inner_le_Lp_mul_Lq + - Real.inner_le_weight_mul_Lp_of_nonneg + - Real.inner_le_Lp_mul_Lq_of_nonneg + - NNReal.Lp_add_le_hasSum + - NNReal.geom_mean_le_arith_mean2_weighted + - Real.geom_mean_le_arith_mean + - NNReal.inner_le_Lp_mul_Lq_tsum + - Real.inner_le_Lp_mul_Lq_tsum_of_nonneg + - Real.inner_le_Lp_mul_Lq + - Real.Lp_add_le_of_nonneg + - Real.geom_mean_le_arith_mean_weighted + - NNReal.inner_le_Lp_mul_Lp_of_norm_eq_zero + - Real.geom_mean_eq_arith_mean_weighted_of_constant + - NNReal.Lp_add_le_tsum + - ENNReal.inner_le_weight_mul_Lp_of_nonneg + - Real.Lp_add_le + - NNReal.inner_le_Lp_mul_Lq + - Real.arith_mean_weighted_of_constant + - NNReal.inner_le_weight_mul_Lp + - NNReal.inner_le_Lp_mul_Lq_hasSum + - Real.geom_mean_le_arith_mean4_weighted + - Real.Lp_add_le_hasSum_of_nonneg + - NNReal.geom_mean_le_arith_mean_weighted + - ENNReal.rpow_sum_le_const_mul_sum_rpow + - path: Mathlib/Topology/MetricSpace/Isometry.lean + theorems: + - IsometryEquiv.image_emetric_closedBall + - Isometry.diam_range + - IsometryEquiv.preimage_closedBall + - Isometry.prod_map + - IsometryEquiv.image_sphere + - IsometryEquiv.image_closedBall + - Isometry.right_inv + - IsometryEquiv.ediam_univ + - Isometry.ediam_range + - isometry_subsingleton + - Isometry.ediam_image + - IsometryEquiv.preimage_emetric_ball + - isometry_dcomp + - IsometryEquiv.image_ball + - IsometryEquiv.preimage_ball + - Isometry.preimage_emetric_closedBall + - Isometry.preimage_setOf_dist + - isometry_iff_dist_eq + - Isometry.antilipschitz + - IsometryEquiv.diam_preimage + - Isometry.preimage_emetric_ball + - Isometry.diam_image + - isometry_iff_nndist_eq + - IsometryEquiv.preimage_emetric_closedBall + - IsometryEquiv.image_emetric_ball + - path: Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean + theorems: + - CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv' + - CategoryTheory.MonoidalCategory.pentagon_inv_inv_hom + - CategoryTheory.MonoidalCategory.pentagon_inv_hom + - CategoryTheory.MonoidalCategory.unitors_equal + - CategoryTheory.MonoidalCategory.id_tensor_rightUnitor_inv + - CategoryTheory.MonoidalCategory.leftUnitor_inv_tensor_id + - CategoryTheory.MonoidalCategory.leftUnitor_tensor'' + - CategoryTheory.MonoidalCategory.leftUnitor_tensor' + - CategoryTheory.MonoidalCategory.pentagon_hom_inv + - CategoryTheory.MonoidalCategory.unitors_inv_equal + - path: Mathlib/RingTheory/Nakayama.lean + theorems: + - Submodule.sup_eq_sup_smul_of_le_smul_of_le_jacobson + - Submodule.eq_bot_of_le_smul_of_le_jacobson_bot + - Submodule.le_of_le_smul_of_le_jacobson_bot + - Submodule.eq_smul_of_le_smul_of_le_jacobson + - path: Mathlib/RingTheory/Multiplicity.lean + theorems: + - multiplicity.Int.coe_nat_multiplicity + - multiplicity.finite_nat_iff + - multiplicity.multiplicity_add_eq_min + - multiplicity.Int.natAbs + - multiplicity.Finset.prod + - multiplicity.pos_of_dvd + - multiplicity.multiplicity_pow_self + - multiplicity.get_multiplicity_self + - multiplicity.exists_eq_pow_mul_and_not_dvd + - multiplicity.pow' + - multiplicity.le_multiplicity_map + - multiplicity.multiplicity_lt_iff_not_dvd + - multiplicity.multiplicity_self + - multiplicity.lt_top_iff_finite + - multiplicity.multiplicity_map_eq + - multiplicity_eq_zero_of_coprime + - multiplicity.is_greatest + - multiplicity.finite_mul_aux + - multiplicity.eq_top_iff + - multiplicity.dvd_iff_multiplicity_pos + - multiplicity.neg + - multiplicity.mul + - multiplicity.finite_of_finite_mul_left + - multiplicity.unique + - multiplicity.dvd_of_multiplicity_pos + - multiplicity.pow_dvd_of_le_multiplicity + - multiplicity.get_one_right + - multiplicity.multiplicity_eq_zero + - multiplicity.multiplicity_add_of_gt + - multiplicity.unique' + - multiplicity.multiplicity_sub_of_gt + - multiplicity.eq_coe_iff + - multiplicity.mul' + - multiplicity.isUnit_right + - multiplicity.multiplicity_le_multiplicity_iff + - multiplicity.ne_top_iff_finite + - multiplicity.pow + - multiplicity.finite_pow + - multiplicity.not_finite_iff_forall + - multiplicity.multiplicity_mk_eq_multiplicity + - multiplicity.ne_zero_of_finite + - multiplicity.is_greatest' + - multiplicity.min_le_multiplicity_add + - multiplicity.pow_multiplicity_dvd + - path: Mathlib/Data/Set/Intervals/Pi.lean + theorems: + - Set.pi_univ_Ioc_update_union + - Set.Icc_diff_pi_univ_Ioc_subset + - Set.image_update_Ioc + - Set.image_update_Ioc_right + - Set.image_update_Ioo_right + - Set.Icc_diff_pi_univ_Ioo_subset + - Set.pi_univ_Ioc_update_left + - Set.image_update_Icc + - Set.image_update_Ico + - Set.image_update_uIcc + - Set.image_update_Ioo + - Set.image_update_Ico_left + - Set.pi_univ_Ioc_update_right + - Set.pi_univ_Iic + - Set.image_update_uIcc_left + - Set.image_update_Icc_left + - Set.image_update_Ioo_left + - Set.image_update_Icc_right + - Set.image_update_Ioc_left + - Set.pi_univ_Ici + - Set.image_update_Ico_right + - Set.pi_univ_Icc + - Set.image_update_uIcc_right + - Set.disjoint_pi_univ_Ioc_update_left_right + - path: Mathlib/Analysis/SpecialFunctions/Exp.lean + theorems: + - Complex.exp_bound_sq + - Real.tendsto_comp_exp_atBot + - Real.isTheta_exp_comp_one + - Real.exp_half + - Real.summable_exp_neg_nat + - Real.range_exp + - Real.tendsto_exp_nhds_zero_nhds_one + - Complex.locally_lipschitz_exp + - Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero + - Complex.comap_exp_cobounded + - Real.isBigO_one_exp_comp + - Real.tendsto_comp_exp_atTop + - Real.tendsto_exp_atTop + - Real.tendsto_exp_comp_atTop + - Complex.comap_exp_nhdsWithin_zero + - Real.comap_exp_nhdsWithin_Ioi_zero + - Real.tendsto_mul_exp_add_div_pow_atTop + - Real.map_exp_atTop + - Real.isBigO_exp_comp_one + - Complex.comap_exp_nhds_zero + - Real.tendsto_exp_div_pow_atTop + - Real.isLittleO_pow_exp_atTop + - Real.isBigO_exp_comp_exp_comp + - Real.isLittleO_exp_comp_exp_comp + - Real.tendsto_div_pow_mul_exp_add_atTop + - Real.isLittleO_one_exp_comp + - Real.isTheta_exp_comp_exp_comp + - Real.tendsto_exp_comp_nhds_zero + - Real.map_exp_atBot + - Real.summable_exp_nat_mul_iff + - Complex.tendsto_exp_nhds_zero_iff + - Real.summable_pow_mul_exp_neg_nat_mul + - Real.comap_exp_atTop + - Complex.continuous_exp + - path: Mathlib/Topology/Order/IntermediateValue.lean + theorems: + - IsPreconnected.mem_intervals + - IsPreconnected.Icc_subset + - IsPreconnected.eq_univ_of_unbounded + - Continuous.strictMonoOn_of_inj_rigidity + - "IsPreconnected.intermediate_value\u2082_eventually\u2082" + - ContinuousOn.surjOn_uIcc + - IsClosed.Icc_subset_of_forall_mem_nhdsWithin + - ContinuousOn.strictMonoOn_of_injOn_Ioo + - IsClosed.Icc_subset_of_forall_exists_gt + - "intermediate_value_univ\u2082" + - IsClosed.mem_of_ge_of_forall_exists_gt + - Continuous.strictMono_of_inj + - Continuous.strictMono_of_inj_boundedOrder + - isPreconnected_Icc + - setOf_isPreconnected_subset_of_ordered + - intermediate_value_uIcc + - "IsPreconnected.intermediate_value\u2082_eventually\u2081" + - ContinuousOn.strictMonoOn_of_injOn_Icc + - setOf_isPreconnected_eq_of_ordered + - isPreconnected_Icc_aux + - path: Mathlib/GroupTheory/Submonoid/Basic.lean + theorems: + - Submonoid.iSup_eq_closure + - Submonoid.closure_singleton_le_iff_mem + - IsUnit.mem_submonoid_iff + - Submonoid.coe_iInf + - Submonoid.mem_iInf + - Submonoid.disjoint_def + - pow_mem + - Submonoid.mem_iSup + - Submonoid.subsingleton_iff + - Submonoid.sup_eq_closure + - Submonoid.dense_induction + - Submonoid.closure_induction' + - path: Mathlib/SetTheory/ZFC/Ordinal.lean + theorems: + - ZFSet.IsTransitive.inter + - ZFSet.IsTransitive.union + - ZFSet.IsTransitive.powerset + - ZFSet.isTransitive_iff_sUnion_subset + - ZFSet.IsTransitive.sUnion + - ZFSet.IsTransitive.sUnion' + - path: Mathlib/Algebra/BigOperators/Multiset/Lemmas.lean + theorems: + - Multiset.prod_eq_one_iff + - CanonicallyOrderedCommSemiring.multiset_prod_pos + - Commute.multiset_sum_right + - Multiset.dvd_prod + - path: Mathlib/RingTheory/Trace.lean + theorems: + - Algebra.traceForm_toMatrix_powerBasis + - Algebra.traceMatrix_eq_embeddingsMatrixReindex_mul_trans + - Algebra.trace_eq_of_algEquiv + - PowerBasis.trace_gen_eq_nextCoeff_minpoly + - IntermediateField.AdjoinSimple.trace_gen_eq_sum_roots + - Algebra.trace_eq_matrix_trace + - Algebra.traceMatrix_reindex + - sum_embeddings_eq_finrank_mul + - Algebra.trace_trace_of_basis + - det_traceForm_ne_zero + - Algebra.trace_ne_zero + - trace_eq_sum_embeddings + - trace_eq_sum_embeddings_gen + - Algebra.trace_surjective + - Algebra.trace_eq_of_equiv_equiv + - Algebra.traceMatrix_of_matrix_vecMul + - Algebra.traceMatrix_of_basis + - Algebra.traceMatrix_eq_embeddingsMatrix_mul_trans + - Algebra.trace_algebraMap_of_basis + - Algebra.trace_algebraMap + - Algebra.trace_prod + - trace_eq_trace_adjoin + - Algebra.embeddingsMatrixReindex_eq_vandermonde + - Algebra.isIntegral_trace + - Algebra.trace_prod_apply + - trace_eq_sum_automorphisms + - IntermediateField.AdjoinSimple.trace_gen_eq_zero + - Algebra.traceForm_toMatrix + - Algebra.trace_comp_trace + - Algebra.traceMatrix_of_matrix_mulVec + - Algebra.traceMatrix_of_basis_mulVec + - PowerBasis.trace_gen_eq_sum_roots + - traceForm_dualBasis_powerBasis_eq + - trace_eq_sum_roots + - Algebra.trace_comp_trace_of_basis + - Algebra.trace_eq_of_ringEquiv + - det_traceMatrix_ne_zero' + - path: Mathlib/NumberTheory/Dioph.lean + theorems: + - Dioph.inject_dummies_lem + - Poly.induction + - Dioph.diophPFun_comp1 + - Dioph.div_dioph + - Dioph.union + - Dioph.DiophList.forall + - Dioph.reindex_dioph + - IsPoly.neg + - Dioph.abs_poly_dioph + - Poly.sumsq_eq_zero + - Dioph.pell_dioph + - Dioph.vec_ex1_dioph + - Dioph.pow_dioph + - Dioph.ex1_dioph + - IsPoly.add + - Dioph.reindex_diophFn + - Dioph.diophFn_comp + - Dioph.diophFn_vec_comp1 + - Dioph.diophFn_iff_pFun + - Dioph.ex_dioph + - Poly.sumsq_nonneg + - Dioph.diophFn_compn + - Dioph.ne_dioph + - Dioph.eq_dioph + - Dioph.sub_dioph + - path: Mathlib/RingTheory/Ideal/Quotient.lean + theorems: + - Ideal.Quotient.eq_zero_iff_dvd + - Ideal.Quotient.factor_comp_mk + - Ideal.Quotient.isDomain_iff_prime + - Ideal.Quotient.lift_surjective_of_surjective + - Ideal.Quotient.quotient_ring_saturate + - Ideal.Quotient.subsingleton_iff + - Ideal.quotEquivOfEq_symm + - Ideal.Quotient.exists_inv + - Ideal.Quotient.mk_eq_mk_iff_sub_mem + - Ideal.map_pi + - Ideal.Quotient.maximal_of_isField + - path: Mathlib/Topology/NoetherianSpace.lean + theorems: + - TopologicalSpace.NoetherianSpace.isCompact + - TopologicalSpace.noetherianSpace_set_iff + - TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible + - TopologicalSpace.NoetherianSpace.finite_irreducibleComponents + - TopologicalSpace.noetherianSpace_iff_opens + - TopologicalSpace.noetherianSpace_TFAE + - TopologicalSpace.NoetherianSpace.iUnion + - TopologicalSpace.NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent + - TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducible + - path: Mathlib/Algebra/Star/Basic.lean + theorems: + - star_ne_zero + - star_bit0 + - star_star_mul + - eq_star_of_eq_star + - Ring.inverse_star + - commute_star_comm + - star_bit1 + - star_div' + - star_mul_star + - semiconjBy_star_star_star + - star_invOf + - path: Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean + theorems: + - Asymptotics.IsEquivalent.neg + - Asymptotics.IsEquivalent.add_isLittleO + - Asymptotics.IsEquivalent.isBigO_symm + - Asymptotics.IsEquivalent.tendsto_atBot + - Asymptotics.isEquivalent_iff_exists_eq_mul + - Asymptotics.IsEquivalent.div + - Asymptotics.isEquivalent_const_iff_tendsto + - Asymptotics.IsEquivalent.inv + - Asymptotics.IsEquivalent.tendsto_const + - Asymptotics.isEquivalent_of_tendsto_one + - Asymptotics.isEquivalent_zero_iff_eventually_zero + - Asymptotics.IsEquivalent.tendsto_nhds + - Asymptotics.IsEquivalent.smul + - Asymptotics.IsEquivalent.sub_isLittleO + - Asymptotics.isEquivalent_iff_tendsto_one + - Asymptotics.IsEquivalent.refl + - Asymptotics.isEquivalent_zero_iff_isBigO_zero + - path: Mathlib/GroupTheory/CoprodI.lean + theorems: + - Monoid.CoprodI.NeWord.mulHead_prod + - Monoid.CoprodI.Word.rcons_eq_smul + - Monoid.CoprodI.lift_injective_of_ping_pong + - FreeGroup.injective_lift_of_ping_pong + - Monoid.CoprodI.Word.equivPair_head + - Monoid.CoprodI.lift_word_ping_pong + - Monoid.CoprodI.lift_word_prod_nontrivial_of_head_eq_last + - Monoid.CoprodI.Word.mem_smul_iff + - Monoid.CoprodI.empty_of_word_prod_eq_one + - Monoid.CoprodI.mrange_eq_iSup + - Monoid.CoprodI.NeWord.replaceHead_head + - Monoid.CoprodI.Word.mem_equivPair_tail_iff + - Monoid.CoprodI.Word.cons_eq_smul + - Monoid.CoprodI.NeWord.mulHead_head + - Monoid.CoprodI.Word.fstIdx_cons + - Monoid.CoprodI.Word.prod_smul + - Monoid.CoprodI.Word.mem_rcons_iff + - Monoid.CoprodI.Word.prod_cons + - Monoid.CoprodI.Word.fstIdx_ne_iff + - Monoid.CoprodI.Word.rcons_inj + - Monoid.CoprodI.NeWord.toList_ne_nil + - Monoid.CoprodI.NeWord.inv_last + - Monoid.CoprodI.NeWord.prod_singleton + - Monoid.CoprodI.Word.prod_rcons + - Monoid.CoprodI.NeWord.inv_prod + - Monoid.CoprodI.ext_hom + - Monoid.CoprodI.of_leftInverse + - Monoid.CoprodI.Word.smul_induction + - Monoid.CoprodI.of_injective + - Monoid.CoprodI.NeWord.append_prod + - Monoid.CoprodI.Word.mem_smul_iff_of_ne + - Monoid.CoprodI.Word.equivPair_smul_same + - Monoid.CoprodI.range_eq_iSup + - Monoid.CoprodI.NeWord.toList_head? + - Monoid.CoprodI.induction_on + - Monoid.CoprodI.lift_of + - Monoid.CoprodI.Word.mem_of_mem_equivPair_tail + - Monoid.CoprodI.lift_mrange_le + - Monoid.CoprodI.Word.equivPair_head_smul_equivPair_tail + - Monoid.CoprodI.NeWord.toList_getLast? + - Monoid.CoprodI.lift_range_le + - Monoid.CoprodI.lift_word_prod_nontrivial_of_other_i + - Monoid.CoprodI.NeWord.inv_head + - Monoid.CoprodI.lift_word_prod_nontrivial_of_head_card + - Monoid.CoprodI.NeWord.of_word + - path: Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean + theorems: + - CircleDeg1Lift.translationNumber_units_inv + - CircleDeg1Lift.coe_pow + - CircleDeg1Lift.iterate_le_of_map_le_add_int + - CircleDeg1Lift.translationNumber_zpow + - CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq + - CircleDeg1Lift.commute_add_nat + - CircleDeg1Lift.map_one_add + - CircleDeg1Lift.iterate_pos_eq_iff + - CircleDeg1Lift.le_iterate_of_add_int_le_map + - CircleDeg1Lift.lt_translationNumber_of_forall_add_lt + - CircleDeg1Lift.translationNumber_lt_of_forall_lt_add + - CircleDeg1Lift.le_translationNumber_of_add_int_le + - CircleDeg1Lift.map_fract_sub_fract_eq + - CircleDeg1Lift.transnumAuxSeq_dist_lt + - CircleDeg1Lift.tendsto_translationNumber + - CircleDeg1Lift.commute_sub_int + - CircleDeg1Lift.translate_iterate + - CircleDeg1Lift.tendsto_translationNumber_of_dist_bounded_aux + - CircleDeg1Lift.continuous_pow + - CircleDeg1Lift.map_lt_add_translationNumber_add_one + - CircleDeg1Lift.commute_add_int + - CircleDeg1Lift.dist_map_map_zero_lt + - CircleDeg1Lift.commute_nat_add + - CircleDeg1Lift.map_int_of_map_zero + - CircleDeg1Lift.translationNumber_mono + - CircleDeg1Lift.iterate_pos_lt_iff + - CircleDeg1Lift.le_iterate_pos_iff + - CircleDeg1Lift.commute_int_add + - CircleDeg1Lift.units_inv_apply_apply + - CircleDeg1Lift.translationNumber_eq_rat_iff + - CircleDeg1Lift.isUnit_iff_bijective + - CircleDeg1Lift.iterate_pos_le_iff + - CircleDeg1Lift.transnumAuxSeq_zero + - "CircleDeg1Lift.tendsto_translation_number\u2080'" + - CircleDeg1Lift.commute_sub_nat + - CircleDeg1Lift.semiconj_of_isUnit_of_translationNumber_eq + - CircleDeg1Lift.tendsto_atTop + - CircleDeg1Lift.units_semiconj_of_translationNumber_eq + - CircleDeg1Lift.translationNumber_of_map_pow_eq_add_int + - CircleDeg1Lift.translationNumber_eq_int_iff + - CircleDeg1Lift.map_lt_add_floor_translationNumber_add_one + - CircleDeg1Lift.translationNumber_translate + - CircleDeg1Lift.translationNumber_pow + - CircleDeg1Lift.dist_map_zero_lt_of_semiconj + - CircleDeg1Lift.pow_mono + - CircleDeg1Lift.mul_floor_map_zero_le_floor_iterate_zero + - CircleDeg1Lift.units_apply_inv_apply + - CircleDeg1Lift.iterate_eq_of_map_eq_add_int + - CircleDeg1Lift.exists_eq_add_translationNumber + - CircleDeg1Lift.translate_zpow + - CircleDeg1Lift.lt_iterate_pos_iff + - path: Mathlib/Data/Set/Sups.lean + theorems: + - Set.image_inf_prod + - upperClosure_sups + - Set.sep_sups_le + - lowerClosure_infs + - Set.sep_infs_le + - path: Mathlib/Analysis/SpecificLimits/Normed.lean + theorems: + - NormedField.continuousAt_inv + - geom_series_mul_neg + - NormedRing.tsum_geometric_of_norm_lt_one + - summable_of_ratio_test_tendsto_lt_one + - mul_neg_geom_series + - Antitone.cauchySeq_alternating_series_of_tendsto_zero + - Monotone.alternating_series_le_tendsto + - summable_powerSeries_of_norm_lt + - summable_of_ratio_norm_eventually_le + - hasSum_coe_mul_geometric_of_norm_lt_one + - summable_of_absolute_convergence_real + - tendsto_pow_atTop_nhds_zero_of_norm_lt_one + - summable_geometric_iff_norm_lt_one + - Monotone.tendsto_le_alternating_series + - cauchy_series_of_le_geometric + - SeminormedAddCommGroup.cauchySeq_of_le_geometric + - summable_norm_pow_mul_geometric_of_norm_lt_one + - not_summable_of_ratio_norm_eventually_ge + - norm_sub_le_of_geometric_bound_of_hasSum + - isLittleO_pow_pow_of_abs_lt_left + - NormedField.tendsto_norm_zpow_nhdsWithin_0_atTop + - hasSum_geometric_of_norm_lt_one + - TFAE_exists_lt_isLittleO_pow + - isLittleO_coe_const_pow_of_one_lt + - norm_sum_neg_one_pow_le + - Monotone.cauchySeq_alternating_series_of_tendsto_zero + - Antitone.alternating_series_le_tendsto + - dist_partial_sum + - dist_partial_sum' + - Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded + - isLittleO_pow_const_const_pow_of_one_lt + - not_summable_of_ratio_test_tendsto_gt_one + - summable_powerSeries_of_norm_lt_one + - tendsto_self_mul_const_pow_of_lt_one + - Antitone.tendsto_le_alternating_series + - NormedField.continuousAt_zpow + - tendsto_self_mul_const_pow_of_abs_lt_one + - Antitone.cauchySeq_series_mul_of_tendsto_zero_of_bounded + - NormedAddCommGroup.cauchy_series_of_le_geometric'' + - NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded + - Real.summable_pow_div_factorial + - exists_norm_le_of_cauchySeq + - path: Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean + theorems: + - Gamma0_det + - Gamma_zero_bot + - Gamma1_mem + - isCongruenceSubgroup_trans + - Gamma1_to_Gamma0_mem + - Gamma_mem + - Gamma1_is_congruence + - Gamma_cong_eq_self + - conj_cong_is_cong + - Gamma_one_top + - path: Mathlib/Analysis/Complex/PhragmenLindelof.lean + theorems: + - PhragmenLindelof.quadrant_III + - PhragmenLindelof.quadrant_II + - PhragmenLindelof.eq_zero_on_right_half_plane_of_superexponential_decay + - PhragmenLindelof.eqOn_right_half_plane_of_superexponential_decay + - PhragmenLindelof.quadrant_IV + - PhragmenLindelof.vertical_strip + - PhragmenLindelof.right_half_plane_of_bounded_on_real + - PhragmenLindelof.isBigO_sub_exp_rpow + - PhragmenLindelof.quadrant_I + - PhragmenLindelof.isBigO_sub_exp_exp + - path: Mathlib/Data/Complex/Module.lean + theorems: + - imaginaryPart_ofReal + - star_mul_self_add_self_mul_star + - realPart_smul + - Complex.finrank_real_complex + - realPart_I_smul + - Complex.coe_realPart + - imaginaryPart_smul + - Real.rank_rat_real + - IsSelfAdjoint.imaginaryPart + - imaginaryPart_surjective + - realPart_apply_coe + - imaginaryPart_comp_subtype_selfAdjoint + - realPart_ofReal + - span_selfAdjoint + - Complex.rank_real_complex + - skewAdjoint.I_smul_neg_I + - Complex.coe_basisOneI + - finrank_real_of_complex + - Complex.rank_rat_complex + - skewAdjointPart_eq_I_smul_imaginaryPart + - Complex.liftAux_apply_I + - Complex.algHom_ext + - realPart_add_I_smul_imaginaryPart + - Complex.real_algHom_eq_id_or_conj + - imaginaryPart_apply_coe + - Complex.coe_selfAdjointEquiv + - Complex.rank_real_complex' + - Complex.nonempty_linearEquiv_real + - Complex.toMatrix_conjAe + - imaginaryPart_I_smul + - path: Mathlib/Algebra/Homology/ShortComplex/Homology.lean + theorems: + - "CategoryTheory.ShortComplex.liftCycles_homology\u03C0_eq_zero_of_boundary" + - CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_descH + - CategoryTheory.ShortComplex.homologyMap_op + - CategoryTheory.ShortComplex.leftRightHomologyComparison'_naturality + - CategoryTheory.ShortComplex.LeftHomologyData.homologyIso_leftHomologyData + - CategoryTheory.ShortComplex.HomologyMapData.comm + - CategoryTheory.ShortComplex.leftRightHomologyComparison'_compatibility + - "CategoryTheory.ShortComplex.\u03C0_homologyMap_\u03B9" + - "CategoryTheory.ShortComplex.homology\u03B9_comp_fromOpcycles" + - CategoryTheory.ShortComplex.epi_homologyMap_of_epi_cyclesMap' + - CategoryTheory.ShortComplex.leftHomologyIso_inv_naturality + - "CategoryTheory.ShortComplex.isIso_homology\u03C0" + - "CategoryTheory.ShortComplex.homology\u03C0_comp_leftHomologyIso_inv" + - "CategoryTheory.ShortComplex.toCycles_comp_homology\u03C0" + - CategoryTheory.ShortComplex.hasHomology_of_isIsoLeftRightHomologyComparison + - CategoryTheory.ShortComplex.leftRightHomologyComparison'_fac + - CategoryTheory.ShortComplex.HomologyData.right_homologyIso_eq_left_homologyIso_trans_iso + - CategoryTheory.ShortComplex.homologyMap'_op + - CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_hom_comp_homologyIso_inv + - "CategoryTheory.ShortComplex.RightHomologyData.homologyIso_inv_comp_homology\u03B9" + - CategoryTheory.ShortComplex.RightHomologyData.homologyIso_hom_comp_rightHomologyIso_inv + - CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap' + - CategoryTheory.ShortComplex.RightHomologyMapData.homologyMap_eq + - CategoryTheory.ShortComplex.isIso_homologyMap_of_epi_of_isIso_of_mono' + - CategoryTheory.ShortComplex.leftRightHomologyComparison_fac + - "CategoryTheory.ShortComplex.LeftHomologyData.\u03C0_comp_homologyIso_inv" + - "CategoryTheory.ShortComplex.isIso_homology\u03B9" + - CategoryTheory.ShortComplex.RightHomologyData.homologyIso_rightHomologyData + - "CategoryTheory.ShortComplex.\u03C0_leftRightHomologyComparison'_\u03B9" + - CategoryTheory.ShortComplex.RightHomologyMapData.homologyMap_comm + - CategoryTheory.ShortComplex.HomologyData.leftRightHomologyComparison'_eq + - CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_hom_naturality + - CategoryTheory.ShortComplex.LeftHomologyData.leftHomologyIso_inv_naturality + - "CategoryTheory.ShortComplex.LeftHomologyData.homology\u03C0_comp_homologyIso_hom" + - CategoryTheory.ShortComplex.LeftHomologyMapData.homologyMap_comm + - "CategoryTheory.ShortComplex.rightHomologyIso_hom_comp_homology\u03B9" + - CategoryTheory.ShortComplex.LeftHomologyData.homologyIso_hom_comp_leftHomologyIso_inv + - CategoryTheory.ShortComplex.RightHomologyData.rightHomologyIso_inv_naturality + - CategoryTheory.ShortComplex.comp_homologyMap_comp + - CategoryTheory.ShortComplex.isIso_homologyMap_of_isIso_cyclesMap_of_epi + - CategoryTheory.ShortComplex.LeftHomologyData.leftHomologyIso_hom_comp_homologyIso_inv + - CategoryTheory.ShortComplex.isIso_homologyMap_of_isIso_opcyclesMap_of_mono + - CategoryTheory.ShortComplex.rightHomologyIso_inv_naturality + - CategoryTheory.ShortComplex.LeftHomologyMapData.homologyMap_eq + - CategoryTheory.ShortComplex.rightHomologyIso_hom_naturality + - CategoryTheory.ShortComplex.LeftHomologyData.leftHomologyIso_hom_naturality + - "CategoryTheory.ShortComplex.HomologyMapData.congr_left_\u03C6H" + - "CategoryTheory.ShortComplex.homology\u03C0_naturality" + - CategoryTheory.ShortComplex.leftHomologyIso_hom_naturality + - "CategoryTheory.ShortComplex.homology\u03B9_naturality" + - "CategoryTheory.ShortComplex.homology\u03B9_descOpcycles_eq_zero_of_boundary" + - CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_liftH + - "CategoryTheory.ShortComplex.homology_\u03C0_\u03B9" + - CategoryTheory.ShortComplex.mono_homologyMap_of_mono_opcyclesMap' + - "CategoryTheory.ShortComplex.RightHomologyData.homologyIso_hom_comp_\u03B9" + - path: .lake/packages/std/Std/Data/List/Basic.lean + theorems: + - List.intercalate_eq_intercalateTR + - List.inits_eq_initsTR + - List.set_eq_setTR + - List.scanl_eq_scanlTR + - List.takeDTR_go_eq + - List.erase_eq_eraseTR + - List.leftpad_eq_leftpadTR + - List.sections_eq_sectionsTR + - List.replaceF_eq_replaceFTR + - List.eraseIdx_eq_eraseIdxTR + - "List.dropSlice_zero\u2082" + - List.takeD_nil + - List.scanlTR_go_eq + - List.range'_eq_range'TR + - List.modifyNthTR_go_eq + - List.intersperse_eq_intersperseTR + - "List.takeWhile\u2082_eq_takeWhile\u2082TR" + - List.takeList_eq_takeListTR + - List.modifyNth_eq_modifyNthTR + - List.removeNth_eq_removeNthTR + - List.bind_eq_bindTR + - List.zipWithLeft'_eq_zipWithLeft'TR + - List.sections_eq_nil_of_isEmpty + - List.fillNones_eq_fillNonesTR + - List.headD_eq_head? + - List.insertNth_eq_insertNthTR + - List.join_eq_joinTR + - List.takeD_succ + - List.foldr_eq_foldrTR + - List.unzip_eq_unzipTR + - List.product_eq_productTR + - List.insertNthTR_go_eq + - List.dropSlice_eq_dropSliceTR + - List.zipWith_eq_zipWithTR + - List.tails_eq_tailsTR + - List.takeD_eq_takeDTR + - List.dropLast_eq_dropLastTR + - List.takeWhile_eq_takeWhileTR + - List.replicateTR_loop_eq + - List.sigma_eq_sigmaTR + - List.zipWithLeft_eq_zipWithLeftTR + - List.filterMap_eq_filterMapTR + - List.enumFrom_eq_enumFromTR + - List.eraseP_eq_erasePTR + - List.replace_eq_replaceTR + - List.splitOnP_eq_splitOnPTR + - List.take_eq_takeTR + - path: Mathlib/CategoryTheory/Products/Bifunctor.lean + theorems: + - CategoryTheory.Bifunctor.diagonal + - CategoryTheory.Bifunctor.map_comp_id + - CategoryTheory.Bifunctor.map_id_comp + - path: Mathlib/Algebra/Group/Semiconj/Units.lean + theorems: + - Units.mk_semiconjBy + - SemiconjBy.units_inv_symm_left + - SemiconjBy.units_inv_right + - SemiconjBy.units_zpow_right + - path: Mathlib/Data/Sign.lean + theorems: + - sign_mul_abs + - SignType.lt_one_iff + - SignType.self_eq_neg_iff + - SignType.nonpos_iff + - sign_nonneg_iff + - SignType.le_antisymm + - SignType.nonneg_iff + - SignType.coe_neg + - sign_neg + - Right.sign_neg + - exists_signed_sum' + - SignType.neg_one_lt_iff + - SignType.neg_eq_self_iff + - sign_nonpos_iff + - Left.sign_neg + - SignType.map_cast' + - sign_sum + - exists_signed_sum_aux + - SignType.mul_comm + - SignType.univ_eq + - Int.sign_eq_sign + - SignType.nonneg_iff_ne_neg_one + - sign_pos + - sign_mul + - exists_signed_sum + - sign_eq_one_iff + - sign_zero + - sign_eq_neg_one_iff + - StrictMono.sign_comp + - sign_eq_zero_iff + - SignType.range_eq + - SignType.map_cast + - SignType.neg_iff + - path: Mathlib/FieldTheory/IsPerfectClosure.lean + theorems: + - IsPRadical.comap_pNilradical + - PerfectRing.liftAux_self_apply + - sub_mem_pNilradical_iff_pow_expChar_pow_eq + - PerfectRing.lift_comp_lift + - pNilradical_le_nilradical + - IsPerfectClosure.equiv_symm_apply + - IsPRadical.injective_comp_of_pNilradical_eq_bot + - pNilradical_eq_bot + - PerfectRing.lift_comp_lift_apply_eq_self + - pNilradical_eq_nilradical + - pNilradical_eq_bot_of_frobenius_inj + - PerfectRing.lift_lift + - PerfectRing.lift_comp_apply + - RingHom.pNilradical_le_ker_of_perfectRing + - PerfectRing.lift_aux + - IsPerfectClosure.equiv_comp_equiv_apply_eq_self + - PerfectRing.liftAux_id_apply + - mem_pNilradical + - IsPRadical.trans + - PerfectRing.liftAux_apply + - path: Mathlib/Algebra/Group/Commute/Units.lean + theorems: + - Commute.div_eq_div_iff_of_isUnit + - Units.pow_ofPowEqOne + - path: Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean + theorems: + - List.Nat.mem_antidiagonalTuple + - List.Nat.antidiagonalTuple_one + - List.Nat.antidiagonalTuple_zero_right + - List.Nat.antidiagonalTuple_two + - List.Nat.antidiagonalTuple_pairwise_pi_lex + - List.Nat.nodup_antidiagonalTuple + - path: Mathlib/Algebra/Lie/IdealOperations.lean + theorems: + - LieSubmodule.inf_lie + - LieSubmodule.map_comap_incl + - LieSubmodule.lie_le_inf + - LieSubmodule.lie_le_iff + - LieIdeal.comap_bracket_incl_of_le + - LieSubmodule.comap_bracket_eq + - LieIdeal.map_comap_bracket_eq + - LieSubmodule.sup_lie + - LieSubmodule.lie_le_left + - LieSubmodule.bot_lie + - LieIdeal.map_comap_incl + - LieSubmodule.lie_inf + - LieSubmodule.lie_comm + - LieSubmodule.lieIdeal_oper_eq_linear_span' + - LieSubmodule.map_comap_eq + - LieSubmodule.lie_coe_mem_lie + - LieIdeal.comap_bracket_incl + - LieIdeal.map_bracket_eq + - LieIdeal.map_bracket_le + - LieSubmodule.comap_map_eq + - LieIdeal.comap_bracket_le + - LieSubmodule.lie_eq_bot_iff + - LieSubmodule.lieIdeal_oper_eq_linear_span + - LieIdeal.comap_bracket_eq + - LieSubmodule.map_bracket_eq + - LieSubmodule.mono_lie + - LieSubmodule.lie_bot + - path: Mathlib/Control/LawfulFix.lean + theorems: + - Part.Fix.mem_iff + - "Part.fix_eq_\u03C9Sup" + - Part.Fix.exists_fix_le_approx + - Part.fix_le + - Part.Fix.approx_mono' + - Pi.continuous_uncurry + - Part.to_unit_cont + - Part.Fix.approx_le_fix + - Part.Fix.le_f_of_mem_approx + - Part.Fix.approx_mono + - Part.fix_eq + - Pi.continuous_curry + - path: Mathlib/RingTheory/WittVector/MulCoeff.lean + theorems: + - WittVector.peval_polyOfInterest + - WittVector.remainder_vars + - WittVector.mul_polyOfInterest_aux4 + - WittVector.peval_polyOfInterest' + - WittVector.wittPolyProd_vars + - WittVector.polyOfInterest_vars_eq + - WittVector.nth_mul_coeff + - WittVector.mul_polyOfInterest_aux1 + - WittVector.wittPolyProdRemainder_vars + - WittVector.mul_polyOfInterest_aux5 + - WittVector.mul_polyOfInterest_aux2 + - WittVector.polyOfInterest_vars + - WittVector.mul_polyOfInterest_vars + - WittVector.nth_mul_coeff' + - WittVector.mul_polyOfInterest_aux3 + - path: Mathlib/Topology/Algebra/MulAction.lean + theorems: + - continuousSMul_inf + - continuousSMul_sInf + - AddTorsor.connectedSpace + - IsCompact.smul_set + - MulAction.continuousSMul_compHom + - Inducing.continuousSMul + - path: Mathlib/LinearAlgebra/Dimension/Finrank.lean + theorems: + - FiniteDimensional.lt_rank_of_lt_finrank + - FiniteDimensional.finrank_eq_of_rank_eq + - FiniteDimensional.finrank_le_of_rank_le + - LinearEquiv.finrank_eq + - finrank_top + - LinearMap.finrank_range_of_inj + - FiniteDimensional.one_lt_rank_of_one_lt_finrank + - FiniteDimensional.finrank_le_finrank_of_rank_le_rank + - FiniteDimensional.finrank_lt_of_rank_lt + - path: Mathlib/Algebra/Category/FGModuleCat/Basic.lean + theorems: + - FGModuleCat.coevaluation_evaluation + - FGModuleCat.evaluation_coevaluation + - path: Mathlib/CategoryTheory/Triangulated/Basic.lean + theorems: + - "CategoryTheory.Pretriangulated.Triangle.eqToHom_hom\u2081" + - CategoryTheory.Pretriangulated.Triangle.isIso_of_isIsos + - "CategoryTheory.Iso.hom_inv_id_triangle_hom\u2082" + - "CategoryTheory.Iso.hom_inv_id_triangle_hom\u2083" + - "CategoryTheory.Iso.hom_inv_id_triangle_hom\u2081" + - "CategoryTheory.Iso.inv_hom_id_triangle_hom\u2083" + - "CategoryTheory.Iso.inv_hom_id_triangle_hom\u2081" + - "CategoryTheory.Iso.inv_hom_id_triangle_hom\u2082" + - "CategoryTheory.Pretriangulated.productTriangle.zero\u2083\u2081" + - "CategoryTheory.Pretriangulated.Triangle.eqToHom_hom\u2083" + - "CategoryTheory.Pretriangulated.Triangle.eqToHom_hom\u2082" + - path: Mathlib/Data/Polynomial/Div.lean + theorems: + - Polynomial.self_mul_modByMonic + - Polynomial.modByMonic_one + - Polynomial.degree_divByMonic_le + - Polynomial.div_modByMonic_unique + - Polynomial.rootMultiplicity_eq_zero_iff + - Polynomial.degree_add_divByMonic + - Polynomial.zero_divByMonic + - Polynomial.modByMonic_X_sub_C_eq_C_eval + - Polynomial.modByMonic_zero + - Polynomial.sub_dvd_eval_sub + - Polynomial.coeff_divByMonic_X_sub_C_rec + - Polynomial.degree_modByMonic_le + - Polynomial.divByMonic_eq_zero_iff + - Polynomial.natDegree_modByMonic_lt + - Polynomial.multiplicity_X_sub_C_finite + - Polynomial.degree_divByMonic_lt + - Polynomial.div_wf_lemma + - Polynomial.degree_modByMonic_lt + - Polynomial.ker_evalRingHom + - Polynomial.sum_modByMonic_coeff + - "Polynomial.eval\u2082_modByMonic_eq_self_of_root" + - Polynomial.X_pow_dvd_iff + - Polynomial.pow_mul_divByMonic_rootMultiplicity_eq + - Polynomial.eval_divByMonic_pow_rootMultiplicity_ne_zero + - Polynomial.dvd_iff_isRoot + - Polynomial.pow_rootMultiplicity_dvd + - Polynomial.map_dvd_map + - Polynomial.mul_divByMonic_eq_iff_isRoot + - Polynomial.rootMultiplicity_C + - Polynomial.not_isField + - Polynomial.X_dvd_iff + - Polynomial.divByMonic_one + - Polynomial.natDegree_divByMonic + - Polynomial.X_sub_C_dvd_sub_C_eval + - Polynomial.rootMultiplicity_eq_nat_find_of_nonzero + - Polynomial.divByMonic_zero + - Polynomial.rootMultiplicity_eq_multiplicity + - Polynomial.X_dvd_sub_C + - Polynomial.mem_span_C_X_sub_C_X_sub_C_iff_eval_eval_eq_zero + - Polynomial.coeff_divByMonic_X_sub_C + - Polynomial.modByMonic_eq_sub_mul_div + - Polynomial.modByMonic_eq_self_iff + - Polynomial.zero_modByMonic + - Polynomial.modByMonic_eq_zero_iff_dvd + - Polynomial.map_mod_divByMonic + - Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd + - Polynomial.modByMonic_X + - Polynomial.mul_self_modByMonic + - Polynomial.rootMultiplicity_pos' + - Polynomial.multiplicity_finite_of_degree_pos_of_monic + - Polynomial.mul_div_mod_by_monic_cancel_left + - path: Mathlib/Geometry/Manifold/ContMDiff/Defs.lean + theorems: + - contMDiffAt_iff_contMDiffOn_nhds + - contMDiffOn_iff_target + - contMDiffAt_iff + - contMDiffAt_iff_of_mem_source + - contMDiffWithinAt_iff_contMDiffOn_nhds + - contDiffWithinAt_localInvariantProp + - contDiffWithinAtProp_self_source + - contMDiffWithinAt_iff_target + - contDiffWithinAtProp_id + - contMDiffOn_iff_source_of_mem_maximalAtlas + - contMDiffAt_iff_target + - contMDiffWithinAt_insert_self + - contMDiffAt_iff_source_of_mem_source + - contMDiff_iff + - contMDiffOn_iff + - contMDiffAt_iff_contMDiffAt_nhds + - contMDiffOn_univ + - contMDiffWithinAt_iff_source_of_mem_maximalAtlas + - contMDiffAt_iff_target_of_mem_source + - contDiffWithinAtProp_mono_of_mem + - contMDiffOn_iff_of_mem_maximalAtlas + - contMDiffWithinAt_iff_of_mem_source' + - contMDiffOn_iff_of_subset_source' + - contMDiffWithinAt_iff_target_of_mem_source + - contMDiff_iff_target + - contMDiffWithinAt_iff_nat + - contMDiffWithinAt_iff_image + - path: Mathlib/Order/Interval.lean + theorems: + - Interval.dual_map + - Interval.mem_pure + - NonemptyInterval.mem_pure + - Interval.coe_iInf + - NonemptyInterval.coe_eq_pure + - Interval.subset_coe_map + - Interval.coe_dual + - Interval.disjoint_coe + - Interval.coe_sInf + - "Interval.coe_iInf\u2082" + - NonemptyInterval.toProd_injective + - Interval.coe_inf + - path: Mathlib/Topology/Category/TopCat/EpiMono.lean + theorems: + - TopCat.mono_iff_injective + - TopCat.epi_iff_surjective + - path: Mathlib/Analysis/Calculus/Deriv/Comp.lean + theorems: + - HasStrictDerivAt.comp + - HasStrictDerivAt.iterate + - HasDerivAtFilter.scomp + - HasDerivAtFilter.iterate + - HasStrictDerivAt.scomp + - HasStrictDerivAt.comp_hasStrictFDerivAt + - HasDerivAtFilter.comp_hasFDerivAtFilter + - HasDerivAtFilter.comp + - HasDerivWithinAt.iterate + - HasDerivWithinAt.comp + - path: Mathlib/Data/List/Sort.lean + theorems: + - List.Sorted.head!_le + - List.perm_mergeSort + - List.mergeSort_singleton + - List.perm_split + - List.Sorted.le_head! + - List.Sorted.insertionSort_eq + - List.mergeSort_nil + - List.sublist_of_subperm_of_sorted + - List.erase_orderedInsert_of_not_mem + - List.Sorted.merge + - List.sublist_orderedInsert + - List.Sorted.rel_of_mem_take_of_mem_drop + - List.split_cons_of_eq + - List.length_split_lt + - List.orderedInsert_length + - List.erase_orderedInsert + - List.Sorted.orderedInsert + - List.sorted_mergeSort + - List.length_split_le + - List.orderedInsert_erase + - List.orderedInsert_eq_take_drop + - List.Sorted.gt_of_ge + - List.Sorted.rel_get_of_le + - List.mem_orderedInsert + - List.orderedInsert_count + - List.mergeSort_cons_cons + - List.perm_insertionSort + - List.eq_of_perm_of_sorted + - List.perm_orderedInsert + - List.sorted_ofFn_iff + - path: Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean + theorems: + - zero_at_infty_of_norm_le + - ZeroAtInftyContinuousMapClass.norm_le + - path: Mathlib/NumberTheory/DirichletCharacter/Basic.lean + theorems: + - DirichletCharacter.factorsThrough_one_iff + - DirichletCharacter.conductor_le_conductor_mem_conductorSet + - DirichletCharacter.Even.toUnitHom_eval_neg_one + - DirichletCharacter.changeLevel_def' + - DirichletCharacter.changeLevel_self_toUnitHom + - DirichletCharacter.toUnitHom_eq_iff + - DirichletCharacter.periodic + - DirichletCharacter.toUnitHom_eq_char' + - DirichletCharacter.changeLevel_self + - DirichletCharacter.primitiveCharacter_isPrimitive + - DirichletCharacter.eval_modulus_sub + - DirichletCharacter.conductor_eq_zero_iff_level_eq_zero + - DirichletCharacter.changeLevel_eq_cast_of_dvd + - DirichletCharacter.changeLevel_one + - DirichletCharacter.conductor_one_dvd + - DirichletCharacter.primitiveCharacter_one + - DirichletCharacter.eq_one_iff_conductor_eq_one + - DirichletCharacter.conductor_one + - DirichletCharacter.Odd.eval_neg + - DirichletCharacter.Odd.toUnitHom_eval_neg_one + - DirichletCharacter.level_one + - DirichletCharacter.level_one' + - DirichletCharacter.even_or_odd + - DirichletCharacter.Even.eval_neg + - path: .lake/packages/std/Std/Data/BinomialHeap/Basic.lean + theorems: + - Std.BinomialHeap.Imp.Heap.realSize_deleteMin + - Std.BinomialHeap.Imp.Heap.WF.deleteMin + - Std.BinomialHeap.Imp.Heap.WF.tail? + - Std.BinomialHeap.Imp.HeapNode.realSize_toHeap + - Std.BinomialHeap.Imp.Heap.realSize_tail? + - Std.BinomialHeap.Imp.Heap.realSize_tail + - Std.BinomialHeap.Imp.Heap.WF.tail + - Std.BinomialHeap.Imp.HeapNode.WF.toHeap + - Std.BinomialHeap.Imp.Heap.WF.merge' + - Std.BinomialHeap.Imp.HeapNode.rankTR_eq + - Std.BinomialHeap.Imp.Heap.realSize_merge + - Std.BinomialHeap.Imp.Heap.WF.singleton + - Std.BinomialHeap.Imp.Heap.realSize_findMin + - path: Mathlib/RingTheory/Valuation/Basic.lean + theorems: + - Valuation.IsEquiv.ne_zero + - Valuation.isEquiv_iff_val_le_one + - Valuation.isEquiv_iff_val_sub_one_lt_one + - Valuation.map_add_supp + - Valuation.map_eq_of_sub_lt + - Valuation.IsEquiv.of_eq + - Valuation.comap_supp + - Valuation.map_one_add_of_lt + - Valuation.map_sum_le + - Valuation.map_add' + - Valuation.isEquiv_iff_val_lt_one + - Valuation.map_add_of_distinct_val + - Valuation.IsEquiv.map + - Valuation.isEquiv_tfae + - Valuation.map_add_eq_of_lt_left + - Valuation.isEquiv_of_val_le_one + - Valuation.map_sum_lt + - AddValuation.map_add' + - Valuation.map_sub + - Valuation.IsEquiv.val_eq + - Valuation.isEquiv_iff_val_eq_one + - path: Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean + theorems: + - "CategoryTheory.MorphismProperty.LeftFraction.comp\u2080_rel" + - CategoryTheory.MorphismProperty.LeftFraction.Localization.StrictUniversalPropertyFixedTarget.fac + - CategoryTheory.MorphismProperty.LeftFractionRel.trans + - CategoryTheory.MorphismProperty.RightFraction.map_ofHom + - CategoryTheory.MorphismProperty.LeftFractionRel.refl + - CategoryTheory.MorphismProperty.LeftFractionRel.symm + - CategoryTheory.MorphismProperty.LeftFraction.map_eq_of_map_eq + - CategoryTheory.MorphismProperty.RightFraction.map_ofInv_hom_id + - CategoryTheory.Localization.exists_leftFraction + - CategoryTheory.MorphismProperty.LeftFraction.Localization.homMk_eq + - CategoryTheory.MorphismProperty.LeftFraction.map_hom_ofInv_id + - CategoryTheory.MorphismProperty.LeftFraction.Localization.homMk_comp_homMk + - CategoryTheory.MorphismProperty.RightFraction.map_hom_ofInv_id + - CategoryTheory.MorphismProperty.LeftFraction.map_comp_map_s + - CategoryTheory.MorphismProperty.RightFraction.map_s_comp_map + - CategoryTheory.MorphismProperty.LeftFraction.map_eq_iff + - CategoryTheory.MorphismProperty.LeftFraction.map_compatibility + - CategoryTheory.MorphismProperty.LeftFraction.Localization.StrictUniversalPropertyFixedTarget.uniq + - CategoryTheory.MorphismProperty.LeftFraction.map_ofHom + - CategoryTheory.MorphismProperty.LeftFraction.Localization.Q_map_comp_Qinv + - CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.mk_surjective + - CategoryTheory.MorphismProperty.map_eq_iff_postcomp + - CategoryTheory.MorphismProperty.LeftFraction.Localization.map_eq_iff + - path: Mathlib/Order/Filter/Pointwise.lean + theorems: + - Filter.isUnit_iff + - Filter.isUnit_iff_singleton + - Filter.mul_top_of_one_le + - Filter.zero_smul_filter + - Filter.one_le_div_iff + - Filter.NeBot.one_le_div + - Filter.pow_mem_pow + - Filter.inv_mem_inv + - Filter.bot_pow + - Filter.map_one + - Filter.mul_eq_one_iff + - Filter.top_pow + - Filter.inv_le_iff_le_inv + - Filter.zero_smul_filter_nonpos + - Filter.top_mul_of_one_le + - path: Mathlib/Algebra/Homology/QuasiIso.lean + theorems: + - quasiIsoAt_of_comp_right + - "HomologicalComplex.Hom.from_single\u2080_exact_f_d_at_zero" + - "ChainComplex.quasiIsoAt\u2080_iff" + - quasiIso_iff_comp_left + - quasiIso_iff_comp_right + - quasiIsoAt_iff_exactAt + - "CochainComplex.quasiIsoAt\u2080_iff" + - quasiIsoAt_iff_comp_right + - HomotopyEquiv.toQuasiIso'_inv + - "HomologicalComplex.Hom.toSingle\u2080CokernelAtZeroIso_hom_eq" + - quasiIsoAt_iff' + - "HomologicalComplex.Hom.fromSingle\u2080KernelAtZeroIso_inv_eq" + - "HomologicalComplex.Hom.to_single\u2080_exact_d_f_at_zero" + - quasiIso_of_comp_left + - quasiIsoAt_of_comp_left + - quasiIsoAt_iff + - HomologicalComplex.quasiIsoAt_map_iff_of_preservesHomology + - CategoryTheory.Functor.quasiIso'_of_map_quasiIso' + - quasiIso_of_arrow_mk_iso + - quasiIsoAt_iff_exactAt' + - "HomologicalComplex.Hom.to_single\u2080_epi_at_zero" + - homotopyEquivalences_subset_quasiIso + - HomologicalComplex.quasiIso_map_iff_of_preservesHomology + - quasiIsoAt_iff_comp_left + - quasiIso_iff_of_arrow_mk_iso + - HomotopyEquiv.toQuasiIso' + - "HomologicalComplex.Hom.from_single\u2080_mono_at_zero" + - quasiIso_of_comp_right + - HomologicalComplex.mem_quasiIso_iff + - quasiIsoAt_iff_isIso_homologyMap + - path: Mathlib/Data/Set/Defs.lean + theorems: + - Set.comp_rangeSplitting + - path: Mathlib/Topology/MetricSpace/Infsep.lean + theorems: + - Set.einfsep_insert_le + - Set.le_einfsep_pair + - Set.einfsep_pair + - Set.infsep_of_fintype + - Set.relatively_discrete_of_finite + - Set.Nontrivial.einfsep_lt_top + - Set.Finite.infsep_of_nontrivial + - Set.einfsep_pos_of_finite + - Set.Finset.coe_einfsep + - Set.einfsep_of_fintype + - Set.Nontrivial.infsep_lt_iff + - Set.einfsep_insert + - Set.Nontrivial.infsep_exists_of_finite + - Set.subsingleton_of_einfsep_eq_top + - Set.infsep_zero_iff_subsingleton_of_finite + - Set.einfsep_pos + - Finset.coe_infsep + - Set.Nontrivial.infsep_eq_iInf + - Set.Nontrivial.einfsep_exists_of_finite + - Set.Nontrivial.infsep_of_fintype + - Set.infsep_zero + - Set.einfsep_eq_iInf + - Set.infsep_triple + - Set.einfsep_pair_le_right + - Set.einfsep_ne_top + - Set.einfsep_lt_iff + - Set.infsep_pair + - Set.infsep_pos_iff_nontrivial_of_finite + - Set.Nontrivial.einfsep_ne_top + - Set.infsep_pos + - Set.Finite.einfsep + - Set.nontrivial_of_einfsep_lt_top + - Set.einfsep_top + - Set.einfsep_iUnion_mem_option + - Set.le_einfsep_pi_of_le + - Set.le_einfsep_iff + - Set.le_edist_of_le_infsep + - Set.le_einfsep_image_iff + - Set.Nontrivial.le_infsep_iff + - Set.einfsep_lt_top + - Finset.coe_infsep_of_offDiag_empty + - Finset.coe_infsep_of_offDiag_nonempty + - Set.Subsingleton.einfsep + - Set.einfsep_triple + - Set.nontrivial_of_infsep_pos + - Set.infsep_eq_iInf + - Set.einfsep_zero + - Set.Finite.infsep + - path: Mathlib/Init/Data/Nat/Bitwise.lean + theorems: + - Nat.binaryRec_decreasing + - Nat.binaryRec_eq + - Nat.bodd_add_div2 + - Nat.shiftLeft'_false + - Nat.bit0_val + - Nat.div2_val + - Nat.bodd_bit + - Nat.shiftLeft_sub + - Nat.bodd_succ + - Nat.div2_succ + - Nat.binaryRec_zero + - Nat.bit_val + - Nat.bodd_mul + - Nat.testBit_bit_zero + - Nat.bodd_eq_and_one_ne_zero + - Nat.shiftLeft_add + - Nat.div2_bit + - Nat.mod_two_of_bodd + - Nat.shiftLeft'_sub + - Nat.testBit_bit_succ + - path: Mathlib/CategoryTheory/Sites/Coverage.lean + theorems: + - CategoryTheory.Presieve.factorsThruAlong_id + - CategoryTheory.Presieve.factorsThru_of_le + - CategoryTheory.Coverage.saturate_of_superset + - CategoryTheory.Presieve.le_of_factorsThru_sieve + - CategoryTheory.Presieve.isSheaf_coverage + - CategoryTheory.Coverage.eq_top_pullback + - CategoryTheory.Presieve.isSheaf_sup + - CategoryTheory.Presieve.isSheafFor_of_factorsThru + - CategoryTheory.Coverage.toGrothendieck_eq_sInf + - path: Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean + theorems: + - BoxIntegral.norm_volume_sub_integral_face_upper_sub_lower_smul_le + - BoxIntegral.hasIntegral_GP_pderiv + - BoxIntegral.hasIntegral_GP_divergence_of_forall_hasDerivWithinAt + - path: Mathlib/Data/List/Sym.lean + theorems: + - List.mem_sym2_cons_iff + - List.mem_sym2_iff + - List.Nodup.sym + - List.mk_mem_sym2 + - List.sym2_eq_nil_iff + - List.Subperm.sym2 + - List.sym2_eq_sym_two + - List.Nodup.sym2 + - List.Perm.sym2 + - List.mem_of_mem_of_mem_sym + - List.left_mem_of_mk_mem_sym2 + - List.Sublist.sym2 + - List.Sublist.sym + - List.length_sym + - List.sym_one_eq + - List.right_mem_of_mk_mem_sym2 + - List.mk_mem_sym2_iff + - List.sym_map + - List.length_sym2 + - path: Mathlib/Topology/UniformSpace/CompareReals.lean + theorems: + - Rat.uniformSpace_eq + - path: Mathlib/Algebra/Lie/Free.lean + theorems: + - FreeLieAlgebra.lift_of_apply + - FreeLieAlgebra.Rel.smulOfTower + - FreeLieAlgebra.Rel.addLeft + - FreeLieAlgebra.Rel.subRight + - FreeLieAlgebra.Rel.neg + - FreeLieAlgebra.liftAux_spec + - FreeLieAlgebra.Rel.subLeft + - FreeLieAlgebra.hom_ext + - FreeLieAlgebra.lift_comp_of + - path: Mathlib/CategoryTheory/Limits/FinallySmall.lean + theorems: + - CategoryTheory.initiallySmall_of_small_weakly_initial_set + - CategoryTheory.initiallySmall_iff_exists_small_weakly_initial_set + - CategoryTheory.InitiallySmall.exists_small_weakly_initial_set + - CategoryTheory.finallySmall_iff_exists_small_weakly_terminal_set + - CategoryTheory.finallySmall_of_small_weakly_terminal_set + - CategoryTheory.FinallySmall.exists_small_weakly_terminal_set + - path: Mathlib/MeasureTheory/Measure/Trim.lean + theorems: + - MeasureTheory.measure_trim_toMeasurable_eq_zero + - MeasureTheory.zero_trim + - MeasureTheory.sigmaFinite_trim_bot_iff + - MeasureTheory.trim_trim + - MeasureTheory.le_trim + - MeasureTheory.sigmaFiniteTrim_mono + - MeasureTheory.restrict_trim + - MeasureTheory.trim_eq_self + - MeasureTheory.toOuterMeasure_trim_eq_trim_toOuterMeasure + - MeasureTheory.trim_measurableSet_eq + - path: Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean + theorems: + - Euclidean.isCompact_closedBall + - Euclidean.closedBall_eq_image + - Euclidean.exists_pos_lt_subset_ball + - Euclidean.nhds_basis_ball + - ContDiff.euclidean_dist + - Euclidean.nhds_basis_closedBall + - path: Mathlib/Data/List/Nodup.lean + theorems: + - List.nodup_append_comm + - List.Nodup.diff_eq_filter + - List.Nodup.sigma + - List.nodup_bind + - List.Nodup.erase_eq_filter + - List.nodup_join + - List.Nodup.insert + - List.nodup_iff_injective_get + - List.Nodup.ne_singleton_iff + - List.Nodup.product + - List.nthLe_eq_of_ne_imp_not_nodup + - List.count_eq_of_nodup + - List.Nodup.mem_diff_iff + - List.nodup_iff_get?_ne_get? + - List.Nodup.map_update + - List.nodup_reverse + - List.Nodup.filterMap + - List.Nodup.pairwise_of_forall_ne + - List.Nodup.concat + - List.nodup_cons + - List.nodup_append + - List.Nodup.mem_erase_iff + - List.Nodup.take_eq_filter_mem + - List.Nodup.nthLe_inj_iff + - List.get_indexOf + - List.Nodup.pmap + - List.nodup_replicate + - List.nodup_iff_sublist + - List.Nodup.filter + - List.Nodup.union + - List.nodup_middle + - List.not_nodup_of_get_eq_of_ne + - List.rel_nodup + - path: Mathlib/Topology/MetricSpace/Dilation.lean + theorems: + - Dilation.ratio_id + - Dilation.ratio_ne_zero + - Dilation.ratio_unique_of_nndist_ne_zero + - Dilation.ratio_of_subsingleton + - Dilation.ratio_unique_of_dist_ne_zero + - Dilation.diam_range + - Dilation.cancel_left + - Dilation.dist_eq + - Dilation.ratio_comp' + - Isometry.toDilation_ratio + - Dilation.ediam_range + - Dilation.ratio_mul + - Dilation.edist_eq + - Dilation.nndist_eq + - Dilation.diam_image + - Dilation.mapsTo_emetric_closedBall + - Dilation.ratio_unique + - Dilation.antilipschitz + - path: Mathlib/MeasureTheory/Decomposition/SignedHahn.lean + theorems: + - MeasureTheory.SignedMeasure.restrictNonposSeq_succ + - MeasureTheory.SignedMeasure.measure_of_restrictNonposSeq + - MeasureTheory.SignedMeasure.someExistsOneDivLT_subset + - MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint' + - MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative + - MeasureTheory.SignedMeasure.restrictNonposSeq_subset + - MeasureTheory.SignedMeasure.findExistsOneDivLT_min + - MeasureTheory.SignedMeasure.exists_compl_positive_negative + - MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint + - MeasureTheory.SignedMeasure.restrictNonposSeq_lt + - MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives + - MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos + - MeasureTheory.SignedMeasure.restrictNonposSeq_measurableSet + - MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos' + - MeasureTheory.SignedMeasure.findExistsOneDivLT_spec + - MeasureTheory.SignedMeasure.someExistsOneDivLT_measurableSet + - MeasureTheory.SignedMeasure.someExistsOneDivLT_spec + - path: Mathlib/Topology/Algebra/Monoid.lean + theorems: + - exists_nhds_one_split4 + - continuous_multiset_prod + - LocallyFinite.exists_finset_mulSupport + - continuous_pow + - continuousOn_list_prod + - le_nhds_mul + - Filter.tendsto_cocompact_mul_left + - exists_open_nhds_one_split + - continuousMul_iInf + - continuousMul_inf + - Filter.tendsto_cocompact_mul_right + - continuous_finprod + - continuousOn_multiset_prod + - continuousMul_of_comm_of_nhds_one + - tendsto_list_prod + - exists_open_nhds_one_mul_subset + - IsCompact.mul + - eventuallyEq_prod + - continuous_finprod_cond + - tendsto_multiset_prod + - ContinuousMul.of_nhds_one + - isClosed_setOf_map_mul + - path: Mathlib/Analysis/VonNeumannAlgebra/Basic.lean + theorems: + - VonNeumannAlgebra.mem_commutant_iff + - VonNeumannAlgebra.coe_commutant + - VonNeumannAlgebra.commutant_commutant + - path: Mathlib/Topology/Sheaves/Presheaf.lean + theorems: + - TopCat.Presheaf.Pushforward.id_hom_app + - TopCat.Presheaf.pushforwardToOfIso_app + - TopCat.Presheaf.id_pushforward + - TopCat.Presheaf.Pullback.id_inv_app + - TopCat.Presheaf.toPushforwardOfIso_app + - TopCat.Presheaf.Pushforward.id_eq + - TopCat.Presheaf.Pushforward.id_inv_app' + - TopCat.Presheaf.pushforward_eq' + - TopCat.Presheaf.ext + - TopCat.Presheaf.pushforwardEq_hom_app + - TopCat.Presheaf.pushforwardEq_rfl + - TopCat.Presheaf.map_restrict + - TopCat.Presheaf.Pushforward.comp_inv_app + - TopCat.Presheaf.restrict_restrict + - TopCat.Presheaf.pushforward_eq'_hom_app + - TopCat.Presheaf.Pushforward.id_hom_app' + - TopCat.Presheaf.Pushforward.comp_hom_app + - path: Mathlib/Algebra/Algebra/Subalgebra/Basic.lean + theorems: + - Subalgebra.range_algebraMap + - Subalgebra.rangeS_algebraMap + - Submodule.toSubalgebra_mk + - Subalgebra.iSupLift_comp_inclusion + - Subalgebra.toSubring_injective + - Subalgebra.iSupLift_of_mem + - Algebra.eq_top_iff + - Subalgebra.toSubsemiring_injective + - Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem + - Subalgebra.iSupLift_inclusion + - Algebra.coe_iInf + - Algebra.sInf_toSubmodule + - Subalgebra.iSupLift_mk + - Subalgebra.prod_top + - Subalgebra.equivOfEq_rfl + - Algebra.mem_sInf + - Algebra.iInf_toSubmodule + - Set.algebraMap_mem_center + - Algebra.sInf_toSubsemiring + - Subalgebra.range_comp_val + - Algebra.mem_iInf + - path: Mathlib/Data/Nat/Factors.lean + theorems: + - Nat.factors_sublist_of_dvd + - Nat.dvd_of_mem_factors + - Nat.factors_add_two + - Nat.four_dvd_or_exists_odd_prime_and_dvd_of_two_lt + - Nat.factors_chain + - Nat.perm_factors_mul_of_coprime + - Nat.factors_unique + - Nat.mem_factors_mul + - Nat.prod_factors + - Nat.factors_prime + - Nat.perm_factors_mul + - Nat.Prime.factors_pow + - Nat.factors_one + - Nat.mem_factors' + - Nat.factors_sublist_right + - Nat.factors_eq_nil + - Nat.prime_of_mem_factors + - Nat.factors_zero + - Nat.le_of_mem_factors + - Nat.coprime_factors_disjoint + - Nat.eq_of_perm_factors + - Nat.dvd_of_factors_subperm + - Nat.mem_factors_mul_right + - Nat.mem_factors_mul_left + - Nat.mem_factors_mul_of_coprime + - Nat.eq_prime_pow_of_unique_prime_dvd + - path: Mathlib/GroupTheory/Subgroup/ZPowers.lean + theorems: + - AddSubgroup.int_cast_mul_mem_zmultiples + - Subgroup.zpowers_le + - Subgroup.center_eq_iInf + - Subgroup.center_eq_infi' + - Subgroup.zpowers_eq_bot + - ofAdd_image_zmultiples_eq_zpowers_ofAdd + - ofMul_image_zpowers_eq_zmultiples_ofMul + - MonoidHom.map_zpowers + - AddSubgroup.int_cast_mem_zmultiples_one + - Int.range_castAddHom + - Int.mem_zmultiples_iff + - Subgroup.zpowers_eq_closure + - Int.zmultiples_one + - path: Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean + theorems: + - BoxIntegral.TaggedPrepartition.forall_mem_single + - BoxIntegral.TaggedPrepartition.IsHenstock.card_filter_tag_eq_le + - BoxIntegral.Prepartition.forall_biUnionTagged + - BoxIntegral.TaggedPrepartition.mem_infPrepartition_comm + - BoxIntegral.TaggedPrepartition.mem_iUnion + - BoxIntegral.Prepartition.tag_biUnionTagged + - BoxIntegral.TaggedPrepartition.IsHenstock.disjUnion + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean + theorems: + - SzemerediRegularity.average_density_near_total_density + - SzemerediRegularity.abs_density_star_sub_density_le_eps + - SzemerediRegularity.sum_density_div_card_le_density_add_eps + - SzemerediRegularity.le_sum_card_subset_chunk_parts + - SzemerediRegularity.eps_le_card_star_div + - SzemerediRegularity.edgeDensity_star_not_uniform + - SzemerediRegularity.sum_card_subset_chunk_parts_le + - SzemerediRegularity.one_sub_le_m_div_m_add_one_sq + - SzemerediRegularity.edgeDensity_chunk_not_uniform + - SzemerediRegularity.card_nonuniformWitness_sdiff_biUnion_star + - SzemerediRegularity.density_sub_eps_le_sum_density_div_card + - SzemerediRegularity.card_chunk + - SzemerediRegularity.one_sub_eps_mul_card_nonuniformWitness_le_card_star + - SzemerediRegularity.m_le_card_of_mem_chunk_parts + - SzemerediRegularity.edgeDensity_chunk_uniform + - SzemerediRegularity.card_eq_of_mem_parts_chunk + - SzemerediRegularity.m_add_one_div_m_le_one_add + - SzemerediRegularity.edgeDensity_chunk_aux + - SzemerediRegularity.card_le_m_add_one_of_mem_chunk_parts + - path: Mathlib/Analysis/SpecialFunctions/Log/Basic.lean + theorems: + - Real.log_abs + - Real.surjOn_log' + - Real.log_neg_nat_cast_nonneg + - Real.log_int_cast_nonneg + - Real.log_neg_of_lt_zero + - Real.log_lt_log + - Real.log_neg_iff + - Real.log_nonpos_iff' + - Mathlib.Meta.Positivity.log_nz_of_isRat_neg + - Real.log_ne_zero + - Real.strictAntiOn_log + - Real.log_nat_eq_sum_factorization + - Real.log_neg_eq_log + - Real.log_mul + - Real.log_lt_log_iff + - Real.le_log_iff_exp_le + - Real.sinh_log + - Real.log_pos_of_lt_neg_one + - Real.isLittleO_pow_log_id_atTop + - Real.cosh_log + - Mathlib.Meta.Positivity.log_nz_of_isRat + - Real.exp_log_eq_abs + - Real.log_zpow + - Real.log_one + - Real.log_inv + - Real.tendsto_log_nhdsWithin_zero + - Real.tendsto_pow_log_div_mul_add_atTop + - Real.abs_log_mul_self_lt + - Real.exp_log + - Real.isLittleO_const_log_atTop + - Real.log_pow + - Real.log_nonpos_iff + - Mathlib.Meta.Positivity.log_pos_of_isRat + - Real.log_pos_iff + - Real.log_le_log_iff + - Real.log_div + - Real.log_lt_iff_lt_exp + - Real.log_nat_cast_nonneg + - Real.log_of_pos + - Mathlib.Meta.Positivity.log_pos_of_isNat + - Mathlib.Meta.Positivity.log_nonneg_of_isNegNat + - Real.log_lt_sub_one_of_pos + - Real.log_eq_zero + - Real.log_nonneg_iff + - Real.continuousOn_log + - Real.exp_log_of_neg + - Real.lt_log_iff_exp_lt + - Mathlib.Meta.Positivity.log_pos_of_isNegNat + - Real.log_sqrt + - Real.log_le_iff_le_exp + - Real.le_exp_log + - Mathlib.Meta.Positivity.log_nonneg_of_isNat + - Real.log_le_sub_one_of_pos + - Real.tendsto_log_comp_add_sub_log + - Real.log_prod + - path: Mathlib/MeasureTheory/Integral/Indicator.lean + theorems: + - MeasureTheory.tendsto_measure_of_tendsto_indicator + - MeasureTheory.tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure + - path: Mathlib/CategoryTheory/Functor/FullyFaithful.lean + theorems: + - CategoryTheory.preimage_map + - CategoryTheory.Faithful.of_iso + - CategoryTheory.isIso_of_fully_faithful + - CategoryTheory.Faithful.div_comp + - CategoryTheory.natIsoOfCompFullyFaithful_inv + - CategoryTheory.natIsoOfCompFullyFaithful_hom + - CategoryTheory.Functor.image_preimage + - CategoryTheory.Functor.preimageIso_mapIso + - CategoryTheory.preimage_id + - path: Mathlib/Data/Rat/Order.lean + theorems: + - Rat.nonneg_mul + - Rat.le_def + - Rat.divInt_nonneg + - Rat.le_iff_Nonneg + - Rat.abs_def + - Rat.le_refl + - Rat.nonneg_antisymm + - Rat.add_le_add_left + - Rat.le_antisymm + - Rat.nonneg_total + - Rat.nonneg_iff_zero_le + - Rat.lt_def + - Rat.le_trans + - Rat.div_lt_div_iff_mul_lt_mul + - Rat.le_def' + - Rat.le_total + - Rat.nonneg_add + - Rat.lt_one_iff_num_lt_denom + - Rat.num_nonpos + - Rat.mul_nonneg + - path: Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean + theorems: + - CategoryTheory.Equalizer.Sieve.w + - CategoryTheory.Equalizer.FirstObj.ext + - CategoryTheory.Equalizer.Sieve.compatible_iff + - CategoryTheory.Equalizer.Sieve.equalizer_sheaf_condition + - CategoryTheory.Equalizer.Presieve.Arrows.sheaf_condition + - CategoryTheory.Equalizer.Presieve.compatible_iff + - CategoryTheory.Equalizer.Presieve.Arrows.FirstObj.ext + - CategoryTheory.Equalizer.Sieve.SecondObj.ext + - CategoryTheory.Equalizer.Presieve.Arrows.compatible_iff + - CategoryTheory.Equalizer.Presieve.Arrows.SecondObj.ext + - CategoryTheory.Equalizer.Presieve.sheaf_condition + - CategoryTheory.Equalizer.Presieve.w + - CategoryTheory.Equalizer.Presieve.Arrows.w + - path: Mathlib/Algebra/Order/Ring/WithTop.lean + theorems: + - WithBot.mul_coe_eq_bind + - WithBot.coe_mul_eq_bind + - WithTop.mul_def + - WithTop.top_mul + - WithTop.coe_mul_eq_bind + - WithBot.mul_eq_bot_iff + - WithBot.unbot'_zero_mul + - WithTop.untop'_zero_mul + - WithTop.mul_eq_top_iff + - WithTop.mul_lt_top' + - WithBot.bot_mul + - WithBot.mul_bot + - WithBot.mul_def + - WithTop.mul_coe_eq_bind + - WithTop.mul_top + - path: Mathlib/Data/Nat/Cast/Order.lean + theorems: + - NeZero.nat_of_injective + - Nat.mono_cast + - Nat.cast_lt_one + - Nat.cast_pos' + - Nat.cast_add_one_pos + - Nat.one_lt_cast + - Nat.one_le_cast + - Nat.cast_le_one + - Nat.cast_tsub + - path: Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean + theorems: + - CategoryTheory.FunctorToTypes.prod_ext' + - CategoryTheory.FunctorToTypes.prodMk_fst + - CategoryTheory.FunctorToTypes.inr_comp_binaryCoproductIso_hom + - CategoryTheory.FunctorToTypes.binaryProductIso_inv_comp_fst + - CategoryTheory.FunctorToTypes.inl_comp_binaryCoproductIso_hom + - CategoryTheory.FunctorToTypes.prodMk_snd + - CategoryTheory.FunctorToTypes.binaryProductIso_inv_comp_snd + - path: Mathlib/NumberTheory/ModularForms/Basic.lean + theorems: + - ModularForm.gradedMonoid_eq_of_cast + - path: Mathlib/Algebra/Star/Module.lean + theorems: + - IsSelfAdjoint.algebraMap + - StarModule.selfAdjointPart_add_skewAdjointPart + - IsSelfAdjoint.coe_selfAdjointPart_apply + - algebraMap_star_comm + - selfAdjointPart_comp_subtype_skewAdjoint + - skewAdjointPart_comp_subtype_skewAdjoint + - IsSelfAdjoint.skewAdjointPart_apply + - path: Mathlib/LinearAlgebra/Lagrange.lean + theorems: + - Lagrange.nodal_empty + - Lagrange.interpolate_empty + - Lagrange.interpolate_singleton + - Lagrange.basis_eq_prod_sub_inv_mul_nodal_div + - Lagrange.basis_singleton + - Lagrange.interpolate_eq_nodalWeight_mul_nodal_div_X_sub_C + - Lagrange.sum_basis + - Lagrange.eval_interpolate_not_at_node + - Lagrange.basisDivisor_inj + - Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq + - Lagrange.basisDivisor_ne_zero_iff + - Polynomial.eq_of_degrees_lt_of_eval_finset_eq + - Lagrange.eval_nodal_not_at_node + - Lagrange.eq_interpolate_of_eval_eq + - Lagrange.eval_basis_not_at_node + - Lagrange.interpolate_one + - Lagrange.eval_nodal_derivative_eval_node_eq + - Lagrange.values_eq_on_of_interpolate_eq + - Lagrange.degree_basisDivisor_self + - Lagrange.interpolate_eq_of_values_eq_on + - Lagrange.basisDivisor_self + - Lagrange.degree_interpolate_lt + - Lagrange.eval_interpolate_at_node + - Lagrange.nodal_ne_zero + - Lagrange.X_sub_C_dvd_nodal + - Lagrange.eval_basisDivisor_right + - Lagrange.nodal_insert_eq_nodal + - Polynomial.eq_zero_of_degree_lt_of_eval_finset_eq_zero + - Lagrange.nodalWeight_eq_eval_nodal_derative + - Lagrange.interpolate_eq_add_interpolate_erase + - Lagrange.derivative_nodal + - Lagrange.nodal_eq_mul_nodal_erase + - Lagrange.natDegree_basis + - Lagrange.basis_ne_zero + - Lagrange.basisDivisor_add_symm + - Polynomial.eq_of_degrees_lt_of_eval_index_eq + - Lagrange.eq_interpolate_iff + - Lagrange.eval_basisDivisor_left_of_ne + - Lagrange.degree_interpolate_le + - Lagrange.natDegree_nodal + - Lagrange.nodal_subgroup_eq_X_pow_card_sub_one + - Lagrange.nodalWeight_ne_zero + - Lagrange.eval_interpolate_not_at_node' + - Lagrange.degree_interpolate_erase_lt + - Lagrange.degree_basis + - Lagrange.degree_basisDivisor_of_ne + - Polynomial.eq_of_degree_sub_lt_of_eval_index_eq + - Polynomial.eq_of_degree_le_of_eval_finset_eq + - Polynomial.eq_of_degree_le_of_eval_index_eq + - Lagrange.nodalWeight_eq_eval_nodal_erase_inv + - Lagrange.basis_pair_right + - Polynomial.eq_zero_of_degree_lt_of_eval_index_eq_zero + - Lagrange.basis_pair_left + - Lagrange.sum_nodalWeight_mul_inv_sub_ne_zero + - Lagrange.eval_nodal + - Lagrange.nodal_erase_eq_nodal_div + - Lagrange.interpolate_eq_sum_interpolate_insert_sdiff + - Lagrange.eval_basis_of_ne + - Lagrange.degree_nodal + - path: Mathlib/Data/Semiquot.lean + theorems: + - Semiquot.pure_isPure + - Semiquot.IsPure.min + - Semiquot.liftOn_ofMem + - Semiquot.blur_eq_blur' + - Semiquot.ext_s + - Semiquot.get_mem + - Semiquot.mem_bind + - Semiquot.eq_pure + - path: Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean + theorems: + - quadraticChar_zero + - quadraticCharFun_eq_one_of_char_two + - quadraticChar_sq_one' + - FiniteField.isSquare_neg_one_iff + - quadraticCharFun_one + - quadraticCharFun_eq_pow_of_char_ne_two + - quadraticCharFun_eq_zero_iff + - quadraticCharFun_mul + - quadraticChar_one_iff_isSquare + - quadraticChar_sq_one + - quadraticChar_neg_one + - quadraticChar_isNontrivial + - quadraticChar_neg_one_iff_not_isSquare + - quadraticChar_card_sqrts + - quadraticChar_eq_neg_one_iff_not_one + - quadraticCharFun_zero + - quadraticChar_isQuadratic + - path: Mathlib/ModelTheory/DirectLimit.lean + theorems: + - FirstOrder.Language.DirectLimit.lift_unique + - FirstOrder.Language.DirectLimit.cg + - FirstOrder.Language.DirectLimit.unify_sigma_mk_self + - FirstOrder.Language.DirectLimit.of_f + - FirstOrder.Language.DirectedSystem.coe_natLERec + - FirstOrder.Language.DirectLimit.exists_quotient_mk'_sigma_mk'_eq + - FirstOrder.Language.DirectLimit.exists_of + - FirstOrder.Language.DirectLimit.exists_unify_eq + - FirstOrder.Language.DirectLimit.comp_unify + - FirstOrder.Language.DirectLimit.relMap_unify_equiv + - FirstOrder.Language.DirectLimit.funMap_quotient_mk'_sigma_mk' + - FirstOrder.Language.DirectLimit.lift_quotient_mk'_sigma_mk' + - FirstOrder.Language.DirectLimit.lift_of + - FirstOrder.Language.DirectLimit.relMap_quotient_mk'_sigma_mk' + - FirstOrder.Language.DirectLimit.funMap_unify_equiv + - path: Mathlib/Algebra/Lie/EngelSubalgebra.lean + theorems: + - LieSubalgebra.normalizer_engel + - LieSubalgebra.isNilpotent_of_forall_le_engel + - LieSubalgebra.normalizer_eq_self_of_engel_le + - LieSubalgebra.mem_engel_iff + - LieSubalgebra.self_mem_engel + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean + theorems: + - Real.arctan_inv_of_neg + - Real.arctan_eq_arccos + - Real.arctan_add_eq_sub_pi + - Real.arctan_zero + - Real.two_mul_arctan + - Real.arctan_ne_mul_pi_div_two + - Real.arctan_neg + - Real.sin_arctan + - Real.arctan_eq_of_tan_eq + - Real.arctan_inv_2_add_arctan_inv_3 + - Real.continuousOn_tan_Ioo + - Real.tan_two_mul + - Real.arccos_eq_arctan + - Real.two_mul_arctan_add_pi + - Real.arctan_one + - Real.tan_int_mul_pi_div_two + - Real.arctan_add_arctan_lt_pi_div_two + - Real.arctan_add + - Real.two_mul_arctan_inv_3_add_arctan_inv_7 + - Real.arctan_inv_of_pos + - Real.tan_add + - Real.two_mul_arctan_sub_pi + - Real.arctan_eq_zero_iff + - Real.continuousOn_tan + - Real.two_mul_arctan_inv_2_sub_arctan_inv_7 + - Real.cos_sq_arctan + - Real.arctan_add_eq_add_pi + - Real.arcsin_eq_arctan + - Real.cos_arctan + - Real.four_mul_arctan_inv_5_sub_arctan_inv_239 + - path: Mathlib/Algebra/GroupPower/NegOnePow.lean + theorems: + - Int.negOnePow_mul_self + - Int.negOnePow_eq_iff + - Int.negOnePow_even + - Int.negOnePow_odd + - Int.negOnePow_sub + - Int.negOnePow_eq_neg_one_iff + - Int.negOnePow_neg + - Int.negOnePow_eq_one_iff + - Int.negOnePow_succ + - path: Mathlib/Analysis/ConstantSpeed.lean + theorems: + - HasConstantSpeedOnWith.hasLocallyBoundedVariationOn + - unique_unit_speed_on_Icc_zero + - hasConstantSpeedOnWith_zero_iff + - hasConstantSpeedOnWith_iff_ordered + - edist_naturalParameterization_eq_zero + - HasConstantSpeedOnWith.Icc_Icc + - HasConstantSpeedOnWith.union + - unique_unit_speed + - HasConstantSpeedOnWith.ratio + - has_unit_speed_naturalParameterization + - hasConstantSpeedOnWith_iff_variationOnFromTo_eq + - path: Mathlib/SetTheory/Cardinal/Finite.lean + theorems: + - Cardinal.natCast_eq_toPartENat_iff + - Nat.card_zmod + - Set.card_singleton_prod + - Nat.card_sum + - PartENat.card_sum + - Nat.card_image_of_injOn + - Nat.card_eq_card_finite_toFinset + - Nat.card_pi + - PartENat.one_lt_card_iff_nontrivial + - Cardinal.toPartENat_lt_natCast_iff + - Nat.card_pos_iff + - Nat.card_le_card_of_injective + - Nat.card_eq_card_toFinset + - Nat.card_of_isEmpty + - Cardinal.natCast_le_toPartENat_iff + - Nat.card_eq_zero + - Nat.card_prod + - Nat.card_preimage_of_injOn + - Cardinal.natCast_lt_toPartENat_iff + - Set.card_prod_singleton + - Nat.card_ne_zero + - PartENat.card_eq_zero_iff_empty + - Nat.card_eq_of_equiv_fin + - Cardinal.toPartENat_le_natCast_iff + - Nat.card_fun + - Nat.card_eq_finsetCard + - Nat.card_le_card_of_surjective + - Nat.card_of_subsingleton + - Cardinal.toPartENat_eq_natCast_iff + - PartENat.card_le_one_iff_subsingleton + - path: Mathlib/Data/Nat/Count.lean + theorems: + - Nat.count_monotone + - Nat.count_eq_card_filter_range + - Nat.count_succ + - Nat.count_le_card + - Nat.count_injective + - Nat.count_mono_left + - Nat.count_succ_eq_succ_count_iff + - Nat.count_eq_card_fintype + - Nat.count_zero + - Nat.count_succ_eq_count_iff + - Nat.count_add' + - Nat.count_succ' + - Nat.count_one + - Nat.count_add + - Nat.count_le_cardinal + - path: Mathlib/CategoryTheory/Sites/Plus.lean + theorems: + - CategoryTheory.GrothendieckTopology.plus_hom_ext + - CategoryTheory.GrothendieckTopology.toPlus_plusLift + - CategoryTheory.GrothendieckTopology.diagramNatTrans_comp + - CategoryTheory.GrothendieckTopology.plusMap_comp + - CategoryTheory.GrothendieckTopology.diagramNatTrans_zero + - CategoryTheory.GrothendieckTopology.plusMap_id + - CategoryTheory.GrothendieckTopology.toPlus_naturality + - CategoryTheory.GrothendieckTopology.isoToPlus_inv + - CategoryTheory.GrothendieckTopology.diagramNatTrans_id + - CategoryTheory.GrothendieckTopology.plusMap_toPlus + - CategoryTheory.GrothendieckTopology.plusMap_zero + - CategoryTheory.GrothendieckTopology.isIso_toPlus_of_isSheaf + - CategoryTheory.GrothendieckTopology.plusMap_plusLift + - CategoryTheory.GrothendieckTopology.plusLift_unique + - path: Mathlib/LinearAlgebra/QuadraticForm/Basic.lean + theorems: + - QuadraticForm.polar_zero_right + - QuadraticForm.polarBilin_injective + - QuadraticForm.isOrtho_polarBilin + - QuadraticForm.associated_isSymm + - QuadraticForm.polar_smul + - QuadraticForm.isSymm_toMatrix' + - LinearMap.BilinForm.polar_toQuadraticForm + - QuadraticForm.map_smul_of_tower + - QuadraticForm.basisRepr_eq_of_iIsOrtho + - QuadraticForm.discr_smul + - QuadraticForm.associated_comp + - QuadraticForm.polar_add + - QuadraticForm.IsOrtho.zero_left + - QuadraticForm.basisRepr_apply + - LinearMap.BilinForm.toQuadraticForm_isOrtho + - QuadraticForm.toQuadraticForm_polarBilin + - QuadraticForm.polar_smul_left + - QuadraticForm.polarBilin_comp + - QuadraticForm.discr_comp + - QuadraticForm.map_sub + - QuadraticForm.polar_smul_right_of_tower + - LinearMap.BilinForm.compQuadraticForm_polar + - QuadraticForm.polar_comp + - QuadraticForm.polar_smul_right + - QuadraticForm.polar_add_right + - QuadraticForm.IsOrtho.zero_right + - QuadraticForm.exists_quadraticForm_ne_zero + - QuadraticForm.polar_zero_left + - QuadraticForm.not_anisotropic_iff_exists + - QuadraticForm.polar_self + - QuadraticForm.map_neg + - QuadraticForm.toMatrix'_smul + - QuadraticForm.associated_linMulLin + - QuadraticForm.polar_neg_left + - LinearMap.BilinForm.exists_bilinForm_self_ne_zero + - QuadraticForm.associated_toQuadraticForm + - QuadraticForm.map_add_self + - QuadraticForm.polar_comm + - QuadraticForm.map_add_add_add_map + - LinearMap.BilinForm.exists_orthogonal_basis + - QuadraticForm.associated_sq + - QuadraticForm.associated_eq_self_apply + - QuadraticForm.two_nsmul_associated + - QuadraticForm.isOrtho_comm + - QuadraticForm.map_zero + - QuadraticForm.polar_smul_left_of_tower + - QuadraticForm.polar_sub_left + - QuadraticForm.PosDef.anisotropic + - QuadraticForm.toMatrix'_comp + - QuadraticForm.associated_isOrtho + - QuadraticForm.polar_add_left_iff + - QuadraticForm.choose_exists_companion + - QuadraticForm.polar_neg + - QuadraticForm.separatingLeft_of_anisotropic + - QuadraticForm.polar_sub_right + - QuadraticForm.polar_neg_right + - path: Mathlib/Probability/Process/HittingTime.lean + theorems: + - MeasureTheory.hitting_le_iff_of_exists + - MeasureTheory.hitting_isStoppingTime + - MeasureTheory.le_hitting_of_exists + - MeasureTheory.hitting_eq_hitting_of_exists + - MeasureTheory.hitting_eq_end_iff + - MeasureTheory.hitting_of_lt + - MeasureTheory.hitting_mem_set_of_hitting_lt + - MeasureTheory.hitting_mono + - MeasureTheory.hitting_mem_set + - MeasureTheory.hitting_lt_iff + - MeasureTheory.hitting_of_le + - MeasureTheory.hitting_eq_sInf + - MeasureTheory.hitting_bot_le_iff + - MeasureTheory.hitting_le + - MeasureTheory.hitting_le_iff_of_lt + - MeasureTheory.not_mem_of_lt_hitting + - MeasureTheory.isStoppingTime_hitting_isStoppingTime + - MeasureTheory.hitting_le_of_mem + - MeasureTheory.stoppedValue_hitting_mem + - MeasureTheory.le_hitting + - path: Mathlib/Algebra/Order/UpperLower.lean + theorems: + - Set.OrdConnected.smul + - lowerClosure_mul + - upperClosure_mul + - mul_lowerClosure + - IsUpperSet.mul_left + - IsUpperSet.div_right + - IsUpperSet.mul_right + - LowerSet.one_mul + - mul_upperClosure + - lowerClosure_mul_distrib + - UpperSet.one_mul + - upperClosure_mul_distrib + - IsUpperSet.div_left + - path: Mathlib/SetTheory/Ordinal/Exponential.lean + theorems: + - Ordinal.opow_succ + - Ordinal.opow_zero + - Ordinal.sup_opow_nat + - Ordinal.zero_opow' + - Ordinal.log_def + - Ordinal.zero_opow + - Ordinal.opow_isLimit_left + - Ordinal.left_le_opow + - Ordinal.opow_log_le_self + - Ordinal.div_opow_log_pos + - Ordinal.log_le_self + - Ordinal.log_of_not_one_lt_left + - Ordinal.opow_mul_add_lt_opow_mul_succ + - Ordinal.opow_one + - Ordinal.one_opow + - Ordinal.opow_lt_opow_left_of_succ + - Ordinal.nat_cast_opow + - Ordinal.log_opow + - Ordinal.div_opow_log_lt + - Ordinal.log_pos + - Ordinal.succ_log_def + - Ordinal.opow_one_add + - Ordinal.opow_le_opow_right + - Ordinal.log_mod_opow_log_lt_log_self + - Ordinal.opow_isNormal + - Ordinal.lt_opow_succ_log_self + - Ordinal.opow_add + - Ordinal.opow_limit + - Ordinal.opow_mul + - Ordinal.log_zero_right + - Ordinal.log_eq_zero + - Ordinal.mod_opow_log_lt_self + - Ordinal.add_log_le_log_mul + - Ordinal.opow_mul_add_lt_opow_succ + - Ordinal.opow_le_opow_left + - Ordinal.log_mono_right + - Ordinal.lt_opow_of_limit + - Ordinal.opow_le_of_limit + - Ordinal.opow_pos + - path: Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean + theorems: + - Real.Gamma_mul_Gamma_add_half_of_pos + - Real.BohrMollerup.f_nat_eq + - Real.doublingGamma_add_one + - Real.BohrMollerup.ge_logGammaSeq + - StrictConcaveOn.congr + - Real.BohrMollerup.f_add_nat_eq + - Real.BohrMollerup.le_logGammaSeq + - Real.BohrMollerup.f_add_nat_ge + - Real.log_doublingGamma_eq + - Real.eq_Gamma_of_log_convex + - Real.BohrMollerup.f_add_nat_le + - StrictConvexOn.congr + - Real.Gamma_two + - Real.doublingGamma_one + - Real.doublingGamma_eq_Gamma + - Real.convexOn_Gamma + - Real.BohrMollerup.tendsto_logGammaSeq_of_le_one + - Real.Gamma_strictMonoOn_Ici + - Real.convexOn_log_Gamma + - Real.BohrMollerup.tendsto_logGammaSeq + - Real.BohrMollerup.tendsto_log_gamma + - ConcaveOn.congr + - Real.Gamma_three_div_two_lt_one + - Real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma + - Real.doublingGamma_log_convex_Ioi + - ConvexOn.congr + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean + theorems: + - CliffordAlgebra.commute_map_mul_map_of_isOrtho_of_mem_evenOdd_zero_right + - CliffordAlgebra.ofProd_comp_toProd + - "CliffordAlgebra.ofProd_\u03B9_mk" + - "CliffordAlgebra.toProd_one_tmul_\u03B9" + - "CliffordAlgebra.toProd_\u03B9_tmul_one" + - CliffordAlgebra.commute_map_mul_map_of_isOrtho_of_mem_evenOdd_zero_left + - CliffordAlgebra.map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one + - CliffordAlgebra.toProd_comp_ofProd + - CliffordAlgebra.map_mul_map_of_isOrtho_of_mem_evenOdd + - path: Mathlib/SetTheory/Game/Birthday.lean + theorems: + - SetTheory.PGame.Relabelling.birthday_congr + - SetTheory.PGame.lt_birthday_iff + - SetTheory.PGame.birthday_add_nat + - SetTheory.PGame.birthday_moveRight_lt + - SetTheory.PGame.neg_birthday_le + - SetTheory.PGame.birthday_add_zero + - SetTheory.PGame.toPGame_birthday + - SetTheory.PGame.birthday_eq_zero + - SetTheory.PGame.birthday_add_one + - SetTheory.PGame.birthday_star + - SetTheory.PGame.le_birthday + - SetTheory.PGame.birthday_add + - SetTheory.PGame.birthday_def + - SetTheory.PGame.birthday_zero + - SetTheory.PGame.birthday_zero_add + - SetTheory.PGame.birthday_moveLeft_lt + - SetTheory.PGame.neg_birthday + - SetTheory.PGame.birthday_one_add + - SetTheory.PGame.birthday_nat_add + - SetTheory.PGame.birthday_one + - SetTheory.PGame.birthday_nat_cast + - path: Mathlib/Data/List/Range.lean + theorems: + - List.ranges_join + - List.mem_finRange + - List.pairwise_gt_iota + - List.prod_range_succ + - List.chain_range_succ + - List.prod_range_succ' + - List.ranges_nodup + - List.unzip_enumFrom_eq_prod + - List.pairwise_le_finRange + - List.pairwise_lt_range + - List.ranges_length + - List.nthLe_range'_1 + - List.mem_mem_ranges_iff_lt_sum + - List.nodup_range + - List.finRange_eq_nil + - List.get_finRange + - List.finRange_map_get + - List.indexOf_finRange + - List.length_finRange + - List.ranges_disjoint + - path: Mathlib/Topology/Algebra/Semigroup.lean + theorems: + - exists_idempotent_of_compact_t2_of_continuous_mul_left + - exists_idempotent_in_compact_subsemigroup + - path: Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean + theorems: + - CategoryTheory.tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight + - CategoryTheory.leftAdjointMate_id + - CategoryTheory.leftDualIso_id + - CategoryTheory.comp_leftAdjointMate + - CategoryTheory.coevaluation_comp_leftAdjointMate + - CategoryTheory.rightDualIso_id + - CategoryTheory.ExactPairing.coevaluation_evaluation'' + - CategoryTheory.leftAdjointMate_comp + - CategoryTheory.tensorLeftHomEquiv_naturality + - CategoryTheory.tensorRightHomEquiv_naturality + - CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_whiskerRight + - CategoryTheory.tensorRightHomEquiv_symm_naturality + - CategoryTheory.coevaluation_comp_rightAdjointMate + - CategoryTheory.tensorRightHomEquiv_symm_coevaluation_comp_whiskerLeft + - CategoryTheory.tensorLeftHomEquiv_symm_naturality + - CategoryTheory.ExactPairing.evaluation_coevaluation'' + - CategoryTheory.rightAdjointMate_comp_evaluation + - CategoryTheory.tensorLeftHomEquiv_tensor + - CategoryTheory.tensorRightHomEquiv_whiskerRight_comp_evaluation + - CategoryTheory.tensorRightHomEquiv_whiskerLeft_comp_evaluation + - CategoryTheory.rightAdjointMate_comp + - CategoryTheory.comp_rightAdjointMate + - CategoryTheory.tensorLeftHomEquiv_whiskerRight_comp_evaluation + - CategoryTheory.tensorLeftHomEquiv_symm_coevaluation_comp_whiskerLeft + - CategoryTheory.rightAdjointMate_id + - CategoryTheory.tensorRightHomEquiv_tensor + - CategoryTheory.leftAdjointMate_comp_evaluation + - path: Mathlib/Topology/UrysohnsLemma.lean + theorems: + - Urysohns.CU.continuous_lim + - Urysohns.CU.lim_of_mem_C + - Urysohns.CU.approx_le_succ + - Urysohns.CU.approx_of_nmem_U + - exists_continuous_nonneg_pos + - exists_continuous_zero_one_of_isClosed + - exists_continuous_one_zero_of_isCompact + - Urysohns.CU.approx_le_approx_of_U_sub_C + - Urysohns.CU.lim_of_nmem_U + - "exists_continuous_one_zero_of_isCompact_of_isG\u03B4" + - exists_continuous_zero_one_of_isCompact + - Urysohns.CU.lim_eq_midpoint + - Urysohns.CU.approx_le_one + - Urysohns.CU.approx_of_mem_C + - Urysohns.CU.approx_mem_Icc_right_left + - Urysohns.CU.approx_nonneg + - path: Mathlib/Analysis/NormedSpace/AddTorsorBases.lean + theorems: + - IsOpen.exists_subset_affineIndependent_span_eq_top + - interior_convexHull_nonempty_iff_affineSpan_eq_top + - AffineBasis.interior_convexHull + - Convex.interior_nonempty_iff_affineSpan_eq_top + - AffineBasis.centroid_mem_interior_convexHull + - IsOpen.exists_between_affineIndependent_span_eq_top + - path: Mathlib/Data/Nat/PSub.lean + theorems: + - Nat.psub_eq_none + - Nat.sub_eq_psub + - Nat.ppred_eq_none + - Nat.pred_eq_ppred + - Nat.psub_add + - Nat.ppred_eq_some + - Nat.psub'_eq_psub + - Nat.psub_eq_some + - path: Mathlib/Computability/Halting.lean + theorems: + - Nat.Partrec'.bind + - Nat.Partrec'.tail + - Partrec.cond + - Partrec.merge + - "Nat.Partrec'.comp\u2081" + - Nat.Partrec'.of_part + - ComputablePred.rice + - ComputablePred.computable_iff_re_compl_re' + - Nat.Partrec'.to_part + - Nat.Partrec'.comp' + - Nat.Partrec'.rfindOpt + - Nat.Partrec'.cons + - "Nat.Partrec'.part_iff\u2081" + - "Nat.Partrec'.part_iff\u2082" + - "ComputablePred.rice\u2082" + - ComputablePred.to_re + - Partrec.dom_re + - Partrec.sum_casesOn + - ComputablePred.not + - Nat.Partrec'.map + - ComputablePred.computable_iff + - Nat.Partrec'.vec_iff + - Nat.Partrec.merge' + - path: Mathlib/Data/Nat/Nth.lean + theorems: + - Nat.count_nth_succ + - Nat.filter_range_nth_eq_insert + - Nat.nth_apply_eq_orderIsoOfNat + - Nat.exists_lt_card_finite_nth_eq + - Nat.filter_range_nth_subset_insert + - Nat.nth_strictMono + - Nat.nth_zero + - Nat.nth_zero_of_exists + - Nat.exists_lt_card_nth_eq + - Nat.count_nth_zero + - Nat.range_nth_of_infinite + - Nat.image_nth_Iio_card + - Nat.le_nth_of_lt_nth_succ + - Nat.nth_lt_of_lt_count + - Nat.nth_eq_zero_mono + - Nat.nth_zero_of_zero + - Nat.nth_count_eq_sInf + - Nat.nth_eq_zero + - Nat.nth_of_card_le + - Nat.nth_eq_orderEmbOfFin + - Nat.count_nth + - Nat.nth_strictMonoOn + - Nat.nth_eq_sInf + - path: Mathlib/Topology/Connected/LocallyConnected.lean + theorems: + - locallyConnectedSpace_of_connected_bases + - IsOpen.connectedComponentIn + - OpenEmbedding.locallyConnectedSpace + - locallyConnectedSpace_iff_connected_basis + - locallyConnectedSpace_iff_open_connected_subsets + - isOpen_connectedComponent + - locallyConnectedSpace_iff_connected_subsets + - connectedComponentIn_mem_nhds + - locallyConnectedSpace_iff_connectedComponentIn_open + - path: Mathlib/Topology/Bornology/Hom.lean + theorems: + - LocallyBoundedMap.cancel_left + - path: Mathlib/Data/Polynomial/Splits.lean + theorems: + - Polynomial.aeval_root_derivative_of_splits + - Polynomial.splits_id_iff_splits + - Polynomial.splits_iff_card_roots + - Polynomial.splits_iff + - Polynomial.eq_prod_roots_of_monic_of_splits_id + - Polynomial.splits_of_splits_id + - Polynomial.exists_root_of_splits' + - Polynomial.degree_eq_card_roots' + - Polynomial.adjoin_rootSet_eq_range + - Polynomial.splits_prod + - Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split + - Polynomial.degree_eq_one_of_irreducible_of_splits + - Polynomial.eq_prod_roots_of_splits + - Polynomial.splits_map_iff + - Polynomial.splits_of_map_degree_eq_one + - Polynomial.image_rootSet + - Polynomial.eq_X_sub_C_of_splits_of_single_root + - Polynomial.splits_of_splits_of_dvd + - Polynomial.natDegree_eq_card_roots' + - Polynomial.splits_prod_iff + - Polynomial.roots_map + - Polynomial.splits_of_exists_multiset + - Polynomial.splits_pow + - Polynomial.degree_eq_card_roots + - Polynomial.eq_prod_roots_of_splits_id + - Polynomial.splits_of_map_eq_C + - Polynomial.splits_of_comp + - Polynomial.splits_of_degree_le_one + - Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split + - Polynomial.roots_ne_zero_of_splits' + - Polynomial.mem_lift_of_splits_of_roots_mem_range + - Polynomial.splits_of_algHom + - path: Mathlib/RingTheory/Subsemiring/Basic.lean + theorems: + - Subsemiring.coe_iSup_of_directed + - Subsemiring.coe_pow + - Subsemiring.closure_induction' + - Subsemiring.mem_iSup_of_directed + - coe_nat_mem + - ofNat_mem + - Subsemiring.coe_sSup_of_directedOn + - RingHom.map_rangeS + - Subsemiring.mem_map_equiv + - SubsemiringClass.coe_pow + - RingHom.rangeS_top_iff_surjective + - natCast_mem + - Subsemiring.closure_addSubmonoid_closure + - Subsemiring.mem_closure_iff_exists_list + - Submonoid.subsemiringClosure_eq_closure + - Subsemiring.top_prod + - Subsemiring.coe_closure_eq + - Subsemiring.prod_top + - RingHom.rangeS_eq_map + - path: Mathlib/GroupTheory/Solvable.lean + theorems: + - Equiv.Perm.fin_5_not_solvable + - Equiv.Perm.not_solvable + - IsSimpleGroup.derivedSeries_succ + - IsSimpleGroup.comm_iff_isSolvable + - map_derivedSeries_le_derivedSeries + - isSolvable_of_comm + - derivedSeries_le_map_derivedSeries + - solvable_of_ker_le_range + - derivedSeries_normal + - path: Mathlib/Algebra/Category/ModuleCat/Basic.lean + theorems: + - ModuleCat.smul_naturality + - ModuleCat.isZero_of_subsingleton + - path: Mathlib/Algebra/Homology/ExactSequence.lean + theorems: + - "CategoryTheory.ComposableArrows.isComplex\u2082_iff" + - "CategoryTheory.ComposableArrows.isComplex\u2080" + - CategoryTheory.ComposableArrows.IsComplex.zero' + - "CategoryTheory.ComposableArrows.exact\u2080" + - "CategoryTheory.ComposableArrows.isComplex\u2081" + - CategoryTheory.ComposableArrows.isComplex_of_iso + - CategoryTheory.ComposableArrows.Exact.exact' + - CategoryTheory.ShortComplex.isComplex_toComposableArrows + - "CategoryTheory.ComposableArrows.exact_of_\u03B4\u2080" + - "CategoryTheory.ComposableArrows.exact_iff_\u03B4\u2080" + - "CategoryTheory.ComposableArrows.exact\u2082_iff" + - "CategoryTheory.ComposableArrows.exact\u2081" + - "CategoryTheory.ComposableArrows.exact_iff_\u03B4last" + - path: Mathlib/Algebra/Algebra/Tower.lean + theorems: + - IsScalarTower.Algebra.ext + - IsScalarTower.of_algebraMap_eq + - IsScalarTower.algebraMap_eq + - IsScalarTower.algebraMap_smul + - Submodule.restrictScalars_span + - Submodule.smul_mem_span_smul' + - Submodule.smul_mem_span_smul + - Submodule.span_smul_of_span_eq_top + - Submodule.map_mem_span_algebraMap_image + - IsScalarTower.of_algebraMap_smul + - AlgHom.map_algebraMap + - Submodule.smul_mem_span_smul_of_mem + - path: Mathlib/MeasureTheory/Group/Action.lean + theorems: + - MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero + - MeasureTheory.measurePreserving_smul + - MeasureTheory.NullMeasurableSet.smul + - MeasureTheory.smulInvariantMeasure_tfae + - MeasureTheory.measure_eq_zero_iff_eq_empty_of_smulInvariant + - MeasureTheory.measure_smul_null + - MeasureTheory.smul_ae_eq_self_of_mem_zpowers + - MeasureTheory.smulInvariantMeasure_map + - MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples + - MeasureTheory.isLocallyFiniteMeasure_of_smulInvariant + - path: Mathlib/Probability/Distributions/Uniform.lean + theorems: + - MeasureTheory.pdf.IsUniform.aemeasurable + - MeasureTheory.pdf.IsUniform.pdf_eq + - PMF.support_uniformOfFintype + - MeasureTheory.pdf.IsUniform.integral_eq + - PMF.ofMultiset_apply_of_not_mem + - MeasureTheory.pdf.IsUniform.isProbabilityMeasure + - PMF.support_ofMultiset + - MeasureTheory.pdf.IsUniform.hasPDF + - MeasureTheory.pdf.IsUniform.cond + - PMF.uniformOfFinset_apply_of_not_mem + - PMF.toOuterMeasure_uniformOfFinset_apply + - MeasureTheory.pdf.uniformPDF_ite + - MeasureTheory.pdf.IsUniform.measure_preimage + - PMF.toOuterMeasure_ofMultiset_apply + - MeasureTheory.pdf.IsUniform.pdf_eq_zero_of_measure_eq_zero_or_top + - PMF.support_uniformOfFinset + - MeasureTheory.pdf.IsUniform.toMeasurable_iff + - PMF.uniformOfFintype_apply + - MeasureTheory.pdf.uniformPDF_eq_pdf + - PMF.mem_support_uniformOfFinset_iff + - PMF.uniformOfFinset_apply_of_mem + - PMF.mem_support_uniformOfFintype + - PMF.toOuterMeasure_uniformOfFintype_apply + - PMF.toMeasure_uniformOfFintype_apply + - MeasureTheory.pdf.IsUniform.mul_pdf_integrable + - PMF.mem_support_ofMultiset_iff + - MeasureTheory.pdf.IsUniform.absolutelyContinuous + - MeasureTheory.pdf.IsUniform.toMeasurable + - path: Mathlib/Order/Filter/Prod.lean + theorems: + - Filter.prod_pure + - Filter.disjoint_prod + - Filter.mem_coprod_iff + - Filter.inf_prod + - Filter.frequently_prod_and + - Filter.prod_iInf_left + - Filter.prod_map_right + - Filter.bot_coprod + - Filter.Tendsto.prod_map + - Filter.pure_prod + - Filter.prod_map_map_eq + - Filter.mem_prod_iff_left + - Filter.prod_inf + - Filter.prod_assoc_symm + - Filter.prod_eq_bot + - Filter.prod_principal_principal + - Filter.map_swap4_prod + - Filter.prod_top + - Filter.mem_prod_top + - Filter.principal_coprod_principal + - Filter.eventually_prod_principal_iff + - Filter.map_const_principal_coprod_map_id_principal + - Filter.mem_prod_principal + - Filter.mem_prod_iff + - Filter.prod_inj + - Filter.coprod_neBot_iff + - Filter.prod_iInf_right + - Filter.Eventually.diag_of_prod_left + - Filter.map_prod_map_coprod_le + - Filter.prod_sup + - Filter.map_prod_map_const_id_principal_coprod_principal + - Filter.Eventually.curry + - Filter.comap_prod + - Filter.Eventually.diag_of_prod + - Filter.eventually_prod_iff + - Filter.Eventually.diag_of_prod_right + - Filter.sup_prod + - Filter.prod_neBot + - Filter.prod_map_left + - Filter.prod_inf_prod + - Filter.mem_prod_iff_right + - Filter.Frequently.uncurry + - Filter.eventually_swap_iff + - Filter.compl_mem_coprod + - Filter.coprod_bot + - Filter.bot_coprod_bot + - Filter.tendsto_prod_iff + - Filter.prod_pure_pure + - Filter.tendsto_prod_iff' + - Filter.map_prod + - Filter.map_fst_prod + - Filter.map_snd_prod + - Filter.map_pure_prod + - Filter.prod_comm + - path: Mathlib/RingTheory/Henselian.lean + theorems: + - HenselianLocalRing.TFAE + - isLocalRingHom_of_le_jacobson_bot + - path: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean + theorems: + - CategoryTheory.Limits.biprod.conePointUniqueUpToIso_inv + - CategoryTheory.Limits.biprod.braiding_map_braiding + - CategoryTheory.Limits.biprod.associator_inv_natural + - CategoryTheory.Limits.biproduct.fromSubtype_eq_lift + - "CategoryTheory.Limits.biproduct.\u03B9_\u03C0" + - CategoryTheory.Limits.biproduct.toSubtype_fromSubtype + - CategoryTheory.Limits.biprod.inl_map + - CategoryTheory.Limits.biproduct.isoProduct_inv + - "CategoryTheory.Limits.bicone_\u03B9_\u03C0_ne" + - CategoryTheory.Limits.biprod_isZero_iff + - CategoryTheory.Limits.BinaryBiconeMorphism.ext + - CategoryTheory.Limits.BiconeMorphism.ext + - "CategoryTheory.Limits.biproduct.fromSubtype_\u03C0_subtype" + - "CategoryTheory.Limits.biproduct.\u03B9_\u03C0_self" + - CategoryTheory.Limits.biproduct.isoCoproduct_hom + - CategoryTheory.Limits.biproduct.map_desc + - CategoryTheory.Limits.biproduct.whiskerEquiv_hom_eq_lift + - CategoryTheory.Limits.biprod.associator_natural + - CategoryTheory.Limits.biprod.symmetry' + - "CategoryTheory.Limits.biproduct.matrix_\u03C0" + - "CategoryTheory.Limits.biproduct.fromSubtype_\u03C0" + - "CategoryTheory.Limits.Bicone.\u03B9_of_isLimit" + - CategoryTheory.isIso_right_of_isIso_biprod_map + - "CategoryTheory.Limits.biproduct.\u03B9_toSubtype" + - CategoryTheory.Limits.biprod.inr_map + - CategoryTheory.Limits.biproduct.lift_map + - "CategoryTheory.Limits.biproduct.\u03B9_\u03C0_ne" + - CategoryTheory.Limits.biprod.isoCoprod_inv + - CategoryTheory.Limits.biproduct.isoCoproduct_inv + - "CategoryTheory.Limits.biproduct.\u03B9_matrix" + - CategoryTheory.Limits.biproduct.whiskerEquiv_inv_eq_lift + - "CategoryTheory.Limits.biproduct.\u03B9_map" + - CategoryTheory.Limits.biprod_isoCoprod_hom + - CategoryTheory.Limits.biprod.map_eq_map' + - CategoryTheory.Limits.biprod.isoProd_hom + - CategoryTheory.Limits.biprod.isoProd_inv + - CategoryTheory.Limits.biprod.braiding'_eq_braiding + - CategoryTheory.Limits.biproduct.map_eq_map' + - "CategoryTheory.Limits.Bicone.\u03C0_of_isColimit" + - "CategoryTheory.Limits.biproduct.eqToHom_comp_\u03B9" + - CategoryTheory.Limits.biproduct.matrix_components + - "CategoryTheory.Limits.bicone_\u03B9_\u03C0_self" + - CategoryTheory.Limits.biproduct.isoProduct_hom + - CategoryTheory.Limits.biproduct.conePointUniqueUpToIso_inv + - CategoryTheory.Limits.biprod.isIso_inl_iff_id_eq_fst_comp_inl + - CategoryTheory.isIso_left_of_isIso_biprod_map + - "CategoryTheory.Limits.biproduct.\u03B9_toSubtype_subtype" + - CategoryTheory.Limits.biproduct.components_matrix + - CategoryTheory.Limits.biproduct.toSubtype_eq_desc + - "CategoryTheory.Limits.biproduct.\u03C0_comp_eqToHom" + - CategoryTheory.Limits.biprod.symmetry + - CategoryTheory.Limits.biproduct.fromSubtype_toSubtype + - path: Mathlib/Algebra/EuclideanDomain/Defs.lean + theorems: + - EuclideanDomain.gcdA_zero_left + - EuclideanDomain.val_dvd_le + - EuclideanDomain.mod_add_div' + - EuclideanDomain.mod_zero + - EuclideanDomain.mod_eq_sub_mul_div + - EuclideanDomain.GCD.induction + - EuclideanDomain.gcdB_zero_left + - EuclideanDomain.gcd_zero_left + - EuclideanDomain.xgcdAux_rec + - EuclideanDomain.xgcd_zero_left + - EuclideanDomain.mul_right_not_lt + - path: Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean + theorems: + - MeasurableSpace.inf_le_invariants_comp + - MeasurableSpace.measurable_invariants_dom + - MeasurableSpace.measurable_invariants_of_semiconj + - MeasurableSpace.le_invariants_iterate + - path: Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean + theorems: + - ProbabilityTheory.isRatCondKernelCDFAux.tendsto_atBot_zero + - ProbabilityTheory.IsCondKernelCDF.toKernel_Iic + - ProbabilityTheory.integrable_stieltjesOfMeasurableRat + - ProbabilityTheory.IsCondKernelCDF.set_lintegral + - ProbabilityTheory.IsCondKernelCDF.integral + - ProbabilityTheory.lintegral_stieltjesOfMeasurableRat + - ProbabilityTheory.lintegral_toKernel_mem + - MeasureTheory.Measure.iInf_rat_gt_prod_Iic + - ProbabilityTheory.isRatCondKernelCDFAux.tendsto_zero_of_antitone + - ProbabilityTheory.set_integral_stieltjesOfMeasurableRat_rat + - ProbabilityTheory.isRatCondKernelCDFAux.bddBelow_range + - ProbabilityTheory.IsRatCondKernelCDF.mono + - ProbabilityTheory.ae_toKernel_eq_one + - ProbabilityTheory.isRatCondKernelCDFAux.mono + - ProbabilityTheory.isRatCondKernelCDFAux.isRatStieltjesPoint_ae + - ProbabilityTheory.integral_stieltjesOfMeasurableRat + - ProbabilityTheory.set_lintegral_stieltjesOfMeasurableRat_rat + - ProbabilityTheory.isRatCondKernelCDFAux.integrable_iInf_rat_gt + - ProbabilityTheory.IsRatCondKernelCDF.tendsto_atBot_zero + - ProbabilityTheory.isRatCondKernelCDFAux.set_integral_iInf_rat_gt + - ProbabilityTheory.isRatCondKernelCDFAux.tendsto_one_of_monotone + - ProbabilityTheory.lintegral_toKernel_univ + - ProbabilityTheory.set_integral_stieltjesOfMeasurableRat + - ProbabilityTheory.IsCondKernelCDF.lintegral + - ProbabilityTheory.set_lintegral_toKernel_univ + - ProbabilityTheory.isRatCondKernelCDFAux.iInf_rat_gt_eq + - ProbabilityTheory.IsRatCondKernelCDF.iInf_rat_gt_eq + - ProbabilityTheory.set_lintegral_toKernel_prod + - ProbabilityTheory.isRatCondKernelCDFAux.measurable_right + - ProbabilityTheory.set_lintegral_stieltjesOfMeasurableRat + - ProbabilityTheory.compProd_toKernel + - ProbabilityTheory.set_lintegral_toKernel_Iic + - ProbabilityTheory.stieltjesOfMeasurableRat_ae_eq + - ProbabilityTheory.IsRatCondKernelCDF.tendsto_atTop_one + - ProbabilityTheory.isRatCondKernelCDFAux.tendsto_atTop_one + - ProbabilityTheory.StieltjesFunction.measurable_measure + - path: Mathlib/Data/Set/Equitable.lean + theorems: + - Set.Subsingleton.equitableOn + - Set.equitableOn_iff_exists_le_le_add_one + - Set.not_equitableOn + - Finset.equitableOn_iff_le_le_add_one + - Set.equitableOn_iff_exists_eq_eq_add_one + - Set.equitableOn_iff_exists_image_subset_icc + - Finset.equitableOn_iff + - path: Mathlib/Analysis/NormedSpace/AffineIsometry.lean + theorems: + - AffineIsometryEquiv.dist_pointReflection_self' + - AffineMap.isOpenMap_linear_iff + - AffineIsometryEquiv.vadd_vsub + - AffineIsometry.dist_map + - AffineIsometry.toAffineMap_injective + - AffineIsometry.coe_toAffineMap + - AffineIsometryEquiv.linearIsometryEquiv_mk' + - AffineIsometryEquiv.dist_pointReflection_self_real + - AffineMap.continuous_linear_iff + - AffineIsometryEquiv.dist_pointReflection_fixed + - AffineIsometryEquiv.dist_pointReflection_self + - AffineIsometryEquiv.range_eq_univ + - AffineIsometry.nndist_map + - AffineIsometryEquiv.symm_constVSub + - AffineIsometryEquiv.linear_eq_linear_isometry + - AffineIsometry.linear_eq_linearIsometry + - AffineIsometry.edist_map + - LinearIsometryEquiv.toAffineIsometryEquiv_linearIsometryEquiv + - LinearIsometryEquiv.coe_toAffineIsometryEquiv + - LinearIsometry.toAffineIsometry_linearIsometry + - path: Mathlib/Analysis/Calculus/FDeriv/Add.lean + theorems: + - differentiableWithinAt_const_sub_iff + - HasFDerivAtFilter.sub + - HasStrictFDerivAt.sub + - differentiableAt_neg_iff + - differentiableWithinAt_neg_iff + - differentiable_neg_iff + - fderiv_sub_const + - differentiableAt_add_const_iff + - HasFDerivAtFilter.sum + - differentiableOn_const_sub_iff + - differentiableOn_sub_const_iff + - fderivWithin_sub_const + - differentiable_const_add_iff + - HasStrictFDerivAt.sum + - fderivWithin_const_sub + - HasStrictFDerivAt.sub_const + - differentiable_add_const_iff + - HasStrictFDerivAt.const_sub + - HasFDerivAtFilter.add + - HasFDerivAtFilter.const_sub + - fderivWithin_neg + - fderivWithin_add_const + - differentiableAt_const_sub_iff + - HasFDerivAtFilter.sub_const + - differentiableOn_neg_iff + - differentiableAt_const_add_iff + - differentiable_sub_const_iff + - differentiableOn_const_add_iff + - HasStrictFDerivAt.add + - differentiableOn_add_const_iff + - differentiableWithinAt_sub_const_iff + - fderiv_neg + - fderiv_add_const + - differentiableWithinAt_add_const_iff + - fderiv_const_sub + - differentiableAt_sub_const_iff + - differentiableWithinAt_const_add_iff + - differentiable_const_sub_iff + - fderivWithin_const_add + - fderiv_const_add + - path: Mathlib/GroupTheory/Perm/Cycle/Basic.lean + theorems: + - Equiv.Perm.IsCycleOn.apply_ne + - Equiv.Perm.IsCycle.of_zpow + - Finset.product_self_eq_disjiUnion_perm_aux + - Equiv.Perm.IsCycle.pow_eq_pow_iff + - Equiv.Perm.sameCycle_inv_apply_right + - Int.addRight_one_isCycle + - Equiv.Perm.IsCycleOn.countable + - Equiv.Perm.IsCycle.ne_one + - Equiv.Perm.SameCycle.apply_eq_self_iff + - Equiv.Perm.sameCycle_extendDomain + - Equiv.Perm.sameCycle_pow_left + - Equiv.Perm.SameCycle.symm + - Equiv.Perm.IsCycle.isConj + - Equiv.Perm.IsCycle.sign + - Equiv.Perm.SameCycle.exists_pow_eq' + - Equiv.Perm.sameCycle_zpow_right + - Equiv.Perm.IsCycleOn.apply_mem_iff + - Equiv.Perm.sameCycle_apply_left + - Equiv.Perm.isCycleOn_swap + - Equiv.Perm.IsCycle.support_pow_eq_iff + - Equiv.Perm.SameCycle.trans + - List.Nodup.isCycleOn_formPerm + - Equiv.Perm.SameCycle.exists_pow_eq'' + - "Equiv.Perm.isCycle_swap_mul_aux\u2082" + - Equiv.Perm.sameCycle_pow_right + - Equiv.Perm.IsCycle.of_pow + - Equiv.Perm.IsCycle.extendDomain + - Equiv.Perm.IsCycle.isCycle_pow_pos_of_lt_prime_order + - Equiv.Perm.IsCycleOn.zpow_apply_eq + - "Equiv.Perm.isCycle_swap_mul_aux\u2081" + - Equiv.Perm.IsCycleOn.subtypePerm + - Eq.sameCycle + - Equiv.Perm.SameCycle.of_pow + - Int.addLeft_one_isCycle + - Equiv.Perm.sameCycle_one + - Equiv.Perm.IsCycle.isConj_iff + - Equiv.Perm.isCycle_iff_exists_isCycleOn + - Set.Countable.exists_cycleOn + - Finset.sum_smul_sum_eq_sum_perm + - Equiv.Perm.SameCycle.of_zpow + - Equiv.Perm.isCycle_swap + - Equiv.Perm.isCycleOn_inv + - Equiv.Perm.IsSwap.isCycle + - Equiv.Perm.IsCycleOn.exists_pow_eq' + - Equiv.Perm.isCycleOn_empty + - Equiv.Perm.IsCycleOn.pow_apply_eq_pow_apply + - Equiv.Perm.IsCycleOn.isCycle_subtypePerm + - Equiv.Perm.IsCycle.eq_swap_of_apply_apply_eq_self + - Equiv.Perm.IsCycleOn.extendDomain + - Equiv.Perm.IsCycleOn.zpow_apply_eq_zpow_apply + - Equiv.Perm.sameCycle_zpow_left + - Equiv.Perm.IsCycle.exists_pow_eq + - Equiv.Perm.sameCycle_inv + - Equiv.Perm.IsCycleOn.pow_apply_eq + - Equiv.Perm.IsCycle.conj + - Finset.exists_cycleOn + - Equiv.Perm.sameCycle_conj + - Equiv.Perm.IsCycle.sameCycle + - Set.prod_self_eq_iUnion_perm + - Equiv.Perm.IsCycle.swap_mul + - Equiv.Perm.isCycle_iff_sameCycle + - Equiv.Perm.isCycleOn_one + - Equiv.Perm.IsCycle.orderOf + - Equiv.Perm.IsCycle.support_congr + - Finset.product_self_eq_disjiUnion_perm + - Equiv.Perm.IsCycle.pow_iff + - Equiv.Perm.sameCycle_apply_right + - Equiv.Perm.IsCycleOn.exists_pow_eq + - Equiv.Perm.IsCycleOn.conj + - Equiv.Perm.IsCycle.pow_eq_one_iff + - Equiv.Perm.sameCycle_inv_apply_left + - Equiv.Perm.sameCycle_subtypePerm + - path: Mathlib/Computability/ContextFreeGrammar.lean + theorems: + - ContextFreeRule.Rewrites.exists_parts + - ContextFreeRule.rewrites_iff + - ContextFreeGrammar.mem_language_iff + - ContextFreeGrammar.Derives.append_right + - ContextFreeGrammar.Derives.append_left + - ContextFreeRule.rewrites_of_exists_parts + - ContextFreeRule.Rewrites.append_right + - ContextFreeRule.Rewrites.append_left + - path: Mathlib/RingTheory/Flat/Basic.lean + theorems: + - Module.Flat.iff_lTensor_injective' + - Module.Flat.lTensor_inj_iff_rTensor_inj + - Module.Flat.of_projective_surjective + - Module.Flat.iff_characterModule_baer + - Module.Flat.injective_characterModule_iff_rTensor_preserves_injective_linearMap + - Module.Flat.iff_rTensor_injective' + - Module.Flat.iff_rTensor_injective + - Module.Flat.of_linearEquiv + - Module.Flat.iff_lTensor_injective + - Module.Flat.iff_rTensor_preserves_injective_linearMap + - path: Mathlib/Topology/DiscreteQuotient.lean + theorems: + - DiscreteQuotient.ofLE_refl_apply + - DiscreteQuotient.comp_finsetClopens + - DiscreteQuotient.LEComap.comp + - DiscreteQuotient.exists_of_compat + - DiscreteQuotient.finsetClopens_inj + - DiscreteQuotient.proj_bot_inj + - DiscreteQuotient.comap_mono + - DiscreteQuotient.map_ofLE + - DiscreteQuotient.map_id + - DiscreteQuotient.ofLE_map + - DiscreteQuotient.ofLE_refl + - DiscreteQuotient.isClopen_setOf_rel + - DiscreteQuotient.fiber_subset_ofLE + - DiscreteQuotient.map_comp + - DiscreteQuotient.eq_of_forall_proj_eq + - DiscreteQuotient.ofLE_ofLE + - DiscreteQuotient.toSetoid_injective + - path: Mathlib/RingTheory/RingHom/Integral.lean + theorems: + - RingHom.isIntegral_respectsIso + - RingHom.isIntegral_stableUnderComposition + - RingHom.isIntegral_stableUnderBaseChange + - path: Mathlib/Tactic/NormNum/BigOperators.lean + theorems: + - Mathlib.Meta.NormNum.Finset.sum_empty + - Mathlib.Meta.Finset.insert_eq_cons + - Mathlib.Meta.List.range_succ_eq_map' + - Mathlib.Meta.Multiset.range_zero' + - Mathlib.Meta.NormNum.Finset.prod_empty + - Mathlib.Meta.Finset.univ_eq_elems + - Mathlib.Meta.Finset.range_zero' + - Mathlib.Meta.Finset.range_succ' + - Mathlib.Meta.Multiset.range_succ' + - Mathlib.Meta.List.range_zero' + - path: Mathlib/Data/Option/Basic.lean + theorems: + - Option.pmap_map + - Option.map_bind' + - Option.pbind_eq_none + - Option.elim_apply + - Option.orElse_eq_none + - Option.mem_pmem + - Option.map_injective + - Option.casesOn'_eq_elim + - Option.casesOn'_none_coe + - Option.orElse_none' + - Option.pmap_eq_none_iff + - Option.elim_none_some + - Option.forall_mem_map + - Option.getD_default_eq_iget + - "Option.elim_comp\u2082" + - Option.elim_comp + - Option.map_pmap + - Option.exists_mem_map + - Option.bind_eq_some' + - Option.orElse_eq_some + - Option.map_injective' + - Option.pmap_eq_map + - Option.bind_pmap + - Option.map_bind + - Option.map_pbind + - Option.pbind_map + - Option.map_comm + - Option.none_orElse' + - Option.pbind_eq_bind + - Option.pmap_eq_some_iff + - Option.join_pmap_eq_pmap_join + - Option.liftOrGet_choice + - Option.mem_map_of_injective + - Option.pmap_bind + - Option.mem_map + - Option.pbind_eq_some + - path: Mathlib/Order/Filter/Ker.lean + theorems: + - Filter.ker_eq_univ + - Filter.ker_comap + - Filter.ker_pure + - path: Mathlib/Order/GaloisConnection.lean + theorems: + - OrderIso.bddAbove_preimage + - GaloisConnection.l_iSup + - GaloisInsertion.l_biInf_u + - sSup_image2_eq_sSup_sSup + - GaloisInsertion.l_biSup_u + - GaloisInsertion.l_inf_u + - GaloisInsertion.l_sSup_u_image + - GaloisInsertion.l_iInf_of_ul_eq_self + - GaloisInsertion.l_sup_u + - GaloisConnection.bddAbove_l_image + - GaloisConnection.l_sup + - GaloisInsertion.l_biInf_of_ul_eq_self + - GaloisConnection.isLUB_l_image + - sInf_image2_eq_sInf_sInf + - GaloisConnection.l_sSup + - GaloisConnection.u_eq + - GaloisInsertion.l_sInf_u_image + - GaloisConnection.upperBounds_l_image + - GaloisConnection.compl + - path: Mathlib/Analysis/BoxIntegral/Basic.lean + theorems: + - BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO + - BoxIntegral.hasIntegral_iff + - BoxIntegral.Integrable.hasIntegral + - BoxIntegral.integralSum_smul + - BoxIntegral.HasIntegral.neg + - BoxIntegral.Integrable.sum_integral_congr + - BoxIntegral.integralSum_add + - BoxIntegral.Integrable.dist_integralSum_sum_integral_le_of_memBaseSet_of_iUnion_eq + - BoxIntegral.integral_smul + - BoxIntegral.integralSum_neg + - BoxIntegral.integrable_of_continuousOn + - BoxIntegral.integralSum_sub_partitions + - BoxIntegral.Integrable.tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity + - BoxIntegral.HasIntegral.sum + - BoxIntegral.integralSum_disjUnion + - BoxIntegral.norm_integral_le_of_le_const + - BoxIntegral.integral_nonneg + - BoxIntegral.Integrable.dist_integralSum_integral_le_of_memBaseSet + - BoxIntegral.Integrable.to_subbox_aux + - BoxIntegral.HasIntegral.add + - BoxIntegral.Integrable.dist_integralSum_le_of_memBaseSet + - BoxIntegral.Integrable.of_smul + - BoxIntegral.Integrable.tendsto_integralSum_sum_integral + - BoxIntegral.HasIntegral.smul + - BoxIntegral.HasIntegral.of_le_Henstock_of_forall_isLittleO + - BoxIntegral.hasIntegral_zero + - BoxIntegral.norm_integral_le_of_norm_le + - BoxIntegral.HasIntegral.of_mul + - BoxIntegral.integralSum_biUnion_partition + - BoxIntegral.integral_neg + - BoxIntegral.HasIntegral.mcShane_of_forall_isLittleO + - BoxIntegral.Integrable.convergenceR_cond + - BoxIntegral.integrable_iff_cauchy_basis + - BoxIntegral.integralSum_biUnionTagged + - BoxIntegral.Integrable.cauchy_map_integralSum_toFilteriUnion + - BoxIntegral.HasIntegral.sub + - path: Mathlib/Topology/VectorBundle/Hom.lean + theorems: + - Pretrivialization.continuousLinearMap_symm_apply' + - Pretrivialization.continuousOn_continuousLinearMapCoordChange + - Pretrivialization.continuousLinearMapCoordChange_apply + - path: Mathlib/Data/Rat/Star.lean + theorems: + - Rat.addSubmonoid_closure_range_mul_self + - Rat.addSubmonoid_closure_range_pow + - NNRat.addSubmonoid_closure_range_mul_self + - path: Mathlib/Geometry/Euclidean/Basic.lean + theorems: + - EuclideanGeometry.inter_eq_singleton_orthogonalProjection + - EuclideanGeometry.reflection_orthogonal_vadd + - EuclideanGeometry.reflection_symm + - EuclideanGeometry.dist_reflection_eq_of_mem + - EuclideanGeometry.reflection_reflection + - EuclideanGeometry.inter_eq_singleton_orthogonalProjectionFn + - EuclideanGeometry.orthogonalProjection_vadd_eq_self + - EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq + - EuclideanGeometry.inner_weightedVSub + - EuclideanGeometry.dist_sq_smul_orthogonal_vadd_smul_orthogonal_vadd + - EuclideanGeometry.reflection_eq_self_iff + - EuclideanGeometry.reflection_mem_of_le_of_mem + - EuclideanGeometry.dist_affineCombination + - EuclideanGeometry.eq_orthogonalProjection_of_eq_subspace + - EuclideanGeometry.dist_smul_vadd_sq + - EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self + - EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff + - EuclideanGeometry.eq_reflection_of_eq_subspace + - EuclideanGeometry.orthogonalProjection_eq_self_iff + - EuclideanGeometry.orthogonalProjection_orthogonalProjection + - EuclideanGeometry.eq_of_dist_eq_of_dist_eq_of_finrank_eq_two + - EuclideanGeometry.orthogonalProjectionFn_mem + - EuclideanGeometry.dist_reflection + - EuclideanGeometry.eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two + - EuclideanGeometry.orthogonalProjectionFn_mem_orthogonal + - EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection + - EuclideanGeometry.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq + - path: Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean + theorems: + - CategoryTheory.Limits.WalkingParallelPairHom.id.sizeOf_spec' + - "CategoryTheory.Limits.coequalizer.\u03C0_colimMap_desc" + - "CategoryTheory.Limits.Fork.app_one_eq_\u03B9_comp_right" + - CategoryTheory.Limits.Cofork.coequalizer_ext + - "CategoryTheory.Limits.Cofork.app_zero_eq_comp_\u03C0_left" + - CategoryTheory.Limits.Cofork.condition + - "CategoryTheory.Limits.Cofork.ofCocone_\u03B9" + - CategoryTheory.Limits.coequalizer.isoTargetOfSelf_hom + - "CategoryTheory.Limits.Fork.hom_comp_\u03B9" + - CategoryTheory.Limits.Fork.condition + - CategoryTheory.Limits.equalizer.isoSourceOfSelf_inv + - CategoryTheory.Limits.Fork.equalizer_ext + - "CategoryTheory.Limits.Fork.ofCone_\u03C0" + - "CategoryTheory.Limits.Cone.ofFork_\u03C0" + - "CategoryTheory.Limits.Fork.\u03C0_comp_hom" + - CategoryTheory.Limits.WalkingParallelPairHom.assoc + - CategoryTheory.Limits.map_lift_equalizerComparison + - "CategoryTheory.Limits.Cofork.app_zero_eq_comp_\u03C0_right" + - "CategoryTheory.Limits.Fork.app_one_eq_\u03B9_comp_left" + - CategoryTheory.Limits.WalkingParallelPairHom.comp_id + - CategoryTheory.Limits.coequalizerComparison_map_desc + - path: Mathlib/Topology/Compactification/OnePoint.lean + theorems: + - OnePoint.not_continuous_cofiniteTopology_of_symm + - OnePoint.isClosed_iff_of_mem + - OnePoint.isOpen_compl_image_coe + - OnePoint.isClosed_image_coe + - OnePoint.tendsto_nhds_infty' + - OnePoint.continuousAt_coe + - OnePoint.not_mem_range_coe_iff + - OnePoint.nhds_infty_eq + - OnePoint.coe_preimage_infty + - OnePoint.le_nhds_infty + - OnePoint.nhdsWithin_compl_infty_eq + - OnePoint.isOpen_iff_of_not_mem + - OnePoint.isOpen_iff_of_mem' + - OnePoint.isOpen_image_coe + - OnePoint.hasBasis_nhds_infty + - OnePoint.isClosed_infty + - OnePoint.ultrafilter_le_nhds_infty + - OnePoint.isOpen_iff_of_mem + - OnePoint.denseRange_coe + - OnePoint.isClosed_iff_of_not_mem + - OnePoint.tendsto_nhds_infty + - OnePoint.compl_image_coe + - OnePoint.comap_coe_nhds_infty + - path: Mathlib/LinearAlgebra/PerfectPairing.lean + theorems: + - Submodule.map_dualCoannihilator_linearEquiv_flip + - Submodule.dualCoannihilator_map_linearEquiv_flip + - LinearEquiv.flip_flip + - LinearEquiv.isReflexive_of_equiv_dual_of_isReflexive + - LinearEquiv.trans_dualMap_symm_flip + - Submodule.map_dualAnnihilator_linearEquiv_flip_symm + - Submodule.dualAnnihilator_map_linearEquiv_flip_symm + - path: Mathlib/RingTheory/LaurentSeries.lean + theorems: + - LaurentSeries.coeff_coe_powerSeries + - LaurentSeries.powerSeriesPart_eq_zero + - LaurentSeries.ofPowerSeries_powerSeriesPart + - LaurentSeries.powerSeriesPart_zero + - LaurentSeries.single_order_mul_powerSeriesPart + - PowerSeries.coeff_coe + - PowerSeries.coe_smul + - path: Mathlib/RingTheory/ClassGroup.lean + theorems: + - FractionalIdeal.canonicalEquiv_mk0 + - card_classGroup_eq_one + - ClassGroup.mk_mk0 + - ClassGroup.mk0_surjective + - ClassGroup.equiv_mk0 + - card_classGroup_eq_one_iff + - coe_toPrincipalIdeal + - ClassGroup.mk0_eq_mk0_inv_iff + - ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring + - ClassGroup.Quot_mk_eq_mk + - ClassGroup.mk0_integralRep + - ClassGroup.mk_eq_one_iff + - ClassGroup.integralRep_mem_nonZeroDivisors + - ClassGroup.mk_eq_mk + - toPrincipalIdeal_eq_iff + - mem_principal_ideals_iff + - ClassGroup.mk0_eq_mk0_iff + - ClassGroup.mk_canonicalEquiv + - ClassGroup.mk_eq_mk_of_coe_ideal + - ClassGroup.equiv_mk + - ClassGroup.induction + - path: Mathlib/Geometry/RingedSpace/Basic.lean + theorems: + - AlgebraicGeometry.RingedSpace.basicOpen_res_eq + - AlgebraicGeometry.RingedSpace.mem_basicOpen + - AlgebraicGeometry.RingedSpace.basicOpen_mul + - AlgebraicGeometry.RingedSpace.basicOpen_res + - AlgebraicGeometry.RingedSpace.isUnit_of_isUnit_germ + - AlgebraicGeometry.RingedSpace.isUnit_res_basicOpen + - AlgebraicGeometry.RingedSpace.mem_top_basicOpen + - AlgebraicGeometry.RingedSpace.basicOpen_le + - AlgebraicGeometry.RingedSpace.basicOpen_of_isUnit + - AlgebraicGeometry.RingedSpace.isUnit_res_of_isUnit_germ + - path: Mathlib/Data/Finset/NoncommProd.lean + theorems: + - Finset.noncommProd_eq_prod + - Finset.noncommProd_mul_distrib_aux + - Finset.noncommProd_cons' + - Multiset.noncommFold_cons + - Multiset.noncommFold_eq_fold + - Finset.noncommProd_map + - Finset.noncommProd_erase_mul + - Finset.noncommProd_eq_pow_card + - Finset.noncommProd_insert_of_not_mem + - Multiset.noncommProd_induction + - Multiset.noncommProd_erase_mul + - Finset.noncommProd_insert_of_not_mem' + - Multiset.noncommFoldr_cons + - Finset.noncommProd_mul_distrib + - Multiset.noncommProd_coe + - MonoidHom.pi_ext + - Multiset.noncommProd_eq_prod + - Multiset.noncommProd_map + - Multiset.noncommProd_cons' + - Finset.noncommProd_cons + - Finset.noncommProd_singleton + - Finset.noncommProd_toFinset + - Multiset.noncommProd_cons + - Finset.noncommProd_union_of_disjoint + - Multiset.mul_noncommProd_erase + - Multiset.noncommProd_map_aux + - Multiset.noncommProd_add + - Finset.noncommProd_induction + - Multiset.noncommFoldr_coe + - Finset.noncommProd_congr + - Multiset.noncommFold_coe + - Finset.noncommProd_commute + - Finset.mul_noncommProd_erase + - Finset.noncommProd_lemma + - Multiset.noncommProd_commute + - Multiset.noncommProd_eq_pow_card + - Finset.noncommProd_mul_single + - path: Mathlib/Analysis/NormedSpace/Int.lean + theorems: + - Int.norm_coe_units + - Int.nnnorm_coe_units + - Int.toNat_add_toNat_neg_eq_norm + - Int.toNat_add_toNat_neg_eq_nnnorm + - path: Mathlib/Data/Finsupp/Multiset.lean + theorems: + - Finsupp.toMultiset_map + - Finsupp.sum_id_lt_of_lt + - Multiset.toFinsupp_inter + - Finsupp.mem_toMultiset + - Finsupp.toMultiset_sum_single + - Finsupp.prod_toMultiset + - Finsupp.toMultiset_inf + - Multiset.toFinsupp_union + - Finsupp.toMultiset_single + - Multiset.toFinsupp_singleton + - Finsupp.count_toMultiset + - Finsupp.toMultiset_strictMono + - Sym.coe_equivNatSumOfFintype_symm_apply + - Finsupp.toFinset_toMultiset + - Multiset.toFinsupp_sum_eq + - Finsupp.card_toMultiset + - path: Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean + theorems: + - "ModuleCat.Free.\u03BC_natural" + - ModuleCat.Free.right_unitality + - CategoryTheory.Free.single_comp_single + - ModuleCat.Free.associativity + - CategoryTheory.Free.lift_map_single + - ModuleCat.Free.left_unitality + - path: Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean + theorems: + - SimpleGraph.end_hom_mk_of_mk + - SimpleGraph.ComponentCompl.disjoint_right + - SimpleGraph.ComponentCompl.mem_of_adj + - SimpleGraph.ComponentCompl.hom_trans + - SimpleGraph.componentComplMk_eq_of_adj + - SimpleGraph.ComponentCompl.pairwise_disjoint + - SimpleGraph.ComponentCompl.ind + - SimpleGraph.ComponentCompl.hom_eq_iff_not_disjoint + - SimpleGraph.ComponentCompl.subset_hom + - SimpleGraph.ComponentCompl.hom_refl + - SimpleGraph.ComponentCompl.infinite_iff_in_all_ranges + - SimpleGraph.infinite_iff_in_eventualRange + - SimpleGraph.ComponentCompl.supp_injective + - path: Mathlib/Algebra/Module/LinearMap/End.lean + theorems: + - LinearMap.smulRight_apply_eq_zero_iff + - LinearMap.iterate_succ + - LinearMap.iterate_bijective + - Module.End_isUnit_inv_apply_apply_of_isUnit + - LinearMap.iterate_injective + - LinearMap.zero_smulRight + - LinearMap.smulRight_zero + - LinearMap.injective_of_iterate_injective + - LinearMap.commute_pow_left_of_commute + - LinearMap.iterate_surjective + - LinearMap.surjective_of_iterate_surjective + - LinearMap.pow_map_zero_of_le + - Module.End_isUnit_apply_inv_apply_of_isUnit + - path: Mathlib/Algebra/CubicDiscriminant.lean + theorems: + - Cubic.degree_of_b_eq_zero + - Cubic.of_a_eq_zero + - Cubic.monic_of_b_eq_one + - Cubic.natDegree_of_a_eq_zero + - Cubic.natDegree_of_b_ne_zero + - Cubic.map_roots + - Cubic.C_mul_prod_X_sub_C_eq + - Cubic.degree_of_a_eq_zero + - Cubic.b_eq_three_roots + - Cubic.degree_of_d_eq_zero + - Cubic.of_d_eq_zero + - Cubic.d_eq_three_roots + - Cubic.c_of_eq + - Cubic.disc_ne_zero_iff_roots_ne + - Cubic.natDegree_of_c_ne_zero + - Cubic.monic_of_c_eq_one + - Cubic.toPoly_eq_zero_iff + - Cubic.map_toPoly + - Cubic.eq_sum_three_roots + - Cubic.d_of_eq + - Cubic.coeffs + - Cubic.disc_ne_zero_iff_roots_nodup + - Cubic.leadingCoeff_of_c_ne_zero + - Cubic.leadingCoeff_of_b_ne_zero + - Cubic.of_b_eq_zero + - Cubic.leadingCoeff_of_c_eq_zero + - Cubic.a_of_eq + - Cubic.degree_of_d_ne_zero + - Cubic.mem_roots_iff + - Cubic.splits_iff_roots_eq_three + - Cubic.card_roots_le + - Cubic.degree_of_c_ne_zero + - Cubic.degree_of_c_eq_zero + - Cubic.splits_iff_card_roots + - Cubic.prod_X_sub_C_eq + - Cubic.b_of_eq + - Cubic.ne_zero + - Cubic.eq_prod_three_roots + - Cubic.degree_of_b_ne_zero + - Cubic.of_c_eq_zero + - Cubic.c_eq_three_roots + - Cubic.natDegree_of_c_eq_zero + - Cubic.card_roots_of_disc_ne_zero + - Cubic.monic_of_a_eq_one + - Cubic.monic_of_d_eq_one + - Cubic.disc_eq_prod_three_roots + - path: Mathlib/Order/Filter/Lift.lean + theorems: + - Filter.HasBasis.lift' + - Filter.comap_lift_eq + - Filter.map_lift'_eq + - Filter.lift'_bot + - Filter.lift'_neBot_iff + - Filter.HasBasis.mem_lift_iff + - Filter.tendsto_lift + - Filter.HasBasis.lift + - Filter.prod_def + - Filter.lift_lift'_assoc + - Filter.mem_lift_sets + - Filter.lift_iInf_of_directed + - Filter.map_lift_eq + - Filter.lift'_iInf + - Filter.lift'_iInf_of_map_univ + - Filter.tendsto_prod_self_iff + - Filter.lift_neBot_iff + - Filter.tendsto_lift' + - Filter.lift'_pure + - Filter.lift'_inf + - Filter.lift_inf + - Filter.lift_principal2 + - Filter.prod_lift_lift + - Filter.lift_iInf + - Filter.lift_top + - Filter.sInter_lift_sets + - path: Mathlib/Geometry/Manifold/ChartedSpace.lean + theorems: + - ChartedSpace.locallyCompactSpace + - closedUnderRestriction_iff_id_le + - ChartedSpace.isOpen_iff + - groupoid_of_pregroupoid_le + - modelProd_range_prod_id + - ChartedSpace.locallyConnectedSpace + - StructureGroupoid.restriction_mem_maximalAtlas_subtype + - ChartedSpaceCore.open_source' + - PartialHomeomorph.singleton_hasGroupoid + - StructureGroupoid.id_mem_maximalAtlas + - ChartedSpaceCore.open_target + - ChartedSpace.secondCountable_of_sigma_compact + - TopologicalSpace.Opens.chartAt_subtype_val_symm_eventuallyEq + - ChartedSpace.secondCountable_of_countable_cover + - StructureGroupoid.restriction_in_maximalAtlas + - StructureGroupoid.mem_maximalAtlas_of_eqOnSource + - TopologicalSpace.Opens.chart_eq' + - mem_pregroupoid_of_eqOnSource + - chartedSpaceSelf_prod + - StructureGroupoid.mem_maximalAtlas_of_mem_groupoid + - StructureGroupoid.compatible_of_mem_maximalAtlas + - TopologicalSpace.Opens.chart_eq + - path: Mathlib/Data/Multiset/Nodup.lean + theorems: + - Multiset.nodup_iff_ne_cons_cons + - Multiset.nodup_iff_count_eq_one + - Multiset.Nodup.sigma + - Multiset.nodup_iff_count_le_one + - Multiset.Nodup.add_iff + - Multiset.nodup_union + - Multiset.map_eq_map_of_bij_of_nodup + - Multiset.Nodup.product + - Multiset.Nodup.mem_erase_iff + - Multiset.count_eq_of_nodup + - Multiset.mem_sub_of_nodup + - Multiset.nodup_bind + - path: Mathlib/RingTheory/Discriminant.lean + theorems: + - Algebra.discr_powerBasis_eq_prod'' + - Algebra.discr_zero_of_not_linearIndependent + - Algebra.discr_eq_discr_of_algEquiv + - Algebra.discr_isIntegral + - Algebra.discr_powerBasis_eq_prod + - Algebra.discr_eq_discr + - Algebra.discr_reindex + - Algebra.discr_powerBasis_eq_prod' + - Algebra.discr_of_matrix_vecMul + - Algebra.discr_powerBasis_eq_norm + - Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two + - path: Mathlib/RingTheory/Polynomial/SeparableDegree.lean + theorems: + - Polynomial.HasSeparableContraction.eq_degree + - Polynomial.contraction_degree_eq_or_insep + - Polynomial.IsSeparableContraction.degree_eq + - Polynomial.IsSeparableContraction.dvd_degree' + - Irreducible.hasSeparableContraction + - path: Mathlib/Analysis/Convex/Intrinsic.lean + theorems: + - intrinsicFrontier_singleton + - intrinsicFrontier_empty + - intrinsicInterior_empty + - AffineIsometry.image_intrinsicClosure + - intrinsicClosure_mono + - intrinsicInterior_nonempty + - intrinsicClosure_idem + - intrinsicClosure_singleton + - aux + - intrinsicFrontier_union_intrinsicInterior + - IsClosed.intrinsicClosure + - AffineIsometry.image_intrinsicFrontier + - Set.Nonempty.intrinsicInterior + - intrinsicClosure_diff_intrinsicFrontier + - intrinsicClosure_eq_closure + - intrinsicInterior_union_intrinsicFrontier + - AffineIsometry.image_intrinsicInterior + - intrinsicInterior_singleton + - intrinsicClosure_nonempty + - path: Mathlib/Data/List/Intervals.lean + theorems: + - List.Ico.succ_singleton + - List.Ico.map_sub + - List.Ico.filter_lt_of_le_bot + - List.Ico.filter_le + - List.Ico.map_add + - List.Ico.eq_cons + - List.Ico.zero_bot + - List.Ico.filter_le_of_le + - List.Ico.filter_lt_of_ge + - List.Ico.filter_lt + - List.Ico.filter_lt_of_top_le + - List.Ico.filter_le_of_bot + - List.Ico.nodup + - List.Ico.not_mem_top + - List.Ico.pred_singleton + - List.Ico.mem + - List.Ico.eq_nil_of_le + - List.Ico.eq_empty_iff + - List.Ico.append_consecutive + - List.Ico.succ_top + - List.Ico.filter_le_of_le_bot + - List.Ico.inter_consecutive + - List.Ico.filter_le_of_top_le + - List.Ico.trichotomy + - List.Ico.chain'_succ + - List.Ico.pairwise_lt + - path: Mathlib/CategoryTheory/Limits/Shapes/Types.lean + theorems: + - CategoryTheory.Limits.Types.Pushout.inl_rel'_inl_iff + - CategoryTheory.Limits.Types.pushoutCocone_inl_eq_inr_iff_of_isColimit + - CategoryTheory.Limits.Types.pullbackIsoPullback_inv_fst_apply + - CategoryTheory.Limits.Types.Pushout.equivalence_rel' + - "CategoryTheory.Limits.Types.pi_lift_\u03C0_apply'" + - CategoryTheory.Limits.Types.Pushout.inl_eq_inr_iff + - CategoryTheory.Limits.Types.Pushout.quot_mk_eq_iff + - CategoryTheory.Limits.Types.initial_iff_empty + - CategoryTheory.Limits.Types.pullbackIsoPullback_inv_snd_apply + - "CategoryTheory.Limits.Types.pi_map_\u03C0_apply'" + - CategoryTheory.Limits.Types.pushoutCocone_inl_eq_inr_iff_of_iso + - CategoryTheory.Limits.Types.coequalizer_preimage_image_eq_of_preimage_eq + - CategoryTheory.Limits.Types.binaryCofan_isColimit_iff + - CategoryTheory.Limits.Types.Pushout.inl_rel'_inr_iff + - CategoryTheory.Limits.Types.unique_of_type_equalizer + - CategoryTheory.Limits.Types.Pushout.condition + - CategoryTheory.Limits.Types.Pushout.Rel'.symm + - CategoryTheory.Limits.Types.Pushout.inr_rel'_inr_iff + - CategoryTheory.Limits.Types.pushoutCocone_inl_eq_inr_imp_of_iso + - path: Mathlib/SetTheory/Ordinal/NaturalOps.lean + theorems: + - Ordinal.nmul_le_iff + - "Ordinal.lt_nmul_iff\u2083" + - Ordinal.succ_nadd + - Ordinal.nmul_le_nmul_of_nonneg_right + - Ordinal.nmul_nadd_lt + - NatOrdinal.mul_le_nmul + - "Ordinal.nmul_nadd_le\u2083" + - Ordinal.nadd_one + - Ordinal.nmul_nadd + - "Ordinal.lt_nmul_iff\u2083'" + - Ordinal.one_nmul + - Ordinal.nmul_nadd_le + - "Ordinal.nmul_le_iff\u2083" + - Ordinal.le_self_nadd + - Ordinal.blsub_nadd_of_mono + - "Ordinal.nmul_nadd_lt\u2083" + - Ordinal.nmul_assoc + - Ordinal.nadd_def + - Ordinal.nmul_le_nmul_of_nonneg_left + - Ordinal.nadd_le_iff + - Ordinal.add_le_nadd + - Ordinal.nadd_succ + - Ordinal.nmul_comm + - Ordinal.nadd_zero + - Ordinal.lt_nadd_iff + - "Ordinal.nmul_le_iff\u2083'" + - "Ordinal.nmul_nadd_le\u2083'" + - Ordinal.nadd_nmul + - Ordinal.nadd_le_nadd_left + - Ordinal.zero_nadd + - Ordinal.toNatOrdinal_cast_nat + - Ordinal.nadd_le_nadd_right + - Ordinal.nadd_comm + - Ordinal.nmul_zero + - Ordinal.one_nadd + - NatOrdinal.toOrdinal_cast_nat + - Ordinal.nadd_assoc + - Ordinal.nmul_one + - Ordinal.nmul_lt_nmul_of_pos_right + - Ordinal.nmul_lt_nmul_of_pos_left + - Ordinal.nmul_succ + - Ordinal.le_nadd_self + - Ordinal.lt_nmul_iff + - Ordinal.zero_nmul + - Ordinal.nmul_def + - Ordinal.succ_nmul + - Ordinal.nat_nadd + - "Ordinal.nmul_nadd_lt\u2083'" + - path: Mathlib/NumberTheory/Rayleigh.lean + theorems: + - Irrational.beattySeq_symmDiff_beattySeq_pos + - compl_beattySeq' + - Irrational.beattySeq'_pos_eq + - beattySeq'_symmDiff_beattySeq_pos + - beattySeq_symmDiff_beattySeq'_pos + - Beatty.no_anticollision + - Beatty.no_collision + - Beatty.hit_or_miss' + - path: Mathlib/LinearAlgebra/Dimension/Basic.lean + theorems: + - rank_map_le + - Algebra.lift_rank_le_of_injective_injective + - LinearMap.lift_rank_le_of_surjective + - lift_rank_eq_of_equiv_equiv + - Algebra.lift_rank_le_of_surjective_injective + - rank_le_of_injective_injective + - lift_rank_range_le + - LinearMap.rank_le_of_surjective + - rank_submodule_le + - LinearEquiv.lift_rank_eq + - rank_le_of_surjective_injective + - rank_subsingleton + - Algebra.rank_le_of_surjective_injective + - rank_eq_of_equiv_equiv + - lift_rank_le_of_injective_injective + - LinearIndependent.cardinal_le_rank + - rank_range_le + - Algebra.lift_rank_eq_of_equiv_equiv + - lift_rank_le_of_surjective_injective + - Algebra.rank_eq_of_equiv_equiv + - rank_range_of_surjective + - lift_rank_map_le + - LinearIndependent.cardinal_lift_le_rank + - Algebra.rank_le_of_injective_injective + - path: Mathlib/Algebra/Lie/Weights/Basic.lean + theorems: + - LieModule.mem_posFittingComp + - LieModule.isCompl_weightSpace_zero_posFittingComp + - LieModule.trace_toEndomorphism_weightSpace + - LieModule.iInf_lowerCentralSeries_eq_posFittingComp + - LieModule.iSup_weightSpace_eq_top + - LieModule.zero_weightSpace_eq_top_of_nilpotent + - LieModule.exists_weightSpace_zero_le_ker_of_isNoetherian + - LieModule.posFittingCompOf_le_lowerCentralSeries + - LieModule.iSup_ucs_le_weightSpace_zero + - LieModule.zero_weightSpace_eq_top_of_nilpotent' + - LieModule.disjoint_weightSpace + - LieModule.independent_weightSpaceOf + - LieModule.mem_weightSpaceOf + - LieModule.map_posFittingComp_eq + - LieModule.comap_weightSpace_eq_of_injective + - LieModule.posFittingCompOf_le_posFittingComp + - LieModule.isNilpotent_toEndomorphism_sub_algebraMap + - LieModule.coe_weightSpaceOf_zero + - LieModule.weightSpace_weightSpaceOf_map_incl + - LieModule.posFittingComp_map_incl_sup_of_codisjoint + - LieModule.map_weightSpace_eq_of_injective + - LieModule.iSup_weightSpaceOf_eq_top + - LieModule.mem_weightSpace + - LieModule.map_weightSpace_eq + - LieModule.map_weightSpace_le + - LieModule.posFittingComp_eq_bot_of_isNilpotent + - LieModule.mem_posFittingCompOf + - LieModule.iSup_ucs_eq_weightSpace_zero + - LieModule.posFittingComp_le_iInf_lowerCentralSeries + - LieModule.coe_weightSpace_of_top + - LieModule.independent_weightSpace + - LieModule.disjoint_weightSpaceOf + - LieModule.weight_vector_multiplication + - LieModule.isNilpotent_toEndomorphism_weightSpace_zero + - LieModule.lie_mem_maxGenEigenspace_toEndomorphism + - LieModule.posFittingCompOf_eq_bot_of_isNilpotent + - LieModule.isCompl_weightSpaceOf_zero_posFittingCompOf + - LieModule.injOn_weightSpace + - LieModule.map_posFittingComp_le + - LieModule.weightSpace_zero_normalizer_eq_self + - LieModule.isCompl_weightSpace_zero_posFittingComp_aux + - path: Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean + theorems: + - TensorPower.toTensorAlgebra_galgebra_toFun + - TensorAlgebra.toDirectSum_comp_ofDirectSum + - TensorAlgebra.toDirectSum_tensorPower_tprod + - TensorPower.list_prod_gradedMonoid_mk_single + - TensorAlgebra.ofDirectSum_comp_toDirectSum + - TensorAlgebra.mk_reindex_fin_cast + - TensorPower.toTensorAlgebra_gMul + - path: Mathlib/Algebra/Module/CharacterModule.lean + theorems: + - CharacterModule.dual_surjective_of_injective + - CharacterModule.int.divByNat_self + - CharacterModule.exists_character_apply_ne_zero_of_ne_zero + - rTensor_injective_iff_lcomp_surjective + - CharacterModule.dual_surjective_iff_injective + - CharacterModule.eq_zero_of_ofSpanSingleton_apply_self + - CharacterModule.eq_zero_of_character_apply + - path: Mathlib/RepresentationTheory/Action/Basic.lean + theorems: + - "Action.Iso.conj_\u03C1" + - "Action.\u03C1_one" + - path: Mathlib/Topology/EMetricSpace/Basic.lean + theorems: + - EMetric.tendsto_nhdsWithin_nhdsWithin + - edist_le_Ico_sum_edist + - EMetric.closedBall_prod_same + - EMetric.tendstoLocallyUniformly_iff + - edist_pi_le_iff + - EMetric.tendsto_nhdsWithin_nhds + - EMetric.countable_closure_of_compact + - EMetric.exists_ball_subset_ball + - edist_triangle_right + - EMetric.mem_ball_comm + - EMetric.tendstoUniformly_iff + - EMetric.subset_countable_closure_of_almost_dense_set + - edist_triangle_left + - EMetric.diam_pos_iff + - uniformity_dist_of_mem_uniformity + - EMetric.mk_uniformity_basis + - EMetric.diam_pi_le_of_le + - EMetric.diam_pos_iff' + - EMetric.diam_le_iff + - EMetric.diam_iUnion_mem_option + - EMetric.ball_zero + - EMetric.tendstoLocallyUniformlyOn_iff + - EMetric.mem_closedBall_comm + - EMetric.inseparable_iff + - EMetric.diam_union' + - EMetric.mem_closure_iff + - EMetric.mem_closedBall_self + - EMetric.diam_pair + - EMetric.mem_ball' + - EMetric.cauchy_iff + - EMetric.secondCountable_of_almost_dense_set + - EMetric.tendstoUniformlyOn_iff + - EMetric.ball_prod_same + - EMetric.diam_insert + - TopologicalSpace.IsSeparable.exists_countable_dense_subset + - EMetric.diam_image_le_iff + - EMetric.tendsto_atTop + - EMetric.mem_ball_self + - EMetric.uniformEmbedding_iff' + - TopologicalSpace.IsSeparable.separableSpace + - EMetric.subset_countable_closure_of_compact + - edist_pos + - EMetric.diam_triple + - EMetric.mem_closedBall' + - EMetric.isOpen_iff + - edist_congr_left + - EMetric.uniformInducing_iff + - EMetric.diam_union + - path: Mathlib/Geometry/Euclidean/PerpBisector.lean + theorems: + - AffineSubspace.perpBisector_self + - AffineSubspace.mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero + - AffineSubspace.perpBisector_eq_top + - AffineSubspace.left_mem_perpBisector + - AffineSubspace.mem_perpBisector_pointReflection_iff_inner_eq_zero + - AffineSubspace.direction_perpBisector + - AffineSubspace.mem_perpBisector_iff_inner_eq_inner + - AffineSubspace.mem_perpBisector_iff_dist_eq + - AffineSubspace.mem_perpBisector_iff_dist_eq' + - AffineSubspace.mem_perpBisector_iff_inner_eq + - EuclideanGeometry.inner_vsub_vsub_of_dist_eq_of_dist_eq + - AffineSubspace.perpBisector_ne_bot + - Isometry.preimage_perpBisector + - AffineSubspace.midpoint_mem_perpBisector + - AffineSubspace.perpBisector_comm + - AffineSubspace.right_mem_perpBisector + - path: Mathlib/LinearAlgebra/AffineSpace/Basis.lean + theorems: + - AffineBasis.sum_coord_apply_eq_one + - AffineBasis.nonempty + - AffineBasis.coord_apply_combination_of_not_mem + - AffineBasis.linear_combination_coord_eq_self + - AffineBasis.ext_elem + - AffineBasis.coord_apply_combination_of_mem + - AffineBasis.coe_coord_of_subsingleton_eq_one + - AffineBasis.coord_apply_eq + - AffineBasis.coord_apply_ne + - AffineBasis.coord_apply_centroid + - AffineBasis.exists_affine_subbasis + - AffineBasis.surjective_coord + - AffineBasis.coord_reindex + - AffineBasis.basisOf_apply + - AffineBasis.basisOf_reindex + - AffineBasis.coord_apply + - path: Mathlib/Algebra/CharZero/Quotient.lean + theorems: + - AddSubgroup.nsmul_mem_zmultiples_iff_exists_sub_div + - QuotientAddGroup.zmultiples_zsmul_eq_zsmul_iff + - QuotientAddGroup.zmultiples_nsmul_eq_nsmul_iff + - AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div + - path: Mathlib/Order/Hom/Order.lean + theorems: + - OrderHom.coe_iSup + - OrderHom.coe_iInf + - OrderHom.iterate_sup_le_sup_iff + - path: Mathlib/Algebra/Order/Floor/Div.lean + theorems: + - floorDiv_one + - floorDiv_zero + - smul_floorDiv + - ceilDiv_one + - Finsupp.support_floorDiv_subset + - floorDiv_le_ceilDiv + - ceilDiv_zero + - Finsupp.support_ceilDiv_subset + - smul_ceilDiv + - path: Mathlib/Data/Nat/Fib/Zeckendorf.lean + theorems: + - Nat.greatestFib_sub_fib_greatestFib_le_greatestFib + - Nat.greatestFib_eq_zero + - List.IsZeckendorfRep.sum_fib_lt + - Nat.isZeckendorfRep_zeckendorf + - Nat.zeckendorf_sum_fib + - Nat.greatestFib_pos + - Nat.sum_zeckendorf_fib + - path: Mathlib/Algebra/SMulWithZero.lean + theorems: + - boole_smul + - MulActionWithZero.subsingleton + - "smul_inv\u2080" + - ite_zero_smul + - path: Mathlib/Probability/CondCount.lean + theorems: + - ProbabilityTheory.finite_of_condCount_ne_zero + - ProbabilityTheory.condCount_inter + - ProbabilityTheory.condCount_empty + - ProbabilityTheory.condCount_eq_one_of + - ProbabilityTheory.condCount_union + - ProbabilityTheory.condCount_compl + - ProbabilityTheory.pred_true_of_condCount_eq_one + - ProbabilityTheory.condCount_empty_meas + - ProbabilityTheory.condCount_univ + - ProbabilityTheory.condCount_add_compl_eq + - ProbabilityTheory.condCount_self + - ProbabilityTheory.condCount_inter_self + - ProbabilityTheory.condCount_isProbabilityMeasure + - ProbabilityTheory.condCount_eq_zero_iff + - ProbabilityTheory.condCount_inter' + - ProbabilityTheory.condCount_disjoint_union + - path: Mathlib/Data/Vector/Mem.lean + theorems: + - Vector.mem_succ_iff + - Vector.get_mem + - Vector.not_mem_nil + - Vector.not_mem_map_zero + - Vector.mem_cons_iff + - Vector.mem_map_iff + - Vector.mem_of_mem_tail + - Vector.mem_iff_get + - Vector.mem_map_succ_iff + - path: Mathlib/FieldTheory/NormalClosure.lean + theorems: + - IntermediateField.normal_iff_forall_fieldRange_le + - normalClosure_eq_iSup_adjoin' + - Algebra.IsAlgebraic.isNormalClosure_normalClosure + - IntermediateField.normal_iff_forall_map_le + - IntermediateField.normalClosure_of_normal + - Algebra.IsAlgebraic.normalClosure_le_iSup_adjoin + - IntermediateField.normalClosure_def' + - IntermediateField.normalClosure_mono + - IntermediateField.normalClosure_def'' + - Algebra.IsAlgebraic.isNormalClosure_iff + - IntermediateField.normal_iff_forall_map_le' + - path: Mathlib/Algebra/CovariantAndContravariant.lean + theorems: + - covariant_le_of_covariant_lt + - Group.covariant_iff_contravariant + - covariant_flip_iff + - contravariant_le_iff_contravariant_lt_and_eq + - Group.covariant_swap_iff_contravariant_swap + - contravariant_flip_iff + - path: Mathlib/Data/Polynomial/Lifts.lean + theorems: + - Polynomial.lifts_and_natDegree_eq_and_monic + - Polynomial.lifts_iff_liftsRing + - Polynomial.C_mem_lifts + - Polynomial.mapAlg_eq_map + - Polynomial.base_mul_mem_lifts + - Polynomial.mem_lifts + - Polynomial.lifts_and_degree_eq_and_monic + - Polynomial.C'_mem_lifts + - Polynomial.mem_lifts_iff_mem_alg + - Polynomial.X_pow_mem_lifts + - Polynomial.lifts_iff_set_range + - Polynomial.monomial_mem_lifts + - Polynomial.lifts_iff_coeff_lifts + - Polynomial.X_mem_lifts + - Polynomial.lifts_iff_ringHom_rangeS + - Polynomial.monomial_mem_lifts_and_degree_eq + - Polynomial.mem_lifts_and_degree_eq + - Polynomial.smul_mem_lifts + - Polynomial.erase_mem_lifts + - path: Mathlib/Algebra/Associated.lean + theorems: + - Associates.isPrimal_iff + - Associated.pow_pow + - eq_of_prime_pow_eq + - Irreducible.dvd_iff + - succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul + - Prime.irreducible + - dvd_prime_pow + - not_irreducible_one + - prime_mul_iff + - Associated.refl + - Prime.dvd_of_dvd_pow + - Associates.isUnit_iff_eq_bot + - associated_isUnit_mul_left_iff + - DvdNotUnit.not_associated + - Associates.mk_dvdNotUnit_mk_iff + - associated_unit_mul_left + - Associates.isUnit_mk + - DvdNotUnit.ne + - dvdNotUnit_of_dvdNotUnit_associated + - eq_of_prime_pow_eq' + - irreducible_units_mul + - Associates.mk_le_mk_of_dvd + - prime_pow_iff + - comap_prime + - Irreducible.prime_of_isPrimal + - Associated.of_pow_associated_of_prime + - irreducible_mul_iff + - Irreducible.dvd_symm + - pow_injective_of_not_unit + - Associated.map + - associated_one_of_associated_mul_one + - Associates.decompositionMonoid_iff + - associated_iff_eq + - Associates.le_of_mul_le_mul_left + - Irreducible.not_square + - Associates.le_mul_left + - associated_zero_iff_eq_zero + - Associates.dvd_of_mk_le_mk + - Associates.irreducible_mk + - Associates.one_or_eq_of_le_of_prime + - Associated.mul_right + - irreducible_mul_units + - Associates.quot_out + - Associated.trans + - irreducible_or_factor + - Associates.mk_isRelPrime_iff + - Associates.irreducible_iff_prime_iff + - Associated.symm + - Associates.coe_unit_eq_one + - Associates.mk_pow + - not_irreducible_pow + - Associated.isUnit + - associated_one_iff_isUnit + - Associates.mul_eq_one_iff + - DvdNotUnit.not_unit + - associated_of_dvd_dvd + - DvdNotUnit.isUnit_of_irreducible_right + - Associated.of_mul_left + - Prime.pow_dvd_of_dvd_mul_left + - Associated.irreducible + - associated_eq_eq + - Associated.eq_zero_iff + - prime_pow_succ_dvd_mul + - Associated.mul_left + - Associates.mul_mono + - Prime.left_dvd_or_dvd_right_of_dvd_mul + - Associates.dvdNotUnit_of_lt + - Irreducible.isRelPrime_iff_not_dvd + - Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd + - Associated.of_mul_right + - isUnit_of_associated_mul + - Associates.prime_mk + - prime_dvd_prime_iff_eq + - Prime.pow_dvd_of_dvd_mul_right + - Associated.prime + - MulEquiv.prime_iff + - Associates.le_one_iff + - path: Mathlib/Data/Fin/VecNotation.lean + theorems: + - Matrix.neg_cons + - Matrix.sub_cons + - Matrix.tail_cons + - Matrix.vecAppend_eq_ite + - Matrix.empty_vecAppend + - Matrix.empty_vecAlt1 + - Matrix.empty_vecAlt0 + - Matrix.vecAlt0_vecAppend + - Matrix.cons_vec_bit0_eq_alt0 + - Matrix.add_cons + - Matrix.cons_vecAlt0 + - Matrix.range_cons_empty + - Matrix.vecHead_vecAlt1 + - Matrix.range_cons_cons_empty + - Matrix.cons_val_succ' + - Matrix.cons_add_cons + - Matrix.cons_zero_zero + - Matrix.vecAlt1_vecAppend + - Matrix.cons_add + - Matrix.range_cons + - Matrix.cons_sub + - Matrix.cons_sub_cons + - Matrix.cons_val_succ + - Matrix.cons_val_fin_one + - Matrix.cons_vecAppend + - Matrix.cons_vec_bit1_eq_alt1 + - Matrix.cons_vecAlt1 + - Matrix.smul_cons + - Matrix.cons_eq_zero_iff + - path: Mathlib/AlgebraicGeometry/Morphisms/Basic.lean + theorems: + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.targetAffineLocallyPullbackFstOfRightOfStableUnderBaseChange + - AlgebraicGeometry.universallyIsLocalAtTargetOfMorphismRestrict + - AlgebraicGeometry.AffineTargetMorphismProperty.diagonalOfTargetAffineLocally + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.affine_openCover_TFAE + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.diagonal_affine_openCover_TFAE + - AlgebraicGeometry.diagonal_targetAffineLocally_eq_targetAffineLocally + - AlgebraicGeometry.AffineTargetMorphismProperty.respectsIso_mk + - AlgebraicGeometry.PropertyIsLocalAtTarget.openCover_iff + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.affine_target_iff + - AlgebraicGeometry.affine_cancel_right_isIso + - AlgebraicGeometry.PropertyIsLocalAtTarget.openCover_TFAE + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.targetAffineLocallyIsLocal + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.affine_openCover_iff + - AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.stableUnderBaseChange + - AlgebraicGeometry.AffineTargetMorphismProperty.diagonal_respectsIso + - AlgebraicGeometry.universallyIsLocalAtTarget + - AlgebraicGeometry.diagonalTargetAffineLocallyOfOpenCover + - AlgebraicGeometry.targetAffineLocally_respectsIso + - AlgebraicGeometry.affine_cancel_left_isIso + - AlgebraicGeometry.AffineTargetMorphismProperty.toProperty_apply + - path: Mathlib/Data/Nat/Factorization/Basic.lean + theorems: + - Nat.Prime.factorization + - Nat.prod_primeFactors_gcd_mul_prod_primeFactors_mul + - Nat.eq_of_factorization_eq + - Nat.factorization_ord_compl + - Nat.factorizationLCMLeft_mul_factorizationLCMRight + - Nat.dvd_iff_div_factorization_eq_tsub + - Nat.exists_factorization_lt_of_lt + - Nat.ord_proj_dvd_ord_proj_iff_dvd + - Nat.factorization_prod_pow_eq_self + - Nat.prod_pow_factorization_eq_self + - Nat.factorization_eq_of_coprime_left + - Nat.multiplicative_factorization' + - Nat.Prime.factorization_pos_of_dvd + - Nat.Prime.factorization_pow + - Nat.dvd_ord_compl_of_dvd_not_dvd + - Nat.factorizationEquiv_apply + - Nat.factorizationLCMLeft_dvd_left + - Nat.factorizationLCMRight_pos + - Nat.ord_proj_of_not_prime + - Nat.Prime.eq_of_factorization_pos + - Nat.factorization_zero + - Nat.eq_iff_prime_padicValNat_eq + - Nat.multiplicity_eq_factorization + - Nat.Icc_factorization_eq_pow_dvd + - Nat.factorization_lcm + - Nat.factorization_eq_of_coprime_right + - Nat.factorization_eq_card_pow_dvd + - Nat.factorization_one + - Nat.factorization_le_of_le_pow + - Nat.factorizationLCMLeft_zero_right + - Nat.eq_pow_of_factorization_eq_single + - Nat.card_multiples' + - Nat.factorization_eq_zero_of_non_prime + - Nat.factorization_gcd + - Nat.factorization_inj + - Nat.Ioc_filter_dvd_card_eq_div + - Nat.factorization_prod + - Nat.factorization_eq_zero_iff + - Nat.factorizationLCMLeft_zero_left + - Nat.ord_proj_mul + - Nat.not_dvd_ord_compl + - Nat.card_multiples + - Nat.ord_proj_pos + - Nat.factorization_eq_zero_iff' + - Nat.ord_compl_mul + - Nat.Prime.pow_dvd_iff_dvd_ord_proj + - Nat.Ico_filter_pow_dvd_eq + - Nat.dvd_iff_prime_pow_dvd_dvd + - Nat.ord_proj_dvd + - Nat.factorizationLCRight_zero_left + - Nat.factors_count_eq + - Nat.eq_factorization_iff + - Nat.prod_pow_prime_padicValNat + - Nat.factorization_le_factorization_mul_right + - Nat.factorization_div + - Nat.setOf_pow_dvd_eq_Icc_factorization + - Nat.factorizationLCMLeft_pos + - Nat.factorization_prime_le_iff_dvd + - Nat.Prime.dvd_iff_one_le_factorization + - Nat.coprime_factorizationLCMLeft_factorizationLCMRight + - Nat.ord_compl_of_not_prime + - Nat.Prime.factorization_self + - Nat.factorization_eq_zero_of_remainder + - Nat.ord_compl_dvd_ord_compl_of_dvd + - Nat.pow_succ_factorization_not_dvd + - Nat.ord_proj_dvd_ord_proj_of_dvd + - Nat.factorization_eq_factors_multiset + - Nat.factorization_pow + - Nat.Prime.pow_dvd_iff_le_factorization + - Nat.factorization_eq_zero_iff_remainder + - Nat.factorization_le_iff_dvd + - Nat.factorization_mul + - Nat.multiplicative_factorization + - Nat.ord_compl_pos + - Nat.factorization_def + - Nat.factorization_mul_apply_of_coprime + - Nat.factorization_mul_of_coprime + - Nat.ord_compl_dvd_ord_compl_iff_dvd + - Nat.factorization_le_factorization_mul_left + - Nat.prod_primeFactors_dvd + - Nat.factorizationLCMRight_zero_right + - Nat.factorization_eq_zero_of_not_dvd + - Nat.factorizationLCMRight_dvd_right + - Nat.factorization_lt + - path: Mathlib/Data/Fintype/Prod.lean + theorems: + - infinite_prod + - Set.toFinset_prod + - Finset.product_eq_univ + - Set.toFinset_off_diag + - path: Mathlib/Order/Filter/SmallSets.lean + theorems: + - Filter.smallSets_eq_generate + - Filter.smallSets_top + - Filter.smallSets_bot + - Filter.eventually_smallSets_eventually + - Filter.eventually_smallSets_forall + - Filter.Tendsto.smallSets_mono + - Filter.tendsto_image_smallSets + - path: Mathlib/CategoryTheory/Limits/Shapes/Countable.lean + theorems: + - CategoryTheory.Limits.sequentialFunctor_initial_aux + - path: Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean + theorems: + - MeasureTheory.FiniteMeasure.tendsto_normalize_testAgainstNN_of_tendsto + - MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto + - MeasureTheory.FiniteMeasure.self_eq_mass_smul_normalize + - MeasureTheory.FiniteMeasure.average_eq_integral_normalize + - MeasureTheory.ProbabilityMeasure.nonempty + - ProbabilityMeasure.toFiniteMeasure_normalize_eq_self + - MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_integral_tendsto + - MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass + - MeasureTheory.FiniteMeasure.tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass + - MeasureTheory.FiniteMeasure.testAgainstNN_eq_mass_mul + - MeasureTheory.FiniteMeasure.normalize_eq_inv_mass_smul_of_nonzero + - MeasureTheory.ProbabilityMeasure.map_apply_of_aemeasurable + - MeasureTheory.FiniteMeasure.tendsto_normalize_iff_tendsto + - MeasureTheory.FiniteMeasure.toMeasure_normalize_eq_of_nonzero + - MeasureTheory.ProbabilityMeasure.eq_of_forall_toMeasure_apply_eq + - MeasureTheory.FiniteMeasure.normalize_eq_of_nonzero + - MeasureTheory.ProbabilityMeasure.toFiniteMeasure_nonzero + - MeasureTheory.ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous + - MeasureTheory.FiniteMeasure.tendsto_normalize_of_tendsto + - MeasureTheory.FiniteMeasure.self_eq_mass_mul_normalize + - MeasureTheory.FiniteMeasure.normalize_testAgainstNN + - MeasureTheory.ProbabilityMeasure.eq_of_forall_apply_eq + - MeasureTheory.ProbabilityMeasure.apply_le_one + - MeasureTheory.ProbabilityMeasure.coeFn_univ_ne_zero + - path: Mathlib/Data/Matrix/Block.lean + theorems: + - Matrix.fromBlocks_map + - Matrix.fromBlocks_multiply + - Matrix.blockDiagonal_conjTranspose + - Matrix.blockDiagonal'_transpose + - Matrix.fromBlocks_smul + - Matrix.fromBlocks_submatrix_sum_swap_left + - Matrix.fromBlocks_zero + - Matrix.blockDiagonal'_add + - Matrix.fromBlocks_toBlocks + - Matrix.fromBlocks_mulVec + - Matrix.fromBlocks_neg + - Matrix.blockDiagonal'_mul + - Matrix.blockDiagonal_mul + - Matrix.blockDiagonal'_one + - Matrix.vecMul_fromBlocks + - Matrix.blockDiagonal_zero + - Matrix.blockDiagonal_map + - Matrix.fromBlocks_conjTranspose + - Matrix.blockDiagonal_one + - "Matrix.toBlocks\u2081\u2081_diagonal" + - "Matrix.toBlocks\u2082\u2082_diagonal" + - Matrix.blockDiag'_diagonal + - Matrix.ext_iff_blocks + - Matrix.toBlock_diagonal_self + - Matrix.fromBlocks_submatrix_sum_swap_sum_swap + - Matrix.fromBlocks_diagonal + - Matrix.blockDiagonal'_smul + - Matrix.blockDiagonal_transpose + - Matrix.blockDiagonal_apply + - Matrix.blockDiagonal_diagonal + - Matrix.blockDiagonal_smul + - Matrix.blockDiagonal'_map + - Matrix.blockDiag_diagonal + - Matrix.fromBlocks_one + - Matrix.toBlock_mul_eq_add + - Matrix.blockDiagonal_add + - Matrix.blockDiagonal'_zero + - Matrix.blockDiagonal'_diagonal + - Matrix.blockDiagonal'_apply + - Matrix.fromBlocks_submatrix_sum_swap_right + - Matrix.blockDiagonal'_conjTranspose + - Matrix.fromBlocks_add + - Matrix.toBlock_mul_eq_mul + - Matrix.fromBlocks_transpose + - path: Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean + theorems: + - MvQPF.Cofix.dest_corec' + - MvQPF.liftR_map + - "MvQPF.Cofix.bisim\u2082" + - MvQPF.Cofix.ext_mk + - "MvQPF.Cofix.dest_corec\u2081" + - MvQPF.Cofix.mk_dest + - MvQPF.Cofix.abs_repr + - MvQPF.Cofix.bisim' + - MvQPF.Cofix.dest_mk + - MvQPF.Cofix.bisim_rel + - MvQPF.Cofix.ext + - MvQPF.Cofix.bisim + - MvQPF.corecF_eq + - MvQPF.Cofix.dest_corec + - MvQPF.corec_roll + - MvQPF.liftR_map_last + - MvQPF.Cofix.bisim_aux + - MvQPF.liftR_map_last' + - path: Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean + theorems: + - MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le + - MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable + - MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge + - MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt + - MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge + - MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal + - MeasureTheory.exists_upperSemicontinuous_le_integral_le + - MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge + - MeasureTheory.exists_upperSemicontinuous_le_lintegral_le + - MeasureTheory.exists_upperSemicontinuous_lt_integral_gt + - path: Mathlib/Order/Estimator.lean + theorems: + - Estimator.improveUntilAux_spec + - path: Mathlib/RingTheory/WittVector/Domain.lean + theorems: + - WittVector.verschiebung_nonzero + - WittVector.eq_iterate_verschiebung + - WittVector.verschiebung_shift + - path: Mathlib/CategoryTheory/Idempotents/Karoubi.lean + theorems: + - CategoryTheory.Idempotents.Karoubi.decompId_i_toKaroubi + - CategoryTheory.Idempotents.Karoubi.p_comp + - CategoryTheory.Idempotents.Karoubi.decompId_p_toKaroubi + - CategoryTheory.Idempotents.Karoubi.decompId_i_naturality + - CategoryTheory.Idempotents.Karoubi.decompId + - CategoryTheory.Idempotents.Karoubi.p_comm + - CategoryTheory.Idempotents.Karoubi.comp_p + - CategoryTheory.Idempotents.Karoubi.eqToHom_f + - CategoryTheory.Idempotents.Karoubi.decomp_p + - CategoryTheory.Idempotents.Karoubi.id_eq + - CategoryTheory.Idempotents.Karoubi.decompId_p_naturality + - CategoryTheory.Idempotents.Karoubi.hom_ext_iff + - CategoryTheory.Idempotents.Karoubi.ext + - CategoryTheory.Idempotents.Karoubi.comp_proof + - CategoryTheory.Idempotents.Karoubi.hom_ext + - path: Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean + theorems: + - IsIntegralClosure.range_le_span_dualBasis + - IsIntegralClosure.isLocalization + - FiniteDimensional.exists_is_basis_integral + - IsIntegralClosure.isNoetherian + - integralClosure_le_span_dualBasis + - IsIntegralClosure.rank + - exists_integral_multiples + - path: Mathlib/Analysis/LocallyConvex/Basic.lean + theorems: + - absorbs_iff_eventually_nhds_zero + - Balanced.zero_insert_interior + - Balanced.smul_mem_mono + - absorbent_nhds_zero + - balanced_empty + - Balanced.interior + - Balanced.convexHull + - balanced_iff_neg_mem + - absorbs_iff_norm + - balanced_iff_closedBall_smul + - absorbs_iff_eventually_nhdsWithin_zero + - Balanced.sub + - Balanced.neg_mem_iff + - Balanced.smul_mono + - absorbent_iff_eventually_nhdsWithin_zero + - Balanced.subset_smul + - path: Mathlib/GroupTheory/Complement.lean + theorems: + - Subgroup.range_mem_rightTransversals + - Subgroup.isComplement_singleton_left + - Subgroup.isComplement'_stabilizer + - Subgroup.exists_right_transversal + - Subgroup.transferTransversal_apply'' + - Subgroup.IsComplement.equiv_fst_eq_iff_leftCosetEquivalence + - Subgroup.isComplement_singleton_right + - Subgroup.smul_apply_eq_smul_apply_inv_smul + - Subgroup.IsComplement.leftCosetEquivalence_equiv_fst + - Subgroup.MemRightTransversals.toEquiv_apply + - Subgroup.transferTransversal_apply' + - Subgroup.isComplement_univ_left + - Subgroup.isComplement'_of_disjoint_and_mul_eq_univ + - Subgroup.IsComplement.equiv_fst_eq_one_of_mem_of_one_mem + - Subgroup.IsComplement.rightCosetEquivalence_equiv_snd + - Subgroup.quotientEquivSigmaZMod_apply + - Subgroup.IsComplement.coe_equiv_snd_eq_one_iff_mem + - Subgroup.mem_rightTransversals_iff_existsUnique_mul_inv_mem + - Subgroup.IsComplement.equiv_snd_eq_one_of_mem_of_one_mem + - Subgroup.IsComplement.equiv_mul_right + - Subgroup.MemLeftTransversals.inv_mul_toFun_mem + - Subgroup.IsComplement.equiv_snd_eq_self_iff_mem + - Subgroup.IsComplement.equiv_fst_eq_self_of_mem_of_one_mem + - Subgroup.MemRightTransversals.toFun_mul_inv_mem + - Subgroup.IsComplement'.isCompl + - Subgroup.mem_rightTransversals_iff_existsUnique_quotient_mk''_eq + - Subgroup.isComplement_univ_right + - Subgroup.range_mem_leftTransversals + - Subgroup.IsComplement.mul_eq + - Subgroup.IsComplement.equiv_snd_eq_self_of_mem_of_one_mem + - Subgroup.mem_leftTransversals_iff_existsUnique_inv_mul_mem + - Subgroup.exists_left_transversal_of_le + - Subgroup.IsComplement.coe_equiv_fst_eq_one_iff_mem + - Subgroup.exists_right_transversal_of_le + - Subgroup.exists_left_transversal + - Subgroup.IsComplement.equiv_one + - Subgroup.IsComplement.equiv_fst_eq_self_iff_mem + - Subgroup.mem_leftTransversals_iff_existsUnique_quotient_mk''_eq + - Subgroup.IsComplement'.symm + - Subgroup.MemLeftTransversals.toEquiv_apply + - path: Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean + theorems: + - aemeasurable_of_tendsto_metrizable_ae + - aemeasurable_of_unif_approx + - nullMeasurableSet_of_tendsto_indicator + - measurable_of_tendsto_metrizable' + - measurable_limit_of_tendsto_metrizable_ae + - measurableSet_of_tendsto_indicator + - path: Mathlib/Order/SupClosed.lean + theorems: + - infClosed_singleton + - InfClosed.supClosure + - Set.Finite.infClosure + - infClosure_singleton + - lowerBounds_infClosure + - latticeClosure_univ + - InfClosed.image + - isLUB_supClosure + - SupClosed.image + - LinearOrder.infClosed + - supClosed_empty + - infClosure_univ + - isGLB_infClosure + - LinearOrder.supClosed + - latticeClosure_empty + - infClosure_empty + - Set.Finite.latticeClosure + - upperBounds_supClosure + - supClosure_singleton + - latticeClosure_singleton + - InfClosed.preimage + - infClosed_range + - SupClosed.preimage + - supClosure_empty + - supClosed_range + - Set.Finite.supClosure + - infClosed_univ + - supClosed_univ + - supClosed_singleton + - SupClosed.infClosure + - supClosure_univ + - path: Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean + theorems: + - Matrix.IsAdjMatrix.toGraph_compl_eq + - SimpleGraph.dotProduct_mulVec_adjMatrix + - SimpleGraph.adjMatrix_vecMul_apply + - SimpleGraph.adjMatrix_mulVec_const_apply_of_regular + - Matrix.IsAdjMatrix.apply_ne_zero_iff + - SimpleGraph.mul_adjMatrix_apply + - SimpleGraph.adjMatrix_dotProduct + - Matrix.IsAdjMatrix.apply_ne_one_iff + - SimpleGraph.isAdjMatrix_adjMatrix + - Matrix.IsAdjMatrix.apply_diag_ne + - SimpleGraph.adjMatrix_mul_self_apply_self + - Matrix.isAdjMatrix_compl + - SimpleGraph.toGraph_adjMatrix_eq + - Matrix.IsAdjMatrix.adjMatrix_toGraph_eq + - SimpleGraph.dotProduct_adjMatrix + - Matrix.compl_apply + - SimpleGraph.one_add_adjMatrix_add_compl_adjMatrix_eq_allOnes + - SimpleGraph.adjMatrix_mul_apply + - Matrix.compl_apply_diag + - SimpleGraph.transpose_adjMatrix + - SimpleGraph.trace_adjMatrix + - SimpleGraph.adjMatrix_mulVec_apply + - Matrix.isSymm_compl + - SimpleGraph.adjMatrix_mulVec_const_apply + - path: Mathlib/RingTheory/Localization/AsSubring.lean + theorems: + - Localization.subalgebra.mem_range_mapToFractionRing_iff_ofField + - Localization.map_isUnit_of_le + - Localization.mem_range_mapToFractionRing_iff + - path: Mathlib/Analysis/Calculus/DiffContOnCl.lean + theorems: + - DiffContOnCl.continuousOn_ball + - path: Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean + theorems: + - CochainComplex.HomComplex.Cochain.leftUnshift_zero + - CochainComplex.HomComplex.Cochain.leftShift_comp + - CochainComplex.HomComplex.Cochain.rightShift_neg + - CochainComplex.HomComplex.Cochain.rightShift_leftShift + - CochainComplex.HomComplex.Cochain.shift_smul + - CochainComplex.HomComplex.Cochain.leftShift_rightShift + - "CochainComplex.HomComplex.Cochain.\u03B4_rightUnshift" + - "CochainComplex.HomComplex.Cochain.\u03B4_leftUnshift" + - CochainComplex.HomComplex.Cochain.rightUnshift_rightShift + - CochainComplex.HomComplex.Cochain.shift_neg + - CochainComplex.HomComplex.Cochain.leftShift_smul + - CochainComplex.HomComplex.Cochain.rightUnshift_comp + - CochainComplex.HomComplex.Cochain.leftShift_add + - CochainComplex.HomComplex.Cochain.leftUnshift_neg + - CochainComplex.HomComplex.Cochain.leftUnshift_leftShift + - CochainComplex.HomComplex.Cochain.rightUnshift_units_smul + - CochainComplex.HomComplex.Cochain.leftUnshift_units_smul + - CochainComplex.HomComplex.Cochain.shift_units_smul + - CochainComplex.HomComplex.Cochain.leftUnshift_v + - CochainComplex.HomComplex.Cochain.leftShift_neg + - CochainComplex.HomComplex.Cochain.shift_add + - CochainComplex.HomComplex.Cochain.rightShift_zero + - CochainComplex.HomComplex.Cochain.rightShift_add + - CochainComplex.HomComplex.Cochain.leftShift_units_smul + - "CochainComplex.HomComplex.Cochain.\u03B4_shift" + - CochainComplex.HomComplex.Cochain.rightUnshift_v + - CochainComplex.HomComplex.Cochain.rightUnshift_add + - CochainComplex.HomComplex.Cochain.rightUnshift_zero + - CochainComplex.HomComplex.Cochain.rightUnshift_neg + - "CochainComplex.HomComplex.Cochain.\u03B4_leftShift" + - CochainComplex.HomComplex.Cochain.rightShift_smul + - CochainComplex.HomComplex.Cochain.leftShift_rightShift_eq_negOnePow_rightShift_leftShift + - CochainComplex.HomComplex.Cochain.shift_v + - CochainComplex.HomComplex.Cochain.leftShift_zero + - CochainComplex.HomComplex.Cochain.rightShift_rightUnshift + - CochainComplex.HomComplex.Cochain.leftUnshift_add + - CochainComplex.HomComplex.Cochain.rightUnshift_smul + - "CochainComplex.HomComplex.Cochain.\u03B4_rightShift" + - CochainComplex.HomComplex.Cochain.leftShift_comp_zero_cochain + - CochainComplex.HomComplex.Cochain.leftShift_v + - CochainComplex.HomComplex.Cochain.rightShift_v + - CochainComplex.HomComplex.Cochain.leftShift_leftUnshift + - CochainComplex.HomComplex.Cochain.rightShift_units_smul + - path: Mathlib/Algebra/Category/GroupCat/EpiMono.lean + theorems: + - GroupCat.surjective_of_epi + - GroupCat.SurjectiveOfEpiAuxs.mul_smul + - GroupCat.SurjectiveOfEpiAuxs.g_apply_fromCoset + - GroupCat.SurjectiveOfEpiAuxs.h_apply_fromCoset + - GroupCat.SurjectiveOfEpiAuxs.one_smul + - GroupCat.SurjectiveOfEpiAuxs.fromCoset_ne_of_nin_range + - AddGroupCat.epi_iff_surjective + - GroupCat.SurjectiveOfEpiAuxs.fromCoset_eq_of_mem_range + - MonoidHom.range_eq_top_of_cancel + - GroupCat.SurjectiveOfEpiAuxs.g_ne_h + - "GroupCat.SurjectiveOfEpiAuxs.\u03C4_symm_apply_fromCoset" + - "GroupCat.SurjectiveOfEpiAuxs.\u03C4_symm_apply_infinity" + - GroupCat.SurjectiveOfEpiAuxs.comp_eq + - GroupCat.SurjectiveOfEpiAuxs.h_apply_infinity + - MonoidHom.ker_eq_bot_of_cancel + - GroupCat.SurjectiveOfEpiAuxs.h_apply_fromCoset_nin_range + - GroupCat.SurjectiveOfEpiAuxs.agree + - CommGroupCat.epi_iff_surjective + - path: Mathlib/Topology/Compactness/Lindelof.lean + theorems: + - isLindelof_of_countable_subcover + - IsLindelof.disjoint_nhdsSet_left + - IsLindelof.elim_countable_subcover_image + - IsLindelof.disjoint_nhdsSet_right + - isLindelof_iff_isLindelof_univ + - isLindelof_singleton + - Inducing.isLindelof_preimage + - IsLindelof.elim_countable_subfamily_closed + - HereditarilyLindelof_LindelofSets + - IsCompact.isLindelof + - LindelofSpace.of_continuous_surjective + - IsLindelof.inter_iInter_nonempty + - Set.Finite.isLindelof_sUnion + - IsLindelof.union + - IsSigmaCompact.isLindelof + - IsLindelof.compl_mem_sets + - isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis + - IsLindelof.elim_nhds_subcover' + - LindelofSpace.elim_nhds_subcover + - mem_coclosed_Lindelof' + - IsLindelof.compl_mem_sets_of_nhdsWithin + - IsLindelof.induction_on + - IsLindelof.inter_right + - cluster_point_of_Lindelof + - isLindelof_range + - isLindelof_of_countable_subfamily_closed + - eq_open_union_countable + - Inducing.isLindelof_iff + - Filter.comap_coLindelof_le + - Tendsto.isLindelof_insert_range_of_coLindelof + - Set.Countable.isLindelof_biUnion + - Set.Countable.isLindelof_sUnion + - IsLindelof.elim_countable_subcover + - mem_coclosedLindelof + - IsLindelof.elim_nhds_subcover + - IsLindelof.image_of_continuousOn + - IsLindelof.countable_of_discrete + - hasBasis_coclosedLindelof + - ClosedEmbedding.LindelofSpace + - path: Mathlib/Topology/Order/PartialSups.lean + theorems: + - Filter.Tendsto.partialSups + - ContinuousAt.partialSups + - Filter.Tendsto.partialSups_apply + - ContinuousWithinAt.partialSups + - path: Mathlib/RingTheory/Kaehler.lean + theorems: + - KaehlerDifferential.kerTotal_mkQ_single_add + - KaehlerDifferential.kerTotal_mkQ_single_mul + - KaehlerDifferential.kerTotal_mkQ_single_algebraMap + - KaehlerDifferential.D_tensorProductTo + - KaehlerDifferential.derivationQuotKerTotal_lift_comp_total + - KaehlerDifferential.mapBaseChange_tmul + - KaehlerDifferential.kerTotal_mkQ_single_algebraMap_one + - KaehlerDifferential.map_surjective_of_surjective + - KaehlerDifferential.kerTotal_map + - Derivation.liftKaehlerDifferential_comp + - KaehlerDifferential.submodule_span_range_eq_ideal + - KaehlerDifferential.kerTotal_eq + - KaehlerDifferential.tensorProductTo_surjective + - KaehlerDifferential.kerTotal_mkQ_single_smul + - KaehlerDifferential.span_range_eq_ideal + - Derivation.tensorProductTo_mul + - KaehlerDifferential.one_smul_sub_smul_one_mem_ideal + - KaehlerDifferential.total_surjective + - KaehlerDifferential.span_range_derivation + - KaehlerDifferential.End_equiv_aux + - KaehlerDifferential.quotKerTotalEquiv_symm_comp_D + - path: Mathlib/Algebra/Module/Submodule/Basic.lean + theorems: + - Submodule.not_mem_of_ortho + - Submodule.sub_mem_iff_left + - Submodule.sub_mem_iff_right + - path: Mathlib/LinearAlgebra/PiTensorProduct.lean + theorems: + - PiTensorProduct.congr_tprod + - PiTensorProduct.lift_comp_map + - PiTensorProduct.reindex_symm + - PiTensorProduct.tmul_apply + - PiTensorProduct.liftAux_tprod + - PiTensorProduct.map_add + - PiTensorProduct.lift_comp_reindex_symm + - PiTensorProduct.congr_symm_tprod + - PiTensorProduct.piTensorHomMap_tprod_tprod + - PiTensorProduct.map_comp_reindex_eq + - PiTensorProduct.lift_comp_reindex + - PiTensorProduct.liftAux.smul + - PiTensorProduct.map_comp_reindex_symm + - PiTensorProduct.ext + - PiTensorProduct.induction_on + - PiTensorProduct.map_comp + - PiTensorProduct.map_range_eq_span_tprod + - PiTensorProduct.map_id + - PiTensorProduct.reindex_trans + - "PiTensorProduct.map\u2082_tprod_tprod" + - PiTensorProduct.reindex_tprod + - PiTensorProduct.smul_tprodCoeff + - "PiTensorProduct.piTensorHomMapFun\u2082_add" + - PiTensorProduct.reindex_refl + - "PiTensorProduct.piTensorHomMap\u2082_tprod_tprod_tprod" + - PiTensorProduct.map_smul + - PiTensorProduct.map_add_smul_aux + - PiTensorProduct.span_tprod_eq_top + - PiTensorProduct.piTensorHomMap_tprod_eq_map + - "PiTensorProduct.piTensorHomMapFun\u2082_smul" + - path: Mathlib/CategoryTheory/Sites/CoverPreserving.lean + theorems: + - CategoryTheory.idCoverPreserving + - CategoryTheory.compatiblePreservingOfDownwardsClosed + - CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward + - CategoryTheory.Functor.isContinuous_comp' + - CategoryTheory.CompatiblePreserving.apply_map + - CategoryTheory.Functor.isContinuous_of_coverPreserving + - CategoryTheory.CoverPreserving.comp + - CategoryTheory.compatiblePreservingOfFlat + - path: Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean + theorems: + - AlgebraicGeometry.quasi_compact_affineProperty_diagonal_eq + - AlgebraicGeometry.quasiSeparated_eq_diagonal_is_quasiCompact + - AlgebraicGeometry.is_localization_basicOpen_of_qcqs + - AlgebraicGeometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux + - AlgebraicGeometry.quasiSeparatedSpace_of_quasiSeparated + - AlgebraicGeometry.QuasiSeparated.affine_openCover_iff + - AlgebraicGeometry.quasiSeparated_eq_affineProperty + - AlgebraicGeometry.quasiSeparated_over_affine_iff + - AlgebraicGeometry.quasi_compact_affineProperty_iff_quasiSeparatedSpace + - AlgebraicGeometry.quasiSeparated_eq_affineProperty_diagonal + - AlgebraicGeometry.quasiSeparatedSpace_iff_affine + - AlgebraicGeometry.QuasiSeparated.affine_openCover_TFAE + - AlgebraicGeometry.exists_eq_pow_mul_of_isCompact_of_isQuasiSeparated + - AlgebraicGeometry.exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux_aux + - AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen + - AlgebraicGeometry.quasiSeparatedOfComp + - AlgebraicGeometry.IsAffineOpen.isQuasiSeparated + - path: Mathlib/Combinatorics/SimpleGraph/Maps.lean + theorems: + - SimpleGraph.Iso.card_eq + - SimpleGraph.induceHom_id + - SimpleGraph.Hom.injective_of_top_hom + - SimpleGraph.comap_monotone + - SimpleGraph.induce_singleton_eq_top + - SimpleGraph.map_adj_apply + - SimpleGraph.induceHom_injective + - SimpleGraph.Iso.card_edgeFinset_eq + - Equiv.simpleGraph_refl + - SimpleGraph.le_comap_of_subsingleton + - SimpleGraph.map_symm + - SimpleGraph.comap_symm + - SimpleGraph.map_le_of_subsingleton + - SimpleGraph.induceHom_comp + - SimpleGraph.Hom.mapEdgeSet.injective + - SimpleGraph.comap_map_eq + - SimpleGraph.induceHomOfLE_toHom + - SimpleGraph.map_monotone + - SimpleGraph.map_comap_le + - path: Mathlib/Data/Set/Pairwise/Basic.lean + theorems: + - pairwise_subtype_iff_pairwise_set + - Set.pairwise_pair + - Set.pairwise_insert_of_not_mem + - Set.pairwise_pair_of_symmetric + - Set.pairwise_union + - exists_ne_mem_inter_of_not_pairwise_disjoint + - Set.pairwise_univ + - Set.injOn_iff_pairwise_ne + - exists_lt_mem_inter_of_not_pairwise_disjoint + - Set.PairwiseDisjoint.range + - Set.pairwiseDisjoint_image_right_iff + - Set.Nonempty.pairwise_iff_exists_forall + - Set.pairwise_iff_exists_forall + - Set.pairwise_insert_of_symmetric + - Set.pairwise_union_of_symmetric + - Set.PairwiseDisjoint.image_of_le + - pairwise_on_bool + - Set.pairwise_insert_of_symmetric_of_not_mem + - Set.exists_ne_mem_inter_of_not_pairwiseDisjoint + - Set.pairwiseDisjoint_image_left_iff + - Set.InjOn.pairwise_image + - Set.exists_lt_mem_inter_of_not_pairwiseDisjoint + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean + theorems: + - CliffordAlgebra.one_le_evenOdd_zero + - "CliffordAlgebra.range_\u03B9_le_evenOdd_one" + - CliffordAlgebra.odd_induction + - CliffordAlgebra.evenOdd_induction + - CliffordAlgebra.evenOdd_isCompl + - CliffordAlgebra.evenOdd_mul_le + - CliffordAlgebra.even_induction + - "CliffordAlgebra.iSup_\u03B9_range_eq_top" + - "CliffordAlgebra.GradedAlgebra.lift_\u03B9_eq" + - "CliffordAlgebra.GradedAlgebra.\u03B9_sq_scalar" + - "CliffordAlgebra.\u03B9_mul_\u03B9_mem_evenOdd_zero" + - path: Mathlib/Algebra/Order/Ring/Lemmas.lean + theorems: + - posMulMono_iff_mulPosMono + - mulPosReflectLT_iff_contravariant_pos + - lt_mul_of_one_lt_right + - lt_mul_iff_one_lt_left + - mul_lt_of_lt_one_left + - Left.mul_nonneg + - posMulStrictMono_iff_mulPosStrictMono + - mulPosMono_iff_covariant_pos + - mul_nonpos_of_nonpos_of_nonneg + - mul_pos_iff_of_pos_right + - le_mul_iff_one_le_right + - posMulReflectLE_iff_mulPosReflectLE + - mul_lt_iff_lt_one_left + - mul_le_iff_le_one_left + - mul_le_of_le_one_left + - Right.mul_pos + - Right.mul_nonneg + - mul_le_of_le_one_right + - le_mul_of_one_le_left + - mul_eq_mul_iff_eq_and_eq_of_pos + - mul_lt_iff_lt_one_right + - mul_neg_of_pos_of_neg + - mul_lt_of_lt_one_right + - posMulMono_iff_covariant_pos + - posMulReflectLT_iff_mulPosReflectLT + - mul_le_iff_le_one_right + - mul_neg_of_neg_of_pos + - mul_nonpos_of_nonneg_of_nonpos + - le_mul_iff_one_le_left + - pos_and_pos_or_neg_and_neg_of_mul_pos + - mul_pos_iff_of_pos_left + - lt_mul_of_one_lt_left + - le_mul_of_one_le_right + - posMulReflectLT_iff_contravariant_pos + - Left.mul_pos + - lt_mul_iff_one_lt_right + - path: Mathlib/Data/NNRat/Defs.lean + theorems: + - NNRat.num_ne_zero + - Rat.toNNRat_pos + - Rat.le_toNNRat_iff_coe_le' + - NNRat.coe_eq_zero + - Rat.toNNRat_add + - Rat.le_toNNRat_iff_coe_le + - Rat.toNNRat_eq_zero + - NNRat.coe_sub + - Rat.toNNRat_lt_toNNRat_iff' + - NNRat.toNNRat_coe_nat + - NNRat.ext_num_den + - NNRat.num_coe + - NNRat.num_pos + - Rat.toNNRat_mul + - Rat.toNNRat_lt_iff_lt_coe + - Rat.toNNRat_le_toNNRat_iff + - NNRat.ext_num_den_iff + - path: Mathlib/Geometry/Euclidean/Sphere/Power.lean + theorems: + - EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical + - EuclideanGeometry.mul_dist_eq_abs_sub_sq_dist + - EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero + - InnerProductGeometry.mul_norm_eq_abs_sub_sq_norm + - path: Mathlib/Logic/Lemmas.lean + theorems: + - dite_dite_distrib_left + - Prop.exists + - Prop.forall + - path: Mathlib/Order/Minimal.lean + theorems: + - minimals_eq_minimals_of_subset_of_forall + - map_mem_minimals_iff + - maximals_mono + - map_mem_minimals + - RelEmbedding.inter_preimage_minimals_eq + - mem_maximals_iff_forall_lt_not_mem' + - inter_preimage_maximals_eq_of_rel_iff_rel_on_of_subset + - RelEmbedding.minimals_preimage_eq + - image_minimals_univ + - minimals_Ico + - map_mem_maximals + - RelEmbedding.maximals_preimage_eq + - inter_minimals_preimage_inter_eq_of_rel_iff_rel_on + - RelEmbedding.image_minimals_eq + - RelEmbedding.inter_preimage_maximals_eq + - map_mem_maximals_iff + - IsAntichain.max_maximals + - mem_minimals_iff_forall_lt_not_mem' + - minimals_Icc + - maximals_union + - maximals_eq_minimals + - image_minimals_of_rel_iff_rel + - IsAntichain.minimals_upperClosure + - IsAntichain.maximals_eq + - IsAntichain.max_minimals + - inter_maximals_preimage_inter_eq_of_rel_iff_rel_on + - maximals_singleton + - mem_maximals_iff + - RelEmbedding.inter_preimage_minimals_eq_of_subset + - RelEmbedding.inter_preimage_maximals_eq_of_subset + - minimals_mono + - OrderIso.map_mem_minimals + - Set.Subsingleton.maximals_eq + - inter_preimage_minimals_eq_of_rel_iff_rel_on_of_subset + - path: Mathlib/Order/ModularLattice.lean + theorems: + - Disjoint.disjoint_sup_left_of_disjoint_sup_right + - covBy_sup_of_inf_covBy_right + - covBy_sup_of_inf_covBy_of_inf_covBy_right + - inf_covBy_of_covBy_sup_right + - Disjoint.disjoint_sup_right_of_disjoint_sup_left + - inf_covBy_of_covBy_sup_of_covBy_sup_right + - isModularLattice_iff_inf_sup_inf_assoc + - inf_sup_assoc_of_le + - path: Mathlib/Data/Option/NAry.lean + theorems: + - "Option.map\u2082_map_right" + - "Option.map\u2082_right_comm" + - "Option.map\u2082_eq_none_iff" + - "Option.map\u2082_left_comm" + - "Option.map\u2082_right_identity" + - "Option.map\u2082_coe_right" + - "Option.map_map\u2082_antidistrib" + - "Option.map\u2082_left_identity" + - "Option.map_map\u2082_antidistrib_left" + - "Option.map\u2082_def" + - "Option.map_map\u2082_distrib_left" + - "Option.map_map\u2082_antidistrib_right" + - "Option.map\u2082_map_left_comm" + - "Option.map_map\u2082_distrib" + - "Option.map\u2082_comm" + - "Option.map\u2082_none_right" + - "Option.mem_map\u2082_iff" + - "Option.map_map\u2082_right_comm" + - "Option.map_map\u2082_right_anticomm" + - "Option.map_map\u2082_distrib_right" + - "Option.map\u2082_map_left" + - "Option.map\u2082_swap" + - Option.map_uncurry + - path: Mathlib/Analysis/Calculus/Implicit.lean + theorems: + - HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_target + - HasStrictFDerivAt.tendsto_implicitFunction + - HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_self + - HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply_ker + - HasStrictFDerivAt.implicitFunctionOfComplemented_apply_image + - HasStrictFDerivAt.to_implicitFunctionOfComplemented + - ImplicitFunctionData.implicitFunction_hasStrictFDerivAt + - HasStrictFDerivAt.implicitFunction_apply_image + - path: Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean + theorems: + - span_gramSchmidtNormed + - gramSchmidt_def' + - gramSchmidtOrthonormalBasis_apply_of_orthogonal + - gramSchmidt_ne_zero_coe + - gramSchmidt_orthogonal + - gramSchmidt_zero + - gramSchmidt_def + - gramSchmidt_orthonormal + - gramSchmidtOrthonormalBasis_det + - gramSchmidt_def'' + - gramSchmidtOrthonormalBasis_inv_triangular' + - gramSchmidt_inv_triangular + - gramSchmidt_mem_span + - span_gramSchmidtNormed_range + - gramSchmidtNormed_unit_length' + - gramSchmidtNormed_unit_length_coe + - gramSchmidtOrthonormalBasis_inv_triangular + - mem_span_gramSchmidt + - gramSchmidt_of_orthogonal + - inner_gramSchmidtOrthonormalBasis_eq_zero + - path: Mathlib/CategoryTheory/Endofunctor/Algebra.lean + theorems: + - CategoryTheory.Endofunctor.Coalgebra.iso_of_iso + - CategoryTheory.Endofunctor.Algebra.iso_of_iso + - CategoryTheory.Endofunctor.Algebra.Initial.right_inv + - CategoryTheory.Endofunctor.Adjunction.Algebra.homEquiv_naturality_str + - CategoryTheory.Endofunctor.Adjunction.Coalgebra.homEquiv_naturality_str_symm + - CategoryTheory.Endofunctor.Algebra.Initial.left_inv' + - path: Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean + theorems: + - CategoryTheory.ShortComplex.Exact.moduleCat_of_range_eq_ker + - CategoryTheory.ShortComplex.moduleCat_exact_iff_range_eq_ker + - CategoryTheory.ShortComplex.toCycles_moduleCatCyclesIso_hom + - CategoryTheory.ShortComplex.moduleCat_exact_iff_ker_sub_range + - CategoryTheory.ShortComplex.exact_iff_surjective_moduleCatToCycles + - CategoryTheory.ShortComplex.Exact.moduleCat_range_eq_ker + - path: Mathlib/Topology/MetricSpace/Bounded.lean + theorems: + - Metric.diam_eq_zero_of_unbounded + - Metric.diam_le_of_subset_closedBall + - Metric.finite_isBounded_inter_isClosed + - Metric.hasBasis_cobounded_compl_closedBall + - Metric.diam_univ_of_noncompact + - Metric.diam_pair + - Metric.mem_cocompact_of_closedBall_compl_subset + - Metric.diam_subsingleton + - IsComplete.nonempty_iInter_of_nonempty_biInter + - Metric.isBounded_range_of_tendsto_cofinite_uniformity + - comap_dist_left_atTop_eq_cocompact + - Metric.hasBasis_cobounded_compl_ball + - Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt + - Metric.isBounded_image_iff + - Metric.diam_union + - Metric.isCompact_of_isClosed_isBounded + - Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt + - Metric.tendsto_dist_right_atTop_iff + - Metric.comap_dist_right_atTop + - Metric.isBounded_range_iff + - Metric.isCobounded_iff_closedBall_compl_subset + - CauchySeq.isBounded_range + - Metric.diam_union' + - Metric.tendsto_dist_left_atTop_iff + - Metric.comap_dist_left_atTop + - Metric.cobounded_eq_cocompact + - path: Mathlib/GroupTheory/Perm/ClosureSwap.lean + theorems: + - surjective_of_isSwap_of_isPretransitive + - Equiv.Perm.IsSwap.finite_compl_fixedBy + - swap_mem_closure_isSwap + - finite_compl_fixedBy_closure_iff + - exists_smul_not_mem_of_subset_orbit_closure + - mem_closure_isSwap + - SubmonoidClass.swap_mem_trans + - closure_of_isSwap_of_isPretransitive + - mem_closure_isSwap' + - finite_compl_fixedBy_swap + - path: Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean + theorems: + - PowerBasis.ofGenMemAdjoin'_gen + - minpoly.isIntegrallyClosed_dvd_iff + - minpoly.isIntegrallyClosed_eq_field_fractions' + - minpoly.IsIntegrallyClosed.degree_le_of_ne_zero + - IsIntegrallyClosed.minpoly.unique + - minpoly.isIntegrallyClosed_dvd + - minpoly.prime_of_isIntegrallyClosed + - minpoly.ker_eval + - minpoly.ToAdjoin.injective + - minpoly.isIntegrallyClosed_eq_field_fractions + - path: Mathlib/GroupTheory/Perm/Support.lean + theorems: + - Equiv.Perm.support_prod_le + - Equiv.Perm.support_swap_mul_ge_support_diff + - Equiv.Perm.disjoint_inv_right_iff + - Equiv.Perm.zpow_apply_eq_of_apply_apply_eq_self + - Equiv.Perm.Disjoint.symm + - Equiv.Perm.eq_on_support_mem_disjoint + - Equiv.Perm.apply_pow_apply_eq_iff + - Equiv.Perm.set_support_mul_subset + - Equiv.Perm.set_support_apply_mem + - Equiv.Perm.Disjoint.commute + - Equiv.Perm.not_mem_support + - Equiv.Perm.support_one + - Equiv.Perm.support_swap + - Equiv.Perm.set_support_zpow_subset + - Equiv.Perm.card_support_ne_one + - Equiv.Perm.card_support_conj + - Equiv.Perm.support_congr + - Equiv.Perm.pow_apply_eq_of_apply_apply_eq_self + - Equiv.Perm.Disjoint.mul_apply_eq_iff + - Equiv.Perm.support_swap_mul_swap + - Equiv.Perm.support_inv + - Equiv.Perm.Disjoint.mul_eq_one_iff + - Equiv.Perm.card_support_le_one + - Equiv.Perm.support_extend_domain + - Equiv.Perm.exists_mem_support_of_mem_support_prod + - Equiv.Perm.set_support_inv_eq + - Equiv.Perm.disjoint_iff_disjoint_support + - Equiv.Perm.apply_mem_support + - Equiv.Perm.Disjoint.mono + - Equiv.Perm.card_support_eq_zero + - Equiv.Perm.support_prod_of_pairwise_disjoint + - Equiv.Perm.one_lt_card_support_of_ne_one + - Equiv.Perm.support_eq_empty_iff + - Equiv.Perm.pow_eq_on_of_mem_support + - Equiv.Perm.Disjoint.mul_right + - Equiv.Perm.disjoint_inv_left_iff + - Equiv.Perm.card_support_swap_mul + - Equiv.Perm.ofSubtype_swap_eq + - Equiv.Perm.support_swap_mul_eq + - Equiv.Perm.disjoint_refl_iff + - Equiv.Perm.support_mul_le + - Equiv.Perm.apply_zpow_apply_eq_iff + - Equiv.Perm.disjoint_prod_right + - Equiv.Perm.Disjoint.mul_left + - Equiv.Perm.card_support_eq_two + - Equiv.Perm.disjoint_conj + - Equiv.Perm.pow_apply_mem_support + - Equiv.Perm.card_support_extend_domain + - Equiv.Perm.coe_support_eq_set_support + - Equiv.Perm.Disjoint.inv_left + - Equiv.Perm.support_subtype_perm + - Equiv.Perm.zpow_apply_mem_support + - Equiv.Perm.Disjoint.support_mul + - Equiv.Perm.mem_support_swap_mul_imp_mem_support_ne + - Equiv.Perm.card_support_prod_list_of_pairwise_disjoint + - Equiv.Perm.support_conj + - Equiv.Perm.mem_support + - Equiv.Perm.IsSwap.of_subtype_isSwap + - Equiv.Perm.card_support_swap + - Equiv.Perm.zpow_apply_eq_self_of_apply_eq_self + - Equiv.Perm.ne_and_ne_of_swap_mul_apply_ne_self + - Equiv.Perm.support_le_prod_of_mem + - Equiv.Perm.support_swap_iff + - Equiv.Perm.fixed_point_card_lt_of_ne_one + - Equiv.Perm.pow_apply_eq_self_of_apply_eq_self + - Equiv.Perm.Disjoint.card_support_mul + - Equiv.Perm.nodup_of_pairwise_disjoint + - path: Mathlib/Order/WellFoundedSet.lean + theorems: + - Set.WellFoundedOn.mono + - Set.PartiallyWellOrderedOn.iff_forall_not_isBadSeq + - Set.wellFoundedOn_range + - Set.WellFoundedOn.induction + - Set.WellFoundedOn.union + - Set.PartiallyWellOrderedOn.image_of_monotone_on + - Set.wellFoundedOn_univ + - Finset.partiallyWellOrderedOn_bUnion + - Finset.wellFoundedOn_bUnion + - Set.partiallyWellOrderedOn_insert + - WellFounded.sigma_lex_of_wellFoundedOn_fiber + - Finset.wellFoundedOn_sup + - Set.PartiallyWellOrderedOn.iff_not_exists_isMinBadSeq + - Set.wellFoundedOn_image + - BddBelow.wellFoundedOn_lt + - Finset.partiallyWellOrderedOn_sup + - Set.wellFoundedOn_iff + - Set.Finite.partiallyWellOrderedOn + - Set.wellFoundedOn_insert + - Set.partiallyWellOrderedOn_iff_exists_monotone_subseq + - Set.wellFoundedOn_iff_no_descending_seq + - Set.WellFoundedOn.sigma_lex_of_wellFoundedOn_fiber + - Set.IsWF.isPWO + - IsAntichain.finite_of_partiallyWellOrderedOn + - Set.partiallyWellOrderedOn_iff_finite_antichains + - WellFounded.prod_lex_of_wellFoundedOn_fiber + - Pi.isPWO + - Set.PartiallyWellOrderedOn.union + - "Set.PartiallyWellOrderedOn.partiallyWellOrderedOn_sublistForall\u2082" + - Set.PartiallyWellOrderedOn.exists_monotone_subseq + - Set.PartiallyWellOrderedOn.prod + - Set.WellFoundedOn.acc_iff_wellFoundedOn + - Set.isWF_insert + - Set.IsPWO.isWF + - Set.isPWO_insert + - Set.IsWF.min_union + - Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber + - Set.PartiallyWellOrderedOn.exists_min_bad_of_exists_bad + - Set.isWF_univ_iff + - path: Mathlib/CategoryTheory/Opposites.lean + theorems: + - CategoryTheory.Iso.unop_hom_inv_id_app + - CategoryTheory.op_inv + - CategoryTheory.isIso_of_op + - CategoryTheory.Iso.op_unop + - CategoryTheory.Functor.rightOp_leftOp_eq + - CategoryTheory.isIso_unop_iff + - CategoryTheory.Iso.unop_inv_hom_id_app + - CategoryTheory.unop_inv + - CategoryTheory.Iso.unop_op + - path: Mathlib/CategoryTheory/Subobject/Basic.lean + theorems: + - CategoryTheory.Subobject.mk_eq_mk_of_comm + - CategoryTheory.Subobject.le_mk_of_comm + - CategoryTheory.Subobject.map_id + - CategoryTheory.Subobject.eq_mk_of_comm + - CategoryTheory.Subobject.ofLEMk_comp_ofMkLE + - CategoryTheory.Subobject.mk_le_of_comm + - CategoryTheory.Subobject.arrow_congr + - "CategoryTheory.Subobject.ind\u2082" + - CategoryTheory.Subobject.pullback_map_self + - CategoryTheory.Subobject.mk_arrow + - CategoryTheory.Subobject.ofLE_mk_le_mk_of_comm + - CategoryTheory.Subobject.ofMkLE_comp_ofLE + - CategoryTheory.Subobject.ofLE_comp_ofLE + - CategoryTheory.Subobject.ind + - CategoryTheory.Subobject.ofLEMk_comp + - CategoryTheory.Subobject.le_of_comm + - CategoryTheory.Subobject.ofLEMk_comp_ofMkLEMk + - CategoryTheory.Subobject.ofMkLEMk_comp_ofMkLEMk + - CategoryTheory.Subobject.ofLE_refl + - CategoryTheory.Subobject.map_pullback + - CategoryTheory.Subobject.map_comp + - CategoryTheory.Subobject.ofMkLEMk_comp_ofMkLE + - CategoryTheory.Subobject.ofMkLEMk_comp + - CategoryTheory.Subobject.ofMkLE_comp_ofLEMk + - CategoryTheory.Subobject.ofMkLE_arrow + - CategoryTheory.Subobject.mk_eq_of_comm + - CategoryTheory.Subobject.pullback_comp + - CategoryTheory.Subobject.pullback_id + - CategoryTheory.Subobject.ofLE_comp_ofLEMk + - path: Mathlib/Algebra/Homology/ImageToKernel.lean + theorems: + - imageToKernel_epi_comp + - imageToKernel_arrow + - homology'.map_desc + - imageSubobjectIso_imageToKernel' + - imageSubobjectMap_comp_imageToKernel + - imageToKernel'_kernelSubobjectIso + - "homology'.\u03C0_desc" + - image_le_kernel + - imageToKernel_comp_left + - factorThruImageSubobject_comp_imageToKernel + - imageToKernel_zero_right + - imageToKernel_comp_hom_inv_comp + - homology'.map_comp + - homology'.comp_right_eq_comp_left + - imageToKernel_comp_right + - homology'.map_id + - "homology'.\u03C0_map" + - "homology'.\u03C0_map_apply" + - imageToKernel_zero_left + - imageToKernel_arrow_apply + - imageToKernel_comp_mono + - path: Mathlib/LinearAlgebra/Matrix/PosDef.lean + theorems: + - QuadraticForm.posDef_toMatrix' + - Matrix.PosSemidef.submatrix + - Matrix.PosSemidef.eq_sqrt_of_sq_eq + - Matrix.PosSemidef.zpow + - Matrix.posSemidef_diagonal_iff + - Matrix.PosDef.eigenvalues_pos + - "Matrix.PosSemidef.toLinearMap\u2082'_zero_iff" + - Matrix.PosSemidef.pow + - Matrix.PosSemidef.mul_mul_conjTranspose_same + - Matrix.posSemidef_conjTranspose_mul_self + - Matrix.PosDef.transpose + - Matrix.PosSemidef.eq_of_sq_eq_sq + - Matrix.PosSemidef.transpose + - QuadraticForm.posDef_of_toMatrix' + - Matrix.PosDef.posSemidef + - Matrix.PosSemidef.sq_sqrt + - Matrix.PosSemidef.posSemidef_sqrt + - Matrix.PosDef.toQuadraticForm' + - Matrix.posSemidef_self_mul_conjTranspose + - Matrix.PosSemidef.inv + - Matrix.PosSemidef.zero + - Matrix.PosSemidef.dotProduct_mulVec_zero_iff + - Matrix.PosSemidef.sqrt_mul_self + - Matrix.IsHermitian.posSemidef_of_eigenvalues_nonneg + - Matrix.PosDef.det_pos + - Matrix.PosDef.of_toQuadraticForm' + - Matrix.posSemidef_submatrix_equiv + - Matrix.posSemidef_iff_eq_transpose_mul_self + - path: Mathlib/Data/Multiset/Bind.lean + theorems: + - Multiset.le_bind + - Multiset.add_product + - Multiset.bind_add + - Multiset.mem_join + - Multiset.product_add + - Multiset.sigma_add + - Multiset.cons_product + - Multiset.product_cons + - Multiset.product_singleton + - Multiset.card_bind + - Multiset.singleton_bind + - Multiset.coe_bind + - Multiset.card_join + - Multiset.bind_zero + - Multiset.bind_cons + - Multiset.rel_bind + - Multiset.count_sum + - Multiset.add_sigma + - Multiset.bind_map_comm + - Multiset.bind_assoc + - Multiset.map_bind + - Multiset.cons_sigma + - Multiset.rel_join + - Multiset.coe_sigma + - Multiset.bind_singleton + - Multiset.cons_bind + - Multiset.bind_bind + - Multiset.product_zero + - Multiset.mem_bind + - Multiset.mem_sigma + - Multiset.add_bind + - Multiset.coe_product + - Multiset.coe_join + - Multiset.bind_hcongr + - Multiset.prod_bind + - Multiset.mem_product + - Multiset.card_product + - Multiset.bind_map + - path: Mathlib/Algebra/Order/Group/PosPart.lean + theorems: + - mabs_mul_eq_oneLePart_sq + - mul_mabs_eq_oneLePart_sq + - le_iff_oneLePart_leOnePart + - oneLePart_eq_ite + - leOnePart_eq_inv + - oneLePart_of_one_lt_oneLePart + - oneLePart_div_leOnePart + - leOnePart_inv + - oneLePart_le_one + - leOnePart_le_one' + - oneLePart_inf_leOnePart_eq_one + - div_mabs_eq_inv_leOnePart_sq + - inf_eq_div_oneLePart_div + - leOnePart_eq_one + - leOnePart_div_oneLePart + - leOnePart_mul_oneLePart + - one_lt_oneLePart + - leOnePart_eq_inv_inf_one + - sup_eq_mul_oneLePart_div + - oneLePart_mul_leOnePart + - leOnePart_eq_ite + - path: Mathlib/Combinatorics/SetFamily/Intersecting.lean + theorems: + - Set.Intersecting.isUpperSet' + - Set.intersecting_singleton + - Set.intersecting_iff_eq_empty_of_subsingleton + - Set.Intersecting.disjoint_map_compl + - Set.Intersecting.is_max_iff_card_eq + - Set.Intersecting.insert + - Set.Intersecting.exists_card_eq + - Set.Intersecting.isUpperSet + - Set.intersecting_iff_pairwise_not_disjoint + - Set.Intersecting.card_le + - path: Mathlib/Data/Matrix/Notation.lean + theorems: + - Matrix.eta_fin_three + - Matrix.eta_fin_two + - Matrix.vec2_eq + - Matrix.one_fin_two + - Matrix.vec3_add + - Matrix.cons_vecMul + - Matrix.one_fin_three + - Matrix.col_cons + - Matrix.cons_dotProduct_cons + - Matrix.smul_vec2 + - Matrix.vec2_dotProduct' + - Matrix.smul_mat_cons + - Matrix.vec3_eq + - Matrix.dotProduct_cons + - Matrix.mul_fin_two + - Matrix.cons_mulVec + - Matrix.natCast_fin_three + - Matrix.vecMulVec_cons + - Matrix.cons_dotProduct + - Matrix.cons_vecMul_cons + - Matrix.cons_mul + - Matrix.tail_val' + - Matrix.mul_fin_three + - Matrix.row_cons + - Matrix.mulVec_cons + - Matrix.submatrix_cons_row + - Matrix.vecMul_cons + - Matrix.smul_vec3 + - Matrix.vec3_dotProduct' + - Matrix.row_empty + - Matrix.cons_val' + - Matrix.cons_vecMulVec + - Matrix.cons_transpose + - Matrix.natCast_fin_two + - Matrix.vec2_add + - path: Mathlib/Data/ENNReal/Inv.lean + theorems: + - ENNReal.exists_mem_Ioc_zpow + - ENNReal.eq_div_iff + - ENNReal.half_pos + - ENNReal.inv_lt_one + - ENNReal.exists_inv_two_pow_lt + - ENNReal.top_div_of_ne_top + - ENNReal.zpow_add + - ENNReal.exists_inv_nat_lt + - ENNReal.exists_nnreal_pos_mul_lt + - ENNReal.div_eq_div_iff + - ENNReal.one_le_inv + - ENNReal.mul_inv + - ENNReal.mul_div_mul_left + - ENNReal.top_div + - ENNReal.zpow_lt_top + - ENNReal.coe_div + - ENNReal.div_zero + - ENNReal.div_eq_zero_iff + - ENNReal.div_lt_of_lt_mul + - ENNReal.inv_le_iff_inv_le + - ENNReal.mul_div_cancel' + - ENNReal.one_sub_inv_two + - ENNReal.add_halves + - ENNReal.le_inv_iff_le_inv + - ENNReal.div_top + - ENNReal.eq_inv_of_mul_eq_one_left + - ENNReal.mul_le_iff_le_inv + - ENNReal.div_eq_one_iff + - ENNReal.inv_eq_one + - ENNReal.div_mul_cancel + - ENNReal.one_lt_inv + - ENNReal.inv_pow + - ENNReal.inv_ne_zero + - ENNReal.zpow_le_of_le + - ENNReal.le_of_forall_nnreal_lt + - ENNReal.Ioo_zero_top_eq_iUnion_Ico_zpow + - ENNReal.coe_inv + - ENNReal.inv_ne_top + - ENNReal.zpow_pos + - ENNReal.inv_lt_iff_inv_lt + - ENNReal.coe_zpow + - ENNReal.mul_div_mul_right + - ENNReal.div_le_of_le_mul + - ENNReal.inv_le_iff_le_mul + - ENNReal.div_lt_of_lt_mul' + - ENNReal.exists_nat_pos_mul_gt + - ENNReal.inv_two_add_inv_two + - ENNReal.div_eq_top + - ENNReal.div_self_le_one + - ENNReal.inv_le_one + - ENNReal.mul_comm_div + - ENNReal.inv_zero + - ENNReal.inv_three_add_inv_three + - ENNReal.sub_div + - ENNReal.coe_inv_two + - ENNReal.inv_lt_top + - ENNReal.sub_half + - ENNReal.mul_div_right_comm + - ENNReal.le_div_iff_mul_le + - ENNReal.inv_top + - ENNReal.exists_mem_Ico_zpow + - ENNReal.mul_le_of_le_div + - ENNReal.coe_inv_le + - ENNReal.exists_nat_pos_inv_mul_lt + - ENNReal.add_thirds + - ENNReal.div_pos_iff + - ENNReal.mul_inv_cancel + - ENNReal.le_inv_iff_mul_le + - ENNReal.lt_inv_iff_lt_inv + - ENNReal.half_lt_self + - ENNReal.div_le_iff_le_mul + - ENNReal.inv_strictAnti + - ENNReal.div_eq_inv_mul + - path: Mathlib/Dynamics/Ergodic/Conservative.lean + theorems: + - MeasureTheory.Conservative.ae_frequently_mem_of_mem_nhds + - MeasureTheory.Conservative.ae_forall_image_mem_imp_frequently_image_mem + - MeasureTheory.Conservative.ae_mem_imp_frequently_image_mem + - MeasureTheory.Conservative.iterate + - MeasureTheory.Conservative.frequently_measure_inter_ne_zero + - MeasureTheory.Conservative.measure_mem_forall_ge_image_not_mem_eq_zero + - path: Mathlib/Algebra/Homology/SingleHomology.lean + theorems: + - ChainComplex.exactAt_succ_single_obj + - HomologicalComplex.isZero_single_obj_homology + - HomologicalComplex.singleObjCyclesSelfIso_hom_singleObjOpcyclesSelfIso_hom + - "HomologicalComplex.singleObjCyclesSelfIso_inv_homology\u03C0" + - "HomologicalComplex.homology\u03C0_singleObjHomologySelfIso_hom" + - HomologicalComplex.singleObjHomologySelfIso_hom_singleObjHomologySelfIso_inv + - HomologicalComplex.singleObjHomologySelfIso_hom_naturality + - HomologicalComplex.singleObjCyclesSelfIso_inv_naturality + - HomologicalComplex.singleObjCyclesSelfIso_hom_naturality + - HomologicalComplex.singleObjHomologySelfIso_hom_singleObjOpcyclesSelfIso_hom + - "HomologicalComplex.singleObjHomologySelfIso_inv_homology\u03B9" + - HomologicalComplex.singleObjHomologySelfIso_inv_naturality + - CochainComplex.exactAt_succ_single_obj + - HomologicalComplex.singleObjOpcyclesSelfIso_inv_naturality + - HomologicalComplex.singleObjOpcyclesSelfIso_hom_naturality + - HomologicalComplex.singleObjCyclesSelfIso_inv_iCycles + - "HomologicalComplex.homology\u03B9_singleObjOpcyclesSelfIso_inv" + - path: Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean + theorems: + - GradedTensorProduct.comm_coe_tmul_coe + - GradedTensorProduct.tmul_coe_mul_one_tmul + - GradedTensorProduct.tmul_coe_mul_zero_coe_tmul + - GradedTensorProduct.tmul_one_mul_coe_tmul + - GradedTensorProduct.auxEquiv_one + - GradedTensorProduct.tmul_zero_coe_mul_coe_tmul + - GradedTensorProduct.tmul_one_mul_one_tmul + - GradedTensorProduct.tmul_coe_mul_coe_tmul + - path: Mathlib/SetTheory/Cardinal/Subfield.lean + theorems: + - Subfield.rangeOfWType_eq_top + - Subfield.cardinal_mk_closure_le_max + - Subfield.surjective_ofWType + - path: Mathlib/GroupTheory/Submonoid/Pointwise.lean + theorems: + - AddSubmonoid.mul_induction_on + - AddSubmonoid.mul_bot + - AddSubmonoid.mul_eq_closure_mul_set + - Submonoid.sup_eq_closure_mul + - AddSubmonoid.closure_mul_closure + - AddSubmonoid.one_eq_closure + - AddSubmonoid.closure_pow + - Submonoid.mem_closure_inv + - AddSubmonoid.bot_mul + - Set.IsPWO.submonoid_closure + - AddSubmonoid.pow_eq_closure_pow_set + - Submonoid.closure_inv + - Submonoid.pow_smul_mem_closure_smul + - Submonoid.coe_mul_self_eq + - AddSubmonoid.mul_mem_mul + - path: Mathlib/RingTheory/Nilpotent.lean + theorems: + - IsNilpotent.pow_of_pos + - nilpotent_iff_mem_prime + - nilpotencyClass_eq_zero_of_subsingleton + - IsNilpotent.of_pow + - Commute.isNilpotent_mul_left + - Commute.isNilpotent_mul_left_iff + - Commute.isNilpotent_mul_right + - Commute.isNilpotent_add + - eq_zero_of_nilpotencyClass_eq_one + - LinearMap.isNilpotent_mulLeft_iff + - Commute.isNilpotent_sub + - IsNilpotent.neg + - RingHom.ker_isRadical_iff_reduced_of_surjective + - Commute.isNilpotent_sum + - IsNilpotent.isUnit_add_one + - IsNilpotent.isUnit_sub_one + - IsNilpotent.map_iff + - IsUnit.isNilpotent_mul_unit_of_commute_iff + - nilpotencyClass_zero + - Commute.isNilpotent_mul_right_iff + - isRadical_iff_span_singleton + - LinearMap.isNilpotent_toMatrix_iff + - IsNilpotent.pow_succ + - isNilpotent_of_pos_nilpotencyClass + - IsRadical.of_dvd + - nilpotencyClass_eq_succ_iff + - LinearMap.isNilpotent_mulRight_iff + - IsNilpotent.smul + - zero_isRadical_iff + - Module.End.IsNilpotent.mapQ + - pos_nilpotencyClass_iff + - nilradical_eq_sInf + - IsNilpotent.isUnit_add_left_of_commute + - isReduced_iff_pow_one_lt + - isReduced_of_injective + - IsNilpotent.map + - NoZeroSMulDivisors.isReduced + - IsNilpotent.isUnit_one_sub + - path: Mathlib/Algebra/Homology/HomologySequence.lean + theorems: + - HomologicalComplex.opcyclesToCycles_naturality + - HomologicalComplex.pOpcycles_opcyclesToCycles + - HomologicalComplex.opcycles_right_exact + - HomologicalComplex.opcyclesToCycles_iCycles + - "CategoryTheory.ShortComplex.ShortExact.\u03B4_eq" + - "HomologicalComplex.HomologySequence.composableArrows\u2083_exact" + - "HomologicalComplex.opcyclesToCycles_homology\u03C0" + - HomologicalComplex.cycles_left_exact + - "HomologicalComplex.homology\u03B9_opcyclesToCycles" + - "CategoryTheory.ShortComplex.ShortExact.homology_exact\u2082" + - HomologicalComplex.pOpcycles_opcyclesToCycles_iCycles + - path: Mathlib/Order/Concept.lean + theorems: + - Concept.ext + - Concept.ext' + - Concept.snd_ssubset_snd_iff + - Concept.snd_subset_snd_iff + - path: Mathlib/AlgebraicGeometry/Gluing.lean + theorems: + - AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'_fst_fst + - AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'_snd_snd + - AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'_snd_fst + - AlgebraicGeometry.Scheme.OpenCover.fromGlued_open_map + - AlgebraicGeometry.Scheme.GlueData.isOpen_iff + - "AlgebraicGeometry.Scheme.OpenCover.\u03B9_glueMorphisms" + - AlgebraicGeometry.Scheme.OpenCover.glued_cover_cocycle + - AlgebraicGeometry.Scheme.OpenCover.glued_cover_cocycle_snd + - "AlgebraicGeometry.Scheme.GlueData.\u03B9_eq_iff" + - AlgebraicGeometry.Scheme.OpenCover.glued_cover_cocycle_fst + - AlgebraicGeometry.Scheme.OpenCover.fromGlued_injective + - AlgebraicGeometry.Scheme.OpenCover.gluedCoverT'_fst_snd + - path: Mathlib/Combinatorics/SimpleGraph/Trails.lean + theorems: + - SimpleGraph.Walk.IsEulerian.mem_edges_iff + - SimpleGraph.Walk.IsEulerian.edgesFinset_eq + - SimpleGraph.Walk.IsTrail.even_countP_edges_iff + - SimpleGraph.Walk.IsEulerian.isTrail + - SimpleGraph.Walk.IsEulerian.card_filter_odd_degree + - SimpleGraph.Walk.IsEulerian.even_degree_iff + - SimpleGraph.Walk.isEulerian_iff + - SimpleGraph.Walk.IsEulerian.card_odd_degree + - path: Mathlib/CategoryTheory/Subobject/Limits.lean + theorems: + - CategoryTheory.Limits.kernelSubobject_comp_le + - CategoryTheory.Limits.image_map_comp_imageSubobjectIso_inv + - CategoryTheory.Limits.factorThruImageSubobject_comp_self_assoc + - CategoryTheory.Limits.kernelSubobject_comp_mono + - CategoryTheory.Limits.imageSubobject_zero_arrow + - CategoryTheory.Limits.imageSubobject_zero + - CategoryTheory.Limits.imageSubobjectMap_arrow + - CategoryTheory.Limits.factorThruKernelSubobject_comp_arrow + - CategoryTheory.Limits.imageSubobject_arrow_comp_eq_zero + - CategoryTheory.Limits.kernelSubobjectIsoComp_hom_arrow + - CategoryTheory.Limits.imageSubobjectCompIso_hom_arrow + - CategoryTheory.Limits.kernelSubobject_factors_iff + - CategoryTheory.Limits.imageSubobject_comp_le + - CategoryTheory.Limits.imageSubobjectCompIso_inv_arrow + - CategoryTheory.Limits.kernelSubobject_arrow + - CategoryTheory.Limits.kernelSubobject_arrow' + - CategoryTheory.Limits.equalizerSubobject_factors + - CategoryTheory.Limits.imageSubobject_factors_comp_self + - CategoryTheory.Limits.equalizerSubobject_arrow + - CategoryTheory.Limits.equalizerSubobject_arrow' + - CategoryTheory.Limits.imageSubobject_le + - CategoryTheory.Limits.equalizerSubobject_factors_iff + - CategoryTheory.Limits.kernelSubobjectMap_comp + - CategoryTheory.Limits.imageSubobject_iso_comp + - CategoryTheory.Limits.imageSubobjectIso_comp_image_map + - CategoryTheory.Limits.imageSubobject_le_mk + - CategoryTheory.Limits.kernel_map_comp_kernelSubobjectIso_inv + - CategoryTheory.Limits.factorThruImageSubobject_comp_self + - CategoryTheory.Limits.imageSubobject_arrow + - CategoryTheory.Limits.kernelSubobjectIsoComp_inv_arrow + - CategoryTheory.Limits.kernelSubobjectIso_comp_kernel_map + - CategoryTheory.Limits.kernelSubobject_zero + - CategoryTheory.Limits.kernelSubobjectMap_arrow + - CategoryTheory.Limits.imageSubobject_arrow_comp + - CategoryTheory.Limits.kernelSubobject_factors + - CategoryTheory.Limits.kernelSubobjectMap_id + - CategoryTheory.Limits.imageSubobject_mono + - CategoryTheory.Limits.imageSubobject_arrow' + - CategoryTheory.Limits.equalizerSubobject_arrow_comp + - CategoryTheory.Limits.kernelSubobject_arrow_comp + - path: Mathlib/MeasureTheory/Group/LIntegral.lean + theorems: + - MeasureTheory.lintegral_div_right_eq_self + - MeasureTheory.lintegral_eq_zero_of_isMulLeftInvariant + - MeasureTheory.lintegral_mul_left_eq_self + - MeasureTheory.lintegral_mul_right_eq_self + - path: Mathlib/Data/List/ProdSigma.lean + theorems: + - List.length_sigma + - List.mem_product + - List.sigma_nil + - List.mem_sigma + - List.product_nil + - List.length_product + - path: Mathlib/ModelTheory/Types.lean + theorems: + - FirstOrder.Language.Theory.CompleteType.not_mem_iff + - FirstOrder.Language.Theory.CompleteType.formula_mem_typeOf + - FirstOrder.Language.Theory.exists_modelType_is_realized_in + - FirstOrder.Language.Theory.CompleteType.setOf_mem_eq_univ_iff + - FirstOrder.Language.Theory.CompleteType.setOf_subset_eq_empty_iff + - FirstOrder.Language.Theory.CompleteType.iInter_setOf_subset + - FirstOrder.Language.Theory.CompleteType.nonempty_iff + - FirstOrder.Language.Theory.CompleteType.toList_foldr_inf_mem + - path: Mathlib/CategoryTheory/Bicategory/Basic.lean + theorems: + - CategoryTheory.Bicategory.pentagon_hom_hom_inv_hom_hom + - CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv + - CategoryTheory.Bicategory.leftUnitor_comp + - CategoryTheory.Bicategory.whiskerRight_iff + - CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv + - CategoryTheory.Bicategory.whiskerLeft_rightUnitor_inv + - CategoryTheory.Bicategory.associator_naturality_left + - CategoryTheory.Bicategory.id_whiskerLeft_symm + - CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_inv + - CategoryTheory.Bicategory.whiskerLeft_iff + - CategoryTheory.Bicategory.leftUnitor_whiskerRight + - CategoryTheory.Bicategory.leftUnitor_naturality + - CategoryTheory.Bicategory.rightUnitor_inv_naturality + - CategoryTheory.Bicategory.unitors_equal + - CategoryTheory.Bicategory.triangle_assoc_comp_right_inv + - CategoryTheory.Bicategory.leftUnitor_inv_whiskerRight + - CategoryTheory.Bicategory.unitors_inv_equal + - CategoryTheory.Bicategory.whisker_assoc_symm + - CategoryTheory.Bicategory.hom_inv_whiskerRight + - CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_hom + - CategoryTheory.Bicategory.rightUnitor_comp + - CategoryTheory.Bicategory.rightUnitor_comp_inv + - CategoryTheory.Bicategory.whiskerRight_id_symm + - CategoryTheory.Bicategory.inv_whiskerLeft + - CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom + - CategoryTheory.Bicategory.pentagon_inv + - CategoryTheory.Bicategory.triangle_assoc_comp_right + - CategoryTheory.Bicategory.comp_whiskerLeft_symm + - CategoryTheory.Bicategory.associator_inv_naturality_left + - CategoryTheory.Bicategory.associator_naturality_right + - CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom + - CategoryTheory.Bicategory.associator_inv_naturality_middle + - CategoryTheory.Bicategory.inv_hom_whiskerRight + - CategoryTheory.Bicategory.whiskerLeft_rightUnitor + - CategoryTheory.Bicategory.associator_naturality_middle + - CategoryTheory.Bicategory.whiskerLeft_inv_hom + - CategoryTheory.Bicategory.whiskerLeft_hom_inv + - CategoryTheory.Bicategory.triangle_assoc_comp_left_inv + - CategoryTheory.Bicategory.rightUnitor_naturality + - CategoryTheory.Bicategory.associator_inv_naturality_right + - CategoryTheory.Bicategory.pentagon_inv_inv_hom_hom_inv + - CategoryTheory.Bicategory.leftUnitor_comp_inv + - CategoryTheory.Bicategory.whiskerRight_comp_symm + - CategoryTheory.Bicategory.inv_whiskerRight + - path: Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean + theorems: + - MeasureTheory.FiniteMeasure.mass_prod + - MeasureTheory.ProbabilityMeasure.prod_swap + - MeasureTheory.FiniteMeasure.prod_apply_symm + - MeasureTheory.FiniteMeasure.prod_prod + - MeasureTheory.FiniteMeasure.prod_zero + - MeasureTheory.FiniteMeasure.prod_swap + - MeasureTheory.FiniteMeasure.prod_apply + - MeasureTheory.ProbabilityMeasure.prod_prod + - MeasureTheory.ProbabilityMeasure.prod_apply_symm + - MeasureTheory.ProbabilityMeasure.map_prod_map + - MeasureTheory.FiniteMeasure.map_snd_prod + - MeasureTheory.FiniteMeasure.zero_prod + - MeasureTheory.ProbabilityMeasure.map_snd_prod + - MeasureTheory.ProbabilityMeasure.prod_apply + - MeasureTheory.FiniteMeasure.map_fst_prod + - MeasureTheory.ProbabilityMeasure.map_fst_prod + - MeasureTheory.FiniteMeasure.map_prod_map + - path: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean + theorems: + - MeasureTheory.StronglyMeasurable.integral_prod_right + - MeasureTheory.integral_prod + - MeasureTheory.AEStronglyMeasurable.prod_swap + - MeasureTheory.set_integral_prod_mul + - MeasureTheory.integral_prod_symm + - MeasureTheory.hasFiniteIntegral_prod_iff' + - MeasureTheory.AEStronglyMeasurable.integral_prod_right' + - MeasureTheory.integral_fn_integral_sub + - MeasureTheory.integral_fun_fst + - MeasureTheory.set_integral_prod + - MeasureTheory.Integrable.prod_smul + - MeasureTheory.AEStronglyMeasurable.prod_mk_left + - MeasureTheory.integral_integral_swap_of_hasCompactSupport + - MeasureTheory.integral_fun_snd + - MeasureTheory.integrable_prod_iff' + - MeasureTheory.integrable_prod_iff + - MeasureTheory.integral_fn_integral_add + - MeasureTheory.continuous_integral_integral + - MeasureTheory.hasFiniteIntegral_prod_iff + - MeasureTheory.integral_prod_smul + - MeasureTheory.StronglyMeasurable.integral_prod_right' + - MeasureTheory.Measure.integrable_measure_prod_mk_left + - path: Mathlib/Data/Nat/Choose/Multinomial.lean + theorems: + - Nat.succ_mul_binomial + - Nat.binomial_spec + - Nat.multinomial_univ_two + - Sym.multinomial_coe_fill_of_not_mem + - Nat.binomial_one + - Finsupp.multinomial_update + - Nat.multinomial_congr + - Nat.multinomial_univ_three + - Nat.multinomial_insert_one + - Nat.binomial_eq_choose + - Nat.multinomial_singleton + - Nat.multinomial_cons + - Finset.sum_pow + - Nat.binomial_succ_succ + - Nat.multinomial_insert + - Finset.sum_pow_of_commute + - Multiset.multinomial_filter_ne + - path: Mathlib/Data/Fintype/Fin.lean + theorems: + - Fin.card_filter_univ_eq_vector_get_eq_count + - Fin.Ioi_succ + - Fin.Iio_castSucc + - Fin.card_filter_univ_succ' + - Fin.Ioi_zero_eq_map + - Fin.map_valEmbedding_univ + - Fin.Iio_last_eq_map + - Fin.card_filter_univ_succ + - path: Mathlib/Analysis/Normed/Field/Basic.lean + theorems: + - List.nnnorm_prod_le' + - Finset.nnnorm_prod_le' + - NNReal.norm_eq + - nnnorm_pow_le' + - nnnorm_pow_le + - NormedField.exists_lt_norm + - NormedField.punctured_nhds_neBot + - Finset.norm_prod_le' + - nnnorm_norm + - Real.nnnorm_mul_toNNReal + - "dist_inv_inv\u2080" + - Finset.norm_prod_le + - Nat.norm_cast_le + - Finset.nnnorm_prod_le + - one_le_norm_one + - nnnorm_inv + - nnnorm_mul_le + - norm_pow_le' + - Real.toNNReal_mul_nnnorm + - NormedField.nhdsWithin_isUnit_neBot + - mulRight_bound + - List.norm_prod_le + - NormedAddCommGroup.tendsto_atTop + - NormedField.exists_norm_lt + - antilipschitzWith_mul_left + - norm_pow_le + - "Filter.inv_cobounded\u2080" + - List.norm_prod_le' + - Filter.inv_nhdsWithin_ne_zero + - NormOneClass.nontrivial + - NormedAddCommGroup.tendsto_atTop' + - List.nnnorm_prod_le + - path: Mathlib/Topology/UniformSpace/UniformConvergence.lean + theorems: + - UniformCauchySeqOnFilter.mono_right + - tendstoUniformly_iff_seq_tendstoUniformly + - tendstoUniformlyOn_univ + - IsOpen.tendstoLocallyUniformlyOn_iff_forall_tendsto + - tendstoUniformlyOn_iff_tendstoUniformlyOnFilter + - tendsto_prod_top_iff + - TendstoLocallyUniformlyOn.mono + - Filter.Tendsto.tendstoUniformlyOnFilter_const + - tendstoLocallyUniformlyOn_iff_filter + - TendstoUniformly.comp + - TendstoUniformlyOn.tendstoLocallyUniformlyOn + - TendstoUniformlyOnFilter.tendsto_at + - "UniformContinuous\u2082.tendstoUniformly" + - TendstoUniformly.tendstoLocallyUniformly + - continuousAt_of_locally_uniform_approx_of_continuousAt + - TendstoLocallyUniformlyOn.comp + - tendsto_comp_of_locally_uniform_limit + - UniformContinuousOn.tendstoUniformly + - TendstoUniformlyOn.congr_right + - TendstoLocallyUniformlyOn.union + - UniformCauchySeqOn.prod_map + - UniformCauchySeqOn.cauchy_map + - tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace + - UniformCauchySeqOn.mono + - UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto + - TendstoLocallyUniformly.comp + - TendstoUniformlyOn.seq_tendstoUniformlyOn + - UniformCauchySeqOnFilter.mono_left + - TendstoUniformlyOn.comp + - tendstoUniformlyOn_iff_tendstoUniformly_comp_coe + - TendstoUniformlyOnFilter.congr + - TendstoUniformlyOnFilter.uniformCauchySeqOnFilter + - TendstoLocallyUniformlyOn.continuousOn + - tendstoUniformlyOn_empty + - tendsto_prod_filter_iff + - tendsto_comp_of_locally_uniform_limit_within + - TendstoUniformlyOnFilter.prod_map + - tendsto_prod_principal_iff + - tendstoUniformly_iff_tendsto + - tendstoUniformlyOn_of_seq_tendstoUniformlyOn + - TendstoUniformly.tendstoUniformlyOnFilter + - tendstoLocallyUniformlyOn_iff_forall_tendsto + - TendstoUniformlyOn.congr + - TendstoUniformlyOn.prod_map + - continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt + - tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe + - tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact + - TendstoUniformlyOnFilter.comp + - tendstoLocallyUniformlyOn_TFAE + - TendstoLocallyUniformlyOn.congr_right + - tendstoLocallyUniformly_iff_forall_tendsto + - TendstoLocallyUniformlyOn.tendsto_at + - tendstoUniformly_iff_tendstoUniformlyOnFilter + - UniformCauchySeqOn.uniformCauchySeqOnFilter + - tendstoLocallyUniformlyOn_sUnion + - uniformCauchySeqOn_iff_uniformCauchySeqOnFilter + - tendstoUniformlyOn_singleton_iff_tendsto + - UniformCauchySeqOn.comp + - UniformContinuousOn.tendstoUniformlyOn + - tendstoLocallyUniformly_iff_filter + - continuous_of_uniform_approx_of_continuous + - UniformCauchySeqOnFilter.comp + - TendstoLocallyUniformlyOn.congr + - tendstoUniformlyOn_iff_tendsto + - path: Mathlib/Data/Multiset/Fintype.lean + theorems: + - Multiset.prod_eq_prod_coe + - Multiset.toEnumFinset_filter_eq + - Multiset.card_toEnumFinset + - Multiset.card_coe + - Multiset.image_toEnumFinset_fst + - Multiset.map_univ + - Multiset.prod_eq_prod_toEnumFinset + - Multiset.prod_toEnumFinset + - Multiset.map_univ_coe + - Multiset.map_univ_coeEmbedding + - Multiset.toEnumFinset_mono + - Multiset.map_toEnumFinset_fst + - Multiset.toEnumFinset_subset_iff + - path: Mathlib/Algebra/Homology/HomotopyCofiber.lean + theorems: + - HomologicalComplex.homotopyCofiber.ext_to_X' + - HomologicalComplex.homotopyCofiber.inlX_sndX + - HomologicalComplex.homotopyCofiber.inlX_d + - HomologicalComplex.homotopyCofiber.d_sndX + - "HomologicalComplex.cylinder.inlX_\u03C0" + - HomologicalComplex.homotopyCofiber.inrCompHomotopy_hom_desc_hom + - HomologicalComplex.homotopyCofiber.inlX_fstX + - HomologicalComplex.homotopyCofiber.inlX_desc_f + - HomologicalComplex.homotopyCofiber.inrX_d + - HomologicalComplex.homotopyCofiber.inlX_d' + - HomologicalComplex.homotopyCofiber.inrX_fstX + - HomologicalComplex.homotopyCofiber.descSigma_ext_iff + - HomologicalComplex.homotopyCofiber.ext_from_X + - HomologicalComplex.homotopyCofiber.ext_from_X' + - HomologicalComplex.homotopyCofiber.desc_f' + - HomologicalComplex.homotopyCofiber.ext_to_X + - HomologicalComplex.homotopyCofiber.inr_desc + - "HomologicalComplex.cylinder.\u03B9\u2080_\u03C0" + - HomologicalComplex.homotopyCofiber.inrX_sndX + - HomologicalComplex.homotopyCofiber.inrX_desc_f + - HomologicalComplex.homotopyCofiber.eq_desc + - "HomologicalComplex.cylinder.\u03C0Comp\u03B9\u2080Homotopy.nullHomotopicMap_eq" + - "HomologicalComplex.cylinder.\u03C0Comp\u03B9\u2080Homotopy.inrX_nullHomotopy_f" + - HomologicalComplex.homotopyCofiber.d_fstX + - "HomologicalComplex.cylinder.\u03C0Comp\u03B9\u2080Homotopy.inlX_nullHomotopy_f" + - HomologicalComplex.homotopyCofiber.desc_f + - "HomologicalComplex.cylinder.\u03C0Comp\u03B9\u2080Homotopy.biprod_lift_id_sub_id" + - "HomologicalComplex.cylinder.\u03B9\u2081_desc" + - Homotopy.map_eq_of_inverts_homotopyEquivalences + - "HomologicalComplex.cylinder.\u03B9\u2080_desc" + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean + theorems: + - Real.pi_div_four_le_arcsin + - Real.range_arcsin + - Real.le_arcsin_iff_sin_le + - Real.arcsin_neg_one + - Real.tan_arccos + - Real.arccos_neg + - Real.cos_arcsin + - Real.sin_arcsin' + - Real.arccos_pos + - Real.arccos_eq_arcsin + - Real.arcsin_lt_pi_div_two + - Real.le_arcsin_iff_sin_le' + - Real.arcsin_projIcc + - Real.arccos_lt_pi_div_two + - Real.arccos_le_pi_div_two + - Real.arcsin_neg + - Real.arccos_nonneg + - Real.sin_arccos + - Real.arccos_one + - Real.arcsin_eq_of_sin_eq + - Real.arcsin_of_one_le + - Real.arcsin_eq_iff_eq_sin + - Real.arccos_le_pi_div_four + - Real.neg_pi_div_two_lt_arcsin + - Real.arcsin_eq_pi_div_two_sub_arccos + - Real.arcsin_nonneg + - Real.arccos_cos + - Real.arcsin_eq_arccos + - Real.arccos_of_one_le + - Real.mapsTo_sin_Ioo + - Real.arcsin_sin' + - Real.arcsin_le_iff_le_sin' + - Real.arcsin_of_le_neg_one + - Real.tan_arcsin + - Real.arccos_le_pi + - Real.arccos_eq_zero + - Real.arccos_zero + - Real.arcsin_le_iff_le_sin + - Real.arccos_eq_of_eq_cos + - Real.arccos_eq_pi + - Real.arcsin_eq_zero_iff + - Real.arccos_eq_pi_div_two + - Real.cos_arccos + - path: Mathlib/Data/Finset/Update.lean + theorems: + - Function.updateFinset_updateFinset + - Function.update_eq_updateFinset + - Function.updateFinset_singleton + - path: Mathlib/CategoryTheory/CofilteredSystem.lean + theorems: + - CategoryTheory.Functor.IsMittagLeffler.subset_image_eventualRange + - CategoryTheory.Functor.eventualRange_mapsTo + - CategoryTheory.Functor.IsMittagLeffler.toPreimages + - CategoryTheory.Functor.eventualRange_eq_range_precomp + - CategoryTheory.Functor.thin_diagram_of_surjective + - nonempty_sections_of_finite_cofiltered_system.init + - CategoryTheory.Functor.surjective_toEventualRanges + - CategoryTheory.Functor.isMittagLeffler_iff_subset_range_comp + - CategoryTheory.Functor.toPreimages_nonempty_of_surjective + - CategoryTheory.Functor.eval_section_surjective_of_surjective + - CategoryTheory.Functor.eval_section_injective_of_eventually_injective + - CategoryTheory.Functor.isMittagLeffler_of_surjective + - nonempty_sections_of_finite_cofiltered_system + - CategoryTheory.Functor.eventualRange_eq_iff + - CategoryTheory.Functor.eventually_injective + - CategoryTheory.Functor.toEventualRanges_nonempty + - CategoryTheory.Functor.isMittagLeffler_of_exists_finite_range + - nonempty_sections_of_finite_inverse_system + - path: Mathlib/Topology/IndicatorConstPointwise.lean + theorems: + - tendsto_indicator_const_apply_iff_eventually' + - tendsto_indicator_const_iff_tendsto_pi_pure + - tendsto_ite + - tendsto_indicator_const_iff_forall_eventually' + - tendsto_indicator_const_iff_forall_eventually + - tendsto_indicator_const_iff_tendsto_pi_pure' + - path: Mathlib/Order/RelClasses.lean + theorems: + - trans_trichotomous_right + - eq_empty_relation + - subset_of_subset_of_eq + - IsOrderConnected.neg_trans + - ssubset_of_eq_of_ssubset + - Set.not_bounded_iff + - trans_trichotomous_left + - IsTrichotomous.swap + - Set.not_unbounded_iff + - ssubset_of_ssubset_of_eq + - subset_of_eq_of_subset + - antisymm_iff + - path: .lake/packages/std/Std/Logic.lean + theorems: + - congr_eqRec + - Eq.congr_left + - cast_eq_iff_heq + - Eq.congr_right + - "congrArg\u2082" + - proof_irrel_heq + - eqRec_heq_self + - eqRec_heq_iff_heq + - heq_eqRec_iff_heq + - Eq.congr + - eq_rec_constant + - eqRec_eq_cast + - path: Mathlib/RepresentationTheory/Action/Monoidal.lean + theorems: + - Action.associator_hom_hom + - "Action.rightDual_\u03C1" + - "Action.functorCategoryMonoidalEquivalence.\u03BC_app" + - Action.associator_inv_hom + - Action.leftUnitor_inv_hom + - "CategoryTheory.MonoidalFunctor.mapAction_\u03BC_inv_hom" + - "Action.leftDual_\u03C1" + - "Action.functorCategoryMonoidalEquivalence.\u03B5_app" + - Action.rightUnitor_inv_hom + - "CategoryTheory.MonoidalFunctor.mapAction_\u03B5_inv_hom" + - Action.leftUnitor_hom_hom + - "Action.functorCategoryMonoidalEquivalence.\u03BCIso_inv_app" + - path: Mathlib/Data/NNRat/BigOperators.lean + theorems: + - NNRat.toNNRat_prod_of_nonneg + - NNRat.toNNRat_sum_of_nonneg + - path: Mathlib/Probability/Distributions/Poisson.lean + theorems: + - ProbabilityTheory.poissonPMFReal_nonneg + - ProbabilityTheory.measurable_poissonPMFReal + - ProbabilityTheory.poissonPMFReal_pos + - ProbabilityTheory.poissonPMFRealSum + - path: Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean + theorems: + - GeneralizedContinuedFraction.of_s_head_aux + - GeneralizedContinuedFraction.IntFractPair.exists_succ_nth_stream_of_fr_zero + - GeneralizedContinuedFraction.get?_of_eq_some_of_succ_get?_intFractPair_stream + - GeneralizedContinuedFraction.of_s_succ + - GeneralizedContinuedFraction.IntFractPair.stream_succ + - GeneralizedContinuedFraction.of_s_of_int + - GeneralizedContinuedFraction.IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some + - GeneralizedContinuedFraction.get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero + - GeneralizedContinuedFraction.of_terminatedAt_n_iff_succ_nth_intFractPair_stream_eq_none + - GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_eq_some_iff + - GeneralizedContinuedFraction.of_s_head + - GeneralizedContinuedFraction.IntFractPair.stream_succ_of_int + - GeneralizedContinuedFraction.IntFractPair.stream_eq_none_of_fr_eq_zero + - GeneralizedContinuedFraction.of_h_eq_floor + - GeneralizedContinuedFraction.convergents'_succ + - GeneralizedContinuedFraction.of_h_eq_intFractPair_seq1_fst_b + - path: Mathlib/FieldTheory/PrimitiveElement.lean + theorems: + - Field.primitive_element_inf_aux + - Field.isAlgebraic_of_adjoin_eq_adjoin + - Field.primitive_element_inf_aux_of_finite_intermediateField + - Field.FiniteDimensional.of_exists_primitive_element + - Field.finite_intermediateField_of_exists_primitive_element + - AlgHom.card_of_splits + - Field.exists_primitive_element_of_finite_intermediateField + - Field.FiniteDimensional.of_finite_intermediateField + - Field.primitive_element_iff_minpoly_degree_eq + - Field.primitive_element_inf_aux_exists_c + - Field.primitive_element_iff_minpoly_natDegree_eq + - Field.primitive_element_iff_algHom_eq_of_eval' + - Field.exists_primitive_element_of_finite_top + - Field.primitive_element_iff_algHom_eq_of_eval + - Field.exists_primitive_element + - AlgHom.card + - path: Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean + theorems: + - contMDiff_model + - isLocalStructomorphOn_contDiffGroupoid_iff + - contMDiffOn_extChartAt_symm + - isLocalStructomorphOn_contDiffGroupoid_iff_aux + - contMDiffOn_model_symm + - contMDiffOn_extend_symm + - path: Mathlib/Data/Prod/TProd.lean + theorems: + - Set.mk_preimage_tprod + - Set.elim_preimage_pi + - List.TProd.elim_mk + - List.TProd.ext + - List.TProd.elim_self + - List.TProd.elim_of_mem + - List.TProd.elim_of_ne + - List.TProd.mk_elim + - path: Mathlib/Topology/MetricSpace/CantorScheme.lean + theorems: + - CantorScheme.VanishingDiam.map_continuous + - CantorScheme.VanishingDiam.dist_lt + - CantorScheme.ClosureAntitone.map_of_vanishingDiam + - CantorScheme.map_mem + - CantorScheme.Disjoint.map_injective + - path: Mathlib/Probability/Martingale/BorelCantelli.lean + theorems: + - MeasureTheory.leastGE_eq_min + - MeasureTheory.BorelCantelli.martingalePart_process_ae_eq + - MeasureTheory.tendsto_sum_indicator_atTop_iff + - MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto + - MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_range + - MeasureTheory.tendsto_sum_indicator_atTop_iff' + - MeasureTheory.Submartingale.stoppedValue_leastGE_snorm_le + - MeasureTheory.Submartingale.stoppedValue_leastGE_snorm_le' + - MeasureTheory.BorelCantelli.process_difference_le + - MeasureTheory.BorelCantelli.process_zero + - MeasureTheory.Submartingale.exists_tendsto_of_abs_bddAbove_aux + - MeasureTheory.norm_stoppedValue_leastGE_le + - MeasureTheory.Submartingale.stoppedValue_leastGE + - MeasureTheory.stoppedValue_stoppedValue_leastGE + - MeasureTheory.Martingale.ae_not_tendsto_atTop_atTop + - MeasureTheory.Martingale.ae_not_tendsto_atTop_atBot + - MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto_aux + - MeasureTheory.BorelCantelli.predictablePart_process_ae_eq + - path: Mathlib/Data/List/EditDistance/Defs.lean + theorems: + - "suffixLevenshtein_cons\u2081_fst" + - suffixLevenshtein_eq + - suffixLevenshtein_cons_cons_fst_get_zero + - Levenshtein.impl_length + - suffixLevenshtein_nil_nil + - "suffixLevenshtein_cons\u2081_aux" + - levenshtein_nil_nil + - Levenshtein.impl_cons + - levenshtein_nil_cons + - "suffixLevenshtein_cons\u2081" + - suffixLevenshtein_eq_tails_map + - List.eq_of_length_one + - suffixLevenshtein_length + - path: Mathlib/Logic/Equiv/Fintype.lean + theorems: + - Equiv.Perm.viaFintypeEmbedding_apply_mem_range + - Equiv.Perm.viaFintypeEmbedding_apply_not_mem_range + - Equiv.extendSubtype_apply_of_mem + - Function.Embedding.toEquivRange_symm_apply_self + - Equiv.extendSubtype_mem + - Equiv.extendSubtype_not_mem + - Equiv.extendSubtype_apply_of_not_mem + - Equiv.Perm.viaFintypeEmbedding_sign + - Function.Embedding.toEquivRange_eq_ofInjective + - Equiv.Perm.viaFintypeEmbedding_apply_image + - path: Mathlib/Order/PropInstances.lean + theorems: + - Prop.isCompl_iff + - Pi.disjoint_iff + - path: Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean + theorems: + - Convex.second_derivative_within_at_symmetric + - second_derivative_symmetric_of_eventually + - Convex.isLittleO_alternate_sum_square + - Convex.second_derivative_within_at_symmetric_of_mem_interior + - Convex.taylor_approx_two_segment + - path: Mathlib/Data/MvPolynomial/Rename.lean + theorems: + - "MvPolynomial.eval\u2082_rename_prod_mk" + - MvPolynomial.map_rename + - MvPolynomial.exists_finset_rename + - MvPolynomial.killCompl_comp_rename + - MvPolynomial.rename_monomial + - MvPolynomial.rename_eq + - "MvPolynomial.rename_eval\u2082" + - MvPolynomial.rename_injective + - "MvPolynomial.exists_finset_rename\u2082" + - MvPolynomial.coeff_rename_eq_zero + - MvPolynomial.support_rename_of_injective + - "MvPolynomial.rename_prod_mk_eval\u2082" + - "MvPolynomial.eval\u2082_cast_comp" + - "MvPolynomial.eval\u2082_rename" + - MvPolynomial.constantCoeff_rename + - MvPolynomial.coeff_rename_mapDomain + - MvPolynomial.rename_rename + - MvPolynomial.coeff_rename_ne_zero + - MvPolynomial.coeff_rename_embDomain + - MvPolynomial.exists_fin_rename + - path: Mathlib/Algebra/Module/Bimodule.lean + theorems: + - Subbimodule.smul_mem' + - Subbimodule.smul_mem + - path: Mathlib/NumberTheory/LucasLehmer.lean + theorems: + - LucasLehmer.X.closed_form + - LucasLehmer.mersenne_coe_X + - LucasLehmer.X.left_distrib + - "LucasLehmer.\u03C9_pow_eq_neg_one" + - LucasLehmer.sZMod_eq_sMod + - LucasLehmer.X.coe_nat + - LucasLehmer.norm_num_ext.testTrueHelper + - LucasLehmer.X.card_units_lt + - LucasLehmer.X.card_eq + - one_lt_mersenne + - succ_mersenne + - LucasLehmer.X.coe_mul + - LucasLehmer.X.ext + - lucas_lehmer_sufficiency + - "LucasLehmer.order_\u03C9" + - "LucasLehmer.X.\u03C9b_mul_\u03C9" + - LucasLehmer.norm_num_ext.sMod'_eq_sMod + - LucasLehmer.Int.coe_nat_pow_pred + - LucasLehmer.two_lt_q + - LucasLehmer.sZMod_eq_s + - LucasLehmer.Int.coe_nat_two_pow_pred + - LucasLehmer.X.right_distrib + - LucasLehmer.sMod_nonneg + - LucasLehmer.sMod_lt + - mersenne_pos + - modEq_mersenne + - LucasLehmer.sMod_mod + - LucasLehmer.norm_num_ext.testFalseHelper + - "LucasLehmer.\u03C9_pow_formula" + - "LucasLehmer.\u03C9_pow_eq_one" + - LucasLehmer.residue_eq_zero_iff_sMod_eq_zero + - path: Mathlib/CategoryTheory/UnivLE.lean + theorems: + - UnivLE.ofEssSurj + - path: Mathlib/MeasureTheory/Group/Prod.lean + theorems: + - MeasureTheory.measurePreserving_div_prod + - MeasureTheory.quasiMeasurePreserving_mul_left + - MeasureTheory.ae_measure_preimage_mul_right_lt_top + - MeasureTheory.measurePreserving_mul_prod + - MeasureTheory.quasiMeasurePreserving_mul_right + - MeasureTheory.measure_mul_right_null + - MeasureTheory.quasiMeasurePreserving_div_left + - MeasureTheory.quasiMeasurePreserving_div_of_right_invariant + - MeasureTheory.measurePreserving_mul_prod_inv_right + - MeasureTheory.measurable_measure_mul_right + - MeasureTheory.quasiMeasurePreserving_div_left_of_right_invariant + - MeasureTheory.measure_mul_measure_eq + - MeasureTheory.ae_measure_preimage_mul_right_lt_top_of_ne_zero + - MeasureTheory.lintegral_lintegral_mul_inv + - MeasureTheory.measure_mul_lintegral_eq + - MeasureTheory.measure_eq_div_smul + - MeasureTheory.measure_inv_null + - MeasureTheory.absolutelyContinuous_of_isMulLeftInvariant + - MeasureTheory.measurePreserving_mul_prod_inv + - MeasureTheory.absolutelyContinuous_inv + - MeasureTheory.quasiMeasurePreserving_inv + - MeasureTheory.quasiMeasurePreserving_inv_of_right_invariant + - MeasureTheory.measure_lintegral_div_measure + - MeasureTheory.absolutelyContinuous_map_div_left + - path: Mathlib/RingTheory/MatrixAlgebra.lean + theorems: + - MatrixEquivTensor.invFun_smul + - MatrixEquivTensor.invFun_zero + - MatrixEquivTensor.invFun_algebraMap + - MatrixEquivTensor.left_inv + - matrixEquivTensor_apply_std_basis + - MatrixEquivTensor.right_inv + - MatrixEquivTensor.invFun_add + - path: Mathlib/CategoryTheory/Limits/Opposites.lean + theorems: + - CategoryTheory.Limits.pushoutIsoUnopPullback_inr_hom + - CategoryTheory.Limits.pullbackIsoUnopPushout_inv_snd + - CategoryTheory.Limits.pushoutIsoUnopPullback_inl_hom + - CategoryTheory.Limits.opProductIsoCoproduct'_inv_comp_lift + - CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct_hom + - CategoryTheory.Limits.opProductIsoCoproduct_inv_comp_lift + - CategoryTheory.Limits.PushoutCocone.op_snd + - CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inr + - CategoryTheory.Limits.PullbackCone.unop_inr + - CategoryTheory.Limits.pullbackIsoUnopPushout_inv_fst + - CategoryTheory.Limits.PushoutCocone.unop_snd + - CategoryTheory.Limits.pullbackIsoUnopPushout_hom_inl + - CategoryTheory.Limits.PushoutCocone.op_fst + - CategoryTheory.Limits.PushoutCocone.unop_fst + - CategoryTheory.Limits.PullbackCone.op_inr + - CategoryTheory.Limits.pushoutIsoUnopPullback_inv_fst + - CategoryTheory.Limits.desc_op_comp_opCoproductIsoProduct'_hom + - CategoryTheory.Limits.PullbackCone.unop_inl + - CategoryTheory.Limits.pushoutIsoUnopPullback_inv_snd + - path: Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean + theorems: + - BoxIntegral.Prepartition.isPartition_splitCenter + - BoxIntegral.Prepartition.exists_tagged_le_isHenstock_isSubordinate_iUnion_eq + - BoxIntegral.Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic + - BoxIntegral.TaggedPrepartition.distortion_unionComplToSubordinate + - BoxIntegral.Box.subbox_induction_on + - BoxIntegral.Prepartition.mem_splitCenter + - path: Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean + theorems: + - Real.rpow_eq_nhds_of_neg + - Complex.continuousAt_cpow_zero_of_re_pos + - ENNReal.eventually_pow_one_div_le + - Real.continuousAt_rpow_of_pos + - zero_cpow_eq_nhds + - NNReal.continuousAt_rpow_const + - NNReal.eventually_pow_one_div_le + - continuousAt_cpow + - Filter.Tendsto.rpow_const + - Real.continuousAt_rpow_const + - ENNReal.continuousAt_rpow_const_of_pos + - Real.rpow_eq_nhds_of_pos + - NNReal.continuousAt_rpow + - ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos + - Real.continuousAt_const_rpow + - Filter.Tendsto.const_cpow + - Real.continuousAt_rpow_of_ne + - ENNReal.continuous_rpow_const + - Complex.continuousAt_ofReal_cpow + - Complex.continuousAt_cpow_of_re_pos + - cpow_eq_nhds + - continuousAt_const_cpow' + - cpow_eq_nhds' + - Real.continuousAt_const_rpow' + - path: Mathlib/MeasureTheory/Measure/Complex.lean + theorems: + - MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff + - path: Mathlib/Order/Birkhoff.lean + theorems: + - OrderEmbedding.birkhoffSet_inf + - OrderIso.supIrredLowerSet_symm_apply + - OrderEmbedding.supIrredLowerSet_surjective + - OrderEmbedding.birkhoffSet_sup + - UpperSet.infIrred_Ici + - OrderEmbedding.birkhoffFinset_sup + - LowerSet.supIrred_Iic + - OrderEmbedding.birkhoffSet_apply + - LowerSet.supIrred_iff_of_finite + - exists_birkhoff_representation + - UpperSet.infIrred_iff_of_finite + - OrderEmbedding.infIrredUpperSet_surjective + - OrderIso.infIrredUpperSet_symm_apply + - OrderEmbedding.coe_birkhoffFinset + - OrderEmbedding.birkhoffFinset_inf + - path: Mathlib/CategoryTheory/Localization/Predicate.lean + theorems: + - CategoryTheory.Functor.IsLocalization.mk' + - CategoryTheory.Localization.comp_liftNatTrans + - CategoryTheory.areEqualizedByLocalization_iff + - CategoryTheory.Localization.natTrans_ext + - CategoryTheory.Functor.IsLocalization.of_isEquivalence + - CategoryTheory.Functor.IsLocalization.of_iso + - CategoryTheory.Functor.IsLocalization.of_equivalence_target + - CategoryTheory.Localization.liftNatTrans_id + - path: Mathlib/Analysis/Convex/Topology.lean + theorems: + - closure_openSegment + - Convex.add_smul_sub_mem_interior' + - stdSimplex_subset_closedBall + - Convex.combo_interior_closure_subset_interior + - Convex.closure_subset_image_homothety_interior_of_one_lt + - Convex.add_smul_mem_interior' + - Convex.isPathConnected + - JoinedIn.of_segment_subset + - segment_subset_closure_openSegment + - Convex.openSegment_closure_interior_subset_interior + - Convex.strictConvex' + - Convex.combo_self_interior_subset_interior + - Convex.strictConvex + - Set.Finite.isCompact_convexHull + - Convex.combo_closure_interior_subset_interior + - path: Mathlib/Analysis/SpecificLimits/Basic.lean + theorems: + - tendsto_nat_ceil_div_atTop + - tendsto_coe_nat_div_add_atTop + - Set.Countable.exists_pos_forall_sum_le + - NNReal.hasSum_geometric + - ENNReal.tendsto_pow_atTop_nhds_zero_iff + - tendsto_factorial_div_pow_self_atTop + - NNReal.tendsto_pow_atTop_nhds_zero_of_lt_one + - summable_one_div_pow_of_le + - Set.Countable.exists_pos_hasSum_le + - tendsto_pow_atTop_nhds_zero_of_lt_one + - ENNReal.exists_pos_sum_of_countable + - hasSum_geometric_of_lt_one + - sum_geometric_two_le + - dist_le_of_le_geometric_of_tendsto + - edist_le_of_edist_le_geometric_of_tendsto + - ENNReal.exists_pos_tsum_mul_lt_of_countable + - NNReal.tendsto_inverse_atTop_nhds_zero_nat + - cauchySeq_of_edist_le_geometric + - le_geom + - NNReal.exists_pos_sum_of_countable + - tendsto_nat_floor_atTop + - ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one + - geom_le + - "edist_le_of_edist_le_geometric_of_tendsto\u2080" + - aux_hasSum_of_le_geometric + - NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat + - tendsto_one_div_add_atTop_nhds_zero_nat + - geom_lt + - tendsto_nat_floor_mul_div_atTop + - lt_geom + - hasSum_geometric_two' + - NNReal.tendsto_pow_atTop_nhds_zero_iff + - NNReal.tendsto_const_div_atTop_nhds_zero_nat + - tendsto_pow_atTop_nhds_zero_iff + - tendsto_nat_ceil_mul_div_atTop + - dist_le_of_le_geometric_two_of_tendsto + - ENNReal.tsum_geometric + - edist_le_of_edist_le_geometric_two_of_tendsto + - tendsto_nat_ceil_atTop + - tendsto_nat_floor_div_atTop + - tendsto_const_div_atTop_nhds_zero_nat + - hasSum_geometric_two + - Nat.tendsto_div_const_atTop + - cauchySeq_of_edist_le_geometric_two + - "edist_le_of_edist_le_geometric_two_of_tendsto\u2080" + - tsum_geometric_inv_two_ge + - path: Mathlib/Data/Polynomial/Smeval.lean + theorems: + - Polynomial.smeval_X_pow + - "Polynomial.eval\u2082_eq_smeval" + - Polynomial.smeval_one + - Polynomial.smeval_at_nat_cast + - Polynomial.smeval_mul_X_pow + - Polynomial.smeval_nat_cast + - Polynomial.smeval_X_pow_assoc + - Polynomial.smeval_mul_X + - Polynomial.smeval_monomial + - Polynomial.smeval_comp + - Polynomial.smeval_X + - Polynomial.smeval_eq_sum + - Polynomial.smeval_smul + - Polynomial.smeval_mul + - Polynomial.smeval_X_pow_mul + - Polynomial.smeval_zero + - Polynomial.smeval_monomial_mul + - Polynomial.smeval_add + - Polynomial.leval_coe_eq_smeval + - Polynomial.aeval_eq_smeval + - Polynomial.eval_eq_smeval + - Polynomial.smeval_assoc_X_pow + - Polynomial.leval_eq_smeval.linearMap + - Polynomial.smeval_at_zero + - Polynomial.smeval_X_mul + - Polynomial.smeval_pow + - Polynomial.smeval_C_mul + - path: Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean + theorems: + - CategoryTheory.Limits.hasCoequalizer_of_common_section + - CategoryTheory.Limits.hasEqualizer_of_common_retraction + - path: Mathlib/RingTheory/PowerSeries/Derivative.lean + theorems: + - PowerSeries.derivativeFun_one + - PowerSeries.derivativeFun_smul + - PowerSeries.derivative.ext + - PowerSeries.derivative_X + - PowerSeries.trunc_derivative' + - PowerSeries.derivative_invOf + - PowerSeries.coeff_derivativeFun + - PowerSeries.derivativeFun_add + - PowerSeries.derivativeFun_C + - PowerSeries.derivativeFun_coe_mul_coe + - PowerSeries.trunc_derivativeFun + - PowerSeries.derivativeFun_mul + - PowerSeries.derivativeFun_coe + - PowerSeries.derivative_inv' + - PowerSeries.derivative_inv + - path: Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean + theorems: + - mem_balancedCore_iff + - balancedCoreAux_balanced + - nhds_basis_closed_balanced + - balancedCore_mem_nhds_zero + - mem_balancedHull_iff + - Balanced.balancedHull_subset_of_subset + - smul_balancedCore_subset + - subset_balancedCore + - balancedHull.balanced + - balancedCoreAux_subset + - balancedCoreAux_maximal + - balancedCoreAux_empty + - IsClosed.balancedCore + - balancedCore_eq_iInter + - path: Mathlib/Order/Filter/Subsingleton.lean + theorems: + - Filter.subsingleton_iff_bot_or_pure + - Filter.subsingleton_iff_exists_le_pure + - Filter.Subsingleton.exists_eq_pure + - Filter.subsingleton_iff_exists_singleton_mem + - Filter.Subsingleton.isCountablyGenerated + - path: Mathlib/NumberTheory/ClassNumber/Finite.lean + theorems: + - ClassGroup.norm_lt + - ClassGroup.prod_finsetApprox_ne_zero + - ClassGroup.norm_le + - ClassGroup.exists_mem_finset_approx' + - ClassGroup.normBound_pos + - ClassGroup.mem_finsetApprox + - ClassGroup.exists_mem_finsetApprox + - ClassGroup.exists_mk0_eq_mk0 + - ClassGroup.exists_min + - path: Mathlib/Probability/Kernel/Disintegration/Density.lean + theorems: + - ProbabilityTheory.kernel.set_integral_densityProcess_of_mem + - ProbabilityTheory.kernel.tendsto_densityProcess_atTop_empty_of_antitone + - ProbabilityTheory.kernel.tendsto_densityProcess_limitProcess + - ProbabilityTheory.kernel.snorm_density_le + - ProbabilityTheory.kernel.density_nonneg + - ProbabilityTheory.kernel.tendsto_snorm_one_densityProcess_limitProcess + - ProbabilityTheory.kernel.measurable_density_left + - ProbabilityTheory.kernel.tendsto_integral_density_of_antitone + - ProbabilityTheory.kernel.snorm_densityProcess_le + - ProbabilityTheory.kernel.tendsto_set_integral_densityProcess + - ProbabilityTheory.kernel.densityProcess_empty + - ProbabilityTheory.kernel.density_fst_univ + - ProbabilityTheory.kernel.densityProcess_fst_univ_ae + - ProbabilityTheory.kernel.memL1_limitProcess_densityProcess + - ProbabilityTheory.kernel.densityProcess_mono_kernel_left + - ProbabilityTheory.kernel.set_integral_densityProcess + - ProbabilityTheory.kernel.measurable_countableFiltration_densityProcess + - ProbabilityTheory.kernel.lintegral_density + - ProbabilityTheory.kernel.tendsto_densityProcess_atTop_of_antitone + - ProbabilityTheory.kernel.set_integral_density_of_measurableSet + - ProbabilityTheory.kernel.set_integral_density + - ProbabilityTheory.kernel.condexp_densityProcess + - ProbabilityTheory.kernel.densityProcess_fst_univ + - ProbabilityTheory.kernel.measurable_densityProcess_aux + - ProbabilityTheory.kernel.tendsto_densityProcess_fst_atTop_univ_of_monotone + - ProbabilityTheory.kernel.integral_density + - ProbabilityTheory.kernel.integrable_density + - ProbabilityTheory.kernel.tendsto_densityProcess_fst_atTop_ae_of_monotone + - ProbabilityTheory.kernel.meas_countablePartitionSet_le_of_fst_le + - ProbabilityTheory.kernel.tendsto_density_fst_atTop_ae_of_monotone + - ProbabilityTheory.kernel.tendsto_integral_density_of_monotone + - ProbabilityTheory.kernel.measurable_densityProcess_countableFiltration_aux + - ProbabilityTheory.kernel.measurable_density_right + - ProbabilityTheory.kernel.densityProcess_mono_set + - ProbabilityTheory.kernel.integrable_densityProcess + - ProbabilityTheory.kernel.tendsto_density_atTop_ae_of_antitone + - ProbabilityTheory.kernel.densityProcess_le_one + - ProbabilityTheory.kernel.density_ae_eq_limitProcess + - ProbabilityTheory.kernel.integral_densityProcess + - ProbabilityTheory.kernel.tendsto_m_density + - ProbabilityTheory.kernel.densityProcess_antitone_kernel_right + - ProbabilityTheory.kernel.density_le_one + - path: Mathlib/Analysis/Calculus/Taylor.lean + theorems: + - taylorWithinEval_self + - taylor_mean_remainder_lagrange + - taylorWithinEval_succ + - hasDerivWithinAt_taylor_coeff_within + - taylorWithin_succ + - taylor_within_zero_eval + - taylor_mean_remainder_cauchy + - exists_taylor_mean_remainder_bound + - taylor_within_apply + - taylor_mean_remainder + - hasDerivWithinAt_taylorWithinEval + - continuousOn_taylorWithinEval + - monomial_has_deriv_aux + - taylor_mean_remainder_bound + - path: Mathlib/Geometry/RingedSpace/OpenImmersion.lean + theorems: + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_inv_ofRestrict + - AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullback_cone_of_left_condition + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_snd + - AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_fac + - AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullback_snd_isIso_of_range_subset + - AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.sheafedSpace_toSheafedSpace + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_fac + - "AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_\u03B9_openEmbedding" + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isIso_of_subset + - AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_uniq + - AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_range + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app' + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.to_iso + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_uniq + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullback_snd_isIso_of_range_subset + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.locallyRingedSpace_toLocallyRingedSpace + - AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality + - path: Mathlib/Geometry/RingedSpace/Stalks.lean + theorems: + - AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_ofRestrict + - AlgebraicGeometry.PresheafedSpace.stalkMap.congr_hom + - AlgebraicGeometry.PresheafedSpace.stalkMap.congr + - AlgebraicGeometry.PresheafedSpace.stalkMap.id + - AlgebraicGeometry.PresheafedSpace.restrictStalkIso_inv_eq_germ + - AlgebraicGeometry.PresheafedSpace.stalkMap.congr_point + - AlgebraicGeometry.PresheafedSpace.stalkMap_germ + - AlgebraicGeometry.PresheafedSpace.stalkMap.stalkSpecializes_stalkMap + - AlgebraicGeometry.PresheafedSpace.stalkMap.comp + - path: Mathlib/Order/Monotone/Union.lean + theorems: + - MonotoneOn.Iic_union_Ici + - StrictMonoOn.Iic_union_Ici + - MonotoneOn.union_right + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean + theorems: + - CliffordAlgebra.contractRight_mul_algebraMap + - CliffordAlgebra.contractRight_contractRight + - "CliffordAlgebra.changeForm_\u03B9_mul" + - CliffordAlgebra.changeFormAux_changeFormAux + - CliffordAlgebra.contractRight_one + - CliffordAlgebra.contractLeftAux_contractLeftAux + - CliffordAlgebra.contractLeft_algebraMap + - CliffordAlgebra.contractLeft_contractLeft + - CliffordAlgebra.contractRight_algebraMap + - CliffordAlgebra.contractRight_comm + - "CliffordAlgebra.changeForm_\u03B9_mul_\u03B9" + - CliffordAlgebra.contractLeft_comm + - CliffordAlgebra.changeForm_contractLeft + - "CliffordAlgebra.contractRight_\u03B9" + - "CliffordAlgebra.contractLeft_\u03B9_mul" + - CliffordAlgebra.contractLeft_one + - CliffordAlgebra.contractLeft_mul_algebraMap + - CliffordAlgebra.contractRight_algebraMap_mul + - CliffordAlgebra.changeForm_self_apply + - CliffordAlgebra.changeForm_one + - "CliffordAlgebra.changeForm_\u03B9" + - "CliffordAlgebra.contractRight_mul_\u03B9" + - "CliffordAlgebra.contractLeft_\u03B9" + - CliffordAlgebra.contractLeft_algebraMap_mul + - path: Mathlib/Probability/Independence/Kernel.lean + theorems: + - ProbabilityTheory.kernel.iIndepFun_iff_measure_inter_preimage_eq_mul + - ProbabilityTheory.kernel.IndepSets.iInter + - ProbabilityTheory.kernel.iIndepSets_singleton_iff + - ProbabilityTheory.kernel.IndepSets.indep_aux + - ProbabilityTheory.kernel.indep_iSup_of_disjoint + - ProbabilityTheory.kernel.iIndepFun.ae_isProbabilityMeasure + - ProbabilityTheory.kernel.iIndepSets.piiUnionInter_of_not_mem + - ProbabilityTheory.kernel.Indep.indepSet_of_measurableSet + - ProbabilityTheory.kernel.indepSets_piiUnionInter_of_disjoint + - ProbabilityTheory.kernel.indep_iSup_of_directed_le + - ProbabilityTheory.kernel.iIndepSets.meas_iInter + - ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_le + - ProbabilityTheory.kernel.iIndepSet.iIndepFun_indicator + - ProbabilityTheory.kernel.iIndepSet_iff_iIndepSets_singleton + - ProbabilityTheory.kernel.IndepSets.union + - ProbabilityTheory.kernel.iIndepFun.indepFun_finset + - ProbabilityTheory.kernel.iIndepFun.indepFun_div_left + - ProbabilityTheory.kernel.iIndepFun.of_subsingleton + - ProbabilityTheory.kernel.indepSet_iff_indepSets_singleton + - ProbabilityTheory.kernel.IndepSets.indep + - ProbabilityTheory.kernel.iIndep.meas_iInter + - ProbabilityTheory.kernel.indepSet_empty_right + - ProbabilityTheory.kernel.iIndepSets.iIndep + - ProbabilityTheory.kernel.IndepFun.comp + - ProbabilityTheory.kernel.iIndep.indep + - ProbabilityTheory.kernel.IndepSets.bInter + - ProbabilityTheory.kernel.iIndepFun.indepFun_finset_prod_of_not_mem + - ProbabilityTheory.kernel.iIndepFun.indepFun_prod_mk + - ProbabilityTheory.kernel.iIndepFun.indepFun_prod_mk_prod_mk + - ProbabilityTheory.kernel.indepSets_singleton_iff + - ProbabilityTheory.kernel.IndepSets.iUnion + - ProbabilityTheory.kernel.IndepFun.ae_eq + - ProbabilityTheory.kernel.IndepSets.bUnion + - ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_of_disjoint + - ProbabilityTheory.kernel.indepFun_iff_indepSet_preimage + - ProbabilityTheory.kernel.iIndepSet.indep_generateFrom_lt + - ProbabilityTheory.kernel.iIndepSets.indepSets + - ProbabilityTheory.kernel.indepFun_iff_measure_inter_preimage_eq_mul + - ProbabilityTheory.kernel.IndepSets.symm + - ProbabilityTheory.kernel.indep_bot_right + - path: Mathlib/Algebra/BigOperators/Associated.lean + theorems: + - Finset.prod_primes_dvd + - Multiset.prod_primes_dvd + - exists_associated_mem_of_dvd_prod + - Associates.prod_eq_one_iff + - Associates.prod_le_prod + - Associates.rel_associated_iff_map_eq_map + - Associated.prod + - Associates.exists_mem_multiset_le_of_prime + - Prime.not_dvd_finset_prod + - Associates.finset_prod_mk + - Prime.exists_mem_multiset_map_dvd + - Prime.exists_mem_multiset_dvd + - path: Mathlib/Order/Filter/Cofinite.lean + theorems: + - Filter.frequently_cofinite_iff_infinite + - Nat.cofinite_eq_atTop + - Set.Finite.cofinite_inf_principal_compl + - Nat.frequently_atTop_iff_infinite + - Filter.Tendsto.exists_within_forall_le + - Filter.cofinite_eq_bot_iff + - Filter.Tendsto.countable_compl_preimage_ker + - Filter.disjoint_cofinite_left + - Filter.le_cofinite_iff_compl_singleton_mem + - Filter.coprod_cofinite + - Set.Finite.cofinite_inf_principal_diff + - Filter.countable_compl_ker + - path: Mathlib/Computability/PartrecCode.lean + theorems: + - Nat.Partrec.Code.evaln_map + - Nat.Partrec.Code.const_inj + - Nat.Partrec.Code.eval_prec_succ + - Nat.Partrec.Code.evaln_mono + - Nat.Partrec.Code.encode_lt_pair + - Nat.Partrec.Code.fixed_point + - Nat.Partrec.Code.eval_const + - Nat.Partrec.Code.eval_id + - Nat.Partrec.Code.eval_eq_rfindOpt + - Nat.Partrec.Code.hG + - Nat.Partrec.Code.rec_prim' + - Nat.Partrec.Code.evaln_sound + - Nat.Partrec.Code.evaln_complete + - Nat.Partrec.Code.const_prim + - "Nat.Partrec.Code.fixed_point\u2082" + - Nat.Partrec.Code.encode_ofNatCode + - Nat.Partrec.Code.encode_lt_comp + - Nat.Partrec.Code.rec_computable + - Nat.Partrec.rfind' + - Nat.Partrec.Code.eval_part + - Nat.Partrec.Code.evaln_prim + - Nat.Partrec.Code.evaln_bound + - Nat.Partrec.Code.curry_inj + - Nat.Partrec.Code.exists_code + - Nat.Partrec.Code.encode_lt_rfind' + - Nat.Partrec.Code.eval_curry + - Nat.Partrec.Code.encode_lt_prec + - path: Mathlib/LinearAlgebra/Basis/VectorSpace.lean + theorems: + - VectorSpace.card_fintype + - Basis.ofVectorSpaceIndex.linearIndependent + - nonzero_span_atom + - LinearMap.exists_leftInverse_of_injective + - atom_iff_nonzero_span + - Basis.range_extend + - Basis.ofVectorSpace_apply_self + - LinearMap.exists_extend + - path: Mathlib/Logic/Equiv/Fin.lean + theorems: + - finRotate_apply_zero + - finSuccAboveEquiv_symm_apply_ne_last + - Fin.preimage_apply_01_prod + - finSuccEquiv'_last_apply + - finSuccEquivLast_last + - coe_finRotate_of_ne_last + - finRotate_succ_apply + - finSuccEquiv'_last_apply_castSucc + - finAddFlip_apply_natAdd + - finSuccEquiv'_below + - finSuccEquiv'_ne_last_apply + - finSuccEquiv'_above + - finAddFlip_apply_mk_left + - finAddFlip_apply_castAdd + - finAddFlip_apply_mk_right + - finSuccEquiv'_at + - finRotate_last' + - finSuccAboveEquiv_symm_apply_last + - coe_finRotate + - Fin.snoc_eq_cons_rotate + - finRotate_of_lt + - path: Mathlib/Init/Data/Nat/Lemmas.lean + theorems: + - Nat.bit0_inj + - Nat.eq_zero_of_mul_eq_zero + - Nat.one_lt_bit0 + - Nat.cond_decide_mod_two + - Nat.one_lt_bit1 + - Nat.bit1_inj + - path: Mathlib/Topology/ContinuousFunction/Ideals.lean + theorems: + - ContinuousMap.not_mem_idealOfSet + - ContinuousMap.idealOfSet_ofIdeal_eq_closure + - ContinuousMap.mem_idealOfSet + - ContinuousMap.not_mem_setOfIdeal + - ContinuousMap.idealOfEmpty_eq_bot + - ContinuousMap.idealOfSet_isMaximal_iff + - ContinuousMap.setOfIdeal_ofSet_eq_interior + - ContinuousMap.setOfIdeal_eq_compl_singleton + - ContinuousMap.mem_idealOfSet_compl_singleton + - ContinuousMap.ideal_isMaximal_iff + - ContinuousMap.setOfIdeal_open + - WeakDual.CharacterSpace.continuousMapEval_bijective + - ContinuousMap.ideal_gc + - ContinuousMap.exists_mul_le_one_eqOn_ge + - ContinuousMap.mem_setOfIdeal + - ContinuousMap.idealOfSet_closed + - path: Mathlib/Data/QPF/Multivariate/Basic.lean + theorems: + - MvQPF.has_good_supp_iff + - MvQPF.suppPreservation_iff_isUniform + - MvQPF.liftpPreservation_iff_uniform + - MvQPF.supp_map + - MvQPF.liftP_iff + - MvQPF.comp_map + - MvQPF.suppPreservation_iff_liftpPreservation + - MvQPF.supp_eq + - MvQPF.id_map + - MvQPF.liftP_iff_of_isUniform + - MvQPF.mem_supp + - MvQPF.supp_eq_of_isUniform + - path: Mathlib/Combinatorics/SetFamily/LYM.lean + theorems: + - Finset.sum_card_slice_div_choose_le_one + - Finset.le_card_falling_div_choose + - Finset.slice_union_shadow_falling_succ + - Finset.card_mul_le_card_shadow_mul + - Finset.card_div_choose_le_card_shadow_div_choose + - Finset.IsAntichain.disjoint_slice_shadow_falling + - Finset.IsAntichain.sperner + - path: Mathlib/LinearAlgebra/SModEq.lean + theorems: + - SModEq.add + - SModEq.bot + - SModEq.eval + - SModEq.zero + - SModEq.smul + - SModEq.sub_mem + - path: Mathlib/Algebra/Module/Basic.lean + theorems: + - two_nsmul_eq_zero + - smul_ne_zero_iff + - Function.support_const_smul_of_ne_zero + - map_rat_cast_smul + - NoZeroSMulDivisors.int_of_charZero + - neg_eq_self + - Units.neg_smul + - Convex.combo_self + - Set.indicator_smul_apply_left + - Set.indicator_smul_apply + - smul_eq_zero_iff_left + - int_smul_eq_zsmul + - Nat.noZeroSMulDivisors + - neg_smul_neg + - smul_ne_zero_iff_right + - self_eq_neg + - smul_add_one_sub_smul + - Function.Injective.noZeroSMulDivisors + - Nat.smul_one_eq_coe + - Convex.combo_eq_smul_sub_add + - Module.ext' + - map_inv_int_cast_smul + - zsmul_eq_smul_cast + - CharZero.of_module + - map_inv_nat_cast_smul + - neg_smul + - smul_eq_zero_iff_right + - map_nat_cast_smul + - nsmul_eq_smul_cast + - Module.eq_zero_of_zero_eq_one + - sub_smul + - map_int_cast_smul + - Int.smul_one_eq_coe + - two_smul + - smul_ne_zero_iff_left + - Function.support_smul_subset_right + - nat_smul_eq_nsmul + - neg_one_smul + - Function.support_smul_subset_left + - CharZero.of_noZeroSMulDivisors + - path: Mathlib/Topology/Instances/Discrete.lean + theorems: + - discreteTopology_iff_orderTopology_of_pred_succ + - LinearOrder.bot_topologicalSpace_eq_generateFrom + - bot_topologicalSpace_eq_generateFrom_of_pred_succOrder + - discreteTopology_iff_orderTopology_of_pred_succ' + - path: Mathlib/Algebra/GCDMonoid/Finset.lean + theorems: + - Finset.lcm_congr + - Finset.dvd_gcd_iff + - Finset.gcd_union + - Finset.lcm_dvd_iff + - Finset.extract_gcd + - Finset.extract_gcd' + - Finset.lcm_insert + - Finset.gcd_insert + - Finset.gcd_eq_gcd_filter_ne_zero + - Finset.gcd_mul_left + - Finset.gcd_mul_right + - Finset.lcm_image + - Finset.gcd_congr + - Finset.normalize_lcm + - Finset.normalize_gcd + - Finset.gcd_eq_of_dvd_sub + - Finset.gcd_image + - Finset.lcm_eq_zero_iff + - Finset.lcm_union + - Finset.gcd_eq_zero_iff + - path: Mathlib/Logic/Nontrivial/Defs.lean + theorems: + - subsingleton_or_nontrivial + - Function.Surjective.nontrivial + - subsingleton_iff + - not_subsingleton_iff_nontrivial + - not_nontrivial_iff_subsingleton + - Decidable.exists_ne + - path: Mathlib/Topology/Category/Compactum.lean + theorems: + - Compactum.cl_cl + - Compactum.basic_inter + - Compactum.lim_eq_str + - Compactum.continuous_of_hom + - Compactum.str_incl + - Compactum.isClosed_cl + - Compactum.isClosed_iff + - Compactum.str_eq_of_le_nhds + - Compactum.join_distrib + - Compactum.cl_eq_closure + - Compactum.subset_cl + - Compactum.le_nhds_of_str_eq + - compactumToCompHaus.faithful + - Compactum.str_hom_commute + - path: Mathlib/Analysis/NormedSpace/HomeomorphBall.lean + theorems: + - PartialHomeomorph.ball_subset_univBall_target + - PartialHomeomorph.continuous_univBall + - PartialHomeomorph.univBall_symm_apply_center + - PartialHomeomorph.univBall_apply_zero + - PartialHomeomorph.univUnitBall_symm_apply_zero + - PartialHomeomorph.univUnitBall_apply_zero + - PartialHomeomorph.univBall_source + - PartialHomeomorph.univBall_target + - path: Mathlib/Data/Finset/Pairwise.lean + theorems: + - List.pairwise_of_coe_toFinset_pairwise + - Finset.pairwiseDisjoint_range_singleton + - Set.PairwiseDisjoint.biUnion_finset + - Set.PairwiseDisjoint.image_finset_of_le + - List.pairwise_iff_coe_toFinset_pairwise + - path: Mathlib/MeasureTheory/Constructions/Prod/Basic.lean + theorems: + - MeasureTheory.Measure.quasiMeasurePreserving_fst + - MeasureTheory.Measure.prod_sum + - IsPiSystem.prod + - MeasureTheory.Measure.measure_prod_compl_eq_zero + - MeasureTheory.lintegral_prod_of_measurable + - measurable_measure_prod_mk_left_finite + - MeasureTheory.Measure.add_prod + - MeasureTheory.Measure.prod_prod + - Measurable.map_prod_mk_right + - MeasureTheory.Measure.AbsolutelyContinuous.prod + - MeasureTheory.Measure.map_prod_map + - MeasureTheory.Measure.dirac_prod + - MeasureTheory.lintegral_prod_symm + - MeasureTheory.Measure.ae_measure_lt_top + - "MeasureTheory.Measure.fst_map_prod_mk\u2080" + - MeasureTheory.Measure.fst_prod + - IsCountablySpanning.prod + - MeasurableEmbedding.prod_mk + - MeasureTheory.QuasiMeasurePreserving.prod_of_right + - MeasureTheory.NullMeasurableSet.of_preimage_snd + - measurable_measure_prod_mk_left + - MeasureTheory.NullMeasurableSet.right_of_prod + - MeasureTheory.MeasurePreserving.skew_product + - MeasureTheory.Measure.prod_swap + - MeasureTheory.Measure.fst_univ + - MeasurableEmbedding.prod_mk_left + - MeasureTheory.NullMeasurableSet.of_preimage_fst + - MeasureTheory.Measure.map_snd_prod + - MeasureTheory.Measure.snd_prod + - MeasureTheory.Measure.snd_zero + - Measurable.map_prod_mk_left + - generateFrom_prod_eq + - MeasureTheory.NullMeasurableSet.left_of_prod + - MeasureTheory.Measure.zero_prod + - MeasureTheory.Measure.quasiMeasurePreserving_snd + - MeasureTheory.Measure.prod_sum_right + - MeasureTheory.Measure.prod_apply_symm + - MeasureTheory.Measure.fst_apply + - MeasureTheory.Measure.measure_ae_null_of_prod_null + - "MeasureTheory.Measure.snd_map_prod_mk\u2080" + - MeasureTheory.Measure.snd_apply + - MeasureTheory.Measure.map_fst_prod + - MeasureTheory.Measure.prodAssoc_prod + - MeasureTheory.Measure.snd_univ + - MeasureTheory.Measure.prod_eq_generateFrom + - MeasureTheory.Measure.nullMeasurableSet_prod + - AEMeasurable.prod_swap + - MeasureTheory.Measure.prod_dirac + - Measurable.lintegral_prod_right' + - MeasureTheory.Measure.restrict_prod_eq_prod_univ + - MeasureTheory.Measure.prod_add + - MeasureTheory.QuasiMeasurePreserving.prod_of_left + - MeasureTheory.Measure.prod_sum_left + - generateFrom_eq_prod + - MeasureTheory.Measure.fst_zero + - MeasurableEmbedding.prod_mk_right + - MeasureTheory.Measure.dirac_prod_dirac + - MeasureTheory.Measure.prod_zero + - MeasureTheory.lintegral_prod + - MeasureTheory.lintegral_prod_mul + - MeasureTheory.Measure.prod_apply + - path: Mathlib/Order/UpperLower/LocallyFinite.lean + theorems: + - Set.Finite.upperClosure + - Set.Finite.lowerClosure + - path: Mathlib/Data/PNat/Factors.lean + theorems: + - PrimeMultiset.factorMultiset_prod + - PrimeMultiset.prod_sup + - PrimeMultiset.prod_dvd_iff + - PrimeMultiset.coe_prod + - PrimeMultiset.prod_ofNatList + - PrimeMultiset.prod_smul + - PrimeMultiset.prod_dvd_iff' + - PrimeMultiset.to_ofNatMultiset + - PrimeMultiset.prod_add + - PNat.factorMultiset_ofPrime + - PrimeMultiset.prod_zero + - PrimeMultiset.prod_ofNatMultiset + - PrimeMultiset.coeNat_prime + - PrimeMultiset.to_ofPNatMultiset + - PrimeMultiset.prod_inf + - PrimeMultiset.prod_ofPNatList + - PNat.prod_factorMultiset + - PNat.factorMultiset_lcm + - PrimeMultiset.coePNat_prime + - PrimeMultiset.coePNat_nat + - PNat.factorMultiset_mul + - PNat.factorMultiset_pow + - PNat.count_factorMultiset + - PNat.factorMultiset_one + - PrimeMultiset.prod_ofPNatMultiset + - PNat.factorMultiset_le_iff + - PNat.factorMultiset_le_iff' + - path: Mathlib/MeasureTheory/Measure/FiniteMeasure.lean + theorems: + - MeasureTheory.FiniteMeasure.testAgainstNN_zero + - MeasureTheory.FiniteMeasure.smul_testAgainstNN_apply + - MeasureTheory.FiniteMeasure.eq_of_forall_toMeasure_apply_eq + - MeasureTheory.FiniteMeasure.zero_testAgainstNN + - MeasureTheory.FiniteMeasure.continuous_map + - MeasureTheory.FiniteMeasure.continuous_mass + - MeasureTheory.FiniteMeasure.restrict_mass + - MeasureTheory.FiniteMeasure.testAgainstNN_add + - MeasureTheory.FiniteMeasure.mass_nonzero_iff + - MeasureTheory.FiniteMeasure.restrict_nonzero_iff + - MeasureTheory.FiniteMeasure.testAgainstNN_one + - MeasureTheory.FiniteMeasure.coeFn_smul + - MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto + - MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto + - MeasureTheory.FiniteMeasure.map_apply_of_aemeasurable + - MeasureTheory.FiniteMeasure.tendsto_zero_testAgainstNN_of_tendsto_zero_mass + - MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval + - MeasureTheory.FiniteMeasure.tendsto_map_of_tendsto_of_continuous + - MeasureTheory.FiniteMeasure.mass_zero_iff + - MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto + - MeasureTheory.FiniteMeasure.coeFn_zero + - MeasureTheory.FiniteMeasure.injective_toWeakDualBCNN + - MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz + - MeasureTheory.FiniteMeasure.map_add + - MeasureTheory.FiniteMeasure.ext_of_forall_lintegral_eq + - MeasureTheory.FiniteMeasure.restrict_apply + - MeasureTheory.FiniteMeasure.apply_le_mass + - MeasureTheory.FiniteMeasure.testAgainstNN_const + - MeasureTheory.FiniteMeasure.tendsto_of_forall_integral_tendsto + - MeasureTheory.FiniteMeasure.restrict_eq_zero_iff + - MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz_estimate + - MeasureTheory.FiniteMeasure.testAgainstNN_mono + - MeasureTheory.FiniteMeasure.testAgainstNN_smul + - MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_filter_of_le_const + - MeasureTheory.FiniteMeasure.tendsto_zero_of_tendsto_zero_mass + - MeasureTheory.FiniteMeasure.zero_testAgainstNN_apply + - MeasureTheory.FiniteMeasure.apply_mono + - MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto + - MeasureTheory.FiniteMeasure.map_smul + - MeasureTheory.FiniteMeasure.coeFn_add + - MeasureTheory.FiniteMeasure.coeFn_smul_apply + - path: Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean + theorems: + - Finset.mul_pluennecke_petridis + - Finset.card_pow_le' + - Finset.card_pow_div_pow_le + - Finset.mul_aux + - Finset.card_add_nsmul_le + - Finset.card_mul_pow_le + - Finset.card_div_mul_le_card_div_mul_card_div + - Finset.card_mul_mul_le_card_mul_mul_card_div + - Finset.card_div_mul_le_card_div_mul_card_mul + - Finset.card_mul_mul_card_le_card_mul_mul_card_mul + - Finset.card_div_mul_le_card_mul_mul_card_div + - Finset.card_pow_le + - Finset.card_div_mul_le_card_mul_mul_card_mul + - Finset.card_mul_mul_le_card_div_mul_card_mul + - Finset.card_pow_div_pow_le' + - path: Mathlib/CategoryTheory/Idempotents/Basic.lean + theorems: + - CategoryTheory.Idempotents.split_iff_of_iso + - CategoryTheory.Idempotents.isIdempotentComplete_iff_idempotents_have_kernels + - CategoryTheory.Idempotents.isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent + - CategoryTheory.Idempotents.isIdempotentComplete_iff_of_equivalence + - CategoryTheory.Idempotents.idem_of_id_sub_idem + - CategoryTheory.Idempotents.isIdempotentComplete_iff_opposite + - CategoryTheory.Idempotents.Equivalence.isIdempotentComplete + - CategoryTheory.Idempotents.isIdempotentComplete_of_isIdempotentComplete_opposite + - CategoryTheory.Idempotents.split_imp_of_iso + - path: Mathlib/Analysis/Normed/Group/HomCompletion.lean + theorems: + - NormedAddGroupHom.extension_unique + - NormedAddGroupHom.ker_completion + - NormedAddGroupHom.completion_comp + - NormedAddGroupHom.completion_toCompl + - NormedAddGroupHom.norm_completion + - NormedAddGroupHom.completion_id + - path: Mathlib/Probability/Independence/ZeroOne.lean + theorems: + - ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop + - ProbabilityTheory.condexp_eq_zero_or_one_of_condIndepSet_self + - ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atBot + - ProbabilityTheory.kernel.indep_iSup_directed_limsup + - ProbabilityTheory.kernel.measure_eq_zero_or_one_or_top_of_indepSet_self + - ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup + - ProbabilityTheory.kernel.indep_limsup_atBot_self + - ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self + - ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup + - ProbabilityTheory.kernel.indep_limsup_atTop_self + - ProbabilityTheory.kernel.indep_iSup_limsup + - ProbabilityTheory.kernel.indep_biSup_limsup + - path: Mathlib/Data/Vector/Snoc.lean + theorems: + - Vector.reverse_cons + - "Vector.map\u2082_snoc" + - Vector.mapAccumr_snoc + - "Vector.mapAccumr\u2082_snoc" + - Vector.replicate_succ_to_snoc + - Vector.reverse_snoc + - path: Mathlib/Data/Nat/Pow.lean + theorems: + - Nat.one_lt_pow + - Nat.mul_lt_mul_pow_succ + - Nat.two_pow_succ + - Nat.not_pos_pow_dvd + - Nat.lt_two_pow + - Nat.pow_mod + - Nat.one_le_pow + - Nat.sq_sub_sq + - Nat.lt_of_pow_dvd_right + - Nat.lt_pow_self + - path: Mathlib/CategoryTheory/Localization/Composition.lean + theorems: + - CategoryTheory.Functor.IsLocalization.of_comp + - CategoryTheory.Functor.IsLocalization.comp + - path: Mathlib/GroupTheory/Perm/Option.lean + theorems: + - Equiv.optionCongr_sign + - Equiv.Perm.decomposeOption_symm_of_none_apply + - Equiv.Perm.decomposeOption_symm_sign + - Equiv.optionCongr_swap + - map_equiv_removeNone + - path: Mathlib/Order/Zorn.lean + theorems: + - "zorn_nonempty_Ici\u2080" + - "zorn_nonempty_preorder\u2080" + - IsChain.exists_maxChain + - exists_maximal_of_chains_bounded + - "zorn_preorder\u2080" + - path: Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean + theorems: + - MeasureTheory.levyProkhorovDist_self + - MeasureTheory.meas_le_of_le_of_forall_le_meas_thickening_add + - MeasureTheory.levyProkhorovEDist_le_of_forall + - MeasureTheory.right_measure_le_of_levyProkhorovEDist_lt + - MeasureTheory.levyProkhorovEDist_lt_top + - MeasureTheory.levyProkhorovEDist_triangle + - MeasureTheory.tendsto_integral_meas_thickening_le + - MeasureTheory.BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt + - MeasureTheory.levyProkhorovEDist_self + - MeasureTheory.levyProkhorovEDist_le_max_measure_univ + - MeasureTheory.levyProkhorovEDist_comm + - MeasureTheory.BoundedContinuousFunction.integral_eq_integral_meas_le_of_hasFiniteIntegral + - MeasureTheory.levyProkhorovDist_comm + - MeasureTheory.levyProkhorovEDist_le_of_forall_add_pos_le + - MeasureTheory.continuous_levyProkhorov_to_probabilityMeasure + - MeasureTheory.left_measure_le_of_levyProkhorovEDist_lt + - path: Mathlib/Analysis/Analytic/Meromorphic.lean + theorems: + - MeromorphicAt.pow + - AnalyticAt.meromorphicAt_order + - MeromorphicAt.inv + - MeromorphicAt.neg_iff + - MeromorphicAt.add + - MeromorphicAt.sub + - MeromorphicAt.iff_eventuallyEq_zpow_smul_analyticAt + - MeromorphicAt.congr + - MeromorphicAt.order_eq_int_iff + - MeromorphicAt.neg + - MeromorphicOn.inv_iff + - MeromorphicAt.smul + - AnalyticAt.meromorphicAt + - MeromorphicOn.congr + - MeromorphicAt.zpow + - MeromorphicAt.inv_iff + - MeromorphicOn.neg_iff + - path: Mathlib/Data/Finset/Pointwise/Interval.lean + theorems: + - Finset.Iic_mul_Iio_subset' + - Finset.Ico_mul_Icc_subset' + - Finset.Icc_mul_Ico_subset' + - Finset.Iic_mul_Iic_subset' + - Finset.Icc_mul_Icc_subset' + - Finset.Ici_mul_Ici_subset' + - Finset.Ico_mul_Ioc_subset' + - Finset.Ici_mul_Ioi_subset' + - Finset.Ioc_mul_Ico_subset' + - Finset.Iio_mul_Iic_subset' + - Finset.Ioi_mul_Ici_subset' + - path: Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean + theorems: + - IsFreeGroupoid.path_nonempty_of_hom + - IsFreeGroupoid.SpanningTree.loopOfHom_eq_id + - IsFreeGroupoid.SpanningTree.endIsFree + - IsFreeGroupoid.SpanningTree.treeHom_eq + - path: Mathlib/Analysis/NormedSpace/Pointwise.lean + theorems: + - "infDist_smul\u2080" + - affinity_unitBall + - smul_unitBall + - ball_sub_ball + - thickening_closedBall + - smul_closedBall + - exists_dist_le_lt + - disjoint_closedBall_ball_iff + - cthickening_ball + - exists_dist_eq + - closure_thickening + - closedBall_sub_ball + - exists_dist_lt_lt + - disjoint_ball_ball_iff + - ball_add_closedBall + - ball_add_ball + - infEdist_cthickening + - thickening_thickening + - "infEdist_smul\u2080" + - set_smul_sphere_zero + - smul_closedBall' + - disjoint_ball_closedBall_iff + - thickening_ball + - exists_dist_le_le + - NormedSpace.sphere_nonempty + - closedBall_add_ball + - cthickening_closedBall + - disjoint_closedBall_closedBall_iff + - cthickening_thickening + - eventually_singleton_add_smul_subset + - smul_closedUnitBall_of_nonneg + - smul_unitBall_of_pos + - infEdist_thickening + - thickening_cthickening + - closedBall_sub_closedBall + - smul_closedUnitBall + - Ioo_smul_sphere_zero + - smul_sphere' + - "ediam_smul\u2080" + - exists_dist_lt_le + - smul_sphere + - affinity_unitClosedBall + - smul_ball + - "diam_smul\u2080" + - ball_sub_closedBall + - path: Mathlib/SetTheory/Ordinal/CantorNormalForm.lean + theorems: + - Ordinal.one_CNF + - Ordinal.CNF_of_lt + - Ordinal.CNF_foldr + - Ordinal.CNF_lt_snd + - Ordinal.CNF_of_le_one + - Ordinal.CNFRec_zero + - Ordinal.CNF_sorted + - Ordinal.CNF_snd_lt + - Ordinal.CNF_fst_le_log + - Ordinal.zero_CNF + - Ordinal.CNFRec_pos + - path: Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean + theorems: + - Matrix.exists_mulVec_eq_zero_iff_aux + - Matrix.det_ne_zero_of_sum_col_pos + - Matrix.det_ne_zero_of_sum_row_pos + - Matrix.exists_vecMul_eq_zero_iff + - Matrix.nondegenerate_iff_det_ne_zero + - Matrix.exists_mulVec_eq_zero_iff' + - path: Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean + theorems: + - intervalIntegrable_sub_inv_iff + - not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter + - not_IntegrableOn_Ioi_inv + - not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton + - not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux + - not_IntegrableOn_Ici_inv + - not_intervalIntegrable_of_sub_inv_isBigO_punctured + - intervalIntegrable_inv_iff + - not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter + - path: Mathlib/Data/Finset/Prod.lean + theorems: + - Finset.diag_insert + - Finset.product_eq_biUnion_right + - Finset.diag_card + - Finset.singleton_product_singleton + - Finset.diag_singleton + - Finset.subset_product_image_snd + - Finset.filter_product + - Finset.product_sdiff_diag + - Finset.mem_diag + - Finset.offDiag_singleton + - Finset.inter_product + - Finset.filter_product_right + - Finset.product_inter + - Finset.diag_union + - Finset.image_swap_product + - Finset.product_image_snd + - Finset.offDiag_insert + - Finset.product_singleton + - Finset.subset_product_image_fst + - Finset.union_product + - Finset.product_eq_biUnion + - Finset.mem_offDiag + - Finset.diag_inter + - Finset.filter_product_card + - Finset.product_sdiff_offDiag + - Finset.filter_product_left + - Finset.product_inter_product + - Finset.product_image_fst + - Finset.map_swap_product + - Finset.singleton_product + - Finset.offDiag_card + - Finset.product_union + - Finset.product_biUnion + - Finset.nontrivial_prod_iff + - Finset.offDiag_inter + - Finset.product_eq_empty + - Finset.disjoint_product + - Finset.offDiag_union + - Finset.diag_union_offDiag + - path: Mathlib/Order/OmegaCompletePartialOrder.lean + theorems: + - OmegaCompletePartialOrder.ContinuousHom.ite_continuous' + - "OmegaCompletePartialOrder.\u03C9Sup_total" + - "Part.mem_chain_of_mem_\u03C9Sup" + - CompleteLattice.bot_continuous + - "Prod.\u03C9Sup_zip" + - OmegaCompletePartialOrder.ContinuousHom.bind_continuous' + - OmegaCompletePartialOrder.continuous_comp + - "OmegaCompletePartialOrder.ContinuousHom.\u03C9Sup_bind" + - Part.eq_of_chain + - "OmegaCompletePartialOrder.ContinuousHom.\u03C9Sup_apply_\u03C9Sup" + - "OmegaCompletePartialOrder.\u03C9Sup_le_\u03C9Sup_of_le" + - OmegaCompletePartialOrder.continuous_id + - CompleteLattice.top_continuous + - CompleteLattice.sSup_continuous' + - OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge + - CompleteLattice.inf_continuous + - OmegaCompletePartialOrder.ContinuousHom.seq_continuous' + - OmegaCompletePartialOrder.isLUB_of_scottContinuous + - OmegaCompletePartialOrder.continuous_const + - "OmegaCompletePartialOrder.\u03C9Sup_le_iff" + - "OmegaCompletePartialOrder.\u03C9Sup_eq_of_isLUB" + - CompleteLattice.sSup_continuous + - OmegaCompletePartialOrder.ScottContinuous.continuous' + - "Pi.OmegaCompletePartialOrder.flip\u2082_continuous'" + - CompleteLattice.sup_continuous + - OmegaCompletePartialOrder.Chain.mem_map_iff + - "OmegaCompletePartialOrder.isLUB_range_\u03C9Sup" + - "Part.mem_\u03C9Sup" + - OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge' + - OmegaCompletePartialOrder.Chain.map_le_map + - path: Mathlib/Algebra/Order/Pointwise.lean + theorems: + - sInf_div + - csInf_div + - csSup_inv + - sSup_inv + - LinearOrderedField.smul_Ioo + - LinearOrderedField.smul_Icc + - sSup_div + - LinearOrderedField.smul_Ioi + - csInf_inv + - csSup_div + - sInf_inv + - LinearOrderedField.smul_Iio + - LinearOrderedField.smul_Ico + - LinearOrderedField.smul_Ioc + - LinearOrderedField.smul_Ici + - path: Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean + theorems: + - GeneralizedContinuedFraction.exists_rat_eq_nth_convergent + - GeneralizedContinuedFraction.coe_of_s_rat_eq + - GeneralizedContinuedFraction.coe_of_rat_eq + - GeneralizedContinuedFraction.coe_of_s_get?_rat_eq + - GeneralizedContinuedFraction.coe_of_h_rat_eq + - GeneralizedContinuedFraction.of_terminates_iff_of_rat_terminates + - GeneralizedContinuedFraction.IntFractPair.coe_stream_nth_rat_eq + - GeneralizedContinuedFraction.IntFractPair.exists_nth_stream_eq_none_of_rat + - GeneralizedContinuedFraction.exists_gcf_pair_rat_eq_of_nth_conts_aux + - GeneralizedContinuedFraction.IntFractPair.coe_stream'_rat_eq + - GeneralizedContinuedFraction.IntFractPair.stream_nth_fr_num_le_fr_num_sub_n_rat + - GeneralizedContinuedFraction.exists_rat_eq_nth_denominator + - GeneralizedContinuedFraction.IntFractPair.coe_of_rat_eq + - GeneralizedContinuedFraction.exists_gcf_pair_rat_eq_nth_conts + - GeneralizedContinuedFraction.exists_rat_eq_of_terminates + - GeneralizedContinuedFraction.IntFractPair.stream_succ_nth_fr_num_lt_nth_fr_num_rat + - path: Mathlib/Probability/BorelCantelli.lean + theorems: + - ProbabilityTheory.measure_limsup_eq_one + - ProbabilityTheory.iIndepSet.condexp_indicator_filtrationOfSet_ae_eq + - ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt + - path: Mathlib/Topology/MetricSpace/Contracting.lean + theorems: + - ContractingWith.dist_le_of_fixedPoint + - ContractingWith.edist_inequality + - ContractingWith.eq_or_edist_eq_top_of_fixedPoints + - ContractingWith.dist_inequality + - ContractingWith.efixedPoint_eq_of_edist_lt_top + - ContractingWith.dist_fixedPoint_fixedPoint_of_dist_le' + - ContractingWith.edist_le_of_fixedPoint + - ContractingWith.aposteriori_dist_iterate_fixedPoint_le + - ContractingWith.edist_efixedPoint_le' + - ContractingWith.edist_efixedPoint_le + - ContractingWith.efixedPoint_eq_of_edist_lt_top' + - ContractingWith.exists_fixedPoint' + - ContractingWith.one_sub_K_pos' + - ContractingWith.one_sub_K_ne_top + - ContractingWith.tendsto_iterate_fixedPoint + - path: Mathlib/Analysis/NormedSpace/Spectrum.lean + theorems: + - SpectrumRestricts.nnreal_iff + - SpectrumRestricts.spectralRadius_eq + - spectrum.pow_norm_pow_one_div_tendsto_nhds_spectralRadius + - spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul + - spectrum.isUnit_one_sub_smul_of_lt_inv_radius + - spectrum.subset_closedBall_norm + - SpectrumRestricts.real_iff + - spectrum.algebraMap_eq_of_mem + - spectrum.spectralRadius_zero + - spectrum.spectralRadius_le_liminf_pow_nnnorm_pow_one_div + - AlgHom.toContinuousLinearMap_norm + - spectrum.limsup_pow_nnnorm_pow_one_div_le_spectralRadius + - spectrum.mem_resolventSet_of_norm_lt_mul + - spectrum.hasDerivAt_resolvent + - spectrum.nonempty + - SpectrumRestricts.nnreal_iff_spectralRadius_le + - spectrum.SpectralRadius.of_subsingleton + - spectrum.spectralRadius_le_nnnorm + - spectrum.resolvent_isBigO_inv + - spectrum.spectralRadius_lt_of_forall_lt_of_nonempty + - spectrum.mem_resolventSet_of_norm_lt + - spectrum.exp_mem_exp + - spectrum.spectralRadius_le_pow_nnnorm_pow_one_div + - spectrum.exists_nnnorm_eq_spectralRadius_of_nonempty + - spectrum.subset_closedBall_norm_mul + - spectrum.eventually_isUnit_resolvent + - spectrum.map_pow + - path: Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean + theorems: + - CategoryTheory.Idempotents.isEquivalence_whiskeringLeft_obj_toKaroubi_aux + - CategoryTheory.Idempotents.natTrans_eq + - CategoryTheory.Idempotents.whiskeringLeft_obj_preimage_app + - path: Mathlib/Data/Int/ConditionallyCompleteOrder.lean + theorems: + - Int.csSup_mem + - Int.csInf_mem + - Int.csInf_eq_least_of_bdd + - Int.csSup_empty + - Int.csSup_of_not_bdd_above + - Int.csInf_of_not_bdd_below + - Int.csInf_empty + - Int.csSup_eq_greatest_of_bdd + - path: Mathlib/Analysis/Normed/Group/AddTorsor.lean + theorems: + - edist_vadd_vadd_le + - nndist_vsub_vsub_le + - edist_vsub_vsub_le + - dist_vadd_left + - IsClosed.vadd_right_of_isCompact + - dist_vsub_cancel_left + - dist_vadd_vadd_le + - dist_vadd_cancel_right + - dist_vadd_right + - dist_vsub_vsub_le + - path: Mathlib/CategoryTheory/Sites/Adjunction.lean + theorems: + - CategoryTheory.Sheaf.adjunctionToTypes_counit_app_val + - CategoryTheory.Sheaf.adjunctionToTypes_unit_app_val + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean + theorems: + - Complex.tendsto_abs_tan_of_cos_eq_zero + - Complex.continuousAt_tan + - Complex.deriv_tan + - Complex.hasStrictDerivAt_tan + - path: Mathlib/ModelTheory/Graph.lean + theorems: + - SimpleGraph.simpleGraphOfStructure + - FirstOrder.Language.structure_simpleGraphOfStructure + - FirstOrder.Language.Theory.simpleGraph_model_iff + - path: Mathlib/Tactic/Positivity/Basic.lean + theorems: + - Mathlib.Meta.Positivity.int_div_self_pos + - Mathlib.Meta.Positivity.ite_nonneg + - Mathlib.Meta.Positivity.max_ne + - Mathlib.Meta.Positivity.ite_ne_zero + - Mathlib.Meta.Positivity.ite_pos + - Mathlib.Meta.Positivity.min_ne + - path: Mathlib/LinearAlgebra/Ray.lean + theorems: + - SameRay.sameRay_nonneg_smul_right + - Module.Ray.units_smul_of_neg + - sameRay_neg_smul_right_iff_of_ne + - SameRay.exists_nonneg_left + - SameRay.map + - Function.Injective.sameRay_map_iff + - Module.Ray.ne_neg_self + - sameRay_neg_swap + - sameRay_or_sameRay_neg_iff_not_linearIndependent + - exists_pos_left_iff_sameRay_and_ne_zero + - SameRay.of_subsingleton + - Module.Ray.neg_units_smul + - exists_nonneg_left_iff_sameRay + - units_inv_smul + - exists_pos_right_iff_sameRay_and_ne_zero + - exists_pos_left_iff_sameRay + - sameRay_smul_right_iff_of_ne + - units_smul_eq_neg_iff + - eq_zero_of_sameRay_neg_smul_right + - exists_nonneg_right_iff_sameRay + - sameRay_of_mem_orbit + - SameRay.add_left + - units_smul_eq_self_iff + - SameRay.refl + - Module.Ray.map_neg + - SameRay.trans + - sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependent + - Module.Ray.units_smul_of_pos + - sameRay_neg_iff + - SameRay.nonneg_smul_right + - sameRay_neg_smul_right_iff + - SameRay.exists_eq_smul_add + - exists_pos_right_iff_sameRay + - SameRay.exists_pos_left + - path: Mathlib/Data/List/BigOperators/Lemmas.lean + theorems: + - List.length_le_sum_of_one_le + - MulOpposite.unop_list_prod + - Commute.list_sum_right + - List.prod_eq_one_iff + - List.dvd_prod + - List.dvd_sum + - List.alternatingProd_append + - MulOpposite.op_list_prod + - List.neg_one_mem_of_prod_eq_neg_one + - List.alternatingProd_reverse + - path: Mathlib/Combinatorics/SetFamily/Compression/Down.lean + theorems: + - Finset.memberSubfamily_memberSubfamily + - Finset.family_induction_on + - Down.mem_compression + - Finset.nonMemberSubfamily_image_insert + - Finset.memberSubfamily_inter + - Finset.memberSubfamily_image_erase + - Down.erase_mem_compression_of_mem_compression + - Finset.memberSubfamily_image_insert + - Finset.nonMemberSubfamily_nonMemberSubfamily + - Down.erase_mem_compression + - Finset.mem_nonMemberSubfamily + - Finset.memberSubfamily_union + - Finset.mem_memberSubfamily + - Down.card_compression + - Finset.memberFamily_induction_on + - Finset.card_memberSubfamily_add_card_nonMemberSubfamily + - Finset.image_insert_memberSubfamily + - Finset.memberSubfamily_nonMemberSubfamily + - Finset.nonMemberSubfamily_memberSubfamily + - Down.mem_compression_of_insert_mem_compression + - Down.compression_idem + - Finset.memberSubfamily_union_nonMemberSubfamily + - path: Mathlib/Geometry/Euclidean/Sphere/Basic.lean + theorems: + - EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere + - EuclideanGeometry.Sphere.center_eq_iff_eq_of_mem + - EuclideanGeometry.inner_nonneg_of_dist_le_radius + - EuclideanGeometry.inner_pos_of_dist_lt_radius + - EuclideanGeometry.cospherical_iff_exists_sphere + - EuclideanGeometry.Sphere.ne_iff + - EuclideanGeometry.cospherical_singleton + - EuclideanGeometry.cospherical_pair + - EuclideanGeometry.Cospherical.subset + - EuclideanGeometry.Cospherical.affineIndependent_of_mem_of_ne + - EuclideanGeometry.Cospherical.affineIndependent + - EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere' + - EuclideanGeometry.Sphere.mk_center_radius + - EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius + - path: Mathlib/Data/Sigma/Lex.lean + theorems: + - Sigma.lex_iff + - Sigma.Lex.mono + - Sigma.lex_swap + - PSigma.Lex.mono + - PSigma.lex_iff + - path: Mathlib/Combinatorics/Pigeonhole.lean + theorems: + - Finset.exists_le_sum_fiber_of_sum_fiber_nonpos_of_nsmul_le_sum + - Finset.exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum + - Finset.exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum + - Finset.exists_card_fiber_lt_of_card_lt_nsmul + - Finset.exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum + - Finset.exists_le_card_fiber_of_nsmul_le_card_of_maps_to + - Finset.exists_lt_card_fiber_of_nsmul_lt_card_of_maps_to + - Finset.exists_card_fiber_le_of_card_le_nsmul + - path: Mathlib/Topology/LocallyFinite.lean + theorems: + - LocallyFinite.isClosed_iUnion + - LocallyFinite.closure_iUnion + - LocallyFinite.continuousOn_iUnion' + - LocallyFinite.exists_forall_eventually_eq_prod + - LocallyFinite.nhdsWithin_iUnion + - locallyFinite_option + - LocallyFinite.eventually_subset + - LocallyFinite.iInter_compl_mem_nhds + - LocallyFinite.comp_injOn + - path: Mathlib/Data/Finset/MulAntidiagonal.lean + theorems: + - Finset.mem_mulAntidiagonal + - Set.IsPWO.mul + - Finset.support_mulAntidiagonal_subset_mul + - Set.IsWF.min_mul + - Finset.swap_mem_mulAntidiagonal + - Finset.mulAntidiagonal_min_mul_min + - path: Mathlib/Data/Set/Intervals/Group.lean + theorems: + - Set.pairwise_disjoint_Ico_mul_zpow + - Set.pairwise_disjoint_Ico_add_int_cast + - Set.pairwise_disjoint_Ioc_mul_zpow + - Set.pairwise_disjoint_Ioo_zpow + - Set.nonempty_Ico_sdiff + - Set.pairwise_disjoint_Ioo_int_cast + - Set.pairwise_disjoint_Ioc_int_cast + - Set.pairwise_disjoint_Ico_zpow + - Set.pairwise_disjoint_Ioc_zpow + - Set.pairwise_disjoint_Ioc_add_int_cast + - Set.pairwise_disjoint_Ico_int_cast + - path: Mathlib/Probability/Integration.lean + theorems: + - ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace + - ProbabilityTheory.indepFun_iff_integral_comp_mul + - ProbabilityTheory.IndepFun.integral_mul_of_integrable + - ProbabilityTheory.IndepFun.integral_mul + - ProbabilityTheory.IndepFun.integrable_left_of_integrable_mul + - ProbabilityTheory.IndepFun.integrable_mul + - ProbabilityTheory.IndepFun.integrable_right_of_integrable_mul + - ProbabilityTheory.IndepFun.integral_mul_of_nonneg + - ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' + - ProbabilityTheory.lintegral_mul_indicator_eq_lintegral_mul_lintegral_indicator + - path: Mathlib/GroupTheory/Subgroup/Pointwise.lean + theorems: + - Subgroup.subgroup_mul_singleton + - Subgroup.closure_toSubmonoid + - Subgroup.inf_mul_assoc + - coe_mul_coe + - Subgroup.mul_normal + - Subgroup.singleton_mul_subgroup + - Subgroup.set_mul_normal_comm + - Subgroup.iSup_induction' + - coe_div_coe + - Subgroup.closure_inv + - Subgroup.iSup_induction + - Subgroup.normal_mul + - Subgroup.conj_smul_subgroupOf + - Subgroup.smul_inf + - op_smul_coe_set + - Subgroup.inv_subset_closure + - Subgroup.closure_induction_right + - Subgroup.closure_induction_left + - Subgroup.mul_inf_assoc + - Subgroup.smul_opposite_image_mul_preimage' + - smul_coe_set + - path: Mathlib/AlgebraicGeometry/Pullbacks.lean + theorems: + - AlgebraicGeometry.Scheme.Pullback.t_fst_snd + - AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_snd + - AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_fst + - AlgebraicGeometry.Scheme.Pullback.cocycle_fst_fst_fst + - AlgebraicGeometry.Scheme.Pullback.gluedLiftPullbackMap_fst + - AlgebraicGeometry.Scheme.Pullback.t_fst_fst + - "AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_hom_\u03B9" + - AlgebraicGeometry.Scheme.Pullback.cocycle_fst_snd + - AlgebraicGeometry.Scheme.Pullback.t_id + - "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03B9ToV_snd" + - AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_snd + - AlgebraicGeometry.Scheme.Pullback.t'_fst_fst_fst + - AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_inv_fst + - AlgebraicGeometry.Scheme.Pullback.cocycle_snd_fst_fst + - "AlgebraicGeometry.Scheme.Pullback.lift_comp_\u03B9" + - AlgebraicGeometry.Scheme.Pullback.t'_fst_fst_snd + - AlgebraicGeometry.Scheme.Pullback.t'_snd_snd + - AlgebraicGeometry.Scheme.Pullback.cocycle_snd_snd + - AlgebraicGeometry.Scheme.Pullback.t'_fst_snd + - AlgebraicGeometry.Scheme.Pullback.cocycle_fst_fst_snd + - "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03B9ToV_fst" + - AlgebraicGeometry.Scheme.Pullback.gluedLiftPullbackMap_snd + - AlgebraicGeometry.Scheme.Pullback.gluedLift_p1 + - AlgebraicGeometry.Scheme.Pullback.cocycle + - AlgebraicGeometry.Scheme.Pullback.gluedLift_p2 + - AlgebraicGeometry.Scheme.Pullback.t_snd + - AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso_inv_snd + - AlgebraicGeometry.Scheme.Pullback.p_comm + - AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_fst + - path: Mathlib/CategoryTheory/Galois/Decomposition.lean + theorems: + - CategoryTheory.PreGaloisCategory.has_decomp_connected_components_aux_conn + - CategoryTheory.PreGaloisCategory.selfProdTermIncl_fib_eq + - CategoryTheory.PreGaloisCategory.subobj_selfProd_trans + - "CategoryTheory.PreGaloisCategory.mkSelfProdFib_map_\u03C0" + - CategoryTheory.PreGaloisCategory.fiber_in_connected_component + - CategoryTheory.PreGaloisCategory.has_decomp_connected_components + - CategoryTheory.PreGaloisCategory.selfProdProj_fiber + - CategoryTheory.PreGaloisCategory.has_decomp_connected_components' + - CategoryTheory.PreGaloisCategory.has_decomp_connected_components_aux + - CategoryTheory.PreGaloisCategory.exists_galois_representative + - CategoryTheory.PreGaloisCategory.connected_component_unique + - CategoryTheory.PreGaloisCategory.exists_hom_from_galois_of_fiber_nonempty + - CategoryTheory.PreGaloisCategory.has_decomp_connected_components_aux_initial + - CategoryTheory.PreGaloisCategory.exists_hom_from_galois_of_fiber + - path: Mathlib/Algebra/GroupPower/CovariantClass.lean + theorems: + - Right.one_le_pow_of_le + - StrictMono.pow_const + - one_lt_pow' + - Left.pow_lt_one_of_lt + - pow_eq_one_iff + - min_le_of_mul_le_sq + - Left.pow_le_one_of_le + - pow_le_pow_right' + - one_le_zpow + - min_lt_of_mul_lt_sq + - Left.one_le_pow_of_le + - Right.pow_lt_one_of_lt + - pow_le_pow_left' + - lt_max_of_sq_lt_mul + - le_max_of_sq_le_mul + - Right.pow_le_one_of_le + - Monotone.pow_const + - one_le_pow_of_one_le' + - pow_lt_pow_right' + - path: Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean + theorems: + - tangentBundle_model_space_chartAt + - TangentBundle.mem_chart_target_iff + - inCoordinates_tangent_bundle_core_model_space + - tangentBundleCore_coordChange_model_space + - hasFDerivWithinAt_tangentCoordChange + - TangentBundle.mem_chart_source_iff + - contDiffOn_fderiv_coord_change + - tangentCoordChange_self + - TangentBundle.symmL_model_space + - tangentCoordChange_comp + - TangentBundle.continuousLinearMapAt_model_space + - tangentBundle_model_space_coe_chartAt_symm + - inTangentCoordinates_model_space + - tangentBundle_model_space_coe_chartAt + - TangentBundle.coordChange_model_space + - continuousOn_tangentCoordChange + - path: Mathlib/Geometry/Manifold/Algebra/LieGroup.lean + theorems: + - ContMDiffOn.div + - "ContMDiffWithinAt.div\u2080" + - ContMDiffAt.div + - "ContMDiffAt.div\u2080" + - ContMDiff.div + - "ContMDiff.div\u2080" + - "ContMDiffOn.div\u2080" + - ContMDiffWithinAt.div + - path: Mathlib/Probability/IdentDistrib.lean + theorems: + - ProbabilityTheory.IdentDistrib.essSup_eq + - ProbabilityTheory.IdentDistrib.aestronglyMeasurable_snd + - ProbabilityTheory.IdentDistrib.measure_mem_eq + - "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib_aux" + - ProbabilityTheory.IdentDistrib.integrable_snd + - ProbabilityTheory.IdentDistrib.evariance_eq + - ProbabilityTheory.IdentDistrib.integral_eq + - ProbabilityTheory.IdentDistrib.snorm_eq + - ProbabilityTheory.IdentDistrib.lintegral_eq + - "ProbabilityTheory.Mem\u2112p.uniformIntegrable_of_identDistrib" + - ProbabilityTheory.IdentDistrib.variance_eq + - "ProbabilityTheory.IdentDistrib.mem\u2112p_snd" + - ProbabilityTheory.IdentDistrib.comp_of_aemeasurable + - path: Mathlib/CategoryTheory/Adjunction/Basic.lean + theorems: + - CategoryTheory.Adjunction.homEquiv_id + - CategoryTheory.Adjunction.homEquiv_naturality_left + - CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm + - CategoryTheory.Adjunction.homEquiv_apply_eq + - CategoryTheory.Adjunction.he'' + - CategoryTheory.Adjunction.eq_homEquiv_apply + - CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left_aux + - CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_right_symm_aux + - CategoryTheory.Adjunction.left_triangle + - CategoryTheory.Adjunction.right_triangle + - CategoryTheory.Adjunction.CoreHomEquiv.homEquiv_naturality_left + - CategoryTheory.Adjunction.homEquiv_symm_id + - CategoryTheory.Adjunction.homEquiv_naturality_right_symm + - CategoryTheory.Adjunction.homEquiv_naturality_right + - CategoryTheory.Adjunction.he' + - CategoryTheory.Adjunction.homEquiv_naturality_left_symm + - path: Mathlib/Data/Multiset/Interval.lean + theorems: + - Multiset.uIcc_eq + - Multiset.card_uIcc + - Multiset.card_Ioc + - Multiset.card_Ioo + - Multiset.card_Icc + - Multiset.card_Iic + - Multiset.card_Ico + - path: Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean + theorems: + - "CategoryTheory.Limits.WidePushout.\u03B9_desc" + - CategoryTheory.Limits.WidePullback.lift_base + - "CategoryTheory.Limits.WidePullback.\u03C0_arrow" + - "CategoryTheory.Limits.WidePushout.arrow_\u03B9" + - CategoryTheory.Limits.WidePullback.eq_lift_of_comp_eq + - CategoryTheory.Limits.WidePullback.hom_ext + - CategoryTheory.Limits.WidePushout.hom_eq_desc + - CategoryTheory.Limits.WidePushout.eq_desc_of_comp_eq + - "CategoryTheory.Limits.WidePullback.lift_\u03C0" + - CategoryTheory.Limits.WidePushout.head_desc + - CategoryTheory.Limits.WidePushout.hom_ext + - CategoryTheory.Limits.WidePullback.hom_eq_lift + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean + theorems: + - FiniteField.trace_pow_card + - ZMod.trace_pow_card + - FiniteField.Matrix.charpoly_pow_card + - ZMod.charpoly_pow_card + - path: Mathlib/Topology/UnitInterval.lean + theorems: + - exists_monotone_Icc_subset_open_cover_Icc + - unitInterval.symm_one + - Set.abs_projIcc_sub_projIcc + - affineHomeomorph_image_I + - unitInterval.mul_pos_mem_iff + - Set.Icc.addNSMul_eq_right + - unitInterval.symm_symm + - Set.Icc.abs_sub_addNSMul_le + - unitInterval.one_minus_le_one + - unitInterval.one_minus_nonneg + - unitInterval.symm_zero + - exists_monotone_Icc_subset_open_cover_unitInterval_prod_self + - Set.Icc.addNSMul_zero + - unitInterval.mem_iff_one_sub_mem + - unitInterval.half_le_symm_iff + - unitInterval.two_mul_sub_one_mem_iff + - exists_monotone_Icc_subset_open_cover_unitInterval + - path: Mathlib/RingTheory/IsAdjoinRoot.lean + theorems: + - IsAdjoinRoot.ext_map + - IsAdjoinRoot.lift_root + - IsAdjoinRootMonic.basis_repr + - IsAdjoinRoot.aequiv_ofEquiv + - IsAdjoinRoot.repr_add_sub_repr_add_repr_mem_span + - IsAdjoinRoot.aequiv_root + - IsAdjoinRootMonic.coeff_root + - IsAdjoinRootMonic.basis_apply + - IsAdjoinRoot.aequiv_trans + - IsAdjoinRoot.liftHom_root + - IsAdjoinRoot.lift_aequiv + - IsAdjoinRoot.ext + - IsAdjoinRootMonic.coeff_root_pow + - IsAdjoinRoot.algebraMap_apply + - IsAdjoinRoot.mem_ker_map + - IsAdjoinRoot.lift_algebraMap + - IsAdjoinRootMonic.coeff_algebraMap + - IsAdjoinRoot.repr_zero_mem_span + - IsAdjoinRootMonic.basis_one + - IsAdjoinRootMonic.deg_pos + - IsAdjoinRoot.aeval_eq + - Algebra.adjoin.powerBasis'_minpoly_gen + - IsAdjoinRootMonic.coeff_one + - IsAdjoinRootMonic.coeff_apply_lt + - IsAdjoinRoot.map_eq_zero_iff + - IsAdjoinRoot.liftHom_map + - IsAdjoinRootMonic.map_modByMonic + - IsAdjoinRoot.aeval_root + - AdjoinRoot.isAdjoinRoot_root_eq_root + - IsAdjoinRoot.aequiv_map + - IsAdjoinRoot.aequiv_symm + - IsAdjoinRoot.aequiv_self + - IsAdjoinRootMonic.map_modByMonicHom + - IsAdjoinRoot.ofEquiv_aequiv + - IsAdjoinRootMonic.coeff_apply + - AdjoinRoot.isAdjoinRootMonic_root_eq_root + - IsAdjoinRootMonic.minpoly_eq + - IsAdjoinRoot.lift_self_apply + - IsAdjoinRootMonic.coeff_apply_le + - IsAdjoinRoot.apply_eq_lift + - IsAdjoinRootMonic.modByMonicHom_root + - "IsAdjoinRoot.eval\u2082_repr_eq_eval\u2082_of_map_eq" + - IsAdjoinRootMonic.modByMonic_repr_map + - IsAdjoinRoot.lift_map + - IsAdjoinRootMonic.ext_elem + - path: Mathlib/CategoryTheory/Monoidal/Functor.lean + theorems: + - CategoryTheory.MonoidalFunctor.map_leftUnitor + - CategoryTheory.MonoidalFunctor.map_whiskerLeft + - CategoryTheory.MonoidalFunctor.map_whiskerRight + - CategoryTheory.LaxMonoidalFunctor.left_unitality_inv + - CategoryTheory.MonoidalFunctor.map_tensor + - "CategoryTheory.LaxMonoidalFunctor.prod'_\u03BC" + - CategoryTheory.LaxMonoidalFunctor.right_unitality_inv + - CategoryTheory.MonoidalFunctor.map_rightUnitor + - CategoryTheory.LaxMonoidalFunctor.associativity_inv + - "CategoryTheory.LaxMonoidalFunctor.prod'_\u03B5" + - "CategoryTheory.LaxMonoidalFunctor.\u03BC_natural" + - path: Mathlib/Probability/Kernel/IntegralCompProd.lean + theorems: + - ProbabilityTheory.kernel.integral_fn_integral_add + - MeasureTheory.AEStronglyMeasurable.compProd_mk_left + - ProbabilityTheory.hasFiniteIntegral_compProd_iff + - ProbabilityTheory.integral_compProd + - ProbabilityTheory.kernel.continuous_integral_integral + - ProbabilityTheory.hasFiniteIntegral_prod_mk_left + - ProbabilityTheory.set_integral_compProd_univ_right + - ProbabilityTheory.kernel.lintegral_fn_integral_sub + - ProbabilityTheory.integrable_kernel_prod_mk_left + - MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd + - ProbabilityTheory.integrable_compProd_iff + - ProbabilityTheory.kernel.integral_fn_integral_sub + - ProbabilityTheory.hasFiniteIntegral_compProd_iff' + - ProbabilityTheory.set_integral_compProd + - path: Mathlib/NumberTheory/NumberField/CanonicalEmbedding.lean + theorems: + - NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt' + - NumberField.canonicalEmbedding.norm_le_iff + - NumberField.canonicalEmbedding.integerLattice.inter_ball_finite + - NumberField.mixedEmbedding.det_basisOfFractionalIdeal_eq_norm + - NumberField.mixedEmbedding.exists_primitive_element_lt_of_isComplex + - NumberField.mixedEmbedding.convexBodySumFun_continuous + - NumberField.mixedEmbedding.det_matrixToStdBasis + - NumberField.mixedEmbedding.fundamentalDomain_stdBasis + - NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_lt' + - NumberField.mixedEmbedding.convexBodySumFun_eq_zero_iff + - NumberField.mixedEmbedding.convexBodyLT'_convex + - NumberField.mixedEmbedding.convexBodySum_neg_mem + - NumberField.mixedEmbedding.latticeBasis_repr_apply + - NumberField.mixedEmbedding.fractionalIdealLatticeBasis_apply + - NumberField.mixedEmbedding.volume_fundamentalDomain_fractionalIdealLatticeBasis + - NumberField.mixedEmbedding.convexBodySum_convex + - NumberField.canonicalEmbedding.mem_span_latticeBasis + - NumberField.mixedEmbedding.minkowskiBound_lt_top + - NumberField.mixedEmbedding.convexBodyLT_volume + - NumberField.mixedEmbedding.disjoint_span_commMap_ker + - NumberField.mixedEmbedding.exists_primitive_element_lt_of_isReal + - NumberField.mixedEmbedding.convexBodySum_isBounded + - NumberField.mixedEmbedding.convexBodyLT_mem + - NumberField.mixedEmbedding.mem_span_latticeBasis + - NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_of_norm_le + - NumberField.mixedEmbedding.convexBodySumFun_add_le + - NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_lt + - NumberField.mixedEmbedding.norm_le_convexBodySumFun + - NumberField.canonicalEmbedding.latticeBasis_apply + - NumberField.mixedEmbedding.convexBodySumFactor_ne_zero + - NumberField.mixedEmbedding.convexBodySumFun_nonneg + - NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_lt + - NumberField.mixedEmbedding.convexBodySum_compact + - NumberField.mixedEmbedding.convexBodyLT_neg_mem + - NumberField.mixedEmbedding.mem_rat_span_latticeBasis + - NumberField.mixedEmbedding_injective + - NumberField.mixedEmbedding.convexBodySumFun_neg + - NumberField.mixedEmbedding.adjust_f + - NumberField.mixedEmbedding.latticeBasis_apply + - NumberField.mixedEmbedding.stdBasis_repr_eq_matrixToStdBasis_mul + - NumberField.mixedEmbedding.convexBodySumFun_smul + - NumberField.mixedEmbedding.convexBodyLT'_neg_mem + - NumberField.mixedEmbedding.mem_span_fractionalIdealLatticeBasis + - NumberField.mixedEmbedding.minkowskiBound_pos + - NumberField.canonicalEmbedding.conj_apply + - NumberField.mixedEmbedding.convexBodyLT'_mem + - NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_of_norm_le + - NumberField.mixedEmbedding.convexBodyLT'_volume + - NumberField.mixedEmbedding.convexBodySum_volume_eq_zero_of_le_zero + - NumberField.mixedEmbedding.commMap_canonical_eq_mixed + - NumberField.mixedEmbedding.convexBodySum_mem + - NumberField.mixedEmbedding.volume_fundamentalDomain_stdBasis + - path: Mathlib/Data/Fintype/Units.lean + theorems: + - Fintype.card_units + - Nat.card_eq_card_units_add_one + - Fintype.card_eq_card_units_add_one + - path: Mathlib/Data/List/AList.lean + theorems: + - AList.insertRec_insert + - AList.mem_replace + - AList.mem_erase + - AList.mk_cons_eq_insert + - AList.insertRec_empty + - AList.lookup_insert_eq_none + - AList.insert_entries_of_neg + - AList.insert_union + - AList.insert_insert_of_ne + - AList.lookup_to_alist + - AList.keys_insert + - AList.perm_insert + - AList.extract_eq_lookup_erase + - AList.insert_singleton_eq + - AList.keys_subset_keys_of_entries_subset_entries + - AList.union_assoc + - AList.union_empty + - AList.union_comm_of_disjoint + - AList.insert_insert + - AList.perm_union + - AList.ext + - path: Mathlib/RingTheory/Artinian.lean + theorems: + - LinearMap.isCompl_iSup_ker_pow_iInf_range_pow + - IsArtinian.eventuallyConst_of_isArtinian + - monotone_stabilizes_iff_artinian + - IsArtinianRing.maximal_ideals_finite + - LinearMap.eventually_isCompl_ker_pow_range_pow + - IsArtinian.surjective_of_injective_endomorphism + - set_has_minimal_iff_artinian + - isArtinian_of_range_eq_ker + - IsArtinianRing.localization_surjective + - LinearMap.eventually_codisjoint_ker_pow_range_pow + - LinearMap.eventually_iInf_range_pow_eq + - Function.Surjective.isArtinianRing + - isArtinian_of_fg_of_artinian + - IsArtinianRing.primeSpectrum_finite + - IsArtinian.finite_of_linearIndependent + - IsArtinianRing.isNilpotent_jacobson_bot + - IsArtinian.disjoint_partial_infs_eventually_top + - IsArtinian.exists_pow_succ_smul_dvd + - IsArtinian.range_smul_pow_stabilizes + - isArtinian_of_tower + - path: Mathlib/Algebra/Order/Pi.lean + theorems: + - Function.one_le_extend + - Function.extend_le_one + - path: Mathlib/CategoryTheory/Extensive.lean + theorems: + - CategoryTheory.finitaryExtensive_of_preserves_and_reflects + - CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism + - CategoryTheory.FinitaryPreExtensive.hasPullbacks_of_is_coproduct + - CategoryTheory.finitaryExtensive_of_reflective + - CategoryTheory.FinitaryExtensive.isVanKampen_finiteCoproducts_Fin + - CategoryTheory.FinitaryExtensive.vanKampen + - CategoryTheory.FinitaryPreExtensive.isUniversal_finiteCoproducts + - CategoryTheory.FinitaryPreExtensive.sigma_desc_iso + - CategoryTheory.finitaryExtensive_iff_of_isTerminal + - CategoryTheory.FinitaryPreExtensive.isUniversal_finiteCoproducts_Fin + - CategoryTheory.FinitaryExtensive.isVanKampen_finiteCoproducts + - path: Mathlib/Topology/Algebra/GroupWithZero.lean + theorems: + - ContinuousOn.div_const + - Continuous.div_const + - "map_mul_right_nhds_one\u2080" + - "nhds_inv\u2080" + - Filter.tendsto_mul_iff_of_ne_zero + - "continuousAt_zpow\u2080" + - Continuous.div + - ContinuousAt.comp_div_cases + - "map_mul_left_nhds_one\u2080" + - Filter.Tendsto.div + - "Continuous.div\u2080" + - "HasContinuousInv\u2080.of_nhds_one" + - Filter.Tendsto.div_const + - "nhds_translation_mul_inv\u2080" + - path: Mathlib/Probability/Martingale/Convergence.lean + theorems: + - MeasureTheory.Submartingale.ae_tendsto_limitProcess + - MeasureTheory.tendsto_of_uncrossing_lt_top + - MeasureTheory.tendsto_snorm_condexp + - MeasureTheory.Submartingale.upcrossings_ae_lt_top + - MeasureTheory.Submartingale.upcrossings_ae_lt_top' + - MeasureTheory.tendsto_ae_condexp + - MeasureTheory.not_frequently_of_upcrossings_lt_top + - MeasureTheory.Submartingale.tendsto_snorm_one_limitProcess + - MeasureTheory.Integrable.tendsto_ae_condexp + - MeasureTheory.Submartingale.exists_ae_tendsto_of_bdd + - MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd + - MeasureTheory.Martingale.eq_condexp_of_tendsto_snorm + - path: Mathlib/Algebra/Module/Projective.lean + theorems: + - Module.Projective.of_basis + - Module.projective_def' + - Module.projective_lifting_property + - path: Mathlib/GroupTheory/FiniteAbelian.lean + theorems: + - AddCommGroup.equiv_free_prod_directSum_zmod + - AddCommGroup.equiv_directSum_zmod_of_finite' + - Module.finite_of_fg_torsion + - path: Mathlib/Probability/Variance.lean + theorems: + - ProbabilityTheory.variance_def' + - ProbabilityTheory.evariance_def' + - ProbabilityTheory.evariance_zero + - ProbabilityTheory.IndepFun.variance_sum + - "MeasureTheory.Mem\u2112p.ofReal_variance_eq" + - ProbabilityTheory.meas_ge_le_evariance_div_sq + - "ProbabilityTheory.evariance_lt_top_iff_mem\u2112p" + - ProbabilityTheory.evariance_mul + - ProbabilityTheory.variance_le_expectation_sq + - ProbabilityTheory.evariance_eq_zero_iff + - "MeasureTheory.Mem\u2112p.variance_eq" + - ProbabilityTheory.variance_mul + - ProbabilityTheory.evariance_eq_lintegral_ofReal + - ProbabilityTheory.IndepFun.variance_add + - ProbabilityTheory.variance_smul' + - ProbabilityTheory.variance_zero + - ProbabilityTheory.meas_ge_le_variance_div_sq + - ProbabilityTheory.evariance_eq_top + - "MeasureTheory.Mem\u2112p.evariance_lt_top" + - "MeasureTheory.Mem\u2112p.variance_eq_of_integral_eq_zero" + - path: Mathlib/Data/Nat/Cast/Defs.lean + theorems: + - three_add_one_eq_four + - Nat.cast_add + - one_add_one_eq_two + - Nat.binCast_eq + - two_add_one_eq_three + - Nat.cast_one + - Nat.cast_bit1 + - Nat.cast_ite + - path: Mathlib/Order/Bounds/Basic.lean + theorems: + - BddAbove.union + - BddAbove.bddAbove_image2_of_bddBelow + - BddBelow.bddAbove_image2_of_bddAbove + - IsGreatest.union + - not_bddAbove_univ + - bddAbove_empty + - IsGLB.insert + - lt_isLUB_iff + - lub_Iio_eq_self_or_Iio_eq_Iic + - isGreatest_univ_iff + - isLUB_le_iff + - Monotone.image_upperBounds_subset_upperBounds_image + - upperBounds_empty + - not_bddAbove_iff + - BddAbove.range_comp + - IsLUB.nonempty + - IsLeast.insert + - isLUB_Ioo + - bddBelow_empty + - BddBelow.range_comp + - ScottContinuous.monotone + - BddAbove.image2 + - isGLB_empty_iff + - OrderTop.upperBounds_univ + - lowerBounds_insert + - bddBelow_bddAbove_iff_subset_Icc + - isLUB_prod + - not_bddAbove_iff' + - upperBounds_insert + - bddAbove_insert + - BddAbove.bddBelow_image2_of_bddAbove + - BddBelow.image2_bddAbove + - BddAbove.image2_bddBelow + - isGLB_Ioo + - bddBelow_insert + - IsLeast.union + - le_isGLB_iff + - IsLUB.insert + - exists_lub_Iio + - isLeast_union_iff + - MonotoneOn.image_upperBounds_subset_upperBounds_image + - bddAbove_range_prod + - BddAbove.range_mono + - bddAbove_range_pi + - isLUB_Ico + - isLUB_pi + - BddBelow.image2 + - bddAbove_iff_exists_ge + - BddBelow.bddBelow_image2_of_bddAbove + - path: Mathlib/Combinatorics/SimpleGraph/Matching.lean + theorems: + - SimpleGraph.Subgraph.IsMatching.support_eq_verts + - SimpleGraph.Subgraph.IsMatching.toEdge_eq_toEdge_of_adj + - SimpleGraph.Subgraph.IsMatching.toEdge.surjective + - SimpleGraph.Subgraph.isMatching_iff_forall_degree + - SimpleGraph.Subgraph.isPerfectMatching_iff + - SimpleGraph.Subgraph.IsMatching.toEdge_eq_of_adj + - SimpleGraph.Subgraph.IsMatching.even_card + - SimpleGraph.Subgraph.IsPerfectMatching.even_card + - SimpleGraph.Subgraph.isPerfectMatching_iff_forall_degree + - path: Mathlib/FieldTheory/SplittingField/Construction.lean + theorems: + - Polynomial.X_sub_C_mul_removeFactor + - Polynomial.SplittingFieldAux.adjoin_rootSet + - Polynomial.irreducible_factor + - Polynomial.natDegree_removeFactor' + - Polynomial.natDegree_removeFactor + - Polynomial.factor_dvd_of_not_isUnit + - Polynomial.SplittingFieldAux.splits + - path: Mathlib/Algebra/Homology/LocalCohomology.lean + theorems: + - localCohomology.Ideal.exists_pow_le_of_le_radical_of_fG + - localCohomology.hasColimitDiagram + - path: Mathlib/Analysis/InnerProductSpace/l2Space.lean + theorems: + - IsHilbertSum.linearIsometryEquiv_symm_apply_single + - Orthonormal.exists_hilbertBasis_extension + - HilbertBasis.orthonormal + - HilbertBasis.hasSum_repr_symm + - IsHilbertSum.linearIsometryEquiv_symm_apply_dfinsupp_sum_single + - Orthonormal.linearIsometryEquiv_symm_apply_single_one + - HilbertBasis.coe_mk + - IsHilbertSum.linearIsometryEquiv_symm_apply + - OrthogonalFamily.range_linearIsometry + - IsHilbertSum.mkInternal + - HilbertBasis.hasSum_orthogonalProjection + - lp.summable_inner + - OrthogonalFamily.summable_of_lp + - HilbertBasis.hasSum_repr + - IsHilbertSum.hasSum_linearIsometryEquiv_symm + - HilbertBasis.dense_span + - IsHilbertSum.linearIsometryEquiv_apply_dfinsupp_sum_single + - OrthogonalFamily.linearIsometry_apply_single + - Submodule.isHilbertSumOrthogonal + - IsHilbertSum.mk + - HilbertBasis.repr_self + - OrthogonalFamily.linearIsometry_apply_dfinsupp_sum_single + - HilbertBasis.repr_apply_apply + - HilbertBasis.finite_spans_dense + - HilbertBasis.hasSum_inner_mul_inner + - path: Mathlib/Topology/Algebra/Order/Archimedean.lean + theorems: + - AddSubgroup.dense_of_no_min + - AddSubgroup.dense_or_cyclic + - AddSubgroup.dense_of_not_isolated_zero + - path: Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean + theorems: + - BilinForm.linearIndependent_of_iIsOrtho + - BilinForm.isOrtho_smul_left + - BilinForm.finrank_add_finrank_orthogonal + - BilinForm.restrictOrthogonalSpanSingletonNondegenerate + - BilinForm.nondegenerateRestrictOfDisjointOrthogonal + - BilinForm.restrict_nondegenerate_of_isCompl_orthogonal + - BilinForm.isOrtho_smul_right + - BilinForm.span_singleton_sup_orthogonal_eq_top + - BilinForm.toLin_restrict_range_dualCoannihilator_eq_orthogonal + - BilinForm.iIsOrtho.not_isOrtho_basis_self_of_nondegenerate + - BilinForm.orthogonal_span_singleton_eq_toLin_ker + - BilinForm.span_singleton_inf_orthogonal_eq_bot + - BilinForm.toLin_restrict_ker_eq_inf_orthogonal + - BilinForm.iIsOrtho.nondegenerate_iff_not_isOrtho_basis_self + - path: Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean + theorems: + - Real.deriv_mul_log + - Real.hasDerivAt_mul_log + - Real.hasDerivAt_negMulLog + - Real.negMulLog_eq_neg + - Real.strictConcaveOn_negMulLog + - Real.deriv2_mul_log + - Real.differentiableOn_negMulLog + - Real.negMulLog_mul + - Real.negMulLog_nonneg + - Real.strictConvexOn_mul_log + - Real.negMulLog_zero + - Real.deriv_negMulLog + - Real.deriv2_negMulLog + - Real.continuous_mul_log + - Real.continuous_negMulLog + - path: Mathlib/CategoryTheory/Preadditive/Injective.lean + theorems: + - CategoryTheory.Injective.Exact.comp_desc + - CategoryTheory.Adjunction.injective_of_map_injective + - CategoryTheory.Injective.of_iso + - CategoryTheory.Adjunction.map_injective + - CategoryTheory.Injective.injective_iff_preservesEpimorphisms_yoneda_obj + - CategoryTheory.Injective.injective_of_adjoint + - path: Mathlib/GroupTheory/Commutator.lean + theorems: + - commutatorElement_eq_one_iff_mul_comm + - Subgroup.map_commutator + - map_commutatorElement + - Subgroup.commutator_commutator_eq_bot_of_rotate + - Subgroup.commutator_prod_prod + - Subgroup.commutator_pi_pi_of_finite + - Subgroup.commutator_eq_bot_iff_le_centralizer + - commutatorElement_inv + - path: Mathlib/Topology/UniformSpace/UniformEmbedding.lean + theorems: + - UniformInducing.cauchy_map_iff + - Filter.HasBasis.uniformEmbedding_iff + - UniformInducing.inducing + - uniformContinuous_uniformly_extend + - IsComplete.completeSpace_coe + - completeSpace_congr + - uniform_extend_subtype + - UniformInducing.comp + - uniformInducing_id + - completeSpace_coe_iff_isComplete + - Filter.HasBasis.uniformEmbedding_iff' + - uniformly_extend_spec + - uniformInducing_of_compose + - isComplete_image_iff + - totallyBounded_preimage + - UniformInducing.uniformContinuousOn_iff + - uniformEmbedding_set_inclusion + - uniformEmbedding_inl + - uniformInducing_iff_uniformSpace + - uniformInducing_iff' + - isComplete_of_complete_image + - completeSpace_iff_isComplete_range + - uniformEmbedding_inr + - UniformInducing.mk' + - uniformEmbedding_of_spaced_out + - Equiv.uniformEmbedding + - closedEmbedding_of_spaced_out + - SeparationQuotient.completeSpace_iff + - Filter.HasBasis.uniformInducing_iff + - comap_uniformity_of_spaced_out + - completeSpace_extension + - UniformInducing.uniformContinuous_iff + - closure_image_mem_nhds_of_uniformInducing + - path: Mathlib/Algebra/Category/GroupCat/Colimits.lean + theorems: + - AddCommGroupCat.Colimits.cocone_naturality + - AddCommGroupCat.Colimits.cocone_naturality_components + - AddCommGroupCat.Colimits.quot_add + - AddCommGroupCat.Colimits.quot_neg + - path: Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean + theorems: + - CategoryTheory.ShortComplex.cyclesMap'_add + - CategoryTheory.ShortComplex.homologyMap'_nullHomotopic + - CategoryTheory.ShortComplex.Homotopy.rightHomologyMap'_congr + - CategoryTheory.ShortComplex.Homotopy.homologyMap'_congr + - CategoryTheory.ShortComplex.leftHomologyMap'_sub + - CategoryTheory.ShortComplex.opcyclesMap'_neg + - CategoryTheory.ShortComplex.cyclesMap'_neg + - CategoryTheory.ShortComplex.opcyclesMap'_add + - CategoryTheory.ShortComplex.rightHomologyMap_nullHomotopic + - CategoryTheory.ShortComplex.leftHomologyMap'_neg + - CategoryTheory.ShortComplex.Homotopy.leftHomologyMap'_congr + - CategoryTheory.ShortComplex.homologyMap_nullHomotopic + - CategoryTheory.ShortComplex.leftHomologyMap'_add + - CategoryTheory.ShortComplex.rightHomologyMap'_add + - CategoryTheory.ShortComplex.Homotopy.eq_add_nullHomotopic + - CategoryTheory.ShortComplex.leftHomologyMap_nullHomotopic + - CategoryTheory.ShortComplex.rightHomologyMap'_sub + - CategoryTheory.ShortComplex.opcyclesMap'_sub + - CategoryTheory.ShortComplex.rightHomologyMap'_neg + - CategoryTheory.ShortComplex.cyclesMap'_sub + - path: Mathlib/Data/Nat/Factorial/Basic.lean + theorems: + - Nat.ascFactorial_le_pow_add + - Nat.add_factorial_succ_lt_factorial_add_succ + - Nat.ascFactorial_eq_div + - Nat.add_factorial_succ_le_factorial_add_succ + - Nat.succ_ascFactorial + - Nat.pow_lt_ascFactorial' + - Nat.zero_ascFactorial + - Nat.add_factorial_lt_factorial_add + - Nat.factorial_eq_one + - Nat.two_pow_mul_factorial_le_factorial_two_mul + - Nat.succ_descFactorial_succ + - Nat.pow_succ_le_ascFactorial + - Nat.descFactorial_eq_div + - Nat.zero_descFactorial_succ + - Nat.descFactorial_lt_pow + - Nat.pow_sub_le_descFactorial + - Nat.factorial_two_mul_le + - Nat.factorial_inj' + - Nat.descFactorial_one + - Nat.add_factorial_le_factorial_add + - Nat.pow_lt_ascFactorial + - Nat.factorial_le_pow + - Nat.factorial_mul_descFactorial + - Nat.factorial_lt + - Nat.factorial_mul_ascFactorial' + - Nat.factorial_dvd_factorial + - Nat.factorial_mul_pow_sub_le_factorial + - Nat.ascFactorial_eq_div' + - Nat.pow_sub_lt_descFactorial' + - Nat.descFactorial_eq_zero_iff_lt + - Nat.add_descFactorial_eq_ascFactorial' + - Nat.one_ascFactorial + - Nat.descFactorial_self + - Nat.lt_factorial_self + - Nat.factorial_mul_pow_le_factorial + - Nat.pow_sub_lt_descFactorial + - Nat.ascFactorial_of_sub + - Nat.succ_descFactorial + - Nat.add_descFactorial_eq_ascFactorial + - Nat.factorial_inj + - Nat.ascFactorial_lt_pow_add + - Nat.descFactorial_le_pow + - path: Mathlib/CategoryTheory/SmallObject/Construction.lean + theorems: + - CategoryTheory.SmallObject.functorMapSrc_functorObjTop + - "CategoryTheory.SmallObject.\u03C1FunctorObj_\u03C0" + - "CategoryTheory.SmallObject.functorMap_\u03C0" + - CategoryTheory.SmallObject.functorMap_comm + - "CategoryTheory.SmallObject.\u03B9_functorMapSrc" + - "CategoryTheory.SmallObject.\u03B9_functorMapTgt" + - "CategoryTheory.SmallObject.\u03B9FunctorObj_extension" + - "CategoryTheory.SmallObject.\u03B9FunctorObj_naturality" + - CategoryTheory.SmallObject.functorMap_id + - CategoryTheory.SmallObject.FunctorObjIndex.comm + - path: Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean + theorems: + - mul_cancel_left_mem_nonZeroDivisors + - coe_nonZeroDivisorsRight_eq + - nonZeroDivisorsLeft_eq_right + - mul_mem_nonZeroDivisors + - prod_zero_iff_exists_zero + - mul_left_mem_nonZeroDivisors_eq_zero_iff + - mul_cancel_right_mem_nonZeroDivisors + - map_le_nonZeroDivisors_of_injective + - coe_nonZeroDivisorsLeft_eq + - dvd_cancel_right_mem_nonZeroDivisors + - dvd_cancel_left_mem_nonZeroDivisors + - path: Mathlib/CategoryTheory/Filtered/Final.lean + theorems: + - CategoryTheory.Functor.initial_of_exists_of_isCofiltered_of_fullyFaithful + - CategoryTheory.Functor.initial_of_exists_of_isCofiltered + - CategoryTheory.Functor.final_iff_isFiltered_structuredArrow + - CategoryTheory.Functor.initial_iff_of_isCofiltered + - CategoryTheory.Functor.final_iff_of_isFiltered + - CategoryTheory.IsFiltered.of_exists_of_isFiltered_of_fullyFaithful + - CategoryTheory.Functor.final_of_exists_of_isFiltered_of_fullyFaithful + - CategoryTheory.Functor.final_of_exists_of_isFiltered + - CategoryTheory.IsCofilteredOrEmpty.of_exists_of_isCofiltered_of_fullyFaithful + - CategoryTheory.isFiltered_structuredArrow_of_isFiltered_of_exists + - CategoryTheory.isCofiltered_costructuredArrow_of_isCofiltered_of_exists + - CategoryTheory.IsFilteredOrEmpty.of_exists_of_isFiltered_of_fullyFaithful + - CategoryTheory.IsCofiltered.of_exists_of_isCofiltered_of_fullyFaithful + - CategoryTheory.Functor.initial_iff_isCofiltered_costructuredArrow + - path: Mathlib/Analysis/Calculus/FDeriv/Measurable.lean + theorems: + - measurableSet_of_differentiableAt_with_param + - RightDerivMeasurableAux.A_mono + - FDerivMeasurableAux.le_of_mem_A + - RightDerivMeasurableAux.B_mem_nhdsWithin_Ioi + - stronglyMeasurable_deriv_with_param + - RightDerivMeasurableAux.mem_A_of_differentiable + - measurable_derivWithin_Ici + - measurable_deriv + - stronglyMeasurable_deriv + - measurable_deriv_with_param + - measurableSet_of_differentiableAt + - FDerivMeasurableAux.isOpen_B + - stronglyMeasurable_derivWithin_Ioi + - FDerivMeasurableAux.mem_A_of_differentiable + - measurableSet_of_differentiableAt_of_isComplete_with_param + - FDerivMeasurableAux.norm_sub_le_of_mem_A + - RightDerivMeasurableAux.differentiable_set_subset_D + - FDerivMeasurableAux.differentiable_set_subset_D + - RightDerivMeasurableAux.le_of_mem_A + - stronglyMeasurable_derivWithin_Ici + - measurable_derivWithin_Ioi + - measurableSet_of_differentiableWithinAt_Ioi + - FDerivMeasurableAux.isOpen_A_with_param + - measurableSet_of_differentiableWithinAt_Ici_of_isComplete + - RightDerivMeasurableAux.D_subset_differentiable_set + - measurableSet_of_differentiableWithinAt_Ici + - FDerivMeasurableAux.isOpen_B_with_param + - measurable_fderiv_with_param + - FDerivMeasurableAux.A_mono + - FDerivMeasurableAux.D_subset_differentiable_set + - measurableSet_of_differentiableAt_of_isComplete + - measurable_fderiv + - RightDerivMeasurableAux.norm_sub_le_of_mem_A + - RightDerivMeasurableAux.A_mem_nhdsWithin_Ioi + - FDerivMeasurableAux.isOpen_A + - path: Mathlib/CategoryTheory/ComposableArrows.lean + theorems: + - "CategoryTheory.ComposableArrows.homMk\u2084_app_four" + - "CategoryTheory.ComposableArrows.mk\u2085_surjective" + - "CategoryTheory.ComposableArrows.ext\u2081" + - "CategoryTheory.ComposableArrows.Mk\u2081.map_id" + - CategoryTheory.ComposableArrows.mkOfObjOfMapSucc_arrow + - "CategoryTheory.ComposableArrows.homMk\u2085_app_five" + - "CategoryTheory.ComposableArrows.mk\u2083_surjective" + - CategoryTheory.ComposableArrows.mkOfObjOfMapSucc_exists + - "CategoryTheory.ComposableArrows.homMk\u2085_app_three" + - "CategoryTheory.ComposableArrows.hom_ext\u2081" + - "CategoryTheory.ComposableArrows.hom_ext\u2080" + - "CategoryTheory.ComposableArrows.homMk\u2083_app_three" + - "CategoryTheory.ComposableArrows.homMk\u2084_app_three" + - "CategoryTheory.ComposableArrows.Mk\u2081.map_comp" + - CategoryTheory.ComposableArrows.Precomp.map_one_succ + - "CategoryTheory.ComposableArrows.mk\u2082_surjective" + - CategoryTheory.ComposableArrows.ext + - "CategoryTheory.ComposableArrows.homMk\u2083_app_two" + - CategoryTheory.ComposableArrows.Precomp.map_id + - CategoryTheory.ComposableArrows.Precomp.map_one_one + - "CategoryTheory.ComposableArrows.mk\u2084_surjective" + - CategoryTheory.ComposableArrows.Precomp.map_zero_one' + - CategoryTheory.ComposableArrows.Precomp.map_zero_succ_succ + - CategoryTheory.ComposableArrows.Precomp.map_zero_one + - CategoryTheory.ComposableArrows.Precomp.map_comp + - CategoryTheory.ComposableArrows.map'_inv_eq_inv_map' + - "CategoryTheory.ComposableArrows.mk\u2081_surjective" + - "CategoryTheory.ComposableArrows.homMk\u2082_app_two" + - "CategoryTheory.ComposableArrows.homMk\u2085_app_two" + - "CategoryTheory.ComposableArrows.homMk\u2084_app_two" + - "CategoryTheory.ComposableArrows.ext\u2080" + - CategoryTheory.ComposableArrows.ext_succ + - CategoryTheory.ComposableArrows.precomp_surjective + - CategoryTheory.ComposableArrows.Precomp.map_zero_zero + - "CategoryTheory.ComposableArrows.homMk\u2085_app_four" + - path: Mathlib/LinearAlgebra/Pi.lean + theorems: + - LinearMap.lsum_single + - LinearEquiv.piRing_symm_apply + - LinearMap.ker_pi + - LinearMap.pi_eq_zero + - Submodule.iSup_map_single + - LinearMap.pi_ext' + - LinearMap.update_apply + - LinearMap.pi_apply_eq_sum_univ + - Submodule.le_comap_single_pi + - Submodule.iInf_comap_proj + - Submodule.biInf_comap_proj + - path: Mathlib/Order/Cover.lean + theorems: + - CovBy.Ioi_eq + - wcovBy_iff_Ioo_eq + - Set.wcovBy_insert + - LT.lt.exists_disjoint_Iio_Ioi + - Prod.fst_eq_or_snd_eq_of_wcovBy + - Prod.mk_covBy_mk_iff_left + - Prod.covBy_iff + - covBy_iff_Ioo_eq + - reflTransGen_wcovBy_eq_reflTransGen_covBy + - CovBy.Ioc_eq + - WCovBy.le_and_le_iff + - Prod.wcovBy_iff + - CovBy.Iio_eq + - WCovBy.Ioc_subset + - wcovBy_eq_reflGen_covBy + - transGen_wcovBy_eq_reflTransGen_covBy + - Set.covBy_iff_exists_sdiff_singleton + - Prod.mk_covBy_mk_iff + - Set.sdiff_singleton_wcovBy + - not_wcovBy_iff + - wcovBy_iff_covBy_or_eq + - WCovBy.Icc_eq + - Prod.mk_wcovBy_mk_iff + - WCovBy.Ico_subset + - Prod.mk_wcovBy_mk_iff_left + - not_covBy_iff + - Set.covBy_iff_exists_insert + - WCovBy.image + - WCovBy.eq_or_eq + - CovBy.Ico_eq + - Prod.mk_covBy_mk_iff_right + - path: Mathlib/ModelTheory/FinitelyGenerated.lean + theorems: + - FirstOrder.Language.Structure.cg_iff + - FirstOrder.Language.Substructure.CG.of_map_embedding + - FirstOrder.Language.Substructure.FG.cg + - FirstOrder.Language.Structure.CG.range + - FirstOrder.Language.Substructure.FG.sup + - FirstOrder.Language.Substructure.fg_def + - FirstOrder.Language.Substructure.fg_bot + - FirstOrder.Language.Substructure.cg_iff_empty_or_exists_nat_generating_family + - FirstOrder.Language.Structure.cg_iff_countable + - FirstOrder.Language.Substructure.cg_iff_structure_cg + - FirstOrder.Language.Structure.FG.map_of_surjective + - FirstOrder.Language.Structure.FG.range + - FirstOrder.Language.Substructure.cg_iff_countable + - FirstOrder.Language.Substructure.CG.sup + - FirstOrder.Language.Structure.fg_iff + - FirstOrder.Language.Substructure.fg_closure + - FirstOrder.Language.Substructure.fg_iff_exists_fin_generating_family + - FirstOrder.Language.Structure.CG.map_of_surjective + - FirstOrder.Language.Substructure.fg_iff_structure_fg + - FirstOrder.Language.Substructure.FG.map + - FirstOrder.Language.Substructure.FG.of_map_embedding + - FirstOrder.Language.Substructure.CG.map + - path: Mathlib/NumberTheory/LegendreSymbol/Basic.lean + theorems: + - legendreSym.sq_one' + - ZMod.mod_four_ne_three_of_sq_eq_neg_sq' + - legendreSym.at_one + - legendreSym.eq_one_iff' + - ZMod.euler_criterion + - legendreSym.eq_one_of_sq_sub_mul_sq_eq_zero' + - ZMod.exists_sq_eq_neg_one_iff + - legendreSym.at_zero + - legendreSym.mul + - legendreSym.eq_zero_mod_of_eq_neg_one + - legendreSym.mod + - ZMod.euler_criterion_units + - legendreSym.at_neg_one + - legendreSym.eq_neg_one_iff' + - legendreSym.eq_pow + - ZMod.pow_div_two_eq_neg_one_or_one + - legendreSym.eq_one_of_sq_sub_mul_sq_eq_zero + - legendreSym.prime_dvd_of_eq_neg_one + - path: Mathlib/RingTheory/IntegrallyClosed.lean + theorems: + - isIntegrallyClosed_iff_isIntegralClosure + - IsIntegralClosure.of_isIntegrallyClosed + - IsIntegrallyClosed.integralClosure_eq_bot_iff + - isIntegrallyClosed_iff + - path: Mathlib/Analysis/Convex/Extrema.lean + theorems: + - IsMinOn.of_isLocalMinOn_of_convexOn_Icc + - IsMinOn.of_isLocalMinOn_of_convexOn + - path: Mathlib/Data/String/Basic.lean + theorems: + - List.asString_inj + - String.toList_nonempty + - List.asString_eq + - String.ltb_cons_addChar + - String.lt_iff_toList_lt + - path: Mathlib/Analysis/Complex/Arg.lean + theorems: + - Complex.sameRay_iff + - Complex.sameRay_iff_arg_div_eq_zero + - path: Mathlib/Analysis/PSeries.lean + theorems: + - NNReal.summable_one_div_rpow + - Finset.le_sum_condensed + - NNReal.summable_rpow + - Real.tendsto_sum_range_one_div_nat_succ_atTop + - Real.summable_one_div_int_add_rpow + - NNReal.summable_rpow_inv + - Real.summable_nat_pow_inv + - ENNReal.le_tsum_condensed + - Real.summable_one_div_nat_rpow + - Real.not_summable_nat_cast_inv + - Finset.sum_condensed_le' + - Real.summable_nat_rpow + - sum_Ioc_inv_sq_le_sub + - sum_Ioo_inv_sq_le + - Real.summable_one_div_nat_add_rpow + - Real.summable_one_div_int_pow + - Finset.sum_condensed_le + - NNReal.summable_condensed_iff + - Real.summable_abs_int_rpow + - summable_condensed_iff_of_nonneg + - Real.not_summable_one_div_nat_cast + - Real.summable_nat_rpow_inv + - ENNReal.tsum_condensed_le + - path: Mathlib/Order/Heyting/Hom.lean + theorems: + - map_bihimp + - BiheytingHom.cancel_left + - map_compl + - map_hnot + - map_symmDiff + - CoheytingHom.cancel_left + - HeytingHom.cancel_left + - BoundedLatticeHomClass.toBiheytingHomClass + - path: Mathlib/LinearAlgebra/Matrix/Gershgorin.lean + theorems: + - det_ne_zero_of_sum_col_lt_diag + - eigenvalue_mem_ball + - det_ne_zero_of_sum_row_lt_diag + - path: Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean + theorems: + - groupCohomology.resolution.d_of + - groupCohomology.resolution.diagonalSucc_hom_single + - Rep.diagonalHomEquiv_symm_partialProd_succ + - "groupCohomology.resolution.quasiIso_forget\u2082_\u03B5ToSingle\u2080" + - "groupCohomology.resolution.d_comp_\u03B5" + - groupCohomology.resolution.actionDiagonalSucc_inv_apply + - Rep.diagonalHomEquiv_apply + - Rep.diagonalHomEquiv_symm_apply + - groupCohomology.resolution.actionDiagonalSucc_hom_apply + - "groupCohomology.resolution.\u03B5ToSingle\u2080_comp_eq" + - groupCohomology.resolution.diagonalSucc_inv_single_left + - groupCohomology.resolution.diagonalSucc_inv_single_right + - groupCohomology.resolution.d_eq + - "groupCohomology.resolution.forget\u2082ToModuleCatHomotopyEquiv_f_0_eq" + - groupCohomology.resolution.diagonalSucc_inv_single_single + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean + theorems: + - CategoryTheory.Limits.PreservesPushout.inl_iso_inv + - CategoryTheory.Limits.PreservesPushout.inr_iso_hom + - CategoryTheory.Limits.PreservesPullback.iso_hom_fst + - CategoryTheory.Limits.PreservesPushout.inr_iso_inv + - CategoryTheory.Limits.PreservesPullback.iso_hom_snd + - CategoryTheory.Limits.PreservesPushout.inl_iso_hom + - CategoryTheory.Limits.PreservesPullback.iso_inv_snd + - CategoryTheory.Limits.PreservesPullback.iso_inv_fst + - path: Mathlib/NumberTheory/Padics/PadicIntegers.lean + theorems: + - PadicInt.valuation_p + - PadicInt.p_nonnunit + - PadicInt.norm_le_pow_iff_mem_span_pow + - PadicInt.valuation_p_pow_mul + - PadicInt.norm_le_pow_iff_le_valuation + - PadicInt.norm_eq_of_norm_add_lt_left + - PadicInt.norm_lt_one_iff_dvd + - PadicInt.norm_lt_one_mul + - PadicInt.valuation_nonneg + - PadicInt.unitCoeff_spec + - PadicInt.mem_nonunits + - PadicInt.norm_eq_pow_val + - PadicInt.mul_inv + - PadicInt.inv_mul + - PadicInt.pow_p_dvd_int_iff + - PadicInt.norm_eq_of_norm_add_lt_right + - PadicInt.norm_lt_pow_iff_norm_le_pow_sub_one + - PadicInt.exists_pow_neg_lt_rat + - PadicInt.norm_mul + - PadicInt.norm_int_cast_eq_padic_norm + - PadicInt.padic_norm_e_of_padicInt + - PadicInt.norm_units + - PadicInt.coe_eq_zero + - PadicInt.isUnit_iff + - PadicInt.mem_span_pow_iff_le_valuation + - PadicInt.norm_int_lt_one_iff_dvd + - PadicInt.norm_pow + - PadicInt.maximalIdeal_eq_span_p + - PadicInt.norm_le_pow_iff_norm_lt_pow_add_one + - PadicInt.exists_pow_neg_lt + - PadicInt.norm_int_le_pow_iff_dvd + - PadicInt.prime_p + - PadicInt.coe_int_eq + - path: Mathlib/RingTheory/Ideal/Cotangent.lean + theorems: + - Ideal.cotangentIdeal_square + - Ideal.mem_toCotangent_ker + - LocalRing.CotangentSpace.span_image_eq_top_iff + - Ideal.toCotangent_eq + - LocalRing.subsingleton_cotangentSpace_iff + - Ideal.cotangent_subsingleton_iff + - Ideal.map_toCotangent_ker + - AlgHom.ker_kerSquareLift + - LocalRing.CotangentSpace.map_eq_top_iff + - LocalRing.finrank_cotangentSpace_eq_zero_iff + - Ideal.cotangentEquivIdeal_symm_apply + - LocalRing.finrank_cotangentSpace_le_one_iff + - path: Mathlib/LinearAlgebra/Dimension/Localization.lean + theorems: + - IsLocalizedModule.rank_eq + - IsLocalizedModule.lift_rank_eq + - rank_quotient_add_rank_of_isDomain + - path: Mathlib/Data/List/Duplicate.lean + theorems: + - List.duplicate_iff_sublist + - List.Duplicate.mem_cons_self + - List.duplicate_iff_two_le_count + - List.Duplicate.mem + - List.duplicate_cons_iff_of_ne + - List.exists_duplicate_iff_not_nodup + - List.nodup_iff_forall_not_duplicate + - List.Duplicate.of_duplicate_cons + - List.duplicate_cons_iff + - List.Duplicate.ne_singleton + - List.Duplicate.mono_sublist + - path: Mathlib/Algebra/Module/Zlattice.lean + theorems: + - Zspan.fundamentalDomain_isBounded + - Zspan.coe_floor_self + - Zspan.span_top + - Zlattice.module_free + - Zspan.fract_zspan_add + - Zspan.fract_add_zspan + - Zspan.repr_fract_apply + - Zspan.ceil_eq_self_of_mem + - Zspan.measure_fundamentalDomain + - Zspan.repr_ceil_apply + - Zspan.fundamentalDomain_measurableSet + - Zspan.floor_eq_self_of_mem + - Zspan.discreteTopology_pi_basisFun + - Zspan.volume_fundamentalDomain + - Zspan.quotientEquiv.symm_apply + - Zspan.isAddFundamentalDomain + - Zspan.exist_unique_vadd_mem_fundamentalDomain + - Zspan.coe_fract_self + - Zspan.norm_fract_le + - Zspan.measure_fundamentalDomain_ne_zero + - Basis.ofZlatticeBasis_apply + - Basis.ofZlatticeBasis_repr_apply + - Zspan.vadd_mem_fundamentalDomain + - Zspan.fract_eq_self + - Zspan.fundamentalDomain_pi_basisFun + - Zspan.fundamentalDomain_reindex + - Zlattice.rank + - Zspan.map_fundamentalDomain + - Zspan.fract_eq_fract + - Zspan.fract_fract + - Basis.ofZlatticeBasis_span + - path: Mathlib/CategoryTheory/Sums/Basic.lean + theorems: + - CategoryTheory.hom_inl_inr_false + - CategoryTheory.hom_inr_inl_false + - path: Mathlib/RingTheory/Ideal/AssociatedPrime.lean + theorems: + - IsAssociatedPrime.annihilator_le + - associatedPrimes.eq_singleton_of_isPrimary + - IsAssociatedPrime.eq_radical + - IsAssociatedPrime.map_of_injective + - not_isAssociatedPrime_of_subsingleton + - associatedPrimes.eq_empty_of_subsingleton + - associatedPrimes.nonempty + - exists_le_isAssociatedPrime_of_isNoetherianRing + - path: Mathlib/Topology/Category/CompHaus/Limits.lean + theorems: + - CompHaus.pullback_snd_eq + - "CompHaus.finiteCoproduct.\u03B9_injective" + - "CompHaus.Sigma.\u03B9_comp_toFiniteCoproduct" + - CompHaus.pullback.hom_ext + - "CompHaus.finiteCoproduct.\u03B9_desc_apply" + - CompHaus.pullback.condition + - CompHaus.pullback_fst_eq + - CompHaus.finiteCoproduct.hom_ext + - path: Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean + theorems: + - GromovHausdorff.one_le_maxVar + - GromovHausdorff.hausdorffDist_optimal_le_HD + - GromovHausdorff.HD_lipschitz_aux1 + - GromovHausdorff.candidates_nonneg + - GromovHausdorff.optimalGHDist_mem_candidatesB + - GromovHausdorff.isCompact_candidatesB + - GromovHausdorff.candidates_dist_bound + - GromovHausdorff.HD_lipschitz_aux2 + - GromovHausdorff.HD_bound_aux1 + - GromovHausdorff.HD_candidatesBDist_le + - GromovHausdorff.candidates_lipschitz_aux + - GromovHausdorff.maxVar_bound + - GromovHausdorff.candidates_lipschitz + - GromovHausdorff.HD_bound_aux2 + - GromovHausdorff.closed_candidatesB + - GromovHausdorff.dist_mem_candidates + - path: Mathlib/MeasureTheory/Measure/OpenPos.lean + theorems: + - IsClosed.measure_eq_univ_iff_eq + - MeasureTheory.Measure.eqOn_open_of_ae_eq + - IsMeagre.of_isSigmaCompact_null + - IsClosed.measure_eq_one_iff_eq_univ + - Metric.measure_closedBall_pos_iff + - Continuous.isOpenPosMeasure_map + - IsOpen.measure_zero_iff_eq_empty + - IsOpen.ae_eq_empty_iff_eq + - IsOpen.measure_eq_zero_iff + - IsClosed.ae_eq_univ_iff_eq + - path: Mathlib/Topology/Algebra/InfiniteSum/Ring.lean + theorems: + - HasSum.div_const + - summable_div_const_iff + - hasSum_mul_left_iff + - summable_mul_left_iff + - tsum_mul_tsum_eq_tsum_sum_antidiagonal + - tsum_mul_left + - hasSum_mul_right_iff + - summable_sum_mul_antidiagonal_of_summable_mul + - tsum_mul_tsum_eq_tsum_sum_range + - summable_sum_mul_range_of_summable_mul + - tsum_mul_right + - HasSum.mul_left + - HasSum.mul_right + - tsum_div_const + - hasSum_div_const_iff + - path: Mathlib/Algebra/Quandle.lean + theorems: + - Quandle.fix_inv + - Quandle.dihedralAct.inv + - Rack.toEnvelGroup.mapAux.well_def + - UnitalShelf.act_idem + - Rack.self_invAct_invAct_eq + - UnitalShelf.act_self_act_eq + - Rack.self_invAct_act_eq + - UnitalShelf.assoc + - UnitalShelf.act_act_self_eq + - Rack.left_cancel_inv + - Rack.assoc_iff_id + - Rack.self_act_invAct_eq + - Rack.self_invAct_eq_iff_eq + - Rack.self_act_eq_iff_eq + - Rack.self_act_act_eq + - Rack.involutory_invAct_eq_act + - Quandle.conj_swap + - Rack.ad_conj + - Rack.left_cancel + - path: Mathlib/ModelTheory/Algebra/Ring/Basic.lean + theorems: + - FirstOrder.Ring.realize_mul + - FirstOrder.Ring.realize_zero + - FirstOrder.Ring.realize_add + - FirstOrder.Ring.realize_neg + - FirstOrder.Ring.card_ring + - path: Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean + theorems: + - ContMDiff.continuous_tangentMap + - ContMDiff.contMDiff_tangentMap + - TangentBundle.tangentMap_tangentBundle_pure + - ContMDiffOn.contMDiffOn_tangentMapWithin_aux + - ContMDiffOn.contMDiffOn_tangentMapWithin + - ContMDiffAt.mfderiv + - ContMDiffOn.continuousOn_tangentMapWithin_aux + - path: Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean + theorems: + - jacobiTheta_T_sq_smul + - "jacobiTheta_eq_jacobiTheta\u2082" + - norm_jacobiTheta_sub_one_le + - norm_exp_mul_sq_le + - exists_summable_bound_exp_mul_sq + - jacobiTheta_S_smul + - differentiableAt_jacobiTheta + - jacobiTheta_eq_tsum_nat + - hasSum_nat_jacobiTheta + - isBigO_at_im_infty_jacobiTheta_sub_one + - jacobiTheta_two_add + - path: Mathlib/Algebra/Category/Ring/Colimits.lean + theorems: + - CommRingCat.Colimits.quot_neg + - CommRingCat.Colimits.quot_add + - CommRingCat.Colimits.cocone_naturality_components + - CommRingCat.Colimits.quot_mul + - CommRingCat.Colimits.cocone_naturality + - path: Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean + theorems: + - AlgebraicTopology.DoldKan.cs_down_0_not_rel_left + - "AlgebraicTopology.DoldKan.map_h\u03C3'" + - "AlgebraicTopology.DoldKan.h\u03C3'_naturality" + - "AlgebraicTopology.DoldKan.h\u03C3'_eq_zero" + - "AlgebraicTopology.DoldKan.H\u03C3_eq_zero" + - "AlgebraicTopology.DoldKan.map_H\u03C3" + - "AlgebraicTopology.DoldKan.h\u03C3'_eq'" + - path: Mathlib/RingTheory/Adjoin/PowerBasis.lean + theorems: + - PowerBasis.repr_mul_isIntegral + - PowerBasis.toMatrix_isIntegral + - PowerBasis.repr_pow_isIntegral + - PowerBasis.repr_gen_pow_isIntegral + - path: Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean + theorems: + - SimpleGraph.card_ConnectedComponent_eq_rank_ker_lapMatrix + - SimpleGraph.mem_ker_toLin'_lapMatrix_of_connectedComponent + - SimpleGraph.linearIndependent_lapMatrix_ker_basis_aux + - SimpleGraph.lapMatrix_toLin'_apply_eq_zero_iff_forall_adj + - SimpleGraph.degree_eq_sum_if_adj + - SimpleGraph.dotProduct_mulVec_degMatrix + - "SimpleGraph.lapMatrix_toLinearMap\u2082'" + - SimpleGraph.posSemidef_lapMatrix + - SimpleGraph.top_le_span_range_lapMatrix_ker_basis_aux + - SimpleGraph.lapMatrix_mulVec_apply + - SimpleGraph.degMatrix_mulVec_apply + - SimpleGraph.lapMatrix_mulVec_const_eq_zero + - "SimpleGraph.lapMatrix_toLinearMap\u2082'_apply'_eq_zero_iff_forall_adj" + - "SimpleGraph.lapMatrix_toLinearMap\u2082'_apply'_eq_zero_iff_forall_reachable" + - path: Mathlib/CategoryTheory/Abelian/Opposite.lean + theorems: + - "CategoryTheory.kernel.\u03B9_op" + - "CategoryTheory.image_\u03B9_op_comp_imageUnopOp_hom" + - CategoryTheory.factorThruImage_comp_imageUnopOp_inv + - "CategoryTheory.kernel.\u03B9_unop" + - CategoryTheory.imageUnopOp_inv_comp_op_factorThruImage + - "CategoryTheory.imageUnopOp_hom_comp_image_\u03B9" + - "CategoryTheory.cokernel.\u03C0_op" + - "CategoryTheory.cokernel.\u03C0_unop" + - path: Mathlib/CategoryTheory/Sites/Sheafification.lean + theorems: + - CategoryTheory.isIso_toSheafify + - CategoryTheory.sheafifyMap_comp + - CategoryTheory.isoSheafify_inv + - CategoryTheory.sheafifyMap_sheafifyLift + - CategoryTheory.toSheafify_sheafifyLift + - CategoryTheory.sheafifyMap_id + - CategoryTheory.sheafifyLift_unique + - CategoryTheory.sheafificationAdjunction_counit_app_val + - CategoryTheory.sheafify_hom_ext + - path: Mathlib/Data/List/Dedup.lean + theorems: + - List.tail_dedup + - List.sum_map_count_dedup_filter_eq_countP + - List.mem_dedup + - List.count_dedup + - List.dedup_append + - List.replicate_dedup + - List.sum_map_count_dedup_eq_length + - List.dedup_eq_cons + - List.dedup_cons_of_mem' + - List.dedup_cons_of_not_mem' + - List.dedup_eq_nil + - List.headI_dedup + - path: Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean + theorems: + - CategoryTheory.regularCoverage.EqualizerCondition.isSheafFor + - CategoryTheory.regularCoverage.equalizerCondition_of_regular + - CategoryTheory.regularCoverage.isSheaf_yoneda_obj + - CategoryTheory.regularCoverage.EqualizerCondition.isSheaf_iff + - CategoryTheory.regularCoverage.isSheafFor_regular_of_projective + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean + theorems: + - EuclideanGeometry.dist_eq_abs_sub_dist_of_angle_eq_zero + - "Sbtw.angle\u2081\u2082\u2083_eq_pi" + - EuclideanGeometry.angle_eq_angle_of_angle_eq_pi + - EuclideanGeometry.angle_sub_const + - EuclideanGeometry.angle_eq_angle_of_angle_eq_pi_of_angle_eq_pi + - EuclideanGeometry.dist_eq_abs_sub_dist_iff_angle_eq_zero + - EuclideanGeometry.angle_self_left + - EuclideanGeometry.angle_midpoint_eq_pi + - EuclideanGeometry.collinear_iff_eq_or_eq_or_sin_eq_zero + - EuclideanGeometry.angle_eq_zero_iff_ne_and_wbtw + - EuclideanGeometry.collinear_of_sin_eq_zero + - EuclideanGeometry.continuousAt_angle + - EuclideanGeometry.angle_const_sub + - EuclideanGeometry.angle_eq_zero_iff_eq_and_ne_or_sbtw + - EuclideanGeometry.left_dist_ne_zero_of_angle_eq_pi + - EuclideanGeometry.angle_right_midpoint_eq_pi_div_two_of_dist_eq + - EuclideanGeometry.dist_eq_add_dist_of_angle_eq_pi + - EuclideanGeometry.dist_eq_add_dist_iff_angle_eq_pi + - "Wbtw.angle\u2083\u2081\u2082_eq_zero_of_ne" + - EuclideanGeometry.angle_eq_zero_of_angle_eq_pi_right + - "Sbtw.angle\u2083\u2082\u2081_eq_pi" + - EuclideanGeometry.collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi + - "Wbtw.angle\u2082\u2081\u2083_eq_zero_of_ne" + - EuclideanGeometry.angle_add_angle_eq_pi_of_angle_eq_pi + - EuclideanGeometry.angle_self_right + - EuclideanGeometry.angle_eq_pi_iff_sbtw + - AffineIsometry.angle_map + - EuclideanGeometry.angle_left_midpoint_eq_pi_div_two_of_dist_eq + - EuclideanGeometry.angle_neg + - path: Mathlib/Topology/Category/TopCat/Opens.lean + theorems: + - TopologicalSpace.Opens.inclusion_top_functor + - TopologicalSpace.Opens.openEmbedding_obj_top + - TopologicalSpace.Opens.mapIso_hom_app + - TopologicalSpace.Opens.map_id_eq + - TopologicalSpace.Opens.map_iSup + - TopologicalSpace.Opens.adjunction_counit_map_functor + - TopologicalSpace.Opens.op_map_id_obj + - TopologicalSpace.Opens.inclusion_map_eq_top + - TopologicalSpace.Opens.set_range_forget_map_inclusion + - TopologicalSpace.Opens.functor_map_eq_inf + - TopologicalSpace.Opens.adjunction_counit_app_self + - TopologicalSpace.Opens.functor_obj_map_obj + - TopologicalSpace.Opens.mapIso_inv_app + - path: Mathlib/GroupTheory/Perm/DomMulAct.lean + theorems: + - DomMulAct.mem_stabilizer_iff + - DomMulAct.stabilizerEquiv_invFun_eq + - DomMulAct.stabilizer_card + - path: Mathlib/Order/CompleteBooleanAlgebra.lean + theorems: + - biSup_inf_biSup + - "iSup\u2082_inf_eq" + - "inf_iSup\u2082_eq" + - compl_sSup + - sSup_inf_sSup + - iSup_inf_of_monotone + - iSup_inf_eq + - iSup_disjoint_iff + - sSup_inf_eq + - "iSup\u2082_disjoint_iff" + - inf_iSup_eq + - compl_iSup + - iInf_iSup_eq + - iSup_iInf_eq + - sSup_disjoint_iff + - iSup_inf_iSup + - compl_sInf + - disjoint_sSup_iff + - disjoint_iSup_iff + - path: Mathlib/GroupTheory/Submonoid/Inverses.lean + theorems: + - Submonoid.leftInv_leftInv_eq + - Submonoid.fromLeftInv_mul + - Submonoid.mul_leftInvEquiv_symm + - Submonoid.fromLeftInv_eq_iff + - Submonoid.leftInvEquiv_mul + - Submonoid.leftInvEquiv_symm_eq_inv + - Submonoid.mul_leftInvEquiv + - Submonoid.leftInv_leftInv_le + - Submonoid.fromLeftInv_eq_inv + - Submonoid.leftInvEquiv_symm_mul + - path: Mathlib/Topology/MetricSpace/Basic.lean + theorems: + - eq_of_nndist_eq_zero + - Metric.secondCountable_of_countable_discretization + - MetricSpace.ext + - dist_le_zero + - Metric.closedEmbedding_of_pairwise_le_dist + - Metric.uniformEmbedding_bot_of_pairwise_le_dist + - Metric.isClosed_of_pairwise_le_dist + - nndist_eq_zero + - MetricSpace.replaceTopology_eq + - Metric.subsingleton_closedBall + - Metric.uniformEmbedding_iff' + - zero_eq_dist + - zero_eq_nndist + - dist_ne_zero + - path: Mathlib/Analysis/NormedSpace/Real.lean + theorems: + - frontier_closedBall' + - closure_ball + - frontier_ball + - interior_closedBall' + - interior_sphere' + - frontier_sphere + - dist_smul_add_one_sub_smul_le + - interior_closedBall + - inv_norm_smul_mem_closed_unit_ball + - interior_sphere + - frontier_sphere' + - exists_norm_eq + - norm_smul_of_nonneg + - frontier_closedBall + - path: Mathlib/Data/W/Constructions.lean + theorems: + - WType.leftInverse_list + - WType.rightInverse_nat + - WType.rightInverse_list + - WType.leftInverse_nat + - path: Mathlib/Order/Monotone/Basic.lean + theorems: + - Monotone.ne_of_lt_of_lt_int + - Int.exists_strictMono + - StrictAnti.prod_map + - Nat.exists_strictMono' + - strictAntiOn_dual_iff + - Antitone.ne_of_lt_of_lt_int + - not_monotone_not_antitone_iff_exists_le_le + - StrictMonoOn.lt_iff_lt + - strictAnti_dual_iff + - Monotone.ne_of_lt_of_lt_nat + - StrictMono.ite' + - Antitone.ne_of_lt_of_lt_nat + - Int.rel_of_forall_rel_succ_of_lt + - not_monotone_not_antitone_iff_exists_lt_lt + - monotoneOn_dual_iff + - antitone_dual_iff + - Nat.rel_of_forall_rel_succ_of_le_of_lt + - StrictMono.prod_map + - strictMono_dual_iff + - strictMonoOn_dual_iff + - injective_of_lt_imp_ne + - antitoneOn_dual_iff + - StrictMonoOn.eq_iff_eq + - path: Mathlib/Analysis/Complex/Circle.lean + theorems: + - expMapCircle_add + - norm_circle_smul + - expMapCircle_zero + - coe_inv_circle_eq_conj + - normSq_eq_of_mem_circle + - mem_circle_iff_normSq + - path: Mathlib/Logic/Denumerable.lean + theorems: + - Denumerable.encode_ofNat + - Nat.Subtype.succ_le_of_lt + - Denumerable.prod_nat_ofNat + - Denumerable.ofEquiv_ofNat + - Nat.Subtype.right_inverse_aux + - Nat.Subtype.coe_comp_ofNat_range + - Denumerable.prod_ofNat_val + - Nat.Subtype.toFunAux_eq + - Nat.Subtype.exists_succ + - Denumerable.sigma_ofNat_val + - path: Mathlib/Data/MvPolynomial/Degrees.lean + theorems: + - MvPolynomial.totalDegree_add_eq_left_of_totalDegree_lt + - MvPolynomial.degrees_add_of_disjoint + - MvPolynomial.totalDegree_le_degrees_card + - MvPolynomial.degreeOf_rename_of_injective + - MvPolynomial.degreeOf_le_iff + - MvPolynomial.degreeOf_mul_le + - MvPolynomial.totalDegree_rename_le + - MvPolynomial.degreeOf_mul_X_ne + - MvPolynomial.degrees_pow + - MvPolynomial.degrees_sum + - MvPolynomial.totalDegree_pow + - MvPolynomial.degrees_rename_of_injective + - MvPolynomial.degreeOf_X + - MvPolynomial.degrees_map + - MvPolynomial.degreeOf_eq_sup + - MvPolynomial.totalDegree_C + - MvPolynomial.degrees_mul + - MvPolynomial.degrees_zero + - MvPolynomial.totalDegree_list_prod + - MvPolynomial.totalDegree_finset_prod + - MvPolynomial.degreeOf_def + - MvPolynomial.totalDegree_multiset_prod + - MvPolynomial.degrees_map_of_injective + - MvPolynomial.totalDegree_finset_sum + - MvPolynomial.totalDegree_X + - MvPolynomial.coeff_eq_zero_of_totalDegree_lt + - MvPolynomial.totalDegree_mul + - MvPolynomial.totalDegree_add_eq_right_of_totalDegree_lt + - MvPolynomial.degreeOf_add_le + - MvPolynomial.monomial_le_degreeOf + - MvPolynomial.totalDegree_eq + - MvPolynomial.le_degrees_add + - MvPolynomial.degrees_monomial + - MvPolynomial.degreeOf_mul_X_eq + - MvPolynomial.degreeOf_zero + - MvPolynomial.mem_degrees + - MvPolynomial.totalDegree_monomial + - MvPolynomial.degrees_add + - MvPolynomial.degrees_def + - MvPolynomial.degrees_prod + - MvPolynomial.totalDegree_X_pow + - MvPolynomial.degreeOf_C_mul_le + - MvPolynomial.degreeOf_C + - MvPolynomial.degreeOf_mul_C_le + - MvPolynomial.exists_degree_lt + - path: Mathlib/RingTheory/Polynomial/Hermite/Basic.lean + theorems: + - Polynomial.hermite_eq_iterate + - Polynomial.degree_hermite + - Polynomial.hermite_one + - Polynomial.coeff_hermite_explicit + - Polynomial.coeff_hermite_self + - Polynomial.coeff_hermite_succ_zero + - Polynomial.coeff_hermite_of_lt + - Polynomial.coeff_hermite + - Polynomial.coeff_hermite_of_even_add + - Polynomial.leadingCoeff_hermite + - Polynomial.hermite_succ + - Polynomial.coeff_hermite_succ_succ + - Polynomial.coeff_hermite_of_odd_add + - path: Mathlib/RingTheory/Localization/Away/Basic.lean + theorems: + - selfZPow_sub_cast_nat + - exists_reduced_fraction' + - selfZPow_of_nonpos + - IsLocalization.Away.mul_invSelf + - selfZPow_neg_mul + - selfZPow_mul_neg + - selfZPow_add + - IsLocalization.away_of_isUnit_of_bijective + - selfZPow_zero + - selfZPow_neg_coe_nat + - path: Mathlib/Algebra/Group/Hom/Defs.lean + theorems: + - map_comp_pow + - ne_one_of_map + - OneHom.comp_one + - map_comp_one + - map_comp_div + - MonoidHom.cancel_left + - MulHom.cancel_left + - map_zpow' + - map_comp_inv + - MonoidHom.map_mul_inv + - map_comp_zpow + - map_comp_mul_inv + - Subsingleton.of_oneHomClass + - map_mul_eq_one + - map_div' + - map_pow + - MonoidHom.comp_one + - map_comp_mul + - map_mul_inv + - map_comp_zpow' + - OneHom.cancel_left + - path: Mathlib/RingTheory/DedekindDomain/Factorization.lean + theorems: + - Ideal.finprod_heightOneSpectrum_factorization + - Ideal.finite_mulSupport_inv + - Ideal.finprod_heightOneSpectrum_factorization_coe + - Associates.finprod_ne_zero + - Ideal.finite_mulSupport_coe + - Ideal.finprod_count + - Associates.finite_factors + - Ideal.finite_mulSupport + - Ideal.finprod_not_dvd + - Ideal.finite_factors + - path: Mathlib/Analysis/Calculus/LineDeriv/Basic.lean + theorems: + - lineDifferentiableWithinAt_smul_iff + - hasLineDerivWithinAt_univ + - HasLineDerivAt.lineDeriv + - HasLineDerivAt.le_of_lip' + - HasLineDerivAt.le_of_lipschitzOn + - hasLineDerivWithinAt_smul_iff + - LineDifferentiableAt.congr_of_eventuallyEq + - lineDifferentiableWithinAt_univ + - HasLineDerivWithinAt.congr_of_eventuallyEq + - lineDerivWithin_congr_set + - norm_lineDeriv_le_of_lip' + - hasLineDerivWithinAt_congr_set + - HasFDerivAt.hasLineDerivAt + - hasLineDerivAt_smul_iff + - Filter.EventuallyEq.lineDeriv_eq + - lineDeriv_smul + - HasLineDerivWithinAt.mono_of_mem + - HasLineDerivWithinAt.smul + - lineDeriv_neg + - Filter.EventuallyEq.lineDerivWithin_eq + - HasLineDerivAt.congr_of_eventuallyEq + - lineDifferentiableAt_smul_iff + - hasLineDerivAt_iff_isLittleO_nhds_zero + - lineDerivWithin_congr + - norm_lineDeriv_le_of_lipschitzOn + - HasLineDerivAt.smul + - lineDerivWithin_of_mem_nhds + - hasLineDerivWithinAt_zero + - Filter.EventuallyEq.hasLineDerivAt_iff + - HasLineDerivWithinAt.hasLineDerivAt + - lineDerivWithin_univ + - Filter.EventuallyEq.hasLineDerivWithinAt_iff + - hasLineDerivAt_zero + - path: Mathlib/Data/PFunctor/Multivariate/W.lean + theorems: + - MvPFunctor.comp_wPathCasesOn + - MvPFunctor.wDest'_wMk + - MvPFunctor.w_map_wMk + - MvPFunctor.wDest'_wMk' + - MvPFunctor.w_ind + - MvPFunctor.map_objAppend1 + - MvPFunctor.wRec_eq + - MvPFunctor.wPathCasesOn_eta + - path: Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean + theorems: + - ModuleCat.ihom_ev_app + - path: Mathlib/Data/Nat/Factorization/PrimePow.lean + theorems: + - Nat.Coprime.isPrimePow_dvd_mul + - exists_ord_compl_eq_one_iff_isPrimePow + - isPrimePow_pow_iff + - isPrimePow_iff_factorization_eq_single + - IsPrimePow.exists_ord_compl_eq_one + - Nat.mul_divisors_filter_prime_pow + - isPrimePow_iff_card_primeFactors_eq_one + - IsPrimePow.minFac_pow_factorization_eq + - isPrimePow_iff_unique_prime_dvd + - isPrimePow_of_minFac_pow_factorization_eq + - path: Mathlib/RingTheory/MvPolynomial/Symmetric.lean + theorems: + - MvPolynomial.psum_zero + - MvPolynomial.esymm_eq_sum_monomial + - MvPolynomial.support_esymm + - MvPolynomial.psum_one + - MvPolynomial.IsSymmetric.map + - MvPolynomial.esymm_isSymmetric + - Finset.esymm_map_val + - MvPolynomial.esymm_eq_multiset_esymm + - MvPolynomial.aeval_esymm_eq_multiset_esymm + - MvPolynomial.rename_esymm + - MvPolynomial.map_esymm + - MvPolynomial.support_esymm' + - MvPolynomial.degrees_esymm + - MvPolynomial.support_esymm'' + - MvPolynomial.esymm_zero + - path: Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean + theorems: + - HomotopyCategory.Pretriangulated.shift_distinguished_triangle + - HomotopyCategory.Pretriangulated.isomorphic_distinguished + - "CochainComplex.mappingCone.rotateHomotopyEquiv_comm\u2083" + - HomotopyCategory.Pretriangulated.distinguished_cocone_triangle + - CochainComplex.mappingCone.map_eq_mapOfHomotopy + - "CochainComplex.mappingCone.inr_triangle\u03B4" + - "CochainComplex.mappingCone.rotateHomotopyEquiv_comm\u2082" + - HomotopyCategory.Pretriangulated.contractible_distinguished + - HomotopyCategory.Pretriangulated.rotate_distinguished_triangle' + - HomotopyCategory.Pretriangulated.rotate_distinguished_triangle + - CochainComplex.mappingCone.map_id + - CochainComplex.mappingCone.map_comp + - "CochainComplex.mappingCone.inl_v_triangle_mor\u2083_f" + - "CochainComplex.mappingCone.inr_f_triangle_mor\u2083_f" + - HomotopyCategory.Pretriangulated.complete_distinguished_triangle_morphism + - "CochainComplex.mappingCone.triangleMapOfHomotopy_comm\u2082" + - "CochainComplex.mappingCone.triangleMapOfHomotopy_comm\u2083" + - path: Mathlib/Analysis/Complex/LocallyUniformLimit.lean + theorems: + - Complex.norm_cderiv_lt + - Complex.tendstoUniformlyOn_deriv_of_cthickening_subset + - Complex.cderiv_sub + - TendstoLocallyUniformlyOn.deriv + - Complex.norm_cderiv_le + - TendstoUniformlyOn.cderiv + - Complex.hasSum_deriv_of_summable_norm + - Complex.exists_cthickening_tendstoUniformlyOn + - TendstoLocallyUniformlyOn.differentiableOn + - Complex.differentiableOn_tsum_of_summable_norm + - path: Mathlib/Topology/Algebra/Order/Compact.lean + theorems: + - ContinuousOn.le_sSup_image_Icc + - atBot_le_cocompact + - IsCompact.sSup_lt_iff_of_continuous + - Continuous.exists_forall_le + - CompactIccSpace.mk'' + - Continuous.exists_forall_le_of_hasCompactMulSupport + - IsCompact.exists_isMinOn + - CompactIccSpace.mk' + - IsCompact.continuous_sSup + - IsCompact.exists_forall_le' + - ContinuousOn.sInf_image_Icc_le + - ContinuousOn.image_uIcc + - IsCompact.bddBelow + - Continuous.exists_forall_le' + - ContinuousOn.exists_isMinOn' + - cocompact_le_atBot + - path: Mathlib/RingTheory/PowerSeries/Order.lean + theorems: + - PowerSeries.order_add_of_order_eq.aux + - PowerSeries.nat_le_order + - PowerSeries.le_order + - PowerSeries.coeff_mul_one_sub_of_lt_order + - PowerSeries.order_X_pow + - PowerSeries.order_eq + - PowerSeries.order_monomial_of_ne_zero + - PowerSeries.coeff_mul_of_lt_order + - PowerSeries.order_eq_multiplicity_X + - PowerSeries.order_eq_top + - PowerSeries.order_monomial + - PowerSeries.order_add_of_order_eq + - PowerSeries.order_finite_iff_ne_zero + - PowerSeries.X_pow_order_dvd + - PowerSeries.order_mul_ge + - PowerSeries.order_eq_nat + - PowerSeries.order_X + - PowerSeries.coeff_mul_prod_one_sub_of_lt_order + - PowerSeries.order_mul + - PowerSeries.order_one + - PowerSeries.exists_coeff_ne_zero_iff_ne_zero + - PowerSeries.order_le + - PowerSeries.le_order_add + - PowerSeries.coeff_order + - path: Mathlib/Probability/Martingale/Centering.lean + theorems: + - MeasureTheory.predictablePart_bdd_difference + - MeasureTheory.predictablePart_add_ae_eq + - MeasureTheory.martingale_martingalePart + - MeasureTheory.martingalePart_bdd_difference + - MeasureTheory.martingalePart_add_ae_eq + - MeasureTheory.predictablePart_zero + - MeasureTheory.martingalePart_eq_sum + - MeasureTheory.integrable_martingalePart + - path: Mathlib/CategoryTheory/Abelian/Exact.lean + theorems: + - CategoryTheory.Functor.preservesEpimorphisms_of_map_exact + - CategoryTheory.Abelian.exact_iff_image_eq_kernel + - CategoryTheory.Abelian.mono_cokernel_desc_of_exact + - CategoryTheory.Abelian.exact_iff + - CategoryTheory.Abelian.Exact.unop_iff + - "CategoryTheory.Abelian.exact_iff_exact_image_\u03B9" + - CategoryTheory.Abelian.Exact.op_iff + - CategoryTheory.Abelian.cokernel.desc.inv + - CategoryTheory.Functor.preservesMonomorphisms_of_map_exact + - CategoryTheory.Abelian.kernel.lift.inv + - CategoryTheory.Abelian.exact_of_is_kernel + - CategoryTheory.Abelian.exact_iff' + - CategoryTheory.Abelian.exact_epi_comp_iff + - CategoryTheory.Abelian.Exact.op + - CategoryTheory.Abelian.Exact.unop + - CategoryTheory.Abelian.IsEquivalence.exact_iff + - CategoryTheory.Abelian.exact_of_is_cokernel + - CategoryTheory.Abelian.tfae_mono + - "CategoryTheory.Abelian.exact_iff_exact_coimage_\u03C0" + - CategoryTheory.Functor.map_exact + - CategoryTheory.Abelian.exact_cokernel + - CategoryTheory.Functor.preservesZeroMorphisms_of_map_exact + - CategoryTheory.Abelian.tfae_epi + - CategoryTheory.Abelian.exact_tfae + - path: Mathlib/Combinatorics/Quiver/Path.lean + theorems: + - Quiver.Path.nil_comp + - Quiver.Path.eq_of_length_zero + - Quiver.Path.toList_injective + - Prefunctor.mapPath_comp + - Quiver.Path.nil_ne_cons + - Quiver.Path.toList_comp + - Quiver.Path.comp_assoc + - Quiver.Path.obj_eq_of_cons_eq_cons + - Quiver.Path.comp_inj' + - Quiver.Path.hom_heq_of_cons_eq_cons + - Quiver.Path.heq_of_cons_eq_cons + - Quiver.Path.cons_ne_nil + - Quiver.Path.comp_inj + - path: Mathlib/GroupTheory/GroupAction/ConjAct.lean + theorems: + - ConjAct.mem_orbit_conjAct + - ConjAct.fixedPoints_eq_center + - ConjAct.orbit_eq_carrier_conjClasses + - ConjAct.orbitRel_conjAct + - MulAut.conjNormal_symm_apply + - path: Mathlib/Data/DFinsupp/NeLocus.lean + theorems: + - DFinsupp.mem_neLocus + - DFinsupp.neLocus_eq_empty + - DFinsupp.neLocus_self_sub_left + - DFinsupp.neLocus_neg + - DFinsupp.neLocus_eq_support_sub + - DFinsupp.subset_mapRange_neLocus + - DFinsupp.neLocus_sub_right + - DFinsupp.neLocus_self_add_left + - DFinsupp.neLocus_sub_left + - DFinsupp.neLocus_self_add_right + - DFinsupp.zipWith_neLocus_eq_left + - DFinsupp.neLocus_zero_right + - DFinsupp.neLocus_comm + - DFinsupp.mapRange_neLocus_eq + - DFinsupp.zipWith_neLocus_eq_right + - DFinsupp.neLocus_self_sub_right + - path: Mathlib/RingTheory/DiscreteValuationRing/Basic.lean + theorems: + - DiscreteValuationRing.associated_of_irreducible + - DiscreteValuationRing.aux_pid_of_ufd_of_unique_irreducible + - DiscreteValuationRing.addVal_uniformizer + - DiscreteValuationRing.of_ufd_of_unique_irreducible + - DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid + - DiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization + - Irreducible.addVal_pow + - DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.of_ufd_of_unique_irreducible + - DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.unique_irreducible + - DiscreteValuationRing.ideal_eq_span_pow_irreducible + - DiscreteValuationRing.unit_mul_pow_congr_unit + - DiscreteValuationRing.addVal_le_iff_dvd + - DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal + - DiscreteValuationRing.addVal_def + - DiscreteValuationRing.exists_irreducible + - DiscreteValuationRing.iff_pid_with_one_nonzero_prime + - DiscreteValuationRing.addVal_eq_top_iff + - DiscreteValuationRing.irreducible_iff_uniformizer + - DiscreteValuationRing.unit_mul_pow_congr_pow + - DiscreteValuationRing.associated_pow_irreducible + - DiscreteValuationRing.eq_unit_mul_pow_irreducible + - path: Mathlib/Topology/Order/ExtrClosure.lean + theorems: + - IsLocalMaxOn.closure + - path: Mathlib/Geometry/Euclidean/Triangle.lean + theorems: + - EuclideanGeometry.angle_eq_angle_of_dist_eq + - InnerProductGeometry.norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi + - EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_sub_two_mul_dist_mul_dist_mul_cos_angle + - InnerProductGeometry.cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle + - InnerProductGeometry.sin_angle_add_angle_sub_add_angle_sub_eq_zero + - InnerProductGeometry.norm_sub_sq_eq_norm_sq_add_norm_sq_sub_two_mul_norm_mul_norm_mul_cos_angle + - EuclideanGeometry.dist_eq_of_angle_eq_angle_of_angle_ne_pi + - InnerProductGeometry.cos_angle_add_angle_sub_add_angle_sub_eq_neg_one + - InnerProductGeometry.sin_angle_sub_add_angle_sub_rev_eq_sin_angle + - InnerProductGeometry.angle_add_angle_sub_add_angle_sub_eq_pi + - InnerProductGeometry.angle_sub_eq_angle_sub_rev_of_norm_eq + - EuclideanGeometry.dist_mul_of_eq_angle_of_dist_mul + - EuclideanGeometry.dist_sq_add_dist_sq_eq_two_mul_dist_midpoint_sq_add_half_dist_sq + - EuclideanGeometry.dist_sq_mul_dist_add_dist_sq_mul_dist + - EuclideanGeometry.oangle_add_oangle_add_oangle_eq_pi + - EuclideanGeometry.angle_add_angle_add_angle_eq_pi + - path: Mathlib/MeasureTheory/Integral/Marginal.lean + theorems: + - MeasureTheory.lmarginal_image + - MeasureTheory.lmarginal_singleton + - MeasureTheory.lmarginal_empty + - MeasureTheory.lintegral_eq_of_lmarginal_eq + - MeasureTheory.lmarginal_insert + - MeasureTheory.lmarginal_update_of_mem + - MeasureTheory.lmarginal_univ + - MeasureTheory.lmarginal_le_of_subset + - MeasureTheory.lmarginal_eq_of_subset + - MeasureTheory.lmarginal_insert' + - MeasureTheory.lmarginal_update_of_not_mem + - MeasureTheory.lintegral_le_of_lmarginal_le + - MeasureTheory.lmarginal_erase + - MeasureTheory.lintegral_eq_lmarginal_univ + - MeasureTheory.lmarginal_erase' + - MeasureTheory.lmarginal_union + - Measurable.lmarginal + - MeasureTheory.lmarginal_union' + - MeasureTheory.lmarginal_congr + - path: Mathlib/Data/Polynomial/Expand.lean + theorems: + - Polynomial.expand_aeval + - Polynomial.coeff_expand + - Polynomial.expand_one + - Polynomial.coeff_expand_mul' + - Polynomial.rootMultiplicity_expand_pow + - Polynomial.coeff_expand_mul + - Polynomial.expand_contract' + - Polynomial.leadingCoeff_expand + - Polynomial.expand_injective + - Polynomial.map_contract + - Polynomial.derivative_expand + - Polynomial.coeff_contract + - Polynomial.expand_expand + - Polynomial.expand_zero + - Polynomial.expand_char + - Polynomial.map_expand_pow_char + - Polynomial.rootMultiplicity_expand + - Polynomial.expand_pow + - Polynomial.isLocalRingHom_expand + - Polynomial.expand_monomial + - Polynomial.expand_contract + - Polynomial.monic_expand_iff + - Polynomial.map_expand + - Polynomial.of_irreducible_expand_pow + - Polynomial.expand_eq_sum + - Polynomial.contract_one + - Polynomial.expand_eval + - Polynomial.expand_eq_C + - path: Mathlib/Data/DList/Defs.lean + theorems: + - Std.DList.toList_append + - Std.DList.toList_singleton + - Std.DList.toList_ofList + - Std.DList.toList_push + - Std.DList.ofList_toList + - Std.DList.toList_cons + - Std.DList.toList_empty + - path: Mathlib/RingTheory/ReesAlgebra.lean + theorems: + - reesAlgebra.monomial_mem + - mem_reesAlgebra_iff_support + - reesAlgebra.fg + - monomial_mem_adjoin_monomial + - adjoin_monomial_eq_reesAlgebra + - path: Mathlib/Tactic/CategoryTheory/Reassoc.lean + theorems: + - CategoryTheory.eq_whisker' + - path: Mathlib/Topology/Order/LeftRightLim.lean + theorems: + - leftLim_eq_of_eq_bot + - Monotone.leftLim + - leftLim_eq_of_tendsto + - Monotone.le_leftLim + - Monotone.tendsto_leftLim + - Monotone.continuousAt_iff_leftLim_eq_rightLim + - Monotone.rightLim_le_leftLim + - Monotone.tendsto_leftLim_within + - Monotone.countable_not_continuousWithinAt_Ioi + - Monotone.countable_not_continuousAt + - Monotone.leftLim_le + - Monotone.continuousWithinAt_Iio_iff_leftLim_eq + - path: Mathlib/Analysis/Normed/Field/UnitBall.lean + theorems: + - unitSphereToUnits_injective + - path: Mathlib/Analysis/Calculus/LocalExtr/Basic.lean + theorems: + - IsLocalMaxOn.fderivWithin_eq_zero + - IsLocalMaxOn.hasFDerivWithinAt_nonpos + - IsLocalMin.hasDerivAt_eq_zero + - IsLocalMaxOn.hasFDerivWithinAt_eq_zero + - mem_posTangentConeAt_of_segment_subset' + - IsLocalMinOn.hasFDerivWithinAt_eq_zero + - IsLocalMinOn.hasFDerivWithinAt_nonneg + - posTangentConeAt_mono + - IsLocalMin.hasFDerivAt_eq_zero + - IsLocalMinOn.fderivWithin_eq_zero + - mem_posTangentConeAt_of_segment_subset + - IsLocalMaxOn.fderivWithin_nonpos + - IsLocalMinOn.fderivWithin_nonneg + - path: Mathlib/Algebra/CharP/Two.lean + theorems: + - CharTwo.bit0_eq_zero + - CharTwo.add_mul_self + - neg_one_eq_one_iff + - CharTwo.bit1_apply_eq_one + - CharTwo.list_sum_mul_self + - CharTwo.bit0_apply_eq_zero + - CharTwo.neg_eq + - CharTwo.add_self_eq_zero + - CharTwo.bit1_eq_one + - CharTwo.sub_eq_add + - orderOf_neg_one + - CharTwo.two_eq_zero + - CharTwo.sum_mul_self + - CharTwo.multiset_sum_mul_self + - path: Mathlib/Data/PNat/Xgcd.lean + theorems: + - PNat.XgcdType.step_wf + - PNat.XgcdType.reduce_isReduced + - PNat.XgcdType.step_v + - PNat.XgcdType.start_isSpecial + - PNat.gcdB'_coe + - PNat.XgcdType.flip_isSpecial + - PNat.gcd_eq + - PNat.XgcdType.qp_eq + - PNat.XgcdType.start_v + - PNat.XgcdType.reduce_v + - PNat.XgcdType.v_eq_succ_vp + - PNat.XgcdType.isSpecial_iff + - PNat.XgcdType.finish_isSpecial + - PNat.gcd_props + - PNat.XgcdType.reduce_isSpecial + - PNat.XgcdType.reduce_a + - PNat.gcdA'_coe + - PNat.XgcdType.finish_isReduced + - PNat.XgcdType.flip_v + - PNat.XgcdType.flip_isReduced + - PNat.XgcdType.step_isSpecial + - PNat.XgcdType.reduce_b + - PNat.XgcdType.finish_v + - path: Mathlib/LinearAlgebra/DFinsupp.lean + theorems: + - CompleteLattice.independent_iff_forall_dfinsupp + - DFinsupp.sum_mapRange_index.linearMap + - CompleteLattice.independent_of_dfinsupp_lsum_injective + - DFinsupp.mapRange_smul + - DFinsupp.mapRange.linearEquiv_trans + - CompleteLattice.Independent.linearIndependent + - CompleteLattice.lsum_comp_mapRange_toSpanSingleton + - Submodule.mem_iSup_iff_exists_dfinsupp' + - Submodule.mem_iSup_iff_exists_finsupp + - Submodule.iSup_eq_range_dfinsupp_lsum + - CompleteLattice.independent_of_dfinsupp_sumAddHom_injective + - DFinsupp.mapRange.linearMap_id + - DFinsupp.coprodMap_apply_single + - DFinsupp.mapRange.linearMap_comp + - CompleteLattice.Independent.dfinsupp_sumAddHom_injective + - CompleteLattice.Independent.dfinsupp_lsum_injective + - DFinsupp.lsum_single + - CompleteLattice.independent_of_dfinsupp_sumAddHom_injective' + - path: Mathlib/Analysis/Calculus/Deriv/Slope.lean + theorems: + - range_derivWithin_subset_closure_span_image + - HasDerivWithinAt.liminf_right_slope_norm_le + - HasDerivWithinAt.limsup_slope_norm_le + - range_deriv_subset_closure_span_image + - HasDerivWithinAt.limsup_norm_slope_le + - isSeparable_range_derivWithin + - hasDerivAt_iff_tendsto_slope_zero + - isSeparable_range_deriv + - hasDerivWithinAt_iff_tendsto_slope + - hasDerivAtFilter_iff_tendsto_slope + - hasDerivWithinAt_iff_tendsto_slope' + - path: Mathlib/Data/Bool/Count.lean + theorems: + - List.Chain'.length_sub_one_le_two_mul_count_bool + - List.Chain'.two_mul_count_bool_eq_ite + - List.count_not_add_count + - List.Chain.count_not + - List.count_add_count_not + - List.Chain'.count_not_le_count_add_one + - List.Chain'.two_mul_count_bool_le_length_add_one + - List.Chain'.count_not_eq_count + - List.Chain'.two_mul_count_bool_of_even + - path: Mathlib/Order/Ideal.lean + theorems: + - Order.Ideal.lt_sup_principal_of_not_mem + - IsCoatom.isMaximal + - Order.Ideal.toLowerSet_injective + - Order.sequenceOfCofinals.monotone + - Order.Ideal.mem_sInf + - Order.Ideal.eq_sup_of_le_sup + - Order.Ideal.IsProper.not_mem_of_compl_mem + - Order.Ideal.IsProper.not_mem_or_compl_not_mem + - Order.Ideal.top_of_top_mem + - Order.Ideal.inter_nonempty + - Order.sequenceOfCofinals.encode_mem + - Order.Ideal.isProper_of_not_mem + - path: Mathlib/GroupTheory/SemidirectProduct.lean + theorems: + - SemidirectProduct.rightHom_comp_inr + - SemidirectProduct.range_inl_eq_ker_rightHom + - SemidirectProduct.map_inr + - SemidirectProduct.rightHom_inl + - SemidirectProduct.lift_comp_inl + - SemidirectProduct.lift_inl + - SemidirectProduct.rightHom_inr + - SemidirectProduct.hom_ext + - SemidirectProduct.rightHom_comp_inl + - SemidirectProduct.mk_eq_inl_mul_inr + - SemidirectProduct.map_comp_inr + - SemidirectProduct.inl_left_mul_inr_right + - SemidirectProduct.inl_aut_inv + - SemidirectProduct.lift_comp_inr + - SemidirectProduct.lift_unique + - SemidirectProduct.map_comp_inl + - SemidirectProduct.inl_aut + - SemidirectProduct.map_inl + - path: Mathlib/Data/Nat/GCD/BigOperators.lean + theorems: + - Nat.coprime_multiset_prod_right_iff + - Nat.coprime_fintype_prod_left_iff + - Nat.coprime_prod_right_iff + - Nat.coprime_fintype_prod_right_iff + - Nat.coprime_list_prod_right_iff + - Nat.coprime_prod_left_iff + - Nat.coprime_list_prod_left_iff + - Nat.coprime_multiset_prod_left_iff + - path: Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean + theorems: + - MvQPF.wEquiv_map + - MvQPF.wEquiv.symm + - MvQPF.Fix.ind + - MvQPF.Fix.dest_mk + - MvQPF.wrepr_wMk + - MvQPF.Fix.rec_eq + - MvQPF.Fix.ind_aux + - MvQPF.Fix.mk_dest + - MvQPF.wEquiv.refl + - MvQPF.Fix.ind_rec + - MvQPF.recF_eq_of_wEquiv + - MvQPF.recF_eq + - MvQPF.recF_eq' + - MvQPF.wEquiv.abs' + - MvQPF.Fix.rec_unique + - path: Mathlib/Topology/Sheaves/Stalks.lean + theorems: + - TopCat.Presheaf.stalkPushforward_germ + - TopCat.Presheaf.germ_eq + - TopCat.Presheaf.stalkSpecializes_refl + - TopCat.Presheaf.stalk_hom_ext + - TopCat.Presheaf.germ_ext + - TopCat.Presheaf.stalkSpecializes_stalkFunctor_map + - TopCat.Presheaf.stalkSpecializes_comp + - TopCat.Presheaf.app_surjective_of_stalkFunctor_map_bijective + - TopCat.Presheaf.app_injective_of_stalkFunctor_map_injective + - TopCat.Presheaf.app_surjective_of_injective_of_locally_surjective + - TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso + - TopCat.Presheaf.isIso_of_stalkFunctor_map_iso + - TopCat.Presheaf.stalkPushforward.id + - TopCat.Presheaf.germ_exist + - TopCat.Presheaf.stalkSpecializes_stalkPushforward + - TopCat.Presheaf.germ_res_apply + - TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding + - TopCat.Presheaf.stalkFunctor_map_injective_of_app_injective + - TopCat.Presheaf.stalkPushforward.comp + - TopCat.Presheaf.section_ext + - path: Mathlib/CategoryTheory/Comma/StructuredArrow.lean + theorems: + - CategoryTheory.CostructuredArrow.map_comp + - CategoryTheory.CostructuredArrow.mkPrecomp_comp + - CategoryTheory.StructuredArrow.homMk'_mk_comp + - CategoryTheory.CostructuredArrow.IsUniversal.existsUnique + - CategoryTheory.StructuredArrow.eqToHom_right + - CategoryTheory.CostructuredArrow.homMk'_mk_id + - CategoryTheory.CostructuredArrow.homMk'_comp + - CategoryTheory.CostructuredArrow.map_id + - CategoryTheory.StructuredArrow.IsUniversal.hom_ext + - CategoryTheory.CostructuredArrow.hom_eq_iff + - CategoryTheory.StructuredArrow.w + - CategoryTheory.StructuredArrow.homMk'_comp + - CategoryTheory.StructuredArrow.map_id + - CategoryTheory.CostructuredArrow.IsUniversal.hom_ext + - CategoryTheory.CostructuredArrow.eqToHom_left + - CategoryTheory.CostructuredArrow.w + - CategoryTheory.StructuredArrow.map_comp + - CategoryTheory.StructuredArrow.mkPostcomp_id + - CategoryTheory.StructuredArrow.IsUniversal.existsUnique + - CategoryTheory.StructuredArrow.homMk'_id + - CategoryTheory.CostructuredArrow.mkPrecomp_id + - CategoryTheory.CostructuredArrow.homMk'_id + - CategoryTheory.StructuredArrow.hom_eq_iff + - CategoryTheory.StructuredArrow.mkPostcomp_comp + - CategoryTheory.StructuredArrow.homMk'_mk_id + - CategoryTheory.CostructuredArrow.homMk'_mk_comp + - path: Mathlib/Geometry/Euclidean/Angle/Sphere.lean + theorems: + - EuclideanGeometry.Sphere.oangle_center_eq_two_zsmul_oangle + - EuclideanGeometry.Sphere.two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi + - EuclideanGeometry.Sphere.dist_div_cos_oangle_center_eq_two_mul_radius + - EuclideanGeometry.Sphere.oangle_eq_pi_sub_two_zsmul_oangle_center_right + - Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq + - EuclideanGeometry.Sphere.tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center + - EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear + - EuclideanGeometry.Sphere.dist_div_sin_oangle_eq_two_mul_radius + - Orientation.oangle_eq_two_zsmul_oangle_sub_of_norm_eq + - EuclideanGeometry.Cospherical.two_zsmul_oangle_eq + - Affine.Triangle.mem_circumsphere_of_two_zsmul_oangle_eq + - EuclideanGeometry.Sphere.dist_div_sin_oangle_div_two_eq_radius + - EuclideanGeometry.Sphere.dist_div_cos_oangle_center_div_two_eq_radius + - EuclideanGeometry.concyclic_or_collinear_of_two_zsmul_oangle_eq + - EuclideanGeometry.Sphere.oangle_eq_pi_sub_two_zsmul_oangle_center_left + - EuclideanGeometry.cospherical_or_collinear_of_two_zsmul_oangle_eq + - EuclideanGeometry.Sphere.two_zsmul_oangle_eq + - EuclideanGeometry.Sphere.inv_tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center + - path: Mathlib/Tactic/Lift.lean + theorems: + - Subtype.exists_pi_extension + - path: Mathlib/Topology/Algebra/Order/Field.lean + theorems: + - TopologicalRing.of_norm + - Filter.Tendsto.mul_atBot + - Filter.Tendsto.neg_mul_atTop + - Filter.Tendsto.atTop_mul + - "inv_atTop\u2080" + - inv_nhdsWithin_Ioi_zero + - tendsto_const_mul_pow_nhds_iff + - Filter.Tendsto.neg_mul_atBot + - Filter.Tendsto.mul_atTop + - Filter.Tendsto.atBot_mul_neg + - tendsto_pow_neg_atTop + - eventually_nhdsWithin_pos_mul_left + - tendsto_const_mul_zpow_atTop_nhds_iff + - tendsto_const_mul_pow_nhds_iff' + - nhdsWithin_pos_comap_mul_left + - tendsto_zpow_atTop_zero + - Filter.Tendsto.atTop_mul_neg + - Filter.Tendsto.atBot_mul + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean + theorems: + - CategoryTheory.Limits.PreservesCokernel.iso_inv + - CategoryTheory.Limits.CokernelCofork.map_condition + - CategoryTheory.Limits.KernelFork.map_condition + - CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map + - CategoryTheory.Limits.PreservesKernel.iso_hom + - CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv + - path: Mathlib/ModelTheory/Substructures.lean + theorems: + - FirstOrder.Language.Hom.range_eq_map + - FirstOrder.Language.Substructure.reduct_withConstants + - FirstOrder.Language.Substructure.lift_card_closure_le_card_term + - FirstOrder.Language.Substructure.closure_induction' + - FirstOrder.Language.Substructure.mem_closure_iff_exists_term + - FirstOrder.Language.Term.realize_mem + - FirstOrder.Language.Hom.range_eq_top + - FirstOrder.Language.Substructure.mem_iInf + - FirstOrder.Language.Substructure.realize_boundedFormula_top + - FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize + - FirstOrder.Language.Substructure.coe_iInf + - FirstOrder.Language.Substructure.realize_formula_top + - FirstOrder.Language.Substructure.subset_closure_withConstants + - FirstOrder.Language.Equiv.toHom_range + - Set.Countable.substructure_closure + - FirstOrder.Language.Hom.range_le_iff_comap + - FirstOrder.Language.Substructure.lift_card_closure_le + - FirstOrder.Language.Substructure.dense_induction + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean + theorems: + - CliffordAlgebra.foldr'Aux_foldr'Aux + - "CliffordAlgebra.foldl_\u03B9" + - CliffordAlgebra.foldl_one + - "CliffordAlgebra.foldr'_\u03B9_mul" + - "CliffordAlgebra.foldr_prod_map_\u03B9" + - CliffordAlgebra.foldl_mul + - CliffordAlgebra.foldl_algebraMap + - CliffordAlgebra.right_induction + - CliffordAlgebra.left_induction + - path: Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean + theorems: + - DomMulAct.smul_Lp_add + - DomMulAct.smul_Lp_neg + - DomMulAct.smul_Lp_sub + - DomMulAct.dist_smul_Lp + - DomMulAct.edist_smul_Lp + - path: Mathlib/CategoryTheory/EssentiallySmall.lean + theorems: + - CategoryTheory.essentiallySmall_of_small_of_locallySmall + - CategoryTheory.essentiallySmall_congr + - CategoryTheory.essentiallySmall_iff + - CategoryTheory.essentiallySmall_iff_of_thin + - path: Mathlib/Data/Num/Bitwise.lean + theorems: + - SNum.bit_zero + - SNum.bit_one + - path: Mathlib/Topology/Algebra/UniformRing.lean + theorems: + - UniformSpace.ring_sep_quot + - UniformSpace.Completion.continuous_mul + - UniformSpace.inseparableSetoid_ring + - UniformSpace.Completion.map_smul_eq_mul_coe + - path: Mathlib/LinearAlgebra/Isomorphisms.lean + theorems: + - LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff + - Submodule.quotientQuotientEquivQuotientAux_mk_mk + - LinearMap.quotientInfEquivSupQuotient_injective + - LinearMap.comap_leq_ker_subToSupQuotient + - LinearMap.quotientInfEquivSupQuotient_surjective + - Submodule.card_quotient_mul_card_quotient + - LinearMap.quotientInfEquivSupQuotient_symm_apply_left + - path: Mathlib/Algebra/Group/UniqueProds.lean + theorems: + - UniqueProds.toTwoUniqueProds_of_group + - uniqueMul_of_twoUniqueMul + - UniqueMul.mulHom_map_iff + - UniqueMul.exists_iff_exists_existsUnique + - UniqueMul.of_image_filter + - UniqueMul.iff_existsUnique + - TwoUniqueProds.of_mulOpposite + - UniqueMul.of_subsingleton + - UniqueMul.iff_card_le_one + - UniqueMul.mulHom_preimage + - UniqueMul.set_subsingleton + - UniqueMul.mulHom_image_iff + - UniqueMul.of_mulHom_image + - UniqueMul.of_mulOpposite + - TwoUniqueProds.of_mulHom + - UniqueProds.toIsLeftCancelMul + - UniqueMul.of_card_le_one + - UniqueMul.mt + - UniqueProds.of_mulHom + - UniqueProds.of_same + - UniqueMul.to_mulOpposite + - path: Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean + theorems: + - spectrum.map_pow_of_pos + - spectrum.map_pow_of_nonempty + - spectrum.map_polynomial_aeval_of_nonempty + - spectrum.nonempty_of_isAlgClosed_of_finiteDimensional + - spectrum.exists_mem_of_not_isUnit_aeval_prod + - spectrum.map_polynomial_aeval_of_degree_pos + - spectrum.pow_image_subset + - spectrum.subset_polynomial_aeval + - path: Mathlib/Data/Nat/Cast/SetInterval.lean + theorems: + - Nat.image_cast_int_Ioi + - Nat.image_cast_int_Iio + - Nat.image_cast_int_Ici + - Nat.image_cast_int_Iic + - Nat.image_cast_int_Ico + - Nat.image_cast_int_Ioc + - Nat.image_cast_int_Icc + - Nat.image_cast_int_Ioo + - path: Mathlib/Data/Finset/PiAntidiagonal.lean + theorems: + - Finset.piAntidiagonal_insert + - Finset.mem_piAntidiagonal_insert + - Finset.mapRange_piAntidiagonal_subset + - "Finset.mem_finAntidiagonal\u2080" + - Finset.mapRange_piAntidiagonal_eq + - Finset.piAntidiagonal_zero + - Finset.piAntidiagonal_empty_of_ne_zero + - Finset.piAntidiagonal_empty + - Finset.piAntidiagonal_empty_of_zero + - Finset.mem_piAntidiagonal' + - "Finset.mem_finAntidiagonal\u2080'" + - path: Mathlib/Order/Compare.lean + theorems: + - cmpLE_eq_cmp + - Ordering.compares_iff_of_compares_impl + - cmpLE_swap + - cmp_swap + - ofDual_compares_ofDual + - eq_iff_eq_of_cmp_eq_cmp + - Ordering.orElse_eq_lt + - cmp_eq_cmp_symm + - Ordering.compares_swap + - Ordering.swap_eq_iff_eq_swap + - toDual_compares_toDual + - Ordering.Compares.eq_lt + - Ordering.Compares.eq_eq + - Ordering.swap_orElse + - Ordering.Compares.ne_lt + - le_iff_le_of_cmp_eq_cmp + - lt_iff_lt_of_cmp_eq_cmp + - cmp_self_eq_eq + - cmp_compares + - path: Mathlib/Algebra/CharZero/Defs.lean + theorems: + - Nat.cast_eq_zero + - Nat.cast_eq_one + - Nat.cast_add_one_ne_zero + - charZero_of_inj_zero + - path: Mathlib/Algebra/Group/Semiconj/Defs.lean + theorems: + - SemiconjBy.mul_left + - SemiconjBy.conj_iff + - SemiconjBy.pow_right + - semiconjBy_iff_eq + - SemiconjBy.conj_mk + - SemiconjBy.mul_right + - SemiconjBy.one_right + - path: Mathlib/Data/Finset/NatAntidiagonal.lean + theorems: + - Finset.Nat.antidiagonal_eq_image' + - Finset.Nat.antidiagonal_filter_snd_le_of_le + - Finset.Nat.antidiagonal_filter_le_fst_of_le + - Finset.Nat.antidiagonal_succ' + - Finset.Nat.antidiagonal_eq_image + - Finset.Nat.antidiagonal_succ_succ' + - Finset.Nat.antidiagonal_filter_le_snd_of_le + - Finset.Nat.antidiagonal_eq_map' + - Finset.Nat.antidiagonal_succ + - Finset.Nat.card_antidiagonal + - Finset.Nat.antidiagonal_filter_fst_le_of_le + - path: Mathlib/Order/Partition/Equipartition.lean + theorems: + - Finpartition.isEquipartition_iff_card_parts_eq_average + - Finpartition.IsEquipartition.card_part_le_average_add_one + - Finpartition.IsEquipartition.average_le_card_part + - Finpartition.indiscrete_isEquipartition + - Finpartition.bot_isEquipartition + - path: Mathlib/Logic/Equiv/Defs.lean + theorems: + - Equiv.exists_unique_congr + - Equiv.arrowCongr_comp + - Equiv.conj_comp + - Equiv.exists_congr_left + - Equiv.equivCongr_trans + - Equiv.Perm.coe_subsingleton + - Equiv.permCongr_trans + - Equiv.refl_trans + - Equiv.self_trans_symm + - Equiv.equivCongr_symm + - Equiv.cast_eq_iff_heq + - Equiv.symm_trans_self + - Equiv.symm_symm + - Equiv.symm_apply_eq + - Equiv.apply_eq_iff_eq_symm_apply + - Equiv.forall_congr + - Equiv.permCongr_refl + - Equiv.forall_congr_left' + - Equiv.equivCongr_refl + - Equiv.exists_unique_congr_left' + - Equiv.trans_refl + - Equiv.cast_trans + - path: Mathlib/Topology/Sheaves/LocalPredicate.lean + theorems: + - TopCat.PrelocalPredicate.sheafifyOf + - TopCat.stalkToFiber_germ + - TopCat.stalkToFiber_surjective + - TopCat.subpresheafToTypes.isSheaf + - TopCat.stalkToFiber_injective + - path: Mathlib/Algebra/DirectSum/Decomposition.lean + theorems: + - DirectSum.decompose_of_mem_same + - DirectSum.degree_eq_of_mem_mem + - DirectSum.sum_support_decompose + - DirectSum.decompose_of_mem_ne + - DirectSum.Decomposition.inductionOn + - DirectSum.decompose_lhom_ext + - DirectSum.decompose_coe + - path: Mathlib/RingTheory/Polynomial/IntegralNormalization.lean + theorems: + - Polynomial.integralNormalization_coeff + - "Polynomial.integralNormalization_eval\u2082_eq_zero" + - Polynomial.integralNormalization_zero + - Polynomial.integralNormalization_coeff_ne_degree + - Polynomial.integralNormalization_coeff_degree + - Polynomial.support_integralNormalization + - path: Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean + theorems: + - uniformEquicontinuous_birkhoffAverage + - dist_birkhoffAverage_birkhoffAverage_le + - tendsto_birkhoffAverage_apply_sub_birkhoffAverage + - dist_birkhoffAverage_apply_birkhoffAverage + - dist_birkhoffAverage_birkhoffAverage + - dist_birkhoffSum_apply_birkhoffSum + - path: Mathlib/ModelTheory/Definability.lean + theorems: + - Set.definable_iff_exists_formula_sum + - Set.definable_univ + - Set.Definable.mono + - Set.definable_empty + - Set.definable_finset_biInter + - Set.Definable.map_expansion + - Set.Definable.image_comp_embedding + - Set.Definable.image_comp_sum_inl_fin + - Set.definable_iff_finitely_definable + - Set.definable_finset_sup + - Set.definable_finset_inf + - Set.Definable.image_comp_equiv + - Set.Definable.compl + - Set.Definable.inter + - Set.definable_finset_biUnion + - Set.Definable.preimage_comp + - Set.Definable.union + - Set.Definable.image_comp + - Set.empty_definable_iff + - path: Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean + theorems: + - integral_comp_neg_Iic + - Real.integrable_of_summable_norm_Icc + - integral_comp_neg_Ioi + - volume_regionBetween_eq_integral' + - integral_comp_abs + - path: Mathlib/CategoryTheory/Comma/Over.lean + theorems: + - CategoryTheory.Under.w + - CategoryTheory.Over.OverMorphism.ext + - CategoryTheory.Under.under_left + - CategoryTheory.Over.w + - CategoryTheory.Under.UnderMorphism.ext + - CategoryTheory.Over.over_right + - path: Mathlib/Data/Matrix/Kronecker.lean + theorems: + - Matrix.det_kroneckerMapBilinear + - Matrix.kroneckerMap_one_one + - Matrix.kroneckerMap_diagonal_left + - Matrix.kroneckerMap_diagonal_diagonal + - Matrix.kroneckerMap_reindex + - Matrix.inv_kronecker + - Matrix.kroneckerMapBilinear_mul_mul + - Matrix.trace_kroneckerMapBilinear + - Matrix.kroneckerMap_diagonal_right + - Matrix.det_kronecker + - Matrix.det_kroneckerTMul + - path: Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean + theorems: + - CategoryTheory.Groupoid.Free.lift_spec + - CategoryTheory.Groupoid.Free.congr_reverse + - CategoryTheory.Groupoid.Free.freeGroupoidFunctor_id + - CategoryTheory.Groupoid.Free.congr_reverse_comp + - CategoryTheory.Groupoid.Free.freeGroupoidFunctor_comp + - CategoryTheory.Groupoid.Free.lift_unique + - CategoryTheory.Groupoid.Free.congr_comp_reverse + - path: Mathlib/Data/Countable/Defs.lean + theorems: + - not_uncountable_iff + - uncountable_iff_forall_not_surjective + - path: Mathlib/Algebra/Module/Submodule/Pointwise.lean + theorems: + - Submodule.mem_set_smul + - Submodule.set_smul_eq_iSup + - Submodule.sup_set_smul + - Submodule.closure_neg + - Submodule.set_smul_le_iff + - Submodule.set_smul_bot + - Submodule.smul_le_self_of_tower + - Submodule.singleton_set_smul + - Submodule.set_smul_eq_map + - Submodule.coe_span_smul + - Submodule.mem_set_smul_of_mem_mem + - Submodule.empty_set_smul + - Submodule.mem_singleton_set_smul + - path: Mathlib/Order/Filter/Extr.lean + theorems: + - IsMaxOn.sub + - IsMinOn.sub + - IsExtrFilter.congr + - IsMinFilter.sub + - Filter.EventuallyLE.isMaxFilter + - IsMaxFilter.sub + - path: Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean + theorems: + - "ExteriorAlgebra.liftAlternating_apply_\u03B9Multi" + - ExteriorAlgebra.liftAlternating_one + - "ExteriorAlgebra.liftAlternating_\u03B9Multi" + - ExteriorAlgebra.liftAlternating_algebraMap + - ExteriorAlgebra.liftAlternating_comp + - "ExteriorAlgebra.liftAlternating_\u03B9" + - "ExteriorAlgebra.liftAlternating_\u03B9_mul" + - path: Mathlib/Data/Set/Intervals/OrderEmbedding.lean + theorems: + - OrderEmbedding.preimage_Ico + - OrderEmbedding.preimage_Ioo + - OrderEmbedding.preimage_Icc + - OrderEmbedding.preimage_Ioc + - OrderEmbedding.preimage_uIoc + - OrderEmbedding.preimage_uIcc + - path: Mathlib/Tactic/NormNum/Eq.lean + theorems: + - Mathlib.Meta.NormNum.isRat_eq_false + - Mathlib.Meta.NormNum.Rat.invOf_denom_swap + - Mathlib.Meta.NormNum.isNat_eq_false + - Mathlib.Meta.NormNum.isInt_eq_false + - path: Mathlib/Analysis/Fourier/FourierTransform.lean + theorems: + - Real.fourierIntegralInv_eq + - Real.fourierIntegralInv_eq_fourierIntegral_neg + - Real.fourierIntegralInv_eq' + - VectorFourier.fourierIntegral_comp_add_right + - VectorFourier.norm_fourierIntegral_le_integral_norm + - VectorFourier.integral_bilin_fourierIntegral_eq_flip + - Real.fourierIntegralInv_comp_linearIsometry + - VectorFourier.fourier_integral_convergent_iff + - Real.vector_fourierIntegral_eq_integral_exp_smul + - VectorFourier.fourierIntegral_continuous + - Real.fourierIntegral_eq' + - Real.fourierIntegral_comp_linearIsometry + - VectorFourier.fourierIntegral_smul_const + - VectorFourier.fourierIntegral_add + - Real.fourierIntegral_real_eq_integral_exp_smul + - path: Mathlib/Topology/List.lean + theorems: + - Vector.continuous_insertNth' + - nhds_cons + - List.tendsto_insertNth' + - Vector.tendsto_insertNth + - List.continuousAt_length + - nhds_list + - List.tendsto_nhds + - nhds_nil + - List.tendsto_cons_iff + - List.tendsto_cons + - List.tendsto_removeNth + - List.continuous_insertNth + - Vector.tendsto_cons + - List.tendsto_prod + - Vector.continuousAt_removeNth + - path: Mathlib/NumberTheory/VonMangoldt.lean + theorems: + - ArithmeticFunction.moebius_mul_log_eq_vonMangoldt + - ArithmeticFunction.log_mul_moebius_eq_vonMangoldt + - ArithmeticFunction.vonMangoldt_apply_pow + - ArithmeticFunction.sum_moebius_mul_log_eq + - ArithmeticFunction.vonMangoldt_nonneg + - ArithmeticFunction.vonMangoldt_le_log + - ArithmeticFunction.vonMangoldt_apply_one + - ArithmeticFunction.zeta_mul_vonMangoldt + - ArithmeticFunction.vonMangoldt_mul_zeta + - ArithmeticFunction.vonMangoldt_sum + - ArithmeticFunction.vonMangoldt_apply_prime + - path: Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean + theorems: + - tfae_of_isNoetherianRing_of_localRing_of_isDomain + - DiscreteValuationRing.TFAE + - exists_maximalIdeal_pow_eq_of_principal + - LocalRing.finrank_CotangentSpace_eq_one_iff + - maximalIdeal_isPrincipal_of_isDedekindDomain + - path: Mathlib/GroupTheory/SchurZassenhaus.lean + theorems: + - Subgroup.eq_one_of_smul_eq_one + - Subgroup.SchurZassenhausInduction.step0 + - Subgroup.SchurZassenhausInduction.step3 + - Subgroup.smul_diff_smul' + - Subgroup.exists_right_complement'_of_coprime + - Subgroup.smul_diff' + - Subgroup.SchurZassenhausInduction.step2 + - Subgroup.SchurZassenhausInduction.step1 + - Subgroup.SchurZassenhausInduction.step7 + - Subgroup.exists_smul_eq + - Subgroup.SchurZassenhausInduction.step6 + - Subgroup.exists_right_complement'_of_coprime_aux' + - path: Mathlib/RingTheory/Finiteness.lean + theorems: + - Submodule.fg_bot + - Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul + - Submodule.fg_biSup + - Submodule.fg_of_fg_map_of_fg_inf_ker + - Submodule.fg_induction + - Module.Finite.of_surjective + - Submodule.FG.sup + - Submodule.fg_iff_compact + - Submodule.fg_of_fg_map_injective + - RingHom.Finite.comp + - Module.Finite.of_restrictScalars_finite + - Submodule.FG.prod + - Submodule.exists_fg_le_eq_rTensor_inclusion + - Ideal.exists_radical_pow_le_of_fg + - Module.Finite.trans + - Submodule.fg_ker_comp + - Submodule.fg_iff_add_subgroup_fg + - Submodule.fg_top + - Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul + - Submodule.fg_def + - Module.Finite.of_equiv_equiv + - Submodule.FG.pow + - RingHom.Finite.of_comp_finite + - Submodule.fg_span + - Submodule.fg_pi + - Submodule.fg_restrictScalars + - Ideal.FG.map + - Submodule.FG.stabilizes_of_iSup_eq + - Subalgebra.fg_bot_toSubmodule + - Submodule.FG.map + - Module.Finite.exists_fin' + - Submodule.fg_iff_addSubmonoid_fg + - Submodule.fg_iSup + - Submodule.fg_iff_exists_fin_generating_family + - Ideal.fg_ker_comp + - path: Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean + theorems: + - GeneralizedContinuedFraction.numerators_recurrence + - GeneralizedContinuedFraction.continuantsAux_recurrence + - GeneralizedContinuedFraction.continuants_recurrenceAux + - GeneralizedContinuedFraction.denominators_recurrence + - GeneralizedContinuedFraction.continuants_recurrence + - path: Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean + theorems: + - tangentMap_chart_symm + - mdifferentiableAt_atlas + - mdifferentiableAt_atlas_symm + - PartialHomeomorph.MDifferentiable.trans + - mfderiv_chartAt_eq_tangentCoordChange + - tangentMap_chart + - path: Mathlib/Tactic/Sat/FromLRAT.lean + theorems: + - Sat.Fmla.reify_or + - Sat.Valuation.mk_implies + - path: Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean + theorems: + - MeasureTheory.snorm_trim_ae + - MeasureTheory.limsup_trim + - MeasureTheory.snorm_trim + - MeasureTheory.essSup_trim + - MeasureTheory.snorm'_trim + - path: Mathlib/Algebra/ContinuedFractions/Translations.lean + theorems: + - GeneralizedContinuedFraction.terminatedAt_iff_part_denom_none + - GeneralizedContinuedFraction.exists_s_a_of_part_num + - GeneralizedContinuedFraction.terminatedAt_iff_part_num_none + - GeneralizedContinuedFraction.second_continuant_aux_eq + - GeneralizedContinuedFraction.part_denom_eq_s_b + - GeneralizedContinuedFraction.part_num_none_iff_s_none + - GeneralizedContinuedFraction.first_continuant_eq + - GeneralizedContinuedFraction.convergents'Aux_succ_none + - GeneralizedContinuedFraction.zeroth_convergent_eq_h + - GeneralizedContinuedFraction.terminatedAt_iff_s_terminatedAt + - GeneralizedContinuedFraction.convergents'Aux_succ_some + - GeneralizedContinuedFraction.exists_conts_a_of_num + - GeneralizedContinuedFraction.zeroth_convergent'_eq_h + - GeneralizedContinuedFraction.part_denom_none_iff_s_none + - GeneralizedContinuedFraction.part_num_eq_s_a + - GeneralizedContinuedFraction.exists_s_b_of_part_denom + - GeneralizedContinuedFraction.terminatedAt_iff_s_none + - GeneralizedContinuedFraction.exists_conts_b_of_denom + - GeneralizedContinuedFraction.first_numerator_eq + - path: Mathlib/MeasureTheory/Constructions/HaarToSphere.lean + theorems: + - MeasureTheory.Measure.toSphere_apply_univ' + - MeasureTheory.integral_fun_norm_addHaar + - MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd + - MeasureTheory.Measure.toSphere_apply' + - MeasureTheory.Measure.toSphere_apply_univ + - MeasureTheory.Measure.volumeIoiPow_apply_Iio + - MeasureTheory.Measure.toSphere_apply_aux + - path: Mathlib/LinearAlgebra/Alternating/DomCoprod.lean + theorems: + - AlternatingMap.domCoprod.summand_eq_zero_of_smul_invariant + - MultilinearMap.domCoprod_alternization_coe + - MultilinearMap.domCoprod_alternization_eq + - AlternatingMap.domCoprod.summand_add_swap_smul_eq_zero + - MultilinearMap.domCoprod_alternization + - path: Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean + theorems: + - ContinuousMap.Homotopy.eq_path_of_eq_image + - ContinuousMap.Homotopy.heq_path_of_eq_image + - ContinuousMap.Homotopy.eq_diag_path + - ContinuousMap.Homotopy.start_path + - ContinuousMap.Homotopy.end_path + - ContinuousMap.Homotopy.apply_one_path + - ContinuousMap.Homotopy.evalAt_eq + - path: Mathlib/CategoryTheory/CommSq.lean + theorems: + - CategoryTheory.CommSq.HasLift.iff_op + - CategoryTheory.CommSq.HasLift.iff + - CategoryTheory.Functor.map_commSq + - CategoryTheory.CommSq.unop + - CategoryTheory.CommSq.HasLift.iff_unop + - CategoryTheory.CommSq.op + - CategoryTheory.CommSq.vert_inv + - path: Mathlib/Algebra/MonoidAlgebra/Division.lean + theorems: + - AddMonoidAlgebra.modOf_apply_of_not_exists_add + - AddMonoidAlgebra.mul_of'_divOf + - AddMonoidAlgebra.of'_mul_modOf + - AddMonoidAlgebra.mul_of'_modOf + - AddMonoidAlgebra.divOf_add_modOf + - AddMonoidAlgebra.divOf_zero + - AddMonoidAlgebra.of'_dvd_iff_modOf_eq_zero + - AddMonoidAlgebra.of'_mul_divOf + - AddMonoidAlgebra.of'_modOf + - AddMonoidAlgebra.of'_divOf + - AddMonoidAlgebra.divOf_add + - AddMonoidAlgebra.modOf_add_divOf + - AddMonoidAlgebra.modOf_apply_of_exists_add + - path: .lake/packages/std/Std/Data/List/Pairwise.lean + theorems: + - List.pairwise_bind + - List.pwFilter_eq_self + - List.pairwise_middle + - List.Pairwise.and_mem + - List.pairwise_filter + - List.pairwise_pair + - List.pairwise_append_comm + - List.pairwise_replicate + - List.pairwise_iff_forall_sublist + - List.forall_mem_pwFilter + - List.pairwise_of_reflexive_on_dupl_of_forall_ne + - List.pairwise_of_forall + - List.pairwise_singleton + - List.pairwise_filterMap + - List.map_get_sublist + - List.pairwise_join + - List.Pairwise.imp_mem + - List.Pairwise.forall_of_forall_of_flip + - List.Pairwise.and + - List.sublist_eq_map_get + - List.pwFilter_map + - List.pairwise_iff_get + - path: Mathlib/LinearAlgebra/TensorProduct/Basic.lean + theorems: + - TensorProduct.exists_eq_tmul_of_forall + - LinearMap.map_comp_rTensor + - "TensorProduct.map\u2082_mk_top_top_eq_top" + - TensorProduct.ite_tmul + - TensorProduct.ext' + - TensorProduct.ext_fourfold' + - LinearMap.rTensor_comp + - TensorProduct.lift.unique + - TensorProduct.ext_threefold + - TensorProduct.zero_smul + - LinearMap.lTensor_comp_map + - LinearMap.lTensor_comp_rTensor + - TensorProduct.tmul_sum + - LinearMap.rTensor_neg + - LinearMap.rTensor_comp_apply + - TensorProduct.map_smul_left + - LinearMap.rTensor_sub + - LinearMap.rTensor_pow + - TensorProduct.map_smul_right + - TensorProduct.add_left_neg + - LinearMap.lTensor_pow + - TensorProduct.sum_tmul + - "TensorProduct.lift_compr\u2082" + - TensorProduct.map_pow + - TensorProduct.ext_fourfold + - TensorProduct.one_smul + - TensorProduct.map_range_eq_span_tmul + - TensorProduct.add_smul + - LinearMap.rTensor_id_apply + - LinearMap.lTensor_comp_apply + - LinearMap.rTensor_comp_lTensor + - "TensorProduct.lift_mk_compr\u2082" + - LinearMap.lTensor_neg + - LinearMap.lTensor_id_apply + - LinearMap.map_comp_lTensor + - TensorProduct.uncurry_apply + - TensorProduct.span_tmul_eq_top + - LinearMap.lTensor_sub + - TensorProduct.liftAux.smul + - TensorProduct.map_add_right + - LinearMap.lTensor_comp + - LinearMap.rTensor_comp_map + - TensorProduct.smul_tmul_smul + - TensorProduct.induction_on + - TensorProduct.tensorTensorTensorComm_symm + - TensorProduct.tmul_ite + - TensorProduct.map_id + - TensorProduct.map_add_left + - TensorProduct.ext + - path: Mathlib/Algebra/Quaternion.lean + theorems: + - QuaternionAlgebra.star_eq_self + - Quaternion.normSq_eq_zero + - Cardinal.mk_univ_quaternionAlgebra_of_infinite + - QuaternionAlgebra.coe_add + - QuaternionAlgebra.self_add_star + - Quaternion.mul_imJ + - QuaternionAlgebra.algebraMap_injective + - Cardinal.mk_quaternionAlgebra_of_infinite + - Quaternion.normSq_neg + - Quaternion.star_mul_self + - Quaternion.normSq_nat_cast + - Quaternion.mul_re + - Quaternion.sq_eq_normSq + - Quaternion.normSq_coe + - QuaternionAlgebra.coe_mul + - Cardinal.mk_univ_quaternionAlgebra + - Quaternion.normSq_smul + - Quaternion.normSq_rat_cast + - QuaternionAlgebra.star_mul_eq_coe + - QuaternionAlgebra.coe_neg + - Quaternion.sq_eq_neg_normSq + - QuaternionAlgebra.star_add_self + - QuaternionAlgebra.rank_eq_four + - Quaternion.normSq_def' + - QuaternionAlgebra.smul_coe + - QuaternionAlgebra.star_coe + - QuaternionAlgebra.finrank_eq_four + - QuaternionAlgebra.eq_re_of_eq_coe + - QuaternionAlgebra.mul_coe_eq_smul + - Quaternion.im_sq + - QuaternionAlgebra.self_add_star' + - Quaternion.normSq_nonneg + - QuaternionAlgebra.star_add_self' + - Quaternion.normSq_int_cast + - Quaternion.normSq_star + - Cardinal.mk_quaternionAlgebra + - Quaternion.mul_imI + - QuaternionAlgebra.star_eq_neg + - Quaternion.mul_imK + - Quaternion.normSq_add + - Cardinal.pow_four + - Quaternion.coe_normSq_add + - QuaternionAlgebra.mul_star_eq_coe + - path: Mathlib/Algebra/GroupWithZero/Semiconj.lean + theorems: + - SemiconjBy.zero_right + - "SemiconjBy.inv_right\u2080" + - "SemiconjBy.inv_symm_left_iff\u2080" + - SemiconjBy.zero_left + - SemiconjBy.div_right + - path: Mathlib/Order/SemiconjSup.lean + theorems: + - IsOrderRightAdjoint.comp_orderIso + - Function.Semiconj.symm_adjoint + - Function.semiconj_of_isLUB + - IsOrderRightAdjoint.orderIso_comp + - path: Mathlib/Data/PEquiv.lean + theorems: + - PEquiv.injective_of_forall_ne_isSome + - PEquiv.isSome_symm_get + - PEquiv.self_trans_symm + - PEquiv.trans_eq_none + - PEquiv.mem_ofSet_iff + - PEquiv.inj + - PEquiv.mem_single_iff + - PEquiv.trans_single_of_eq_none + - PEquiv.trans_symm_eq_iff_forall_isSome + - PEquiv.mem_ofSet_self_iff + - PEquiv.trans_bot + - PEquiv.trans_refl + - PEquiv.refl_trans + - PEquiv.single_trans_of_mem + - PEquiv.symm_trans_self + - PEquiv.ofSet_eq_refl + - PEquiv.bot_trans + - PEquiv.symm_symm + - PEquiv.single_subsingleton_eq_refl + - path: Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean + theorems: + - CategoryTheory.Limits.IsZero.of_mono + - CategoryTheory.Limits.HasZeroObject.zeroIsoInitial_hom + - CategoryTheory.Limits.epi_of_target_iso_zero + - CategoryTheory.Limits.IsZero.iff_isSplitEpi_eq_zero + - CategoryTheory.Limits.zero_of_to_zero + - CategoryTheory.Limits.IsZero.of_epi_eq_zero + - CategoryTheory.Limits.HasZeroMorphisms.ext + - CategoryTheory.Limits.IsZero.iff_id_eq_zero + - CategoryTheory.Limits.IsZero.of_epi + - CategoryTheory.Limits.IsZero.of_mono_zero + - CategoryTheory.Limits.HasZeroObject.zeroIsoInitial_inv + - CategoryTheory.Limits.id_zero + - CategoryTheory.Limits.mono_of_source_iso_zero + - CategoryTheory.Limits.zero_of_comp_mono + - CategoryTheory.Limits.hasZeroObject_of_hasTerminal_object + - CategoryTheory.Limits.zero_of_source_iso_zero + - CategoryTheory.Limits.zero_of_target_iso_zero + - CategoryTheory.Limits.HasZeroObject.zeroIsoTerminal_hom + - "CategoryTheory.Limits.image.\u03B9_zero" + - CategoryTheory.Limits.zero_of_from_zero + - CategoryTheory.Limits.HasZeroMorphisms.ext_aux + - CategoryTheory.Limits.IsZero.of_epi_zero + - CategoryTheory.Limits.eq_zero_of_image_eq_zero + - CategoryTheory.Limits.HasZeroObject.zeroIsoIsInitial_hom + - CategoryTheory.Limits.comp_factorThruImage_eq_zero + - CategoryTheory.Limits.hasZeroObject_of_hasInitial_object + - CategoryTheory.Limits.zero_of_epi_comp + - CategoryTheory.Limits.HasZeroObject.zeroIsoIsTerminal_inv + - CategoryTheory.Limits.IsZero.of_mono_eq_zero + - CategoryTheory.Limits.IsZero.iff_isSplitMono_eq_zero + - path: Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean + theorems: + - "summable_jacobiTheta\u2082_term_bound" + - "jacobiTheta\u2082_functional_equation" + - "jacobiTheta\u2082_term_bound" + - "differentiableAt_jacobiTheta\u2082" + - "jacobiTheta\u2082_add_right" + - "jacobiTheta\u2082_neg_left" + - "jacobiTheta\u2082_add_left'" + - path: Mathlib/Data/Multiset/Lattice.lean + theorems: + - Multiset.nodup_sup_iff + - Multiset.sup_ndunion + - Multiset.inf_ndinsert + - Multiset.inf_add + - Multiset.sup_ndinsert + - Multiset.sup_le + - Multiset.le_inf + - Multiset.sup_add + - Multiset.inf_union + - Multiset.sup_union + - Multiset.inf_ndunion + - path: Mathlib/RingTheory/Valuation/Quotient.lean + theorems: + - Valuation.supp_quot + - Valuation.comap_onQuot_eq + - Valuation.supp_quot_supp + - Valuation.self_le_supp_comap + - path: Mathlib/Order/SuccPred/Limit.lean + theorems: + - Order.mem_range_succ_of_not_isSuccLimit + - Order.IsPredLimit.isMin + - Order.not_isPredLimit_pred_of_not_isMin + - Order.not_isPredLimit_iff_exists_covBy + - Order.IsPredLimit.pred_ne + - Order.not_isSuccLimit_succ_of_not_isMax + - Order.isSuccLimit_toDual_iff + - Order.IsSuccLimit.succ_lt + - Order.not_isPredLimit + - Order.not_isSuccLimit + - Order.not_isSuccLimit_iff_exists_covBy + - Order.IsSuccLimit.isMin + - Order.not_isPredLimit_iff + - Order.isPredLimit_toDual_iff + - Order.not_isSuccLimit_iff' + - Order.IsSuccLimit.succ_ne + - Order.IsSuccLimit.isMin_of_noMax + - Order.not_isSuccLimit_iff + - Order.isSuccLimitRecOn_limit + - Order.not_isPredLimit_of_noMin + - Order.IsSuccLimit.isMax + - Order.isSuccLimitRecOn_succ' + - path: Mathlib/CategoryTheory/Sites/SheafHom.lean + theorems: + - CategoryTheory.presheafHom_isSheafFor + - CategoryTheory.Presheaf.IsSheaf.hom + - CategoryTheory.PresheafHom.IsSheafFor.exists_app + - CategoryTheory.presheafHom_map_app + - CategoryTheory.presheafHom_map_app_op_mk_id + - CategoryTheory.PresheafHom.isAmalgamation_iff + - path: Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean + theorems: + - aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet + - aeSeq.mk_eq_fun_of_mem_aeSeqSet + - aeSeq.iSup + - aeSeq.prop_of_mem_aeSeqSet + - aeSeq.measure_compl_aeSeqSet_eq_zero + - aeSeq.aeSeq_eq_mk_ae + - aeSeq.aeSeq_eq_mk_of_mem_aeSeqSet + - aeSeq.fun_prop_of_mem_aeSeqSet + - path: Mathlib/SetTheory/Cardinal/Continuum.lean + theorems: + - Cardinal.continuum_power_aleph0 + - Cardinal.beth_one + - Cardinal.lift_continuum + - Cardinal.continuum_lt_lift + - Cardinal.mk_set_nat + - Cardinal.continuum_le_lift + - Cardinal.aleph_one_le_continuum + - Cardinal.lift_le_continuum + - Cardinal.lift_lt_continuum + - path: Mathlib/Algebra/Periodic.lean + theorems: + - Function.Periodic.sub_const + - Function.Antiperiodic.add + - Function.Periodic.neg + - "Function.Periodic.const_smul\u2080" + - "Function.Antiperiodic.const_smul\u2080" + - Function.Periodic.zsmul + - Function.Antiperiodic.const_sub + - Function.Antiperiodic.eq + - Function.Antiperiodic.periodic + - Function.Antiperiodic.int_odd_mul_antiperiodic + - Function.Periodic.int_mul + - Function.Antiperiodic.sub_eq' + - Function.Antiperiodic.const_inv_smul + - Function.Antiperiodic.div + - Function.Periodic.mul_const' + - Function.Periodic.smul + - Function.Antiperiodic.nat_odd_mul_antiperiodic + - Function.Antiperiodic.smul + - Function.periodic_with_period_zero + - Function.Periodic.eq + - Function.Periodic.map_vadd_zmultiples + - Function.Periodic.add_period + - "Function.Periodic.const_inv_smul\u2080" + - Function.Periodic.div_const + - Function.Periodic.sub_nsmul_eq + - Function.Periodic.nat_mul + - Function.Periodic.const_sub + - Function.Periodic.const_add + - Function.Antiperiodic.mul_const' + - Function.Antiperiodic.sub_eq + - Function.Periodic.sub_period + - Finset.periodic_prod + - Function.Periodic.mul + - Function.Periodic.comp + - Function.Antiperiodic.sub + - "Function.Antiperiodic.const_inv_smul\u2080" + - Function.Periodic.nat_mul_sub_eq + - Function.Periodic.add_const + - List.periodic_prod + - Function.Antiperiodic.nat_mul_eq_of_eq_zero + - Function.Antiperiodic.const_smul + - Function.Periodic.add_antiperiod + - Function.Antiperiodic.neg + - Function.Antiperiodic.const_add + - Function.Antiperiodic.mul + - Function.Periodic.const_smul + - Function.Periodic.nsmul + - Function.Periodic.comp_addHom + - Function.Periodic.sub_eq' + - Function.Antiperiodic.add_const + - Function.Periodic.div + - Function.Antiperiodic.int_mul_eq_of_eq_zero + - Function.Periodic.map_vadd_multiples + - Function.Periodic.sub_nat_mul_eq + - Function.Antiperiodic.div_inv + - Function.Antiperiodic.sub_const + - Function.Periodic.sub_eq + - Function.Periodic.image_uIcc + - Function.Periodic.sub_antiperiod + - path: Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean + theorems: + - dirichletLSeries_eulerProduct + - riemannZeta_eulerProduct + - summable_dirichletSummand + - summable_riemannZetaSummand + - path: Mathlib/Analysis/Convex/PartitionOfUnity.lean + theorems: + - exists_continuous_forall_mem_convex_of_local + - path: Mathlib/Data/Finset/Grade.lean + theorems: + - Multiset.covBy_cons + - Finset.isAtom_iff + - Finset.isCoatom_iff + - Finset.covBy_iff_exists_insert + - Finset.ordConnected_range_coe + - CovBy.card_multiset + - Finset.erase_wcovBy + - CovBy.exists_finset_erase + - Finset.covBy_iff_card_sdiff_eq_one + - Finset.ordConnected_range_val + - Finset.covBy_iff_exists_cons + - Finset.wcovBy_insert + - CovBy.exists_finset_insert + - Multiset.covBy_iff + - Finset.covBy_iff_exists_erase + - Finset.covBy_cons + - Multiset.isAtom_iff + - path: Mathlib/Data/Multiset/LocallyFinite.lean + theorems: + - Multiset.Ico_filter_lt + - Multiset.map_add_left_Ioo + - Multiset.Ioo_eq_zero_iff + - Multiset.map_add_right_Ioc + - Multiset.map_add_left_Ioc + - Multiset.Ico_eq_zero_iff + - Multiset.map_add_right_Ico + - Multiset.Ico_cons_right + - Multiset.Icc_self + - Multiset.map_add_right_Ioo + - Multiset.Icc_eq_zero_iff + - Multiset.Ico_disjoint_Ico + - Multiset.Ioc_self + - Multiset.Ioo_cons_left + - Multiset.Ioo_self + - Multiset.map_add_right_Icc + - Multiset.Ico_self + - Multiset.Ico_sub_Ico_right + - Multiset.Ico_filter_le_of_left_le + - Multiset.Ioc_eq_zero_iff + - Multiset.Ico_filter_lt_of_le_left + - Multiset.Ico_add_Ico_eq_Ico + - Multiset.Ico_filter_lt_of_right_le + - Multiset.Ico_filter_le_of_right_le + - Multiset.Ico_filter_le_of_le_left + - Multiset.Ico_inter_Ico + - Multiset.map_add_left_Ico + - Multiset.Ico_filter_le_left + - Multiset.map_add_left_Icc + - Multiset.Ico_filter_le + - Multiset.Ico_sub_Ico_left + - path: Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean + theorems: + - Polynomial.card_roots_le_derivative + - Polynomial.card_roots_toFinset_le_card_roots_derivative_diff_roots_succ + - Polynomial.card_rootSet_le_derivative + - path: Mathlib/RepresentationTheory/Maschke.lean + theorems: + - LinearMap.equivariantProjection_condition + - LinearMap.conjugate_i + - LinearMap.equivariantProjection_apply + - MonoidAlgebra.Submodule.exists_isCompl + - MonoidAlgebra.exists_leftInverse_of_injective + - path: Mathlib/CategoryTheory/Functor/Category.lean + theorems: + - CategoryTheory.NatTrans.congr_app + - CategoryTheory.NatTrans.exchange + - CategoryTheory.map_hom_inv_app + - CategoryTheory.NatTrans.hcomp_id_app + - CategoryTheory.NatTrans.id_hcomp_app + - CategoryTheory.NatTrans.epi_of_epi_app + - CategoryTheory.map_inv_hom_app + - CategoryTheory.NatTrans.mono_of_mono_app + - path: Mathlib/MeasureTheory/Group/Integral.lean + theorems: + - MeasureTheory.integral_mul_left_eq_self + - MeasureTheory.integral_div_left_eq_self + - MeasureTheory.integral_smul_eq_self + - MeasureTheory.integral_eq_zero_of_mul_right_eq_neg + - MeasureTheory.integral_inv_eq_self + - MeasureTheory.integrable_comp_div_left + - MeasureTheory.integral_div_right_eq_self + - MeasureTheory.integral_mul_right_eq_self + - MeasureTheory.integral_eq_zero_of_mul_left_eq_neg + - path: Mathlib/Analysis/Calculus/LagrangeMultipliers.lean + theorems: + - IsLocalExtrOn.exists_linear_map_of_hasStrictFDerivAt + - IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt + - IsLocalExtrOn.range_ne_top_of_hasStrictFDerivAt + - IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt_1d + - IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt + - path: Mathlib/CategoryTheory/Limits/MonoCoprod.lean + theorems: + - CategoryTheory.Limits.MonoCoprod.mono_of_injective_aux + - CategoryTheory.Limits.MonoCoprod.mono_inj + - CategoryTheory.Limits.MonoCoprod.mono_binaryCofanSum_inl' + - CategoryTheory.Limits.MonoCoprod.mono_inl_iff + - CategoryTheory.Limits.MonoCoprod.mk' + - CategoryTheory.Limits.MonoCoprod.binaryCofan_inr + - CategoryTheory.Limits.MonoCoprod.mono_map'_of_injective + - path: Mathlib/Geometry/Euclidean/Inversion/Basic.lean + theorems: + - EuclideanGeometry.dist_inversion_mul_dist_center_eq + - EuclideanGeometry.inversion_dist_center' + - EuclideanGeometry.inversion_zero_radius + - EuclideanGeometry.dist_inversion_inversion + - EuclideanGeometry.inversion_self + - EuclideanGeometry.inversion_inversion + - EuclideanGeometry.inversion_dist_center + - EuclideanGeometry.inversion_eq_center' + - EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist + - EuclideanGeometry.inversion_mul + - EuclideanGeometry.dist_inversion_center + - EuclideanGeometry.dist_center_inversion + - path: Mathlib/Analysis/SpecialFunctions/Arsinh.lean + theorems: + - Real.arsinh_nonneg_iff + - Real.arsinh_zero + - Real.arsinh_neg + - Real.exp_arsinh + - Real.hasStrictDerivAt_arsinh + - Real.arsinh_nonpos_iff + - Real.cosh_arsinh + - Real.sinh_arsinh + - path: Mathlib/FieldTheory/Finite/GaloisField.lean + theorems: + - GaloisField.card + - GaloisField.splits_zmod_X_pow_sub_X + - GaloisField.finrank + - galois_poly_separable + - path: Mathlib/FieldTheory/Finiteness.lean + theorems: + - IsNoetherian.iff_fg + - IsNoetherian.iff_rank_lt_aleph0 + - IsNoetherian.range_finsetBasis + - path: Mathlib/Tactic/NormNum/OfScientific.lean + theorems: + - Mathlib.Meta.NormNum.isNat_ofScientific_of_false + - path: Mathlib/MeasureTheory/Measure/MutuallySingular.lean + theorems: + - MeasureTheory.Measure.MutuallySingular.sum_left + - MeasureTheory.Measure.MutuallySingular.restrict_nullSet + - MeasureTheory.Measure.MutuallySingular.self_iff + - MeasureTheory.Measure.MutuallySingular.restrict_compl_nullSet + - MeasureTheory.Measure.MutuallySingular.mk + - MeasureTheory.Measure.MutuallySingular.add_left_iff + - path: Mathlib/Logic/Encodable/Basic.lean + theorems: + - Encodable.encode_injective + - Encodable.decode_ge_two + - Encodable.Subtype.encode_eq + - "Encodable.decode\u2082_ne_none_iff" + - Directed.rel_sequence + - "Encodable.decode\u2082_encode" + - Encodable.decode_ofEquiv + - Encodable.decode_prod_val + - ULower.up_down + - Encodable.surjective_decode_iget + - Directed.sequence_mono_nat + - "Encodable.mem_decode\u2082'" + - path: Mathlib/Data/Pi/Lex.lean + theorems: + - Pi.toLex_strictMono + - Pi.toLex_monotone + - Pi.isTrichotomous_lex + - Pi.lt_toLex_update_self_iff + - Pi.le_toLex_update_self_iff + - Pi.lex_desc + - Pi.lex_lt_of_lt + - Pi.Lex.noMaxOrder' + - Pi.toLex_update_lt_self_iff + - Pi.toLex_update_le_self_iff + - path: Mathlib/Algebra/Module/Torsion.lean + theorems: + - Ideal.Quotient.torsionBy_eq_span_singleton + - Submodule.torsionBySet_singleton_eq + - Submodule.torsionBy_isInternal + - Submodule.torsionBySet_univ + - AddMonoid.isTorsion_iff_isTorsion_nat + - Submodule.torsionBy_one + - Submodule.noZeroSMulDivisors_iff_torsion_eq_bot + - Submodule.isTorsion'_powers_iff + - Submodule.supIndep_torsionBySet_ideal + - Submodule.mem_torsionBySet_iff + - Submodule.torsionBySet_isInternal + - Module.isTorsionBy_iff_torsionBy_eq_top + - Submodule.torsion_int + - Submodule.torsionBy_le_torsionBy_of_dvd + - Module.isTorsionBySet_iff_torsionBySet_eq_top + - Ideal.CompleteLattice.Independent.linear_independent' + - Module.isTorsionBySet_iff_is_torsion_by_span + - Submodule.torsionBySet_eq_torsionBySet_span + - Submodule.iSup_torsionBy_eq_torsionBy_prod + - Submodule.annihilator_top_inter_nonZeroDivisors + - Submodule.exists_isTorsionBy + - Submodule.QuotientTorsion.torsion_eq_bot + - Ideal.torsionOf_zero + - Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors + - Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf + - Submodule.coe_torsion_eq_annihilator_ne_bot + - Ideal.torsionOf_eq_top_iff + - Submodule.isTorsion'_iff_torsion'_eq_top + - Submodule.supIndep_torsionBy + - AddMonoid.isTorsion_iff_isTorsion_int + - path: Mathlib/Topology/Algebra/Algebra.lean + theorems: + - Subalgebra.topologicalClosure_comap_homeomorph + - continuous_algebraMap_iff_smul + - Subalgebra.isClosed_topologicalClosure + - path: Mathlib/Algebra/FreeMonoid/Count.lean + theorems: + - FreeMonoid.count_of + - FreeAddMonoid.countP_of + - FreeMonoid.countP_of' + - FreeMonoid.countP_of + - FreeAddMonoid.count_of + - path: Mathlib/Combinatorics/SimpleGraph/Metric.lean + theorems: + - SimpleGraph.nonempty_of_pos_dist + - SimpleGraph.Connected.pos_dist_of_ne + - SimpleGraph.Connected.dist_eq_zero_iff + - SimpleGraph.Reachable.dist_eq_zero_iff + - SimpleGraph.dist_comm + - SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable + - SimpleGraph.dist_comm_aux + - SimpleGraph.dist_eq_zero_of_not_reachable + - SimpleGraph.Reachable.pos_dist_of_ne + - SimpleGraph.Connected.dist_triangle + - SimpleGraph.dist_self + - path: Mathlib/Computability/EpsilonNFA.lean + theorems: + - "\u03B5NFA.evalFrom_append_singleton" + - "NFA.to\u03B5NFA_evalFrom_match" + - "\u03B5NFA.mem_stepSet_iff" + - "\u03B5NFA.\u03B5Closure_empty" + - "NFA.to\u03B5NFA_\u03B5Closure" + - "NFA.to\u03B5NFA_correct" + - "\u03B5NFA.evalFrom_empty" + - "\u03B5NFA.stepSet_empty" + - path: Mathlib/CategoryTheory/Shift/ShiftSequence.lean + theorems: + - CategoryTheory.Functor.shiftIso_add' + - CategoryTheory.Functor.shiftIso_add + - CategoryTheory.Functor.shiftIso_zero_inv_app + - CategoryTheory.Functor.shiftIso_add_hom_app + - CategoryTheory.Functor.shiftIso_hom_app_comp + - CategoryTheory.Functor.shiftIso_zero_hom_app + - CategoryTheory.Functor.shiftIso_add'_hom_app + - CategoryTheory.Functor.shiftIso_add_inv_app + - CategoryTheory.Functor.shiftIso_add'_inv_app + - path: Mathlib/AlgebraicGeometry/Restrict.lean + theorems: + - AlgebraicGeometry.isPullback_morphismRestrict + - AlgebraicGeometry.image_morphismRestrict_preimage + - AlgebraicGeometry.Scheme.map_basicOpen' + - AlgebraicGeometry.Scheme.restrictFunctor_map_base + - "AlgebraicGeometry.morphismRestrict_\u03B9" + - AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_restrict + - AlgebraicGeometry.Scheme.restrictFunctor_map_left + - AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst + - AlgebraicGeometry.morphismRestrict_c_app + - AlgebraicGeometry.Scheme.map_basicOpen_map + - AlgebraicGeometry.morphismRestrict_comp + - AlgebraicGeometry.Scheme.restrictFunctor_map_ofRestrict + - AlgebraicGeometry.Scheme.restrictFunctor_map_app + - AlgebraicGeometry.Scheme.ofRestrict_val_c_app_self + - AlgebraicGeometry.Scheme.restrictFunctor_map_app_aux + - AlgebraicGeometry.Scheme.map_basicOpen + - "AlgebraicGeometry.\u0393_map_morphismRestrict" + - path: Mathlib/Topology/Irreducible.lean + theorems: + - nonempty_preirreducible_inter + - subset_closure_inter_of_isPreirreducible_of_isOpen + - isClosed_of_mem_irreducibleComponents + - isPreirreducible_iff_closed_union_closed + - IsPreirreducible.subset_irreducible + - isIrreducible_iff_sUnion_closed + - IsPreirreducible.preimage + - isPreirreducible_iff_closure + - irreducibleComponents_eq_maximals_closed + - IsPreirreducible.image + - isIrreducible_iff_sInter + - Subtype.preirreducibleSpace + - path: Mathlib/Algebra/BigOperators/Order.lean + theorems: + - Finset.prod_lt_prod_of_subset' + - Fintype.one_lt_prod + - Finset.sum_card_le + - Fintype.prod_eq_one_iff_of_one_le + - Finset.mul_card_image_le_card_of_maps_to + - Finset.prod_add_prod_le' + - Finset.prod_lt_prod_of_nonempty' + - Finset.abs_sum_of_nonneg' + - Fintype.prod_lt_one_iff_of_le_one + - Finset.le_prod_of_submultiplicative + - Finset.prod_le_pow_card + - Finset.prod_le_one' + - Finset.one_lt_prod + - Finset.prod_sdiff_lt_prod_sdiff + - Fintype.prod_lt_one + - Finset.prod_le_prod_of_subset_of_one_le' + - Finset.exists_le_of_prod_le' + - Finset.card_le_card_biUnion_add_card_fiber + - Finset.one_le_prod' + - Finset.prod_le_one + - Finset.single_lt_prod' + - Finset.prod_eq_prod_iff_of_le + - Fintype.one_lt_prod_iff_of_one_le + - Finset.mul_le_prod + - Finset.abs_sum_of_nonneg + - Finset.prod_lt_prod + - Finset.prod_le_prod_of_ne_one' + - Finset.sum_card_inter_le + - Finset.le_prod_nonempty_of_submultiplicative_on_pred + - Finset.prod_add_prod_le + - Fintype.prod_eq_one_iff_of_le_one + - Finset.sum_mul_sq_le_sq_mul_sq + - Finset.prod_eq_one_iff_of_one_le' + - Finset.le_sum_card + - Finset.card_le_card_biUnion + - Finset.exists_lt_of_prod_lt' + - Finset.le_prod_of_submultiplicative_on_pred + - Finset.prod_lt_one + - Finset.exists_one_lt_of_prod_one_of_exists_ne_one' + - Finset.prod_lt_prod_of_nonempty + - Finset.card_le_mul_card_image_of_maps_to + - Finset.le_sum_card_inter + - Finset.prod_sdiff_le_prod_sdiff + - Finset.prod_le_prod + - Finset.sum_card + - path: Mathlib/Tactic/Rify.lean + theorems: + - Mathlib.Tactic.Rify.rat_cast_lt + - Mathlib.Tactic.Rify.rat_cast_ne + - Mathlib.Tactic.Rify.rat_cast_eq + - Mathlib.Tactic.Rify.rat_cast_le + - path: Mathlib/Geometry/Manifold/VectorBundle/Basic.lean + theorems: + - smoothOn_symm_coordChangeL + - FiberBundle.extChartAt_target + - Trivialization.contMDiffOn_symm_trans + - Trivialization.smoothOn_symm + - Bundle.contMDiffWithinAt_totalSpace + - ContMDiffWithinAt.change_section_trivialization + - Trivialization.contMDiffOn_iff + - Bundle.contMDiff_proj + - Trivialization.smoothOn + - FiberBundle.chartedSpace_chartAt + - FiberBundle.extChartAt + - VectorPrebundle.smoothVectorBundle + - Bundle.contMDiffAt_section + - Bundle.smooth_zeroSection + - ContMDiffWithinAt.coordChange + - FiberBundle.chartedSpace_chartAt_symm_fst + - VectorPrebundle.mk_smoothCoordChange + - path: Mathlib/Algebra/Order/Monoid/MinMax.lean + theorems: + - le_or_lt_of_mul_le_mul + - le_or_le_of_mul_le_mul + - lt_or_le_of_mul_le_mul + - fn_max_mul_fn_min + - lt_or_lt_of_mul_lt_mul + - mul_lt_mul_iff_of_le_of_le + - path: Mathlib/Data/Real/Pointwise.lean + theorems: + - Real.sInf_smul_of_nonneg + - Real.sInf_smul_of_nonpos + - Real.iSup_mul_of_nonneg + - Real.iInf_mul_of_nonneg + - Real.iInf_mul_of_nonpos + - Real.iSup_mul_of_nonpos + - path: Mathlib/Algebra/Order/Hom/Ring.lean + theorems: + - OrderRingIso.refl_apply + - OrderRingHom.cancel_left + - OrderRingHom.cancel_right + - OrderRingIso.toOrderRingHom_injective + - path: Mathlib/Order/PartialSups.lean + theorems: + - partialSups_eq_sup_range + - partialSups_iff_forall + - partialSups_apply + - iSup_eq_iSup_of_partialSups_eq_partialSups + - partialSups_eq_sup'_range + - Monotone.partialSups_eq + - partialSups_eq_ciSup_Iic + - upperBounds_range_partialSups + - ciSup_partialSups_eq + - iSup_le_iSup_of_partialSups_le_partialSups + - partialSups_eq_biSup + - path: Mathlib/Analysis/Calculus/Deriv/Star.lean + theorems: + - HasDerivAtFilter.star + - HasStrictDerivAt.star + - path: Mathlib/Topology/Order/LowerUpperTopology.lean + theorems: + - Topology.IsUpper.isOpen_iff_generate_Iic_compl + - Topology.IsLower.isLowerSet_of_isOpen + - Topology.isUpper_orderDual + - Topology.IsLower.isOpen_iff_generate_Ici_compl + - Topology.IsLower.isTopologicalBasis + - sInfHom.continuous + - Topology.IsLower.continuous_iff_Ici + - path: Mathlib/Algebra/Homology/HomotopyCategory.lean + theorems: + - HomotopyCategory.quotient_map_out_comp_out + - HomotopyCategory.quotient_inverts_homotopyEquivalences + - CategoryTheory.NatTrans.mapHomotopyCategory_id + - HomotopyCategory.isZero_quotient_obj_iff + - CategoryTheory.NatTrans.mapHomotopyCategory_comp + - path: Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean + theorems: + - CategoryTheory.ShortComplex.quasiIso_iff_isIso_leftHomologyMap' + - CategoryTheory.ShortComplex.quasiIso_iff_isIso_rightHomologyMap' + - CategoryTheory.ShortComplex.quasiIso_of_epi_of_isIso_of_mono + - CategoryTheory.ShortComplex.quasiIso_iff_comp_left + - CategoryTheory.ShortComplex.quasiIso_of_comp_left + - CategoryTheory.ShortComplex.quasiIso_of_comp_right + - CategoryTheory.ShortComplex.quasiIso_opMap + - CategoryTheory.ShortComplex.quasiIso_iff + - CategoryTheory.ShortComplex.RightHomologyMapData.quasiIso_iff + - CategoryTheory.ShortComplex.quasiIso_iff_isIso_liftCycles + - CategoryTheory.ShortComplex.quasiIso_unopMap + - CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff + - CategoryTheory.ShortComplex.quasiIso_of_arrow_mk_iso + - CategoryTheory.ShortComplex.quasiIso_iff_comp_right + - path: Mathlib/Data/Set/Pairwise/Lattice.lean + theorems: + - pairwiseDisjoint_unique + - Set.biUnion_diff_biUnion_eq + - Set.PairwiseDisjoint.biUnion + - Set.pairwiseDisjoint_prod_left + - Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion + - Set.pairwise_iUnion + - Set.PairwiseDisjoint.prod_left + - path: Mathlib/Analysis/Normed/Order/Lattice.lean + theorems: + - dual_solid + - lipschitzWith_negPart + - norm_sup_sub_sup_le_add_norm + - lipschitzWith_sup_right + - isClosed_nonneg + - norm_inf_sub_inf_le_add_norm + - isClosed_le_of_isClosed_nonneg + - norm_sup_le_add + - norm_inf_le_add + - path: Mathlib/Data/Seq/Parallel.lean + theorems: + - Computation.terminates_parallel.aux + - Computation.parallel_congr_left + - Computation.mem_parallel + - Computation.terminates_parallel + - Computation.map_parallel + - Computation.exists_of_mem_parallel + - Computation.parallel_empty + - path: Mathlib/Data/List/Pairwise.lean + theorems: + - List.pairwise_of_forall_mem_list + - List.Pairwise.forall + - List.Pairwise.forall_of_forall + - List.Pairwise.pmap + - List.pairwise_map' + - List.pairwise_pmap + - List.pairwise_of_reflexive_of_forall_ne + - path: Mathlib/FieldTheory/IntermediateField.lean + theorems: + - IntermediateField.coe_sum + - IntermediateField.eq_of_le_of_finrank_le'' + - IntermediateField.toSubalgebra_eq_iff + - IntermediateField.map_injective + - toSubalgebra_toIntermediateField' + - IntermediateField.fieldRange_le + - toSubalgebra_toIntermediateField + - IntermediateField.coe_nat_mem + - toIntermediateField_toSubalgebra + - IntermediateField.neg_mem + - IntermediateField.restrictScalars_injective + - IntermediateField.isIntegral_iff + - IntermediateField.aeval_coe + - IntermediateField.coe_prod + - IntermediateField.eq_of_le_of_finrank_le' + - IntermediateField.toSubalgebra_injective + - toIntermediateField'_toSubalgebra + - IntermediateField.coe_isIntegral_iff + - path: Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean + theorems: + - Basis.addHaar_reindex + - parallelepiped_eq_convexHull + - parallelepiped_comp_equiv + - parallelepiped_orthonormalBasis_one_dim + - parallelepiped_single + - Basis.prod_parallelepiped + - parallelepiped_eq_sum_segment + - convex_parallelepiped + - Basis.addHaar_self + - image_parallelepiped + - mem_parallelepiped_iff + - Basis.prod_addHaar + - path: Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean + theorems: + - AlgebraicGeometry.universallyClosed_is_local_at_target + - AlgebraicGeometry.universallyClosed_stableUnderComposition + - AlgebraicGeometry.universallyClosed_eq + - path: Mathlib/AlgebraicTopology/DoldKan/Normalized.lean + theorems: + - AlgebraicTopology.DoldKan.factors_normalizedMooreComplex_PInfty + - AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_naturality + - AlgebraicTopology.DoldKan.inclusionOfMooreComplexMap_comp_PInfty + - AlgebraicTopology.DoldKan.HigherFacesVanish.inclusionOfMooreComplexMap + - AlgebraicTopology.DoldKan.PInfty_comp_PInftyToNormalizedMooreComplex + - AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap + - path: Mathlib/Analysis/Calculus/SmoothSeries.lean + theorems: + - iteratedFDeriv_tsum + - differentiable_tsum' + - hasFDerivAt_tsum + - summable_of_summable_hasFDerivAt + - iteratedFDeriv_tsum_apply + - summable_of_summable_hasDerivAt_of_isPreconnected + - deriv_tsum + - contDiff_tsum_of_eventually + - hasDerivAt_tsum + - summable_of_summable_hasFDerivAt_of_isPreconnected + - contDiff_tsum + - hasDerivAt_tsum_of_isPreconnected + - fderiv_tsum + - hasFDerivAt_tsum_of_isPreconnected + - summable_of_summable_hasDerivAt + - path: Mathlib/Order/Partition/Finpartition.lean + theorems: + - Finpartition.parts_top_subset + - Finpartition.biUnion_filter_atomise + - Finpartition.exists_mem + - Finpartition.parts_eq_empty_iff + - Finpartition.ne_bot + - Finpartition.card_bind + - Finpartition.mem_avoid + - Finpartition.mem_atomise + - Finpartition.parts_nonempty_iff + - Finpartition.mem_bind + - Finpartition.atomise_empty + - Finpartition.card_mono + - Finpartition.sum_card_parts + - Finpartition.card_parts_le_card + - Finpartition.exists_le_of_le + - Finpartition.card_filter_atomise_le_two_pow + - path: Mathlib/Logic/Function/Conjugate.lean + theorems: + - Function.Semiconj.trans + - Function.Semiconj.comp_right + - "Function.Semiconj\u2082.isIdempotent_right" + - "Function.Semiconj\u2082.isAssociative_left" + - "Function.Semiconj\u2082.isIdempotent_left" + - "Function.Semiconj\u2082.isAssociative_right" + - "Function.Semiconj\u2082.comp" + - Function.Semiconj.inverse_left + - path: Mathlib/Data/Polynomial/UnitTrinomial.lean + theorems: + - Polynomial.isUnitTrinomial_iff'' + - Polynomial.trinomial_natTrailingDegree + - Polynomial.IsUnitTrinomial.irreducible_of_coprime + - Polynomial.trinomial_monic + - Polynomial.isUnitTrinomial_iff + - Polynomial.trinomial_leading_coeff' + - Polynomial.IsUnitTrinomial.irreducible_aux1 + - Polynomial.IsUnitTrinomial.irreducible_aux2 + - Polynomial.IsUnitTrinomial.coeff_isUnit + - Polynomial.isUnitTrinomial_iff' + - Polynomial.IsUnitTrinomial.not_isUnit + - Polynomial.IsUnitTrinomial.irreducible_aux3 + - Polynomial.trinomial_mirror + - Polynomial.trinomial_trailingCoeff + - Polynomial.trinomial_trailing_coeff' + - Polynomial.IsUnitTrinomial.irreducible_of_coprime' + - Polynomial.trinomial_leadingCoeff + - Polynomial.trinomial_natDegree + - Polynomial.IsUnitTrinomial.card_support_eq_three + - Polynomial.IsUnitTrinomial.ne_zero + - Polynomial.trinomial_middle_coeff + - path: Mathlib/NumberTheory/LSeries/Linearity.lean + theorems: + - LSeries.term_smul_apply + - LSeriesHasSum.smul + - LSeries_smul + - LSeries_neg + - LSeriesSummable.smul + - LSeries.term_sub_apply + - LSeries.term_add + - LSeries.term_add_apply + - LSeriesSummable.of_smul + - LSeriesHasSum.add + - LSeries_sub + - LSeries.term_sub + - LSeriesSummable.neg + - LSeries_add + - LSeriesHasSum.sub + - LSeriesSummable.sub + - LSeriesSummable.add + - LSeries.term_smul + - LSeries.term_neg + - LSeries.term_neg_apply + - LSeriesHasSum.neg + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean + theorems: + - Real.one_le_cosh + - Complex.hasStrictDerivAt_sinh + - Real.sinh_pos_iff + - Real.sinh_le_self_iff + - Real.sinh_sub_id_strictMono + - Complex.hasStrictDerivAt_cosh + - Real.sinh_eq_zero + - Real.sinh_nonpos_iff + - Real.cosh_strictMonoOn + - Real.self_le_sinh_iff + - Real.one_lt_cosh + - Complex.hasStrictDerivAt_cos + - Complex.hasStrictDerivAt_sin + - Real.sinh_nonneg_iff + - Real.abs_sinh + - Real.sinh_strictMono + - Real.sinh_neg_iff + - path: Mathlib/Geometry/Manifold/InteriorBoundary.lean + theorems: + - ModelWithCorners.Boundaryless.boundary_eq_empty + - ModelWithCorners.isInteriorPoint_or_isBoundaryPoint + - ModelWithCorners.disjoint_interior_boundary + - range_mem_nhds_isInteriorPoint + - path: Mathlib/NumberTheory/Liouville/Basic.lean + theorems: + - Liouville.exists_one_le_pow_mul_dist + - Liouville.irrational + - Liouville.transcendental + - Liouville.exists_pos_real_of_irrational_root + - path: Mathlib/Algebra/Lie/Killing.lean + theorems: + - LieModule.traceForm_apply_lie_apply + - LieIdeal.le_killingCompl_top_of_isLieAbelian + - LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm + - LieModule.traceForm_eq_zero_if_mem_lcs_of_mem_ucs + - LieAlgebra.IsKilling.rootSpaceProductNegSelf_zero_eq_bot + - LieModule.trace_toEndomorphism_eq_zero_of_mem_lcs + - LieModule.traceForm_eq_zero_of_isNilpotent + - LieModule.traceForm_apply_eq_zero_of_mem_lcs_of_mem_center + - LieAlgebra.killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero + - LieIdeal.killingForm_eq + - LieAlgebra.IsKilling.ker_killingForm_eq_bot + - LieAlgebra.IsKilling.ker_weight_inf_rootSpaceProductNegSelf_eq_bot + - LieModule.traceForm_eq_zero_of_isTrivial + - LieSubmodule.traceForm_eq_of_le_idealizer + - LieSubmodule.trace_eq_trace_restrict_of_le_idealizer + - LieAlgebra.IsKilling.iInf_ker_weight_eq_bot + - LieIdeal.mem_killingCompl + - LieAlgebra.IsKilling.eq_zero_of_apply_eq_zero_of_mem_rootSpaceProductNegSelf + - LieAlgebra.IsKilling.span_weight_eq_top + - LieSubmodule.traceForm_eq_zero_of_isTrivial + - LieModule.traceForm_eq_sum_finrank_nsmul_mul + - LieModule.range_traceForm_le_span_weight + - LieModule.lie_traceForm_eq_zero + - LieModule.traceForm_weightSpace_eq + - LieModule.traceForm_eq_sum_weightSpaceOf + - LieModule.isLieAbelian_of_ker_traceForm_eq_bot + - LieModule.traceForm_apply_lie_apply' + - LieModule.eq_zero_of_mem_weightSpace_mem_posFitting + - LieModule.traceForm_eq_sum_finrank_nsmul + - path: Mathlib/Data/Setoid/Partition.lean + theorems: + - IndexedPartition.piecewise_inj + - Setoid.eqv_class_mem' + - IndexedPartition.iUnion + - IndexedPartition.proj_fiber + - Setoid.IsPartition.sUnion_eq_univ + - IndexedPartition.piecewise_bij + - Setoid.classes_eqv_classes + - Setoid.rel_iff_exists_classes + - Setoid.classes_mkClasses + - Setoid.card_classes_ker_le + - Setoid.eqv_classes_of_disjoint_union + - Setoid.classes_inj + - path: Mathlib/Analysis/Normed/Group/Quotient.lean + theorems: + - AddSubgroup.norm_normedMk + - QuotientAddGroup.norm_mk + - NormedAddGroupHom.IsQuotient.norm_lift + - quotient_norm_add_le + - NormedAddGroupHom.isQuotientQuotient + - quotient_norm_mk_eq + - norm_mk_zero + - NormedAddGroupHom.IsQuotient.norm_le + - AddSubgroup.norm_trivial_quotient_mk + - QuotientAddGroup.norm_eq_infDist + - quotient_nhd_basis + - QuotientAddGroup.norm_lift_apply_le + - norm_mk_eq_zero + - quotient_norm_sub_rev + - QuotientAddGroup.norm_lt_iff + - NormedAddGroupHom.lift_unique + - NormedAddGroupHom.lift_normNoninc + - norm_mk_lt' + - AddSubgroup.norm_normedMk_le + - quotient_norm_neg + - quotient_norm_eq_zero_iff + - path: Mathlib/MeasureTheory/Integral/TorusIntegral.lean + theorems: + - torusIntegral_succAbove + - TorusIntegrable.function_integrable + - torusIntegral_succ + - TorusIntegrable.torusIntegrable_zero_radius + - torusMap_eq_center_iff + - TorusIntegrable.torusIntegrable_const + - torusIntegral_sub + - torusIntegral_add + - torusIntegral_dim0 + - torusIntegral_radius_zero + - torusMap_sub_center + - torusIntegral_smul + - torusIntegral_dim1 + - norm_torusIntegral_le_of_norm_le_const + - torusIntegral_neg + - path: Mathlib/Topology/Partial.lean + theorems: + - rtendsto'_nhds + - open_dom_of_pcontinuous + - path: Mathlib/LinearAlgebra/AffineSpace/Slope.lean + theorems: + - sub_smul_slope + - slope_same + - sub_smul_slope_vadd + - slope_vadd_const + - slope_comm + - lineMap_slope_slope_sub_div_sub + - slope_sub_smul + - sub_div_sub_smul_slope_add_sub_div_sub_smul_slope + - lineMap_slope_lineMap_slope_lineMap + - eq_of_slope_eq_zero + - path: Mathlib/RingTheory/TensorProduct/Basic.lean + theorems: + - LinearMap.baseChange_neg + - Algebra.TensorProduct.mul_assoc + - Algebra.TensorProduct.mul_one + - LinearMap.baseChange_id + - Algebra.TensorProduct.tmul_pow + - Algebra.TensorProduct.basisAux_tmul + - Algebra.TensorProduct.productMap_left_apply + - Subalgebra.finiteDimensional_sup + - Algebra.TensorProduct.one_mul + - Algebra.TensorProduct.basisAux_map_smul + - LinearMap.baseChange_zero + - Algebra.TensorProduct.productMap_right_apply + - Algebra.TensorProduct.algebraMap_apply' + - Algebra.TensorProduct.lift_comp_includeLeft + - Algebra.TensorProduct.basis_repr_symm_apply + - LinearMap.baseChange_add + - LinearMap.toMatrix_baseChange + - LinearMap.baseChange_comp + - Algebra.TensorProduct.natCast_def' + - LinearMap.baseChange_sub + - Algebra.TensorProduct.map_restrictScalars_comp_includeRight + - Algebra.TensorProduct.map_range + - Algebra.TensorProduct.basis_repr_symm_apply' + - Algebra.TensorProduct.comm_symm + - LinearMap.baseChange_smul + - Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap + - Algebra.TensorProduct.ext + - Algebra.TensorProduct.productMap_range + - Algebra.TensorProduct.productMap_eq_comp_map + - Algebra.TensorProduct.intCast_def' + - path: Mathlib/Analysis/Calculus/Deriv/Polynomial.lean + theorems: + - Polynomial.hasStrictDerivAt_aeval + - Polynomial.derivWithin + - Polynomial.hasStrictDerivAt + - Polynomial.derivWithin_aeval + - path: Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean + theorems: + - volume_regionBetween_eq_lintegral' + - nullMeasurableSet_region_between_co + - measurableSet_region_between_co + - Real.volume_Iio + - Real.volume_pi_Ioc_toReal + - Real.volume_emetric_closedBall + - regionBetween_subset + - nullMeasurableSet_region_between_cc + - Real.volume_pi_Ico_toReal + - Real.volume_Ico + - Real.volume_emetric_ball + - measurableSet_regionBetween + - measurableSet_region_between_cc + - Real.map_matrix_volume_pi_eq_smul_volume_pi + - Real.volume_le_diam + - Real.map_volume_mul_right + - Real.volume_Icc_pi_toReal + - nullMeasurableSet_regionBetween + - measurableSet_graph + - Real.volume_Ioo + - Real.volume_singleton + - Real.map_volume_mul_left + - Real.volume_closedBall + - Real.volume_preimage_mul_right + - nullMeasurableSet_region_between_oc + - ae_restrict_of_ae_restrict_inter_Ioo + - Real.volume_Ioi + - Real.volume_pi_ball + - Real.volume_pi_le_diam_pow + - ae_of_mem_of_ae_of_mem_inter_Ioo + - Real.volume_pi_Ioo_toReal + - Real.smul_map_volume_mul_right + - Real.volume_Ici + - Real.volume_interval + - Real.smul_map_diagonal_volume_pi + - Real.volume_preserving_transvectionStruct + - Real.map_linearMap_volume_pi_eq_smul_volume_pi + - Real.volume_Icc_pi + - Real.volume_Icc + - Real.volume_pi_closedBall + - volume_regionBetween_eq_lintegral + - Real.volume_val + - Real.smul_map_volume_mul_left + - Real.volume_Iic + - measurableSet_region_between_oc + - Real.volume_univ + - Real.volume_preimage_mul_left + - Real.volume_ball + - Filter.Eventually.volume_pos_of_nhds_real + - Real.volume_Ioc + - path: Mathlib/RingTheory/Polynomial/Vieta.lean + theorems: + - Multiset.prod_X_add_C_coeff + - Finset.prod_X_add_C_coeff + - Multiset.prod_X_sub_X_eq_sum_esymm + - Multiset.prod_X_add_C_eq_sum_esymm + - Multiset.prod_X_sub_C_coeff + - Multiset.esymm_neg + - Polynomial.coeff_eq_esymm_roots_of_card + - MvPolynomial.prod_X_add_C_coeff + - Multiset.prod_X_add_C_coeff' + - path: Mathlib/Analysis/NormedSpace/AddTorsor.lean + theorems: + - dist_left_midpoint + - dist_midpoint_right + - AffineSubspace.isClosed_direction_iff + - DilationEquiv.smulTorsor_ratio + - eventually_homothety_mem_of_mem_interior + - dist_pointReflection_left + - dist_self_homothety + - eventually_homothety_image_subset_of_finite_subset_interior + - dist_midpoint_midpoint_le' + - dist_lineMap_lineMap + - dist_homothety_self + - dist_pointReflection_right + - DilationEquiv.smulTorsor_preimage_ball + - dist_right_midpoint + - dist_lineMap_right + - dist_homothety_center + - dist_center_homothety + - dist_midpoint_midpoint_le + - dist_lineMap_left + - antilipschitzWith_lineMap + - dist_midpoint_left + - path: Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean + theorems: + - CategoryTheory.Limits.Cotrident.condition + - CategoryTheory.Limits.Cotrident.app_one + - CategoryTheory.Limits.Trident.equalizer_ext + - CategoryTheory.Limits.Cotrident.coequalizer_ext + - CategoryTheory.Limits.Trident.app_zero + - "CategoryTheory.Limits.Cocone.ofCotrident_\u03B9" + - "CategoryTheory.Limits.Cotrident.ofCocone_\u03B9" + - "CategoryTheory.Limits.Trident.ofCone_\u03C0" + - CategoryTheory.Limits.Trident.condition + - "CategoryTheory.Limits.Cone.ofTrident_\u03C0" + - path: Mathlib/Data/ENat/Basic.lean + theorems: + - ENat.succ_def + - ENat.add_one_le_iff + - ENat.nat_induction + - ENat.toNat_add + - ENat.toNat_sub + - ENat.toNat_eq_iff + - ENat.coe_toNat_eq_self + - path: Mathlib/Tactic/Qify.lean + theorems: + - Mathlib.Tactic.Qify.int_cast_ne + - Mathlib.Tactic.Qify.int_cast_eq + - path: Mathlib/Algebra/Pointwise/Stabilizer.lean + theorems: + - MulAction.stabilizer_coe_finset + - MulAction.stabilizer_mul_self + - MulAction.stabilizer_finset_singleton + - MulAction.mem_stabilizer_set_iff_smul_set_subset + - MulAction.mem_stabilizer_set + - MulAction.mem_stabilizer_set' + - MulAction.stabilizer_singleton + - MulAction.mem_stabilizer_finset_iff_subset_smul_finset + - MulAction.mem_stabilizer_set_iff_subset_smul_set + - MulAction.stabilizer_finset_univ + - MulAction.stabilizer_subgroup_op + - MulAction.mem_stabilizer_finset_iff_smul_finset_subset + - MulAction.map_stabilizer_le + - MulAction.mem_stabilizer_finset + - MulAction.stabilizer_subgroup + - MulAction.stabilizer_univ + - MulAction.stabilizer_op_subgroup + - MulAction.mul_stabilizer_self + - MulAction.mem_stabilizer_finset' + - MulAction.stabilizer_image_coe_quotient + - path: Mathlib/Data/Finset/LocallyFinite/Box.lean + theorems: + - Int.mem_box + - Finset.Icc_neg_mono + - Finset.box_zero + - Int.existsUnique_mem_box + - Finset.box_succ_disjUnion + - Prod.card_box_succ + - Finset.zero_mem_box + - Finset.disjoint_box_succ_prod + - Int.card_box + - path: Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean + theorems: + - Real.strictConcaveOn_rpow + - Real.strictConcaveOn_sqrt + - NNReal.strictConcaveOn_sqrt + - NNReal.concaveOn_rpow + - NNReal.strictConcaveOn_rpow + - Real.concaveOn_rpow + - path: Mathlib/CategoryTheory/GradedObject/Bifunctor.lean + theorems: + - "CategoryTheory.GradedObject.\u03B9_mapBifunctorMapMap" + - path: Mathlib/Algebra/Star/NonUnitalSubalgebra.lean + theorems: + - NonUnitalStarSubalgebra.iSupLift_mk + - NonUnitalStarSubalgebra.toNonUnitalSubring_injective + - NonUnitalStarSubalgebra.iSupLift_of_mem + - NonUnitalStarAlgebra.mem_iInf + - NonUnitalStarAlgebra.eq_top_iff + - NonUnitalStarSubalgebra.mem_centralizer_iff + - NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot + - NonUnitalStarAlgebra.gc + - NonUnitalStarSubalgebra.prod_top + - NonUnitalStarAlgebra.mem_sup_right + - NonUnitalStarSubalgebra.centralizer_univ + - NonUnitalStarAlgebra.top_toNonUnitalSubalgebra + - NonUnitalStarAlgebra.map_bot + - NonUnitalStarAlgebra.mem_bot + - NonUnitalStarAlgebra.coe_iInf + - NonUnitalSubalgebra.star_mem_star_iff + - NonUnitalStarAlgebra.iInf_toNonUnitalSubalgebra + - NonUnitalStarAlgebra.coe_bot + - NonUnitalSubalgebra.star_adjoin_comm + - NonUnitalStarSubalgebra.toNonUnitalSubalgebra_injective + - NonUnitalStarSubalgebra.iSupLift_comp_inclusion + - NonUnitalStarAlgebra.mem_sInf + - NonUnitalStarAlgHom.coe_range + - NonUnitalStarAlgebra.mem_sup_left + - NonUnitalStarAlgebra.sInf_toNonUnitalSubalgebra + - NonUnitalStarSubalgebra.iSupLift_inclusion + - path: Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean + theorems: + - IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible + - Polynomial.cyclotomic.irreducible + - Polynomial.cyclotomic_eq_minpoly + - Polynomial.roots_cyclotomic_nodup + - Polynomial.isRoot_of_unity_of_root_cyclotomic + - isRoot_of_unity_iff + - Polynomial.isRoot_cyclotomic_iff' + - Polynomial.cyclotomic.irreducible_rat + - Polynomial.cyclotomic.isCoprime_rat + - Polynomial.cyclotomic.roots_eq_primitiveRoots_val + - IsPrimitiveRoot.minpoly_dvd_cyclotomic + - Polynomial.cyclotomic_injective + - Polynomial.cyclotomic_eq_minpoly_rat + - Polynomial.cyclotomic.roots_to_finset_eq_primitiveRoots + - Polynomial.isRoot_cyclotomic_iff + - IsPrimitiveRoot.isRoot_cyclotomic + - path: Mathlib/Logic/Equiv/PartialEquiv.lean + theorems: + - PartialEquiv.trans_refl_restr + - PartialEquiv.exists_mem_target + - PartialEquiv.IsImage.iff_preimage_eq + - PartialEquiv.EqOnSource.symm' + - PartialEquiv.self_trans_symm + - PartialEquiv.restr_trans + - PartialEquiv.trans_refl_restr' + - PartialEquiv.copy_eq + - PartialEquiv.refl_restr_source + - PartialEquiv.EqOnSource.source_inter_preimage_eq + - PartialEquiv.EqOnSource.trans' + - PartialEquiv.trans_transEquiv + - PartialEquiv.source_inter_preimage_inv_preimage + - PartialEquiv.eq_of_eqOnSource_univ + - PartialEquiv.trans_source'' + - PartialEquiv.IsImage.inter_eq_of_inter_eq_of_eqOn + - Equiv.transPartialEquiv_trans + - PartialEquiv.image_source_inter_eq' + - PartialEquiv.mem_symm_trans_source + - PartialEquiv.image_eq_target_inter_inv_preimage + - PartialEquiv.image_source_inter_eq + - PartialEquiv.refl_trans + - Equiv.trans_transPartialEquiv + - Equiv.trans_toPartialEquiv + - PartialEquiv.eq_symm_apply + - PartialEquiv.ext + - PartialEquiv.trans_assoc + - PartialEquiv.refl_prod_refl + - PartialEquiv.trans_symm_eq_symm_trans_symm + - PartialEquiv.EqOnSource.restr + - PartialEquiv.IsImage.symm_eq_on_of_inter_eq_of_eqOn + - PartialEquiv.forall_mem_target + - PartialEquiv.prod_trans + - PartialEquiv.isImage_source_target_of_disjoint + - PartialEquiv.trans_refl + - PartialEquiv.restr_eq_of_source_subset + - PartialEquiv.transEquiv_transEquiv + - PartialEquiv.isImage_source_target + - PartialEquiv.IsImage.symm_apply_mem_iff + - PartialEquiv.pi_trans + - PartialEquiv.EqOnSource.target_eq + - PartialEquiv.trans_source' + - PartialEquiv.pi_refl + - path: Mathlib/AlgebraicGeometry/Scheme.lean + theorems: + - AlgebraicGeometry.basicOpen_eq_of_affine' + - AlgebraicGeometry.Scheme.Spec_map_presheaf_map_eqToHom + - AlgebraicGeometry.basicOpen_eq_of_affine + - AlgebraicGeometry.Scheme.inv_val_c_app_top + - AlgebraicGeometry.Scheme.id_app + - AlgebraicGeometry.Scheme.mem_basicOpen_top' + - AlgebraicGeometry.Scheme.app_eq + - AlgebraicGeometry.Scheme.inv_val_c_app + - AlgebraicGeometry.Scheme.presheaf_map_eqToHom_op + - AlgebraicGeometry.Scheme.congr_app + - AlgebraicGeometry.Scheme.comp_val_base_apply + - path: Mathlib/Topology/Algebra/Order/LiminfLimsup.lean + theorems: + - Antitone.map_limsSup_of_continuousAt + - tendsto_of_no_upcrossings + - limsup_eq_tendsto_sum_indicator_atTop + - limsup_eq_bot + - limsup_eq_tendsto_sum_indicator_nat_atTop + - iInf_eq_of_forall_le_of_tendsto + - eventually_le_limsup + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean + theorems: + - Real.two_div_pi_mul_le_sin + - Real.sin_le_two_div_pi_mul + - Real.lt_sin + - Real.cos_le_one_div_sqrt_sq_add_one + - Real.cos_quadratic_upper_bound + - Real.deriv_tan_sub_id + - Real.le_sin + - Real.one_sub_sq_div_two_le_cos + - Real.one_sub_two_div_pi_mul_le_cos + - Real.cos_lt_one_div_sqrt_sq_add_one + - Real.sin_gt_sub_cube + - Real.sin_lt + - Real.sin_le + - Real.lt_tan + - Real.le_tan + - path: Mathlib/Probability/Independence/Conditional.lean + theorems: + - ProbabilityTheory.condIndep_iff + - ProbabilityTheory.condIndepFun_iff_condexp_inter_preimage_eq_mul + - ProbabilityTheory.condIndepFun_iff + - ProbabilityTheory.condIndep_iff_condIndepSets + - ProbabilityTheory.condIndepSets_iff + - ProbabilityTheory.iCondIndepFun_iff_condexp_inter_preimage_eq_mul + - ProbabilityTheory.condIndepSet_iff_condIndep + - ProbabilityTheory.iCondIndepFun.condIndepFun_prod_mk_prod_mk + - ProbabilityTheory.condIndepSets_singleton_iff + - ProbabilityTheory.condIndepFun_iff_condIndepSet_preimage + - ProbabilityTheory.iCondIndepSets_singleton_iff + - ProbabilityTheory.condIndepSet_iff + - ProbabilityTheory.iCondIndepSet_iff + - ProbabilityTheory.iCondIndep_iff_iCondIndepSets + - ProbabilityTheory.iCondIndep_iff + - ProbabilityTheory.iCondIndepSets_iff + - ProbabilityTheory.iCondIndepSet_iff_iCondIndep + - ProbabilityTheory.condIndepFun_iff_condIndep + - ProbabilityTheory.iCondIndepFun_iff_iCondIndep + - ProbabilityTheory.iCondIndepFun_iff + - path: Mathlib/Analysis/Calculus/FDeriv/Analytic.lean + theorems: + - CPolynomialOn.iteratedFDeriv + - HasFPowerSeriesAt.hasStrictFDerivAt + - HasFPowerSeriesOnBall.hasSum_iteratedFDeriv + - AnalyticOn.iterated_deriv + - AnalyticOn.iteratedFDeriv + - ContinuousMultilinearMap.hasFiniteFPowerSeriesOnBall + - HasFPowerSeriesOnBall.iteratedFDeriv_zero_apply_diag + - AnalyticAt.contDiffAt + - AnalyticOn.fderiv + - HasFPowerSeriesOnBall.factorial_smul + - ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries + - CPolynomialOn.fderiv + - HasFPowerSeriesOnBall.factorial_smul' + - HasFiniteFPowerSeriesOnBall.fderiv + - HasFiniteFPowerSeriesOnBall.fderiv' + - CPolynomialOn.iterated_deriv + - HasFPowerSeriesOnBall.fderiv + - ContinuousMultilinearMap.hasFDerivAt + - FormalMultilinearSeries.derivSeries_apply_diag + - ContinuousMultilinearMap.changeOriginSeries_support + - path: Mathlib/Topology/Order/ExtendFrom.lean + theorems: + - continuousOn_Ioc_extendFrom_Ioo + - continuousOn_Icc_extendFrom_Ioo + - eq_lim_at_right_extendFrom_Ioo + - continuousOn_Ico_extendFrom_Ioo + - eq_lim_at_left_extendFrom_Ioo + - path: Mathlib/NumberTheory/Cyclotomic/Gal.lean + theorems: + - IsPrimitiveRoot.autToPow_injective + - IsCyclotomicExtension.fromZetaAut_spec + - path: Mathlib/Analysis/Calculus/BumpFunction/Normed.lean + theorems: + - ContDiffBump.integral_pos + - ContDiffBump.normed_neg + - ContDiffBump.support_normed_eq + - ContDiffBump.hasCompactSupport_normed + - ContDiffBump.normed_sub + - ContDiffBump.normed_le_div_measure_closedBall_rIn + - ContDiffBump.integral_normed + - ContDiffBump.tendsto_support_normed_smallSets + - ContDiffBump.tsupport_normed_eq + - ContDiffBump.integral_normed_smul + - ContDiffBump.normed_le_div_measure_closedBall_rOut + - ContDiffBump.measure_closedBall_le_integral + - ContDiffBump.integral_le_measure_closedBall + - path: Mathlib/Algebra/Regular/SMul.lean + theorems: + - IsSMulRegular.not_zero_iff + - IsSMulRegular.mul_iff + - IsUnit.isSMulRegular + - isSMulRegular_of_group + - IsSMulRegular.of_mul_eq_one + - IsSMulRegular.of_smul_eq_one + - IsSMulRegular.one + - IsSMulRegular.of_smul + - IsSMulRegular.pow_iff + - IsSMulRegular.subsingleton + - IsSMulRegular.of_mul + - IsSMulRegular.mul_and_mul_iff + - IsSMulRegular.pow + - path: Mathlib/Analysis/NormedSpace/IsROrC.lean + theorems: + - IsROrC.norm_coe_norm + - LinearMap.bound_of_sphere_bound + - ContinuousLinearMap.opNorm_bound_of_ball_bound + - norm_smul_inv_norm' + - norm_smul_inv_norm + - path: Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean + theorems: + - CategoryTheory.MonoidalOfChosenFiniteProducts.associator_naturality + - CategoryTheory.MonoidalOfChosenFiniteProducts.tensor_id + - CategoryTheory.MonoidalOfChosenFiniteProducts.rightUnitor_naturality + - CategoryTheory.MonoidalOfChosenFiniteProducts.tensor_comp + - CategoryTheory.MonoidalOfChosenFiniteProducts.pentagon + - CategoryTheory.MonoidalOfChosenFiniteProducts.triangle + - CategoryTheory.MonoidalOfChosenFiniteProducts.leftUnitor_naturality + - path: Mathlib/Algebra/GroupPower/Identities.lean + theorems: + - sq_add_sq_mul_sq_add_sq + - sum_eight_sq_mul_sum_eight_sq + - pow_four_add_four_mul_pow_four + - sq_add_mul_sq_mul_sq_add_mul_sq + - sum_four_sq_mul_sum_four_sq + - pow_four_add_four_mul_pow_four' + - path: Mathlib/Data/List/Join.lean + theorems: + - List.drop_take_succ_join_eq_nthLe + - List.join_eq_nil + - List.join_singleton + - List.join_filter_not_isEmpty + - List.join_drop_length_sub_one + - List.drop_take_succ_eq_cons_get + - List.bind_eq_nil + - List.sublist_join + - List.append_join_map_append + - List.join_reverse + - List.take_sum_join + - List.drop_take_succ_eq_cons_nthLe + - List.reverse_join + - List.drop_sum_join + - List.drop_take_succ_join_eq_get + - List.join_join + - List.eq_iff_join_eq + - List.sum_take_map_length_lt1 + - List.join_append + - List.length_bind + - List.join_concat + - List.join_filter_ne_nil + - List.nthLe_join + - List.sum_take_map_length_lt2 + - path: Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean + theorems: + - TensorProduct.gradedComm_of_zero_tmul + - TensorProduct.gradedComm_gradedMul + - TensorProduct.gradedComm_of_tmul_of + - TensorProduct.one_gradedMul + - TensorProduct.gradedMul_assoc + - TensorProduct.algebraMap_gradedMul + - TensorProduct.gradedMul_algebraMap + - TensorProduct.gradedComm_tmul_of_zero + - TensorProduct.gradedCommAux_comp_gradedCommAux + - TensorProduct.tmul_of_gradedMul_of_tmul + - TensorProduct.gradedComm_symm + - TensorProduct.gradedMul_one + - TensorProduct.gradedCommAux_lof_tmul + - path: Mathlib/Order/Filter/Ultrafilter.lean + theorems: + - Filter.tendsto_iff_ultrafilter + - Ultrafilter.le_of_inf_neBot' + - Ultrafilter.coe_injective + - Ultrafilter.eq_pure_of_finite_mem + - Ultrafilter.comap_pure + - Filter.mem_iff_ultrafilter + - Ultrafilter.finite_sUnion_mem_iff + - Ultrafilter.union_mem_iff + - Ultrafilter.ofComapInfPrincipal_eq_of_map + - Ultrafilter.comap_comap + - Ultrafilter.eventually_imp + - Ultrafilter.disjoint_iff_not_le + - Filter.iSup_ultrafilter_le_eq + - Ultrafilter.finite_biUnion_mem_iff + - Ultrafilter.comap_id + - Ultrafilter.le_sup_iff + - Ultrafilter.ofComapInfPrincipal_mem + - Filter.atTop_eq_pure_of_isTop + - Ultrafilter.compl_mem_iff_not_mem + - Ultrafilter.compl_not_mem_iff + - Filter.forall_neBot_le_iff + - Filter.NeBot.eq_pure_iff + - path: Mathlib/Geometry/Manifold/VectorBundle/Hom.lean + theorems: + - smoothOn_continuousLinearMapCoordChange + - hom_chart + - path: Mathlib/NumberTheory/FLT/Basic.lean + theorems: + - not_fermatLastTheoremFor_one + - fermatLastTheoremFor_zero + - not_fermatLastTheoremFor_two + - FermatLastTheoremFor.mono + - fermatLastTheoremWith_of_fermatLastTheoremWith_coprime + - FermatLastTheoremWith.mono + - fermatLastTheoremWith_nat_int_rat_tfae + - path: Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean + theorems: + - MeasureTheory.snorm'_le_snorm'_of_exponent_le + - MeasureTheory.snorm_le_snorm_mul_rpow_measure_univ + - MeasureTheory.snorm_le_snorm_top_mul_snorm + - MeasureTheory.snorm'_le_snormEssSup + - "MeasureTheory.Mem\u2112p.smul_of_top_right" + - MeasureTheory.snorm'_le_snormEssSup_mul_rpow_measure_univ + - "MeasureTheory.Mem\u2112p.smul_of_top_left" + - MeasureTheory.snorm_le_snorm_of_exponent_le + - MeasureTheory.snorm'_le_snorm'_mul_rpow_measure_univ + - "MeasureTheory.Mem\u2112p.mem\u2112p_of_exponent_le" + - MeasureTheory.snorm'_le_snorm'_mul_snorm' + - MeasureTheory.snorm_le_snorm_mul_snorm_top + - MeasureTheory.snorm'_lt_top_of_snorm'_lt_top_of_exponent_le + - path: Mathlib/Topology/MetricSpace/GromovHausdorff.lean + theorems: + - GromovHausdorff.ghDist_eq_hausdorffDist + - GromovHausdorff.totallyBounded + - GromovHausdorff.ghDist_le_hausdorffDist + - GromovHausdorff.dist_ghDist + - GromovHausdorff.ghDist_le_nonemptyCompacts_dist + - GromovHausdorff.eq_toGHSpace_iff + - GromovHausdorff.GHSpace.toGHSpace_rep + - GromovHausdorff.hausdorffDist_optimal + - GromovHausdorff.toGHSpace_eq_toGHSpace_iff_isometryEquiv + - GromovHausdorff.ghDist_le_of_approx_subsets + - path: Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean + theorems: + - MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets + - ENNReal.aemeasurable_of_exist_almost_disjoint_supersets + - path: Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean + theorems: + - ContinuousAffineMap.to_affine_map_contLinear + - ContinuousAffineMap.coe_contLinear_eq_linear + - ContinuousAffineMap.norm_eq + - ContinuousAffineMap.contLinear_eq_zero_iff_exists_const + - ContinuousAffineMap.decomp + - ContinuousAffineMap.norm_comp_le + - path: Mathlib/RingTheory/Ideal/MinimalPrime.lean + theorems: + - IsLocalization.AtPrime.prime_unique_of_minimal + - Ideal.minimalPrimes_eq_subsingleton + - minimalPrimes_eq_minimals + - Localization.AtPrime.nilpotent_iff_mem_maximal_of_minimal + - Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective + - Ideal.radical_minimalPrimes + - Ideal.exists_minimalPrimes_comap_eq + - Ideal.minimalPrimes_eq_subsingleton_self + - Ideal.exists_comap_eq_of_mem_minimalPrimes + - Ideal.minimal_primes_comap_of_surjective + - Ideal.sInf_minimalPrimes + - Ideal.minimalPrimes_eq_comap + - Localization.AtPrime.nilpotent_iff_not_unit_of_minimal + - Ideal.exists_minimalPrimes_le + - Ideal.comap_minimalPrimes_eq_of_surjective + - path: Mathlib/Analysis/Convex/Cone/InnerDual.lean + theorems: + - pointed_innerDualCone + - innerDualCone_insert + - isClosed_innerDualCone + - ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem + - innerDualCone_iUnion + - innerDualCone_sUnion + - innerDualCone_eq_iInter_innerDualCone_singleton + - innerDualCone_univ + - ConvexCone.innerDualCone_of_innerDualCone_eq_self + - ConvexCone.pointed_of_nonempty_of_isClosed + - path: Mathlib/Algebra/MonoidAlgebra/Degree.lean + theorems: + - AddMonoidAlgebra.supDegree_zero + - AddMonoidAlgebra.supDegree_single + - AddMonoidAlgebra.sup_support_pow_le + - AddMonoidAlgebra.le_inf_support_pow + - AddMonoidAlgebra.le_inf_support_list_prod + - AddMonoidAlgebra.supDegree_neg + - AddMonoidAlgebra.apply_eq_zero_of_not_le_supDegree + - AddMonoidAlgebra.sup_support_list_prod_le + - AddMonoidAlgebra.supDegree_single_ne_zero + - AddMonoidAlgebra.le_inf_support_finset_prod + - AddMonoidAlgebra.supDegree_withBot_some_comp + - AddMonoidAlgebra.supDegree_sub_le + - AddMonoidAlgebra.sup_support_multiset_prod_le + - AddMonoidAlgebra.supDegree_prod_le + - AddMonoidAlgebra.le_inf_support_multiset_prod + - AddMonoidAlgebra.infDegree_withTop_some_comp + - AddMonoidAlgebra.apply_add_of_supDegree_le + - path: Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean + theorems: + - "HomogeneousIdeal.toIdeal_iInf\u2082" + - Ideal.IsHomogeneous.toIdeal_homogeneousHull_eq_self + - Ideal.IsHomogeneous.iInf + - Ideal.IsHomogeneous.iSup + - Ideal.IsHomogeneous.toIdeal_homogeneousCore_eq_self + - Ideal.mul_homogeneous_element_mem_of_mem + - Ideal.IsHomogeneous.sInf + - Ideal.homogeneousHull_eq_iSup + - Ideal.homogeneousHull_mono + - Ideal.IsHomogeneous.sSup + - Ideal.IsHomogeneous.sup + - Ideal.toIdeal_homogeneousHull_eq_iSup + - HomogeneousIdeal.toIdeal_iInf + - Ideal.IsHomogeneous.mul + - "HomogeneousIdeal.toIdeal_iSup\u2082" + - Ideal.IsHomogeneous.bot + - Ideal.le_toIdeal_homogeneousHull + - Ideal.IsHomogeneous.iff_exists + - Ideal.IsHomogeneous.top + - Ideal.homogeneous_span + - HomogeneousIdeal.toIdeal_injective + - HomogeneousIdeal.toIdeal_homogeneousCore_eq_self + - Ideal.homogeneousCore'_eq_sSup + - HomogeneousIdeal.toIdeal_iSup + - path: Mathlib/Dynamics/Ergodic/AddCircle.lean + theorems: + - AddCircle.ergodic_nsmul + - AddCircle.ergodic_zsmul_add + - AddCircle.ergodic_nsmul_add + - AddCircle.ergodic_zsmul + - AddCircle.ae_empty_or_univ_of_forall_vadd_ae_eq_self + - path: Mathlib/CategoryTheory/Localization/FiniteProducts.lean + theorems: + - CategoryTheory.Localization.HasProductsOfShapeAux.adj_counit_app + - path: Mathlib/Combinatorics/Additive/SalemSpencer.lean + theorems: + - le_mulRothNumber_product + - mulRothNumber_singleton + - mulSalemSpencer_mul_right_iff + - mulRothNumber_map_mul_left + - mulSalemSpencer_mul_left_iff + - "mulSalemSpencer_mul_right_iff\u2080" + - "MulSalemSpencer.mul_left\u2080" + - mulSalemSpencer_pair + - mulRothNumber_union_le + - MulSalemSpencer.image + - mulRothNumber_map_mul_right + - addSalemSpencer_frontier + - mulSalemSpencer_insert + - "mulSalemSpencer_mul_left_iff\u2080" + - MulSalemSpencer.mul_left + - mulRothNumber_spec + - "MulSalemSpencer.mul_right\u2080" + - addRothNumber_Ico + - mulRothNumber_lt_of_forall_not_mulSalemSpencer + - addSalemSpencer_iff_eq_right + - rothNumberNat_isBigOWith_id + - addSalemSpencer_sphere + - mulSalemSpencer_insert_of_lt + - rothNumberNat_add_le + - path: Mathlib/Algebra/Group/Conj.lean + theorems: + - isConj_iff_conjugatesOf_eq + - isConj_iff_eq + - conj_zpow + - ConjClasses.mem_carrier_iff_mk_eq + - conj_pow + - isConj_one_right + - MonoidHom.map_isConj + - ConjClasses.map_surjective + - "isConj_iff\u2080" + - path: Mathlib/Analysis/Convex/GaugeRescale.lean + theorems: + - gaugeRescale_gaugeRescale + - gauge_gaugeRescale' + - gauge_gaugeRescale + - exists_homeomorph_image_eq + - gaugeRescale_self_apply + - gaugeRescale_smul + - image_gaugeRescaleHomeomorph_closure + - continuous_gaugeRescale + - mapsTo_gaugeRescale_interior + - gauge_gaugeRescale_le + - exists_homeomorph_image_interior_closure_frontier_eq_unitBall + - path: Mathlib/Data/Fintype/Sigma.lean + theorems: + - Set.biInter_finsetSigma_univ' + - Set.biUnion_finsetSigma_univ + - Set.biInter_finsetSigma_univ + - Set.biUnion_finsetSigma_univ' + - path: Mathlib/Combinatorics/SimpleGraph/Acyclic.lean + theorems: + - SimpleGraph.IsAcyclic.path_unique + - SimpleGraph.IsTree.card_edgeFinset + - SimpleGraph.isAcyclic_iff_forall_edge_isBridge + - SimpleGraph.isTree_iff_existsUnique_path + - SimpleGraph.isAcyclic_of_path_unique + - SimpleGraph.isAcyclic_iff_forall_adj_isBridge + - path: Mathlib/Deprecated/Group.lean + theorems: + - IsAddGroupHom.sub + - IsMulHom.mul + - IsMulHom.comp + - IsMulHom.to_isMonoidHom + - IsMonoidHom.comp + - IsGroupHom.map_div + - IsGroupHom.injective_iff + - IsGroupHom.map_inv + - path: Mathlib/Topology/Compactness/Paracompact.lean + theorems: + - refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set + - precise_refinement_set + - ClosedEmbedding.paracompactSpace + - precise_refinement + - path: Mathlib/Probability/Kernel/Condexp.lean + theorems: + - ProbabilityTheory.condexp_ae_eq_integral_condexpKernel + - ProbabilityTheory.condexpKernel_apply_eq_condDistrib + - MeasureTheory.Integrable.comp_snd_map_prod_id + - MeasureTheory.Integrable.integral_norm_condexpKernel + - ProbabilityTheory.integrable_toReal_condexpKernel + - MeasureTheory.AEStronglyMeasurable.integral_condexpKernel + - ProbabilityTheory.condexp_ae_eq_integral_condexpKernel' + - MeasureTheory.Integrable.integral_condexpKernel + - ProbabilityTheory.aestronglyMeasurable'_integral_condexpKernel + - MeasureTheory.Integrable.condexpKernel_ae + - MeasureTheory.Integrable.norm_integral_condexpKernel + - ProbabilityTheory.condexpKernel_ae_eq_condexp + - ProbabilityTheory.condexpKernel_ae_eq_condexp' + - ProbabilityTheory.condexpKernel_ae_eq_trim_condexp + - ProbabilityTheory.measurable_condexpKernel + - path: Mathlib/Analysis/Calculus/Deriv/Pi.lean + theorems: + - hasDerivAt_update + - path: Mathlib/Topology/ContinuousFunction/UniqueCFC.lean + theorems: + - ContinuousMap.toNNReal_mul_add_neg_mul_add_mul_neg_eq + - IsROrC.uniqueContinuousFunctionalCalculus_of_compactSpace_spectrum + - StarAlgHom.continuous_realContinuousMapOfNNReal + - ContinuousMap.toNNReal_add_add_neg_add_neg_eq + - ContinuousMap.toNNReal_neg_algebraMap + - ContinuousMap.toNNReal_algebraMap + - StarAlgHom.realContinuousMapOfNNReal_apply_comp_toReal + - StarAlgHom.realContinuousMapOfNNReal_injective + - path: Mathlib/Combinatorics/SimpleGraph/Operations.lean + theorems: + - SimpleGraph.card_edgeFinset_sup_edge + - SimpleGraph.edgeFinset_replaceVertex_of_not_adj + - SimpleGraph.replaceVertex_self + - SimpleGraph.edge_adj + - SimpleGraph.edgeSet_replaceVertex_of_adj + - SimpleGraph.edgeFinset_replaceVertex_of_adj + - SimpleGraph.edge_self_eq_bot + - SimpleGraph.disjoint_sdiff_neighborFinset_image + - SimpleGraph.edgeSet_replaceVertex_of_not_adj + - SimpleGraph.adj_replaceVertex_iff_of_ne_right + - SimpleGraph.sup_edge_of_adj + - SimpleGraph.not_adj_replaceVertex_same + - SimpleGraph.card_edgeFinset_replaceVertex_of_not_adj + - SimpleGraph.adj_replaceVertex_iff_of_ne_left + - SimpleGraph.card_edgeFinset_replaceVertex_of_adj + - SimpleGraph.adj_replaceVertex_iff_of_ne + - SimpleGraph.edge_edgeSet_of_ne + - SimpleGraph.sup_edge_self + - SimpleGraph.edgeFinset_sup_edge + - path: Mathlib/AlgebraicGeometry/Properties.lean + theorems: + - AlgebraicGeometry.isIntegralOfIsAffineIsDomain + - AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot + - AlgebraicGeometry.isIntegralOfIsIrreducibleIsReduced + - AlgebraicGeometry.reduce_to_affine_global + - AlgebraicGeometry.isReducedOfStalkIsReduced + - AlgebraicGeometry.map_injective_of_isIntegral + - AlgebraicGeometry.isIntegralOfOpenImmersion + - AlgebraicGeometry.isReducedOfOpenImmersion + - AlgebraicGeometry.isReducedOfIsAffineIsReduced + - AlgebraicGeometry.affine_isReduced_iff + - AlgebraicGeometry.reduce_to_affine_nbhd + - path: Mathlib/Topology/StoneCech.lean + theorems: + - ultrafilter_comap_pure_nhds + - ultrafilter_pure_injective + - ultrafilterBasis_is_basis + - ultrafilter_converges_iff + - ultrafilter_extend_extends + - ultrafilter_isClosed_basic + - Ultrafilter.tendsto_pure_self + - convergent_eqv_pure + - ultrafilter_extend_eq_iff + - continuous_ultrafilter_extend + - induced_topology_pure + - stoneCech_hom_ext + - continuous_stoneCechUnit + - path: Mathlib/Analysis/Normed/MulAction.lean + theorems: + - BoundedSMul.of_norm_smul_le + - "edist_smul\u2080" + - norm_smul_le + - dist_smul_le + - "dist_smul\u2080" + - norm_smul + - path: Mathlib/Algebra/GroupPower/IterateHom.lean + theorems: + - smul_iterate + - smul_iterate_apply + - SemiconjBy.function_semiconj_mul_left + - SemiconjBy.function_semiconj_mul_right_swap + - mul_right_iterate_apply_one + - hom_coe_pow + - path: Mathlib/Order/SuccPred/CompleteLinearOrder.lean + theorems: + - sSup_mem_of_not_isSuccLimit + - csInf_mem_of_not_isPredLimit + - csSup_mem_of_not_isSuccLimit' + - IsLUB.mem_of_not_isSuccLimit + - csSup_mem_of_not_isSuccLimit + - sInf_mem_of_not_isPredLimit + - path: Mathlib/Topology/Algebra/Module/LocallyConvex.lean + theorems: + - Disjoint.exists_open_convexes + - locallyConvexSpace_induced + - locallyConvexSpace_iInf + - locallyConvexSpace_inf + - locallyConvexSpace_sInf + - LocallyConvexSpace.ofBasisZero + - path: Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean + theorems: + - AbsoluteValue.IsAdmissible.exists_approx_aux + - AbsoluteValue.IsAdmissible.exists_partition + - AbsoluteValue.IsAdmissible.exists_approx + - path: Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean + theorems: + - norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin + - iteratedDeriv_succ' + - norm_iteratedFDeriv_eq_norm_iteratedDeriv + - iteratedDeriv_eq_equiv_comp + - iteratedDeriv_eq_iterate + - iteratedDerivWithin_succ + - iteratedDeriv_zero + - contDiff_iff_iteratedDeriv + - iteratedFDeriv_apply_eq_iteratedDeriv_mul_prod + - iteratedFDerivWithin_apply_eq_iteratedDerivWithin_mul_prod + - iteratedDerivWithin_eq_iterate + - iteratedDeriv_one + - iteratedFDeriv_eq_equiv_comp + - ContDiffOn.differentiableOn_iteratedDerivWithin + - iteratedDerivWithin_succ' + - contDiffOn_of_differentiableOn_deriv + - contDiffOn_iff_continuousOn_differentiableOn_deriv + - iteratedDeriv_succ + - contDiffOn_of_continuousOn_differentiableOn_deriv + - iteratedFDerivWithin_eq_equiv_comp + - iteratedDerivWithin_zero + - ContDiffOn.continuousOn_iteratedDerivWithin + - ContDiffWithinAt.differentiableWithinAt_iteratedDerivWithin + - iteratedDerivWithin_eq_equiv_comp + - iteratedDerivWithin_univ + - iteratedDerivWithin_one + - path: Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean + theorems: + - ae_eq_of_integral_smooth_smul_eq + - ae_eq_zero_of_integral_smooth_smul_eq_zero + - IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero' + - path: Mathlib/Analysis/Complex/Tietze.lean + theorems: + - Metric.instTietzeExtensionClosedBall + - BoundedContinuousFunction.exists_norm_eq_restrict_eq + - path: Mathlib/Analysis/Complex/TaylorSeries.lean + theorems: + - Complex.taylorSeries_eq_on_ball' + - Complex.hasSum_taylorSeries_on_ball + - Complex.taylorSeries_eq_of_entire' + - Complex.hasSum_taylorSeries_on_emetric_ball + - Complex.taylorSeries_eq_on_emetric_ball' + - path: Mathlib/CategoryTheory/Monoidal/Tor.lean + theorems: + - CategoryTheory.isZero_Tor'_succ_of_projective + - CategoryTheory.Tor'_map_app' + - CategoryTheory.isZero_Tor_succ_of_projective + - path: Mathlib/Algebra/Ring/Defs.lean + theorems: + - boole_mul + - mul_neg_one + - mul_one_add + - neg_mul_neg + - neg_one_mul + - mul_sub_one + - neg_eq_neg_one_mul + - mul_ite + - ite_mul + - neg_mul_comm + - ite_zero_mul + - add_mul_self_eq + - sub_one_mul + - mul_boole + - mul_ite_zero + - one_add_mul + - ite_add_ite + - distrib_three_right + - add_one_mul + - mul_one_sub + - mul_two + - mul_sub_right_distrib + - mul_sub_left_distrib + - one_sub_mul + - two_mul + - ite_zero_mul_ite_zero + - mul_add_one + - path: Mathlib/Algebra/Homology/Additive.lean + theorems: + - CategoryTheory.NatTrans.mapHomologicalComplex_naturality + - CategoryTheory.NatTrans.mapHomologicalComplex_comp + - CategoryTheory.NatTrans.mapHomologicalComplex_id + - HomologicalComplex.singleMapHomologicalComplex_inv_app_self + - HomologicalComplex.singleMapHomologicalComplex_hom_app_self + - ChainComplex.map_chain_complex_of + - HomologicalComplex.singleMapHomologicalComplex_inv_app_ne + - path: Mathlib/RingTheory/Subring/Basic.lean + theorems: + - Subring.coe_sSup_of_directedOn + - Subring.coe_iSup_of_directed + - Subring.mem_map + - Subring.prod_top + - Subring.mem_iSup_of_directed + - Subring.top_prod + - Subring.exists_list_of_mem_closure + - Subring.mem_closure_iff + - AddSubgroup.int_mul_mem + - "Subring.closure_induction\u2082" + - RingHom.range_top_iff_surjective + - Subring.closure_induction' + - Subring.mem_sSup_of_directedOn + - Subring.mem_iInf + - RingHom.range_eq_map + - RingHom.map_range + - Subring.coe_iInf + - Subring.InClosure.recOn + - path: Mathlib/Topology/UniformSpace/Completion.lean + theorems: + - CauchyFilter.comp_gen + - CauchyFilter.symm_gen + - CauchyFilter.mem_uniformity' + - CauchyFilter.uniformContinuous_extend + - CauchyFilter.inseparable_iff_of_le_nhds + - CauchyFilter.nonempty_cauchyFilter_iff + - UniformSpace.Completion.extension_map + - CauchyFilter.extend_pureCauchy + - CauchyFilter.denseRange_pureCauchy + - CauchyFilter.cauchyFilter_eq + - path: Mathlib/Data/Fintype/Option.lean + theorems: + - Fintype.induction_empty_option + - Finite.induction_empty_option + - Fintype.card_option + - path: Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean + theorems: + - BoxIntegral.Box.upper_sub_lower_splitCenterBox + - BoxIntegral.Box.mem_splitCenterBox + - BoxIntegral.Box.iUnion_coe_splitCenterBox + - BoxIntegral.Box.subbox_induction_on' + - BoxIntegral.Box.disjoint_splitCenterBox + - path: Mathlib/RingTheory/Polynomial/Chebyshev.lean + theorems: + - Polynomial.Chebyshev.U_add_two + - Polynomial.Chebyshev.T_eq_X_mul_T_sub_pol_U + - Polynomial.Chebyshev.T_derivative_eq_U + - Polynomial.Chebyshev.U_two + - Polynomial.Chebyshev.T_two + - Polynomial.Chebyshev.T_mul + - Polynomial.Chebyshev.U_eq_X_mul_U_add_T + - Polynomial.Chebyshev.T_add_two + - Polynomial.Chebyshev.map_U + - Polynomial.Chebyshev.add_one_mul_T_eq_poly_in_U + - Polynomial.Chebyshev.one_sub_X_sq_mul_U_eq_pol_in_T + - Polynomial.Chebyshev.mul_T + - Polynomial.Chebyshev.T_eq_U_sub_X_mul_U + - Polynomial.Chebyshev.U_of_two_le + - Polynomial.Chebyshev.T_of_two_le + - Polynomial.Chebyshev.one_sub_X_sq_mul_derivative_T_eq_poly_in_T + - Polynomial.Chebyshev.map_T + - path: .lake/packages/std/Std/Data/UnionFind/Basic.lean + theorems: + - Std.UnionFind.linkAux_size + - Std.UnionFind.rank'_lt_rankMax + - Std.UnionFind.lt_rank_root + - Std.UnionFind.rankD_lt_rankMax + - Std.UnionFind.parentD_findAux_lt + - Std.UnionFind.push_rankD + - Std.UnionFind.lt_rankD_findAux + - Std.UnionFind.findAux_root + - Std.UnionFind.setParentBump_rankD_lt + - Std.UnionFind.rankD_set + - Std.UnionFind.rankD_findAux + - Std.UnionFind.find_root_1 + - Std.UnionFind.rank'_lt + - Std.UnionFind.parent'_lt + - Std.UnionFind.parentD_findAux_rootD + - Std.UnionFind.find_size + - Std.UnionFind.find_root_2 + - Std.UnionFind.findAux_s + - Std.UnionFind.parentD_set + - Std.UnionFind.rootD_ext + - Std.UnionFind.rootD_lt + - Std.UnionFind.parentD_findAux + - Std.UnionFind.parentD_findAux_or + - Std.UnionFind.parent_rootD + - Std.UnionFind.setParent_rankD_lt + - Std.UnionFind.rank_lt + - Std.UnionFind.parent_root + - Std.UnionFind.rootD_parent + - Std.UnionFind.le_rank_root + - Std.UnionFind.push_parentD + - Std.UnionFind.parent_lt + - Std.UnionFind.rootD_eq_self + - Std.UnionFind.find_parent_1 + - path: Mathlib/MeasureTheory/Integral/Pi.lean + theorems: + - MeasureTheory.integral_fintype_prod_eq_prod + - MeasureTheory.Integrable.fintype_prod_dep + - MeasureTheory.integral_fintype_prod_eq_pow + - MeasureTheory.integral_fin_nat_prod_eq_prod + - MeasureTheory.Integrable.fin_nat_prod + - path: Mathlib/Algebra/Lie/Engel.lean + theorems: + - LieModule.isNilpotent_iff_forall' + - LieSubmodule.lie_top_eq_of_span_sup_eq_top + - LieSubmodule.exists_smul_add_of_span_sup_eq_top + - LieSubmodule.isNilpotentOfIsNilpotentSpanSupEqTop + - LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer + - LieAlgebra.isEngelian_of_isNoetherian + - LieSubmodule.lcs_le_lcs_of_is_nilpotent_span_sup_eq_top + - LieAlgebra.isEngelian_of_subsingleton + - Function.Surjective.isEngelian + - path: Mathlib/RingTheory/Localization/Integer.lean + theorems: + - IsLocalization.finsetIntegerMultiple_image + - IsLocalization.exist_integer_multiples + - IsLocalization.isInteger_smul + - IsLocalization.exists_integer_multiple + - IsLocalization.exist_integer_multiples_of_finite + - path: Mathlib/Data/Bundle.lean + theorems: + - Bundle.TotalSpace.mk_inj + - Bundle.TotalSpace.mk_cast + - Bundle.TotalSpace.range_mk + - path: Mathlib/Analysis/Convex/Join.lean + theorems: + - convexJoin_iUnion_left + - convexJoin_singleton_right + - convexJoin_union_left + - convexHull_insert + - Convex.convexJoin + - convexJoin_comm + - convexJoin_iUnion_right + - convexJoin_empty_right + - convexHull_union + - convexJoin_singleton_left + - convexJoin_singleton_segment + - convexJoin_assoc + - convexJoin_segments + - convexJoin_assoc_aux + - convexJoin_empty_left + - convexJoin_right_comm + - convexJoin_union_right + - convexJoin_singletons + - mem_convexJoin + - convexJoin_convexJoin_convexJoin_comm + - convexJoin_segment_singleton + - convexJoin_left_comm + - path: Mathlib/RingTheory/OreLocalization/Basic.lean + theorems: + - OreLocalization.add''_char + - OreLocalization.universalMulHom_commutes + - OreLocalization.div_eq_one' + - OreLocalization.div_one_mul + - OreLocalization.ind + - OreLocalization.left_distrib + - OreLocalization.numeratorHom_inj + - OreLocalization.one_mul + - OreLocalization.mul'_char + - OreLocalization.mul_inv_cancel + - OreLocalization.nontrivial_of_nonZeroDivisors + - OreLocalization.add_left_neg + - OreLocalization.oreDiv_mul_oreDiv_comm + - OreLocalization.mul_cancel' + - OreLocalization.zero_add + - OreLocalization.add_oreDiv + - OreLocalization.expand' + - OreLocalization.add_assoc + - OreLocalization.add'_comm + - OreLocalization.eq_of_num_factor_eq + - OreLocalization.mul_one_div + - OreLocalization.mul_one + - OreLocalization.zero_mul + - OreLocalization.universalMulHom_unique + - OreLocalization.mul_assoc + - OreLocalization.right_distrib + - OreLocalization.inv_zero + - OreLocalization.mul_zero + - OreLocalization.universalHom_commutes + - OreLocalization.mul_inv + - OreLocalization.add_comm + - OreLocalization.expand + - OreLocalization.mul_cancel + - OreLocalization.zero_div_eq_zero + - path: Mathlib/MeasureTheory/Measure/Tilted.lean + theorems: + - MeasureTheory.toReal_rnDeriv_tilted_right + - MeasureTheory.set_lintegral_tilted' + - MeasureTheory.tilted_apply + - MeasureTheory.rnDeriv_tilted_left + - MeasureTheory.lintegral_tilted + - MeasureTheory.integral_tilted + - MeasureTheory.set_lintegral_tilted + - MeasureTheory.tilted_zero + - MeasureTheory.toReal_rnDeriv_tilted_left + - MeasureTheory.tilted_of_not_integrable + - MeasureTheory.isProbabilityMeasure_tilted + - MeasureTheory.tilted_eq_withDensity_nnreal + - MeasureTheory.log_rnDeriv_tilted_left_self + - MeasureTheory.set_integral_tilted' + - MeasureTheory.tilted_const + - MeasureTheory.tilted_apply_eq_ofReal_integral' + - MeasureTheory.integral_exp_tilted + - MeasureTheory.tilted_comm + - MeasureTheory.tilted_zero' + - MeasureTheory.rnDeriv_tilted_left_self + - MeasureTheory.tilted_neg_same' + - MeasureTheory.tilted_zero_measure + - MeasureTheory.tilted_of_not_aemeasurable + - MeasureTheory.absolutelyContinuous_tilted + - MeasureTheory.tilted_apply_eq_ofReal_integral + - MeasureTheory.set_integral_tilted + - MeasureTheory.tilted_tilted + - MeasureTheory.rnDeriv_tilted_right + - MeasureTheory.tilted_congr + - MeasureTheory.tilted_const' + - MeasureTheory.tilted_apply' + - path: Mathlib/Algebra/Homology/BifunctorHomotopy.lean + theorems: + - "HomologicalComplex.mapBifunctorMapHomotopy.\u03B9MapBifunctor_hom\u2081" + - "HomologicalComplex.mapBifunctorMapHomotopy.comm\u2081" + - "HomologicalComplex.mapBifunctorMapHomotopy.comm\u2081_aux" + - "HomologicalComplex.mapBifunctorMapHomotopy.zero\u2081" + - path: Mathlib/Probability/ProbabilityMassFunction/Monad.lean + theorems: + - PMF.bindOnSupport_bindOnSupport + - PMF.toMeasure_pure + - PMF.bindOnSupport_eq_bind + - PMF.pure_bindOnSupport + - PMF.mem_support_pure_iff + - PMF.toOuterMeasure_pure_apply + - PMF.mem_support_bindOnSupport_iff + - PMF.toOuterMeasure_bindOnSupport_apply + - PMF.bind_pure + - PMF.support_bindOnSupport + - PMF.toOuterMeasure_bind_apply + - PMF.toPMF_dirac + - PMF.pure_bind + - PMF.bind_bind + - PMF.support_bind + - PMF.toMeasure_bindOnSupport_apply + - PMF.bind_const + - PMF.bindOnSupport_eq_zero_iff + - PMF.bindOnSupport_comm + - PMF.support_pure + - PMF.bindOnSupport_pure + - PMF.bind_comm + - PMF.mem_support_bind_iff + - path: Mathlib/CategoryTheory/Sites/Sheaf.lean + theorems: + - CategoryTheory.Presheaf.w + - CategoryTheory.isSheaf_iff_isSheaf_of_type + - CategoryTheory.Presheaf.isSheaf_iff_isLimit_pretopology + - CategoryTheory.Presheaf.isLimit_iff_isSheafFor + - CategoryTheory.Presheaf.isSheaf_iff_isSheaf' + - CategoryTheory.Presheaf.isSheaf_comp_of_isSheaf + - CategoryTheory.Presheaf.isSheaf_of_isSheaf_comp + - CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget + - CategoryTheory.Presheaf.isSheaf_iff_multifork + - CategoryTheory.Presheaf.IsSheaf.amalgamate_map + - CategoryTheory.Presheaf.subsingleton_iff_isSeparatedFor + - path: Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean + theorems: + - SimpleGraph.singletonFinsubgraph_le_adj_right + - SimpleGraph.nonempty_hom_of_forall_finite_subgraph_hom + - SimpleGraph.singletonFinsubgraph_le_adj_left + - path: Mathlib/RingTheory/Localization/FractionRing.lean + theorems: + - IsFractionRing.mul_inv_cancel + - FractionRing.mk_eq_div + - IsFractionRing.lift_mk' + - IsFractionRing.isFractionRing_iff_of_base_ringEquiv + - IsFractionRing.mk'_eq_zero_iff_eq_zero + - IsFractionRing.div_surjective + - IsFractionRing.mk'_eq_one_iff_eq + - IsFractionRing.nontrivial + - path: Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean + theorems: + - RingHom.PropertyIsLocal.affine_openCover_TFAE + - AlgebraicGeometry.isOpenImmersionCat_comp_of_sourceAffineLocally + - RingHom.PropertyIsLocal.openCover_TFAE + - "RingHom.StableUnderBaseChange.\u0393_pullback_fst" + - AlgebraicGeometry.sourceAffineLocally_isLocal + - RingHom.RespectsIso.basicOpen_iff_localization + - RingHom.PropertyIsLocal.source_openCover_iff + - RingHom.PropertyIsLocal.sourceAffineLocally_of_source_openCover + - RingHom.RespectsIso.basicOpen_iff + - RingHom.PropertyIsLocal.affineLocally_of_comp + - AlgebraicGeometry.scheme_restrict_basicOpen_of_localizationPreserves + - RingHom.RespectsIso.ofRestrict_morphismRestrict_iff + - RingHom.PropertyIsLocal.sourceAffineLocally_comp_of_isOpenImmersion + - RingHom.PropertyIsLocal.source_affine_openCover_iff + - AlgebraicGeometry.sourceAffineLocally_respectsIso + - RingHom.PropertyIsLocal.affineLocally_of_isOpenImmersion + - AlgebraicGeometry.affineLocally_iff_affineOpens_le + - RingHom.PropertyIsLocal.affineLocally_stableUnderComposition + - AlgebraicGeometry.sourceAffineLocally_of_source_open_cover_aux + - path: Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean + theorems: + - "ZMod.eisenstein_lemma_aux\u2081" + - ZMod.eisenstein_lemma_aux + - ZMod.sum_mul_div_add_sum_mul_div_eq_mul + - ZMod.sum_Ico_eq_card_lt + - "ZMod.gauss_lemma_aux\u2081" + - ZMod.div_eq_filter_card + - ZMod.eisenstein_lemma + - ZMod.gauss_lemma + - ZMod.gauss_lemma_aux + - ZMod.Ico_map_valMinAbs_natAbs_eq_Ico_map_id + - path: Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean + theorems: + - ContinuousMultilinearMap.mkPiRing_eq_zero_iff + - ContinuousMultilinearMap.linearDeriv_apply + - ContinuousMultilinearMap.mkPiRing_zero + - ContinuousMultilinearMap.mkPiRing_eq_iff + - ContinuousMultilinearMap.ext_iff + - ContinuousLinearMap.compContinuousMultilinearMap_coe + - ContinuousMultilinearMap.toMultilinearMap_injective + - path: Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean + theorems: + - AffineMap.homothety_neg_one_apply + - AffineEquiv.toAffineMap_injective + - AffineEquiv.injective_pointReflection_left_of_module + - path: Mathlib/Data/Nat/Order/Lemmas.lean + theorems: + - Nat.div_eq_iff_eq_of_dvd_dvd + - Nat.div_ne_zero_iff + - Nat.dvd_div_of_mul_dvd + - Nat.div_lt_div_of_lt_of_dvd + - Nat.dvd_left_iff_eq + - Nat.not_two_dvd_bit1 + - Nat.div_div_div_eq_div + - Nat.not_dvd_iff_between_consec_multiples + - Nat.lt_div_iff_mul_lt + - Nat.div_pos_iff + - Nat.set_eq_univ + - Nat.succ_div_of_not_dvd + - Nat.le_of_lt_add_of_dvd + - Nat.mul_div_eq_iff_dvd + - Nat.succ_div_of_dvd + - Nat.dvd_sub' + - Nat.succ_div + - path: Mathlib/Combinatorics/SetFamily/Compression/UV.lean + theorems: + - UV.mem_compression + - UV.compress_sdiff_sdiff + - UV.aux + - UV.mem_of_mem_compression + - sup_sdiff_injOn + - UV.card_compress + - UV.compress_mem_compression + - UV.sup_sdiff_mem_of_mem_compression_of_not_mem + - UV.compress_of_disjoint_of_le' + - UV.compress_self + - UV.compress_idem + - Set.Sized.uvCompression + - UV.sup_sdiff_mem_of_mem_compression + - UV.card_compression + - UV.disjoint_of_mem_compression_of_not_mem + - UV.compress_injOn + - UV.le_of_mem_compression_of_not_mem + - UV.compression_idem + - UV.compress_mem_compression_of_mem_compression + - path: Mathlib/Analysis/InnerProductSpace/Orthogonal.lean + theorems: + - Submodule.mem_orthogonal_singleton_iff_inner_left + - Submodule.orthogonal_eq_inter + - Submodule.orthogonal_eq_top_iff + - Submodule.IsOrtho.map_iff + - Submodule.inner_left_of_mem_orthogonal + - Submodule.mem_orthogonal_singleton_iff_inner_right + - Submodule.IsOrtho.comap + - Submodule.isOrtho_sSup_right + - Submodule.isOrtho_span + - Submodule.orthogonal_disjoint + - Submodule.inf_orthogonal_eq_bot + - Submodule.bot_orthogonal_eq_top + - Submodule.IsOrtho.map + - Submodule.sub_mem_orthogonal_of_inner_right + - Submodule.top_orthogonal_eq_bot + - Submodule.isClosed_orthogonal + - Submodule.isOrtho_iSup_right + - Submodule.IsOrtho.comap_iff + - Submodule.mem_orthogonal' + - path: Mathlib/Algebra/Module/PID.lean + theorems: + - Module.torsion_by_prime_power_decomposition + - Submodule.isInternal_prime_power_torsion_of_pid + - Module.equiv_directSum_of_isTorsion + - Module.p_pow_smul_lift + - Ideal.torsionOf_eq_span_pow_pOrder + - Submodule.exists_isInternal_prime_power_torsion_of_pid + - Module.equiv_free_prod_directSum + - Module.exists_smul_eq_zero_and_mk_eq + - path: Mathlib/LinearAlgebra/Dimension/LinearMap.lean + theorems: + - LinearMap.rank_comp_le_right + - LinearMap.rank_add_le + - LinearMap.le_rank_iff_exists_linearIndependent_finset + - LinearMap.rank_zero + - LinearMap.rank_comp_le + - LinearMap.le_rank_iff_exists_linearIndependent + - LinearMap.lift_rank_comp_le_right + - LinearMap.rank_comp_le_left + - path: Mathlib/CategoryTheory/Adjunction/Opposites.lean + theorems: + - CategoryTheory.Adjunction.rightAdjointUniq_hom_app_counit + - CategoryTheory.Adjunction.leftAdjointUniq_hom_counit + - CategoryTheory.Adjunction.unit_rightAdjointUniq_hom + - CategoryTheory.Adjunction.unit_leftAdjointUniq_hom + - CategoryTheory.Adjunction.unit_leftAdjointUniq_hom_app + - CategoryTheory.Adjunction.unit_rightAdjointUniq_hom_app + - CategoryTheory.Adjunction.rightAdjointUniq_refl + - CategoryTheory.Adjunction.rightAdjointUniq_trans + - CategoryTheory.Adjunction.leftAdjointUniq_trans + - CategoryTheory.Adjunction.leftAdjointUniq_hom_app_counit + - CategoryTheory.Adjunction.rightAdjointUniq_trans_app + - CategoryTheory.Adjunction.leftAdjointUniq_refl + - CategoryTheory.Adjunction.homEquiv_symm_rightAdjointUniq_hom_app + - CategoryTheory.Adjunction.rightAdjointUniq_hom_counit + - CategoryTheory.Adjunction.leftAdjointUniq_trans_app + - path: Mathlib/Combinatorics/SimpleGraph/Dart.lean + theorems: + - SimpleGraph.dartOfNeighborSet_injective + - SimpleGraph.dart_edge_eq_mk'_iff' + - SimpleGraph.dart_edge_eq_mk'_iff + - SimpleGraph.Dart.ext_iff + - SimpleGraph.dart_edge_eq_iff + - path: Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean + theorems: + - CategoryTheory.exact_f_d + - "CategoryTheory.InjectiveResolution.homotopyEquiv_inv_\u03B9" + - CategoryTheory.InjectiveResolution.descFOne_zero_comm + - CategoryTheory.InjectiveResolution.ofCocomplex_exactAt_succ + - "CategoryTheory.InjectiveResolution.homotopyEquiv_hom_\u03B9" + - CategoryTheory.InjectiveResolution.desc_commutes + - CategoryTheory.InjectiveResolution.iso_hom_naturality + - CategoryTheory.InjectiveResolution.desc_commutes_zero + - CategoryTheory.InjectiveResolution.iso_inv_naturality + - CategoryTheory.InjectiveResolution.ofCocomplex_d_0_1 + - path: Mathlib/CategoryTheory/Abelian/LeftDerived.lean + theorems: + - CategoryTheory.ProjectiveResolution.pOpcycles_comp_fromLeftDerivedZero' + - CategoryTheory.ProjectiveResolution.isoLeftDerivedObj_inv_naturality + - CategoryTheory.ProjectiveResolution.leftDerived_app_eq + - CategoryTheory.Functor.isZero_leftDerived_obj_projective_succ + - CategoryTheory.NatTrans.leftDerived_id + - CategoryTheory.ProjectiveResolution.fromLeftDerivedZero'_naturality + - CategoryTheory.ProjectiveResolution.fromLeftDerivedZero_eq + - CategoryTheory.Functor.leftDerived_map_eq + - CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_hom_naturality + - CategoryTheory.NatTrans.leftDerived_comp + - CategoryTheory.ProjectiveResolution.isoLeftDerivedToHomotopyCategoryObj_inv_naturality + - CategoryTheory.ProjectiveResolution.leftDerivedToHomotopyCategory_app_eq + - path: Mathlib/GroupTheory/Torsion.lean + theorems: + - IsTorsion.of_surjective + - ExponentExists.isTorsion + - isTorsionFree_iff_noZeroSMulDivisors_int + - Monoid.isTorsionFree_iff_torsion_eq_bot + - IsTorsionFree.subgroup + - IsTorsionFree.not_torsion + - CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow + - CommMonoid.primaryComponent.disjoint + - Monoid.IsTorsion.torsion_eq_top + - Monoid.not_isTorsionFree_iff + - IsTorsionFree.quotient_torsion + - isTorsionFree_iff_noZeroSMulDivisors_nat + - IsTorsion.extension_closed + - IsTorsion.not_torsion_free + - CommMonoid.torsion.isTorsion + - AddMonoid.IsTorsion.module_of_torsion + - Monoid.not_isTorsion_iff + - path: Mathlib/MeasureTheory/Group/Convolution.lean + theorems: + - MeasureTheory.Measure.dirac_one_mconv + - MeasureTheory.Measure.mconv_comm + - MeasureTheory.Measure.zero_mconv + - MeasureTheory.Measure.add_mconv + - MeasureTheory.Measure.mconv_dirac_one + - MeasureTheory.Measure.mconv_zero + - path: Mathlib/Topology/MetricSpace/Polish.lean + theorems: + - ClosedEmbedding.polishSpace + - CompletePseudometrizable.iInf + - IsOpen.isClopenable + - PolishSpace.iInf + - IsOpen.polishSpace + - IsClosed.isClopenable + - PolishSpace.IsClopenable.iUnion + - PolishSpace.IsClopenable.compl + - complete_polishSpaceMetric + - path: Mathlib/Data/MvPolynomial/Variables.lean + theorems: + - MvPolynomial.vars_mul + - MvPolynomial.vars_add_of_disjoint + - MvPolynomial.mem_vars_rename + - MvPolynomial.mem_vars + - MvPolynomial.vars_monomial_single + - MvPolynomial.mem_support_not_mem_vars_zero + - MvPolynomial.vars_def + - MvPolynomial.vars_monomial + - MvPolynomial.vars_X + - MvPolynomial.vars_add_subset + - MvPolynomial.vars_C + - MvPolynomial.hom_congr_vars + - MvPolynomial.vars_map + - MvPolynomial.vars_rename + - MvPolynomial.vars_prod + - MvPolynomial.vars_0 + - MvPolynomial.vars_eq_support_biUnion_support + - "MvPolynomial.eval\u2082Hom_eq_constantCoeff_of_vars" + - MvPolynomial.vars_map_of_injective + - MvPolynomial.vars_pow + - MvPolynomial.vars_sum_of_disjoint + - MvPolynomial.vars_sum_subset + - MvPolynomial.vars_C_mul + - MvPolynomial.exists_rename_eq_of_vars_subset_range + - "MvPolynomial.eval\u2082Hom_congr'" + - path: Mathlib/Analysis/Fourier/Inversion.lean + theorems: + - Real.tendsto_integral_cexp_sq_smul + - Real.tendsto_integral_gaussian_smul + - Real.tendsto_integral_gaussian_smul' + - Continuous.fourier_inversion + - path: Mathlib/GroupTheory/QuotientGroup.lean + theorems: + - QuotientGroup.eq_iff_div_mem + - QuotientGroup.kerLift_injective + - QuotientGroup.map_mk'_self + - QuotientGroup.comap_comap_center + - QuotientGroup.eq_one_iff + - QuotientGroup.preimage_image_coe + - QuotientGroup.subsingleton_quotient_top + - QuotientGroup.image_coe_inj + - QuotientGroup.equivQuotientZPowOfEquiv_refl + - QuotientGroup.mk'_eq_mk' + - QuotientGroup.ker_le_range_iff + - QuotientGroup.map_map + - QuotientGroup.rangeKerLift_surjective + - QuotientGroup.rangeKerLift_injective + - QuotientGroup.equivQuotientZPowOfEquiv_trans + - QuotientGroup.congr_refl + - QuotientGroup.subgroup_eq_top_of_subsingleton + - path: Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean + theorems: + - Complex.arctan_tan + - Real.hasSum_arctan + - Complex.hasSum_arctan + - Complex.cos_ne_zero_of_arctan_bounds + - Complex.arg_one_add_mem_Ioo + - Complex.hasSum_arctan_aux + - Complex.tan_arctan + - Complex.ofReal_arctan + - path: Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean + theorems: + - CategoryTheory.Equivalence.preregular + - CategoryTheory.Equivalence.preregular_eq + - CategoryTheory.Equivalence.preregular_isSheaf_iff + - CategoryTheory.Equivalence.precoherent_isSheaf_iff + - CategoryTheory.Equivalence.precoherent + - path: Mathlib/Probability/Independence/Basic.lean + theorems: + - ProbabilityTheory.iIndepFun.isProbabilityMeasure + - ProbabilityTheory.IndepFun_iff_Indep + - ProbabilityTheory.Indep_iff + - ProbabilityTheory.indepFun_iff_indepSet_preimage + - ProbabilityTheory.iIndepFun_iff_iIndep + - ProbabilityTheory.iIndepSet_iff_meas_biInter + - ProbabilityTheory.iIndepFun.indepFun_prod_mk_prod_mk + - ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul + - ProbabilityTheory.iIndep_iff_iIndepSets + - ProbabilityTheory.iIndepFun_iff + - ProbabilityTheory.IndepSet_iff + - ProbabilityTheory.IndepSet_iff_Indep + - ProbabilityTheory.IndepFun.ae_eq + - ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul + - ProbabilityTheory.indepSets_singleton_iff + - ProbabilityTheory.iIndepSets_iff + - ProbabilityTheory.Indep_iff_IndepSets + - ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map + - ProbabilityTheory.iIndep.meas_iInter + - ProbabilityTheory.iIndepSets_singleton_iff + - ProbabilityTheory.IndepSets_iff + - ProbabilityTheory.IndepFun_iff + - ProbabilityTheory.iIndepSet_iff_iIndep + - ProbabilityTheory.iIndepSets.meas_iInter + - path: Mathlib/Order/CompactlyGenerated/Basic.lean + theorems: + - Directed.disjoint_iSup_right + - Directed.disjoint_iSup_left + - CompleteLattice.IsCompactElement.exists_finset_of_le_iSup + - CompleteLattice.setIndependent_iUnion_of_directed + - CompleteLattice.WellFounded.isSupFiniteCompact + - CompleteLattice.isCompactElement_finsetSup + - DirectedOn.disjoint_sSup_right + - DirectedOn.inf_sSup_eq + - CompleteLattice.WellFounded.finite_of_setIndependent + - CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le + - Directed.iSup_inf_eq + - CompleteLattice.Iic_coatomic_of_compact_element + - CompleteLattice.wellFounded_characterisations + - CompleteLattice.isCompactlyGenerated_of_wellFounded + - CompleteLattice.independent_iff_supIndep_of_injOn + - sSup_compact_le_eq + - le_iff_compact_le_imp + - CompleteLattice.IsSupFiniteCompact.isSupClosedCompact + - exists_setIndependent_isCompl_sSup_atoms + - CompleteLattice.setIndependent_iff_finite + - DirectedOn.sSup_inf_eq + - CompleteLattice.isCompactElement_iff + - sSup_compact_eq_top + - inf_sSup_eq_iSup_inf_sup_finset + - CompleteLattice.WellFounded.finite_ne_bot_of_independent + - CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt + - CompleteLattice.independent_sUnion_of_directed + - complementedLattice_iff_isAtomistic + - Directed.inf_iSup_eq + - CompleteLattice.isSupFiniteCompact_iff_all_elements_compact + - CompleteLattice.IsSupClosedCompact.wellFounded + - path: Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean + theorems: + - Matrix.det_sum_le + - Matrix.det_sum_smul_le + - Matrix.det_le + - path: Mathlib/NumberTheory/KummerDedekind.lean + theorems: + - comap_map_eq_map_adjoin_of_coprime_conductor + - KummerDedekind.multiplicity_factors_map_eq_multiplicity + - KummerDedekind.Ideal.irreducible_map_of_irreducible_minpoly + - KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map + - conductor_eq_top_of_adjoin_eq_top + - prod_mem_ideal_map_of_mem_conductor + - mem_coeSubmodule_conductor + - conductor_subset_adjoin + - path: Mathlib/Data/Set/Opposite.lean + theorems: + - Set.op_mem_op + - Set.unop_mem_unop + - Set.singleton_unop_op + - Set.singleton_op + - Set.singleton_unop + - Set.singleton_op_unop + - path: Mathlib/Geometry/Manifold/Algebra/Monoid.lean + theorems: + - smooth_pow + - R_mul + - contMDiffWithinAt_finset_prod + - ContMDiffWithinAt.div_const + - ContMDiffAt.prod + - L_mul + - ContMDiffWithinAt.prod + - contMDiff_finprod_cond + - path: Mathlib/GroupTheory/FixedPointFree.lean + theorems: + - MonoidHom.FixedPointFree.odd_card_of_involutive + - MonoidHom.FixedPointFree.orderOf_ne_two_of_involutive + - MonoidHom.FixedPointFree.coe_eq_inv_of_sq_eq_one + - MonoidHom.FixedPointFree.commutatorMap_injective + - MonoidHom.FixedPointFree.commute_all_of_involutive + - path: Mathlib/Probability/Kernel/MeasurableIntegral.lean + theorems: + - ProbabilityTheory.kernel.measurable_kernel_prod_mk_left' + - Measurable.set_lintegral_kernel + - ProbabilityTheory.measurableSet_kernel_integrable + - Measurable.lintegral_kernel_prod_right + - Measurable.lintegral_kernel_prod_right'' + - MeasureTheory.StronglyMeasurable.integral_kernel_prod_right' + - Measurable.set_lintegral_kernel_prod_right + - MeasureTheory.StronglyMeasurable.integral_kernel_prod_right + - ProbabilityTheory.kernel.measurable_kernel_prod_mk_left + - Measurable.set_lintegral_kernel_prod_left + - MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'' + - ProbabilityTheory.kernel.measurable_lintegral_indicator_const + - ProbabilityTheory.kernel.measurable_kernel_prod_mk_left_of_finite + - path: Mathlib/Data/List/GetD.lean + theorems: + - List.getI_zero_eq_headI + - List.getD_replicate_default_eq + - List.getD_eq_getD_get? + - List.getD_append_right + - List.getD_eq_default + - List.getD_singleton_default_eq + - List.getD_append + - List.getI_eq_iget_get? + - List.getD_eq_get + - List.getD_map + - path: Mathlib/SetTheory/Game/Ordinal.lean + theorems: + - Ordinal.toPGame_add + - Ordinal.toPGame_def + - Ordinal.toPGame_le + - Ordinal.one_toPGame_moveLeft + - Ordinal.toPGame_equiv_iff + - Ordinal.toPGame_lf + - Ordinal.to_leftMoves_one_toPGame_symm + - Ordinal.toPGame_moveLeft_hEq + - Ordinal.toPGame_rightMoves + - Ordinal.toPGame_moveLeft + - Ordinal.toPGame_leftMoves + - Ordinal.toPGame_le_iff + - Ordinal.toPGame_lt_iff + - Ordinal.toPGame_lf_iff + - path: Mathlib/Topology/Order/Bounded.lean + theorems: + - Filter.isBounded_ge_map_of_bounded_range + - Filter.isBounded_le_map_of_bounded_range + - path: Mathlib/LinearAlgebra/AffineSpace/Restrict.lean + theorems: + - AffineSubspace.nonempty_map + - AffineMap.restrict.injective + - AffineMap.restrict.surjective + - AffineMap.restrict.linear_aux + - path: Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean + theorems: + - CategoryTheory.Limits.IsInitial.hasInitial + - CategoryTheory.Limits.IsInitial.hom_ext + - CategoryTheory.Limits.IsTerminal.hom_ext + - "CategoryTheory.Limits.isIso_\u03B9_of_isTerminal" + - CategoryTheory.Limits.IsInitial.mono_from + - CategoryTheory.Limits.IsTerminal.hasTerminal + - CategoryTheory.Limits.hasTerminalChangeDiagram + - CategoryTheory.Limits.IsInitial.epi_to + - "CategoryTheory.Limits.isIso_\u03B9_of_isInitial" + - CategoryTheory.Limits.terminal.comp_from + - "CategoryTheory.Limits.isIso_\u03C0_of_isTerminal" + - "CategoryTheory.Limits.isIso_\u03C0_of_isInitial" + - CategoryTheory.Limits.hasInitialChangeDiagram + - CategoryTheory.Limits.initial.to_comp + - "CategoryTheory.Limits.limitConstTerminal_inv_\u03C0" + - CategoryTheory.Limits.InitialMonoClass.of_isInitial + - CategoryTheory.Limits.IsTerminal.mono_from + - path: Mathlib/Data/Nat/Choose/Sum.lean + theorems: + - Finset.sum_powerset_neg_one_pow_card + - Commute.add_pow' + - Nat.sum_range_choose + - Finset.sum_choose_succ_mul + - Nat.choose_middle_le_pow + - Finset.prod_pow_choose_succ + - Finset.sum_powerset_neg_one_pow_card_of_nonempty + - Nat.four_pow_le_two_mul_add_one_mul_central_binom + - Int.alternating_sum_range_choose + - Nat.sum_range_choose_halfway + - Finset.sum_antidiagonal_choose_succ_mul + - Commute.add_pow + - Finset.sum_powerset_apply_card + - Nat.sum_Icc_choose + - Finset.prod_antidiagonal_pow_choose_succ + - Int.alternating_sum_range_choose_of_ne + - path: Mathlib/Tactic/NormNum/Pow.lean + theorems: + - Mathlib.Meta.NormNum.IsNatPowT.bit1 + - Mathlib.Meta.NormNum.IsNatPowT.bit0 + - Mathlib.Meta.NormNum.isInt_pow + - Mathlib.Meta.NormNum.isNat_zpow_neg + - Mathlib.Meta.NormNum.isRat_pow + - Mathlib.Meta.NormNum.intPow_negOfNat_bit1 + - Mathlib.Meta.NormNum.isInt_zpow_pos + - Mathlib.Meta.NormNum.intPow_negOfNat_bit0 + - Mathlib.Meta.NormNum.isRat_zpow_neg + - Mathlib.Meta.NormNum.isNat_pow + - Mathlib.Meta.NormNum.intPow_ofNat + - Mathlib.Meta.NormNum.isInt_zpow_neg + - Mathlib.Meta.NormNum.isRat_zpow_pos + - path: Mathlib/LinearAlgebra/Matrix/Spectrum.lean + theorems: + - Matrix.IsHermitian.eigenvectorMatrixInv_apply + - Matrix.IsHermitian.eigenvalues_eq + - Matrix.IsHermitian.det_eq_prod_eigenvalues + - Matrix.IsHermitian.eigenvectorMatrix_mul_inv + - Matrix.IsHermitian.rank_eq_rank_diagonal + - Matrix.IsHermitian.spectral_theorem + - Matrix.IsHermitian.conjTranspose_eigenvectorMatrix + - Matrix.IsHermitian.exists_eigenvector_of_ne_zero + - Matrix.IsHermitian.mulVec_eigenvectorBasis + - Matrix.IsHermitian.conjTranspose_eigenvectorMatrixInv + - Matrix.IsHermitian.rank_eq_card_non_zero_eigs + - path: Mathlib/Algebra/Group/Defs.lean + theorems: + - MulOneClass.ext + - inv_mul_cancel_left + - mul_inv_cancel_right + - mul_right_inv + - inv_mul_cancel_right + - inv_mul_cancel_comm + - negSucc_zsmul + - inv_eq_of_mul_eq_one_left + - left_inv_eq_right_inv + - zpow_negSucc + - mul_inv_cancel_left + - path: Mathlib/Topology/IsLocalHomeomorph.lean + theorems: + - IsLocalHomeomorphOn.of_comp_left + - IsLocalHomeomorphOn.comp + - IsLocalHomeomorph.isLocallyInjective + - isLocalHomeomorphOn_iff_openEmbedding_restrict + - IsLocalHomeomorphOn.mk + - IsLocalHomeomorphOn.of_comp_right + - isLocalHomeomorph_iff_openEmbedding_restrict + - IsLocalHomeomorph.isTopologicalBasis + - path: Mathlib/Data/Polynomial/Mirror.lean + theorems: + - Polynomial.mirror_zero + - Polynomial.mirror_natDegree + - Polynomial.natTrailingDegree_mul_mirror + - Polynomial.mirror_neg + - Polynomial.mirror_eval_one + - Polynomial.coeff_mirror + - Polynomial.mirror_natTrailingDegree + - Polynomial.mirror_trailingCoeff + - Polynomial.coeff_mul_mirror + - Polynomial.mirror_monomial + - Polynomial.irreducible_of_mirror + - Polynomial.mirror_mirror + - Polynomial.mirror_smul + - Polynomial.mirror_eq_zero + - Polynomial.natDegree_mul_mirror + - Polynomial.mirror_leadingCoeff + - Polynomial.mirror_mul_of_domain + - path: Mathlib/Analysis/Convex/Extreme.lean + theorems: + - isExtreme_iInter + - isExtreme_biInter + - Convex.mem_extremePoints_iff_mem_diff_convexHull_diff + - mem_extremePoints_iff_forall_segment + - Convex.mem_extremePoints_iff_convex_diff + - isExtreme_sInter + - extremePoints_prod + - extremePoints_convexHull_subset + - IsExtreme.inter + - IsExtreme.trans + - image_extremePoints + - IsExtreme.extremePoints_subset_extremePoints + - isExtreme_singleton + - extremePoints_pi + - path: Mathlib/Data/W/Basic.lean + theorems: + - WType.elim_injective + - WType.depth_pos + - WType.infinite_of_nonempty_of_isEmpty + - path: Mathlib/Init/Data/Ordering/Lemmas.lean + theorems: + - Ordering.ite_eq_eq_distrib + - cmpUsing_eq_eq + - cmpUsing_eq_lt + - Ordering.ite_eq_gt_distrib + - cmpUsing_eq_gt + - Ordering.ite_eq_lt_distrib + - path: Mathlib/Algebra/Invertible/Basic.lean + theorems: + - Commute.invOf_left + - invOf_div + - invOf_two_add_invOf_two + - invOf_neg + - map_invOf + - Commute.invOf_right + - one_sub_invOf_two + - path: Mathlib/Algebra/Squarefree/Basic.lean + theorems: + - Squarefree.pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right + - Int.squarefree_coe_nat + - Irreducible.squarefree + - Squarefree.isRadical + - squarefree_iff_irreducible_sq_not_dvd_of_ne_zero + - Squarefree.ne_zero + - Int.squarefree_natAbs + - UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors + - not_squarefree_zero + - IsRadical.squarefree + - exists_squarefree_dvd_pow_of_ne_zero + - irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree + - Squarefree.dvd_of_squarefree_of_mul_dvd_mul_right + - squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible + - Squarefree.pow_dvd_of_squarefree_of_pow_succ_dvd_mul_left + - isRadical_iff_squarefree_or_zero + - multiplicity.squarefree_iff_multiplicity_le_one + - squarefree_iff_no_irreducibles + - Squarefree.eq_zero_or_one_of_pow_of_not_isUnit + - path: Mathlib/Data/Fintype/CardEmbedding.lean + theorems: + - Fintype.card_embedding_eq + - path: Mathlib/CategoryTheory/Sites/Canonical.lean + theorems: + - CategoryTheory.Sheaf.sheaf_for_finestTopology + - CategoryTheory.Sheaf.isSheafFor_bind + - CategoryTheory.Sheaf.Subcanonical.of_yoneda_isSheaf + - CategoryTheory.Sheaf.le_finestTopology + - CategoryTheory.Sheaf.isSheafFor_trans + - path: Mathlib/SetTheory/Ordinal/Topology.lean + theorems: + - Ordinal.enumOrd_isNormal_iff_isClosed + - Ordinal.isClosed_iff_sup + - Ordinal.mem_closed_iff_sup + - Ordinal.nhds_left_eq_nhds + - Ordinal.isClosed_iff_bsup + - Ordinal.mem_closure_tfae + - Ordinal.isOpen_iff + - Ordinal.mem_closed_iff_bsup + - Ordinal.mem_closure_iff_sup + - Ordinal.isNormal_iff_strictMono_and_continuous + - Ordinal.isLimit_of_mem_frontier + - Ordinal.isOpen_singleton_iff + - Ordinal.mem_closure_iff_bsup + - Ordinal.nhds_left'_eq_nhds_ne + - path: Mathlib/MeasureTheory/Measure/Dirac.lean + theorems: + - MeasureTheory.ae_dirac_eq + - MeasureTheory.Measure.map_eq_sum + - MeasureTheory.Measure.dirac_apply_of_mem + - MeasureTheory.Measure.sum_smul_dirac + - MeasureTheory.Measure.restrict_singleton + - MeasureTheory.Measure.map_dirac + - MeasureTheory.Measure.dirac_apply + - MeasureTheory.ae_eq_dirac + - MeasureTheory.mutuallySingular_dirac + - MeasureTheory.mem_ae_dirac_iff + - MeasureTheory.restrict_dirac' + - MeasureTheory.Measure.map_const + - MeasureTheory.restrict_dirac + - path: Mathlib/NumberTheory/SumTwoSquares.lean + theorems: + - Nat.eq_sq_add_sq_of_isSquare_mod_neg_one + - ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime + - ZMod.isSquare_neg_one_of_dvd + - Nat.Prime.sq_add_sq + - Nat.Prime.mod_four_ne_three_of_dvd_isSquare_neg_one + - ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_coprime + - ZMod.isSquare_neg_one_iff' + - sq_add_sq_mul + - ZMod.isSquare_neg_one_mul + - Nat.eq_sq_add_sq_iff_eq_sq_mul + - Nat.sq_add_sq_mul + - Nat.eq_sq_add_sq_iff + - ZMod.isSquare_neg_one_iff + - path: Mathlib/LinearAlgebra/Dimension/Free.lean + theorems: + - FiniteDimensional.finrank_eq_card_chooseBasisIndex + - FiniteDimensional.basisUnique_repr_eq_zero_iff + - Module.finite_of_finrank_eq_succ + - FiniteDimensional.nonempty_linearEquiv_of_finrank_eq + - rank_mul_rank + - Module.Free.rank_eq_mk_of_infinite_lt + - lift_rank_mul_lift_rank + - Module.finite_of_finrank_pos + - FiniteDimensional.finrank_mul_finrank + - FiniteDimensional.finrank_of_not_finite + - Module.rank_lt_alpeh0_iff + - Module.finite_iff_of_rank_eq_nsmul + - path: Mathlib/Topology/Order/ScottTopology.lean + theorems: + - Topology.IsScott.scottContinuous_iff_continuous + - Topology.IsScott.closure_singleton + - Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInacc + - dirSupClosed_compl + - DirSupInacc.union + - Topology.IsScottHausdorff.dirSupInacc_of_isOpen + - Topology.IsScott.lowerClosure_subset_closure + - dirSupClosed_iff_forall_sSup + - Topology.IsScott.monotone_of_continuous + - dirSupInacc_iff_forall_sSup + - Topology.IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open + - Topology.IsScott.scottHausdorff_le + - Topology.IsScottHausdorff.isOpen_iff + - dirSupInacc_compl + - Topology.IsScott.isClosed_iff_isLowerSet_and_dirSupClosed + - path: Mathlib/CategoryTheory/Monoidal/Center.lean + theorems: + - CategoryTheory.Center.rightUnitor_inv_f + - CategoryTheory.Center.associator_inv_f + - CategoryTheory.Center.ext + - CategoryTheory.Center.whiskerLeft_comm + - CategoryTheory.Center.whiskerRight_comm + - CategoryTheory.Center.leftUnitor_inv_f + - path: Mathlib/Algebra/Module/LinearMap/Basic.lean + theorems: + - LinearMap.comp_zero + - DistribMulActionHom.toLinearMap_injective + - SemilinearMapClass.map_smul_inv + - Function.Injective.injective_linearMapComp_left + - LinearMap.isScalarTower_of_injective + - LinearMap.ext_ring + - AddMonoidHom.toRatLinearMap_injective + - IsLinearMap.isLinearMap_smul + - AddMonoidHom.toNatLinearMap_injective + - LinearMap.ext_ring_op + - AddMonoidHom.toIntLinearMap_injective + - path: Mathlib/Algebra/Ring/Semiconj.lean + theorems: + - SemiconjBy.neg_right + - SemiconjBy.sub_left + - SemiconjBy.add_left + - SemiconjBy.add_right + - SemiconjBy.neg_left + - path: Mathlib/LinearAlgebra/Matrix/Basis.lean + theorems: + - Basis.toMatrix_map + - Basis.toMatrix_smul + - basis_toMatrix_basisFun_mul + - Basis.toMatrix_reindex + - Basis.toMatrix_mul_toMatrix + - Basis.toMatrix_eq_toMatrix_constr + - basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix + - basis_toMatrix_mul_linearMap_toMatrix + - Basis.toMatrix_self + - Basis.toMatrix_unitsSMul + - mul_basis_toMatrix + - Basis.toMatrix_mul_toMatrix_flip + - linearMap_toMatrix_mul_basis_toMatrix + - Basis.toMatrix_update + - Basis.toLin_toMatrix + - basis_toMatrix_mul + - LinearMap.toMatrix_id_eq_basis_toMatrix + - Basis.restrictScalars_toMatrix + - Basis.toMatrix_reindex' + - Basis.toMatrix_map_vecMul + - Basis.sum_toMatrix_smul_self + - path: Mathlib/MeasureTheory/Integral/LebesgueNormedSpace.lean + theorems: + - aemeasurable_withDensity_iff + - path: Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean + theorems: + - MaximalSpectrum.iInf_localization_eq_bot + - MaximalSpectrum.toPrimeSpectrum_range + - PrimeSpectrum.iInf_localization_eq_bot + - MaximalSpectrum.toPrimeSpectrum_injective + - path: Mathlib/RingTheory/PowerSeries/WellKnown.lean + theorems: + - PowerSeries.map_cos + - PowerSeries.exp_pow_sum + - PowerSeries.invUnitsSub_mul_sub + - PowerSeries.invUnitsSub_mul_X + - PowerSeries.exp_pow_eq_rescale_exp + - PowerSeries.constantCoeff_exp + - PowerSeries.exp_mul_exp_neg_eq_one + - PowerSeries.coeff_sin_bit1 + - PowerSeries.coeff_cos_bit1 + - PowerSeries.map_invUnitsSub + - PowerSeries.map_sin + - PowerSeries.coeff_sin_bit0 + - PowerSeries.exp_mul_exp_eq_exp_add + - PowerSeries.constantCoeff_invUnitsSub + - PowerSeries.coeff_cos_bit0 + - path: Mathlib/Init/Order/Defs.lean + theorems: + - compare_le_iff_le + - compare_lt_iff_lt + - compare_ge_iff_ge + - compare_iff + - compare_eq_iff_eq + - compare_gt_iff_gt + - path: Mathlib/CategoryTheory/Functor/EpiMono.lean + theorems: + - CategoryTheory.Functor.preservesEpimorphsisms_of_adjunction + - CategoryTheory.Functor.epi_map_iff_epi + - CategoryTheory.Functor.mono_map_iff_mono + - CategoryTheory.Functor.isSplitEpi_iff + - CategoryTheory.Functor.preservesMonomorphisms.of_iso + - CategoryTheory.Adjunction.strongEpi_map_of_strongEpi + - CategoryTheory.Functor.reflectsMonomorphisms.of_iso + - CategoryTheory.Functor.isSplitMono_iff + - CategoryTheory.Functor.splitEpiCategoryImpOfIsEquivalence + - CategoryTheory.Functor.strongEpi_map_iff_strongEpi_of_isEquivalence + - CategoryTheory.Functor.reflectsEpimorphisms.of_iso + - CategoryTheory.Functor.preservesEpimorphisms.of_iso + - CategoryTheory.Functor.preservesMonomorphisms_of_adjunction + - path: Mathlib/GroupTheory/Subsemigroup/Operations.lean + theorems: + - Subsemigroup.top_prod + - MulHom.srange_top_iff_surjective + - Subsemigroup.prod_eq_top_iff + - MulHom.subsemigroupMap_surjective + - Subsemigroup.bot_prod_bot + - Subsemigroup.closure_closure_coe_preimage + - Subsemigroup.prod_top + - Subsemigroup.le_prod_iff + - Subsemigroup.comap_id + - MulHom.map_srange + - path: Mathlib/Topology/Algebra/Valuation.lean + theorems: + - Valued.hasBasis_uniformity + - Valued.mem_nhds_zero + - Valued.cauchy_iff + - Valuation.subgroups_basis + - Valued.hasBasis_nhds_zero + - Valued.loc_const + - Valued.mem_nhds + - Valued.integer_isOpen + - path: Mathlib/Algebra/TrivSqZeroExt.lean + theorems: + - TrivSqZeroExt.lift_apply_inl + - TrivSqZeroExt.map_id + - TrivSqZeroExt.mul_inl_eq_op_smul + - TrivSqZeroExt.inl_mul_inr + - TrivSqZeroExt.inl_mul_eq_smul + - TrivSqZeroExt.inl_pow + - TrivSqZeroExt.snd_pow_of_smul_comm' + - TrivSqZeroExt.inr_mul_inl + - TrivSqZeroExt.map_inl + - TrivSqZeroExt.map_comp_map + - TrivSqZeroExt.fst_map + - TrivSqZeroExt.snd_pow_of_smul_comm + - TrivSqZeroExt.inr_mul_inr + - TrivSqZeroExt.lift_apply_inr + - TrivSqZeroExt.snd_map + - TrivSqZeroExt.snd_list_prod + - TrivSqZeroExt.map_inr + - TrivSqZeroExt.inl_mul + - path: Mathlib/RingTheory/Localization/NumDen.lean + theorems: + - IsFractionRing.num_mul_den_eq_num_iff_eq' + - IsFractionRing.num_mul_den_eq_num_iff_eq + - IsFractionRing.isInteger_of_isUnit_den + - IsFractionRing.num_mul_den_eq_num_mul_den_iff_eq + - IsFractionRing.mk'_num_den' + - IsFractionRing.eq_zero_of_num_eq_zero + - IsFractionRing.exists_reduced_fraction + - path: Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean + theorems: + - CategoryTheory.OplaxNatTrans.whiskerLeft_naturality_naturality + - CategoryTheory.OplaxNatTrans.whiskerLeft_naturality_id + - CategoryTheory.OplaxNatTrans.Modification.whiskerLeft_naturality + - CategoryTheory.OplaxNatTrans.whiskerRight_naturality_comp + - CategoryTheory.OplaxNatTrans.Modification.whiskerRight_naturality + - CategoryTheory.OplaxNatTrans.ext + - CategoryTheory.OplaxNatTrans.whiskerRight_naturality_naturality + - CategoryTheory.OplaxNatTrans.whiskerLeft_naturality_comp + - CategoryTheory.OplaxNatTrans.whiskerRight_naturality_id + - path: Mathlib/Analysis/Complex/OpenMapping.lean + theorems: + - DiffContOnCl.ball_subset_image_closedBall + - AnalyticOn.is_constant_or_isOpen + - AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux + - AnalyticAt.eventually_constant_or_nhds_le_map_nhds + - path: Mathlib/Combinatorics/Catalan.lean + theorems: + - catalan_one + - gosper_catalan_sub_eq_central_binom_div + - catalan_eq_centralBinom_div + - catalan_zero + - Tree.treesOfNumNodesEq_card_eq_catalan + - Tree.treesOfNumNodesEq_zero + - catalan_three + - catalan_succ' + - catalan_succ + - catalan_two + - Tree.coe_treesOfNumNodesEq + - Tree.mem_treesOfNumNodesEq + - Tree.treesOfNumNodesEq_succ + - gosper_trick + - path: Mathlib/Topology/Category/LightProfinite/IsLight.lean + theorems: + - Profinite.isLight_of_mono + - path: Mathlib/NumberTheory/ADEInequality.lean + theorems: + - ADEInequality.admissible_of_one_lt_sumInv_aux + - ADEInequality.lt_six + - ADEInequality.Admissible.one_lt_sumInv + - ADEInequality.sumInv_pqr + - ADEInequality.lt_three + - ADEInequality.admissible_of_one_lt_sumInv + - ADEInequality.lt_four + - path: Mathlib/ModelTheory/Syntax.lean + theorems: + - FirstOrder.Language.Term.relabel_relabel + - FirstOrder.Language.LHom.id_onTerm + - FirstOrder.Language.LHom.comp_onBoundedFormula + - FirstOrder.Language.BoundedFormula.not_ex_isAtomic + - FirstOrder.Language.BoundedFormula.not_ex_isQF + - FirstOrder.Language.BoundedFormula.isPrenex_toPrenexImpRight + - FirstOrder.Language.BoundedFormula.relabel_not + - FirstOrder.Language.BoundedFormula.not_all_isAtomic + - FirstOrder.Language.BoundedFormula.relabel_sum_inl + - FirstOrder.Language.BoundedFormula.relabel_all + - FirstOrder.Language.distinctConstantsTheory_eq_iUnion + - FirstOrder.Language.BoundedFormula.castLE_castLE + - FirstOrder.Language.BoundedFormula.relabelAux_sum_inl + - FirstOrder.Language.BoundedFormula.sum_elim_comp_relabelAux + - FirstOrder.Language.LHom.id_onBoundedFormula + - FirstOrder.Language.BoundedFormula.IsPrenex.relabel + - FirstOrder.Language.BoundedFormula.mapTermRel_mapTermRel + - FirstOrder.Language.BoundedFormula.not_all_isQF + - FirstOrder.Language.LHom.comp_onTerm + - FirstOrder.Language.BoundedFormula.mapTermRel_id_id_id + - FirstOrder.Language.BoundedFormula.isPrenex_toPrenexImp + - FirstOrder.Language.Term.relabel_id + - FirstOrder.Language.BoundedFormula.relabel_ex + - FirstOrder.Language.BoundedFormula.castLE_rfl + - path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean + theorems: + - ApproximatesLinearOn.inverse_continuousOn + - ApproximatesLinearOn.surjective + - ApproximatesLinearOn.map_nhds_eq + - ApproximatesLinearOn.lipschitz + - ApproximatesLinearOn.image_mem_nhds + - ApproximatesLinearOn.antilipschitz + - approximatesLinearOn_empty + - ApproximatesLinearOn.approximatesLinearOn_iff_lipschitzOnWith + - ApproximatesLinearOn.surjOn_closedBall_of_nonlinearRightInverse + - ApproximatesLinearOn.to_inv + - path: Mathlib/Analysis/Convex/Integral.lean + theorems: + - ae_eq_const_or_norm_integral_lt_of_norm_le_const + - Convex.average_mem_interior_of_set + - StrictConcaveOn.ae_eq_const_or_lt_map_average + - ae_eq_const_or_norm_average_lt_of_norm_le_const + - ConcaveOn.set_average_mem_hypograph + - Convex.integral_mem + - ConvexOn.average_mem_epigraph + - ae_eq_const_or_exists_average_ne_compl + - ConcaveOn.le_map_integral + - StrictConvex.ae_eq_const_or_average_mem_interior + - Convex.average_mem + - StrictConvexOn.ae_eq_const_or_map_average_lt + - ae_eq_const_or_norm_set_integral_lt_of_norm_le_const + - ConvexOn.map_integral_le + - ConcaveOn.average_mem_hypograph + - path: Mathlib/Geometry/Manifold/ContMDiff/Basic.lean + theorems: + - contMDiff_one + - contMDiffOn_openEmbedding_symm + - contMDiffWithinAt_of_not_mem_mulTSupport + - ContMDiffWithinAt.comp + - contMDiff_const + - ContMDiffWithinAt.comp_of_eq + - ContMDiff.extend_one + - ContMDiffAt.comp_of_eq + - contMDiff_of_mulTSupport + - ContMDiff.of_comp_openEmbedding + - smooth_one + - contMDiff_openEmbedding + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean + theorems: + - Real.deriv_arcsin + - Real.differentiableWithinAt_arcsin_Ici + - Real.differentiableWithinAt_arcsin_Iic + - Real.contDiffAt_arccos_iff + - Real.deriv_arccos + - Real.deriv_arcsin_aux + - Real.hasDerivWithinAt_arcsin_Ici + - Real.hasDerivWithinAt_arcsin_Iic + - path: Mathlib/Probability/ProbabilityMassFunction/Constructions.lean + theorems: + - PMF.filter_apply_eq_zero_of_not_mem + - PMF.support_ofFinset + - PMF.mem_support_seq_iff + - PMF.filter_apply_eq_zero_iff + - PMF.mem_support_normalize_iff + - PMF.map_const + - PMF.support_normalize + - PMF.seq_apply + - PMF.map_apply + - PMF.support_bernoulli + - PMF.mem_support_map_iff + - PMF.mem_support_ofFinset_iff + - PMF.mem_support_bernoulli_iff + - PMF.toOuterMeasure_map_apply + - PMF.filter_apply + - PMF.support_map + - PMF.support_seq + - PMF.toMeasure_map_apply + - PMF.filter_apply_ne_zero_iff + - path: Mathlib/RingTheory/Localization/LocalizationLocalization.lean + theorems: + - IsFractionRing.isFractionRing_of_isDomain_of_isLocalization + - IsLocalization.localization_localization_surj + - IsLocalization.localization_localization_exists_of_eq + - IsLocalization.localization_localization_map_units + - IsLocalization.isLocalization_of_submonoid_le + - IsLocalization.localization_localization_isLocalization_of_has_all_units + - IsLocalization.isLocalization_isLocalization_atPrime_isLocalization + - IsLocalization.isLocalization_of_is_exists_mul_mem + - IsFractionRing.isFractionRing_of_isLocalization + - IsLocalization.mem_localizationLocalizationSubmodule + - path: Mathlib/Topology/Algebra/OpenSubgroup.lean + theorems: + - OpenSubgroup.isClosed + - Subgroup.isOpen_of_mem_nhds + - path: Mathlib/Data/Int/Cast/Lemmas.lean + theorems: + - Int.cast_commute + - Int.nneg_mul_add_sq_of_abs_le_one + - Int.cast_lt_zero + - AddMonoidHom.eq_int_castAddHom + - MonoidHom.apply_mint + - Int.cast_le + - Int.cast_nonpos + - zsmul_one + - Int.cast_pos + - Int.toNat_lt' + - AddMonoidHom.apply_int + - Int.cast_abs + - zsmul_eq_mul + - Int.cast_natAbs + - Int.coe_nat_pow + - zsmul_eq_mul' + - MonoidHom.ext_int + - Int.cast_mono + - path: Mathlib/Analysis/NormedSpace/LinearIsometry.lean + theorems: + - LinearIsometryEquiv.range_eq_univ + - LinearIsometryEquiv.coe_ofSurjective + - LinearIsometry.coe_injective + - path: Mathlib/MeasureTheory/Function/UniformIntegrable.lean + theorems: + - MeasureTheory.unifIntegrable_zero_meas + - "MeasureTheory.Mem\u2112p.snorm_indicator_le" + - "MeasureTheory.Mem\u2112p.snormEssSup_indicator_norm_ge_eq_zero" + - MeasureTheory.snorm_sub_le_of_dist_bdd + - "MeasureTheory.Mem\u2112p.snorm_indicator_le'" + - MeasureTheory.unifIntegrable_of' + - "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_le" + - "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_nonneg_le" + - MeasureTheory.unifIntegrable_of_tendsto_Lp_zero + - MeasureTheory.UniformIntegrable.spec + - MeasureTheory.tendsto_indicator_ge + - "MeasureTheory.Mem\u2112p.integral_indicator_norm_ge_nonneg_le_of_meas" + - MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas + - MeasureTheory.UniformIntegrable.spec' + - MeasureTheory.unifIntegrable_finite + - MeasureTheory.unifIntegrable_subsingleton + - MeasureTheory.unifIntegrable_of_tendsto_Lp + - MeasureTheory.uniformIntegrable_of + - MeasureTheory.snorm_indicator_le_of_bound + - MeasureTheory.UnifIntegrable.neg + - MeasureTheory.UniformIntegrable.ae_eq + - "MeasureTheory.Mem\u2112p.snorm_indicator_norm_ge_le" + - MeasureTheory.uniformIntegrable_of' + - "MeasureTheory.Mem\u2112p.snorm_indicator_le_of_meas" + - MeasureTheory.unifIntegrable_const + - MeasureTheory.UnifIntegrable.add + - MeasureTheory.unifIntegrable_of + - MeasureTheory.UnifIntegrable.sub + - MeasureTheory.uniformIntegrable_average_real + - MeasureTheory.uniformIntegrable_finite + - "MeasureTheory.Mem\u2112p.snorm_indicator_norm_ge_pos_le" + - MeasureTheory.unifIntegrable_fin + - MeasureTheory.UnifIntegrable.ae_eq + - MeasureTheory.tendsto_Lp_of_tendstoInMeasure + - MeasureTheory.uniformIntegrable_average + - path: Mathlib/AlgebraicTopology/DoldKan/Projections.lean + theorems: + - AlgebraicTopology.DoldKan.P_f_0_eq + - AlgebraicTopology.DoldKan.P_f_idem + - AlgebraicTopology.DoldKan.P_add_Q + - AlgebraicTopology.DoldKan.comp_P_eq_self_iff + - AlgebraicTopology.DoldKan.P_idem + - AlgebraicTopology.DoldKan.Q_succ + - AlgebraicTopology.DoldKan.map_Q + - AlgebraicTopology.DoldKan.map_P + - AlgebraicTopology.DoldKan.Q_f_0_eq + - AlgebraicTopology.DoldKan.Q_f_naturality + - AlgebraicTopology.DoldKan.HigherFacesVanish.of_P + - AlgebraicTopology.DoldKan.HigherFacesVanish.comp_P_eq_self + - path: Mathlib/Analysis/InnerProductSpace/PiL2.lean + theorems: + - OrthonormalBasis.span_apply + - OrthonormalBasis.coe_ofRepr + - OrthonormalBasis.orthonormal + - EuclideanSpace.single_apply + - OrthonormalBasis.det_to_matrix_orthonormalBasis_real + - OrthonormalBasis.coe_mk + - Complex.isometryOfOrthonormal_apply + - EuclideanSpace.ball_zero_eq + - OrthonormalBasis.repr_apply_apply + - orthonormalBasis_one_dim + - Orthonormal.exists_orthonormalBasis_extension + - OrthonormalBasis.coe_toBasis + - EuclideanSpace.inner_single_left + - OrthonormalBasis.det_to_matrix_orthonormalBasis + - LinearIsometry.extend_apply + - DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply + - EuclideanSpace.orthonormal_single + - EuclideanSpace.inner_single_right + - OrthonormalBasis.repr_symm_single + - OrthonormalBasis.repr_self + - finrank_euclideanSpace_fin + - EuclideanSpace.sphere_zero_eq + - inner_matrix_row_row + - Orthonormal.exists_orthonormalBasis_extension_of_card_eq + - OrthonormalBasis.coe_toBasis_repr_apply + - OrthonormalBasis.sum_inner_mul_inner + - DirectSum.IsInternal.collectedOrthonormalBasis_mem + - OrthonormalBasis.sum_repr + - DirectSum.IsInternal.subordinateOrthonormalBasis_subordinate + - OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary + - Complex.map_isometryOfOrthonormal + - Complex.coe_orthonormalBasisOneI + - OrthonormalBasis.repr_injective + - Basis.toBasis_toOrthonormalBasis + - EuclideanSpace.norm_eq + - Basis.coe_toOrthonormalBasis + - Complex.isometryOfOrthonormal_symm_apply + - finrank_euclideanSpace + - OrthonormalBasis.reindex_apply + - OrthonormalBasis.repr_reindex + - EuclideanSpace.closedBall_zero_eq + - path: Mathlib/Data/Finsupp/Interval.lean + theorems: + - Finsupp.card_Iic + - Finsupp.card_Ico + - Finsupp.card_Ioo + - Finsupp.card_Ioc + - Finsupp.card_Icc + - Finsupp.card_Iio + - Finsupp.card_uIcc + - path: Mathlib/Algebra/Lie/SkewAdjoint.lean + theorems: + - LinearMap.BilinForm.isSkewAdjoint_bracket + - Matrix.isSkewAdjoint_bracket + - mem_skewAdjointMatricesLieSubalgebra_unit_smul + - Matrix.lie_transpose + - skewAdjointLieSubalgebraEquiv_apply + - skewAdjointLieSubalgebraEquiv_symm_apply + - skewAdjointMatricesLieSubalgebraEquiv_apply + - path: Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean + theorems: + - HasDerivWithinAt.clog_real + - HasStrictDerivAt.clog + - HasStrictDerivAt.clog_real + - HasDerivWithinAt.clog + - HasDerivAt.clog_real + - HasDerivAt.clog + - Complex.hasStrictDerivAt_log + - path: Mathlib/Algebra/Homology/ShortComplex/Linear.lean + theorems: + - CategoryTheory.ShortComplex.rightHomologyMap'_smul + - CategoryTheory.ShortComplex.cyclesMap'_smul + - CategoryTheory.ShortComplex.opcyclesMap'_smul + - CategoryTheory.ShortComplex.leftHomologyMap'_smul + - path: Mathlib/GroupTheory/Submonoid/MulOpposite.lean + theorems: + - Submonoid.op_closure + - Submonoid.unop_closure + - path: Mathlib/Data/Multiset/Fold.lean + theorems: + - Multiset.fold_cons'_right + - Multiset.fold_dedup_idem + - Multiset.fold_add + - Multiset.coe_fold_l + - Multiset.fold_cons_right + - Multiset.fold_hom + - Multiset.fold_cons'_left + - Multiset.fold_distrib + - Multiset.le_smul_dedup + - Multiset.fold_bind + - Multiset.fold_union_inter + - Multiset.max_le_of_forall_le + - path: Mathlib/RingTheory/Norm.lean + theorems: + - Algebra.norm_eq_one_of_not_module_finite + - Algebra.norm_eq_one_of_not_exists_basis + - Algebra.isIntegral_norm + - Algebra.norm_eq_zero_iff + - Algebra.prod_embeddings_eq_finrank_pow + - Algebra.norm_eq_prod_automorphisms + - IntermediateField.AdjoinSimple.norm_gen_eq_one + - Algebra.norm_eq_prod_roots + - Algebra.norm_norm + - IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots + - Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly + - Algebra.norm_algebraMap + - Algebra.norm_eq_matrix_det + - Algebra.norm_eq_prod_embeddings + - Algebra.norm_eq_of_algEquiv + - Algebra.norm_zero + - Algebra.norm_eq_norm_adjoin + - Algebra.PowerBasis.norm_gen_eq_prod_roots + - Algebra.norm_eq_of_equiv_equiv + - Algebra.norm_algebraMap_of_basis + - Algebra.norm_eq_zero_iff_of_basis + - Algebra.norm_eq_of_ringEquiv + - path: Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean + theorems: + - CategoryTheory.strongEpi_comp + - CategoryTheory.StrongMono.iff_of_arrow_iso + - CategoryTheory.isIso_of_epi_of_strongMono + - CategoryTheory.strongMono_comp + - CategoryTheory.StrongEpi.of_arrow_iso + - CategoryTheory.strongEpi_of_strongEpi + - CategoryTheory.StrongEpi.iff_of_arrow_iso + - CategoryTheory.StrongMono.of_arrow_iso + - CategoryTheory.isIso_of_mono_of_strongEpi + - CategoryTheory.strongMono_of_strongMono + - path: Mathlib/RingTheory/JacobsonIdeal.lean + theorems: + - Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot + - Ideal.mem_jacobson_bot + - Ideal.isPrimary_of_isMaximal_radical + - Ideal.isUnit_of_sub_one_mem_jacobson_bot + - Ideal.comap_jacobson_of_surjective + - Ideal.eq_jacobson_iff_not_mem + - Ideal.IsLocal.mem_jacobson_or_exists_inv + - Ideal.jacobson_mono + - Ideal.mem_jacobson_iff + - Ideal.map_jacobson_of_surjective + - Ideal.jacobson_eq_iff_jacobson_quotient_eq_bot + - Ideal.jacobson_bot_polynomial_of_jacobson_bot + - Ideal.jacobson_bot_polynomial_le_sInf_map_maximal + - path: Mathlib/Topology/Algebra/InfiniteSum/Module.lean + theorems: + - tsum_const_smul'' + - ContinuousLinearEquiv.hasSum + - ContinuousLinearEquiv.tsum_eq_iff + - ContinuousLinearMap.hasSum + - tsum_const_smul' + - ContinuousLinearEquiv.map_tsum + - MulAction.automorphize_smul_left + - ContinuousLinearEquiv.hasSum' + - AddAction.automorphize_smul_left + - path: Mathlib/CategoryTheory/GradedObject/Unitor.lean + theorems: + - "CategoryTheory.GradedObject.\u03B9_mapBifunctorLeftUnitor_hom" + - CategoryTheory.GradedObject.mapBifunctorLeftUnitor_naturality + - CategoryTheory.GradedObject.mapBifunctorLeftUnitor_inv_naturality + - CategoryTheory.GradedObject.mapBifunctorLeftUnitorCofan_inj + - path: Mathlib/RingTheory/Localization/Submodule.lean + theorems: + - IsLocalization.coeSubmodule_isPrincipal + - IsLocalization.isNoetherianRing + - IsLocalization.mem_span_iff + - IsLocalization.mem_span_map + - IsLocalization.coeSubmodule_span_singleton + - IsLocalization.coeSubmodule_top + - IsLocalization.coeSubmodule_bot + - IsLocalization.coeSubmodule_span + - path: Mathlib/Algebra/Order/CauSeq/BigOperators.lean + theorems: + - IsCauSeq.of_decreasing_bounded + - IsCauSeq.geo_series_const + - cauchy_product + - IsCauSeq.of_mono_bounded + - IsCauSeq.geo_series + - path: Mathlib/ModelTheory/Fraisse.lean + theorems: + - FirstOrder.Language.exists_countable_is_age_of_iff + - FirstOrder.Language.IsUltrahomogeneous.amalgamation_age + - FirstOrder.Language.exists_cg_is_age_of + - FirstOrder.Language.age_directLimit + - FirstOrder.Language.age.countable_quotient + - path: Mathlib/Algebra/Category/Ring/Constructions.lean + theorems: + - CommRingCat.pushoutCocone_pt + - CommRingCat.pushoutCocone_inr + - CommRingCat.subsingleton_of_isTerminal + - path: Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean + theorems: + - Path.Homotopy.transAssocReparamAux_zero + - Path.Homotopy.trans_refl_reparam + - Path.Homotopy.transReflReparamAux_one + - Path.Homotopy.continuous_transAssocReparamAux + - Path.Homotopy.transAssocReparamAux_mem_I + - Path.Homotopy.continuous_reflTransSymmAux + - Path.Homotopy.reflTransSymmAux_mem_I + - Path.Homotopy.transReflReparamAux_mem_I + - Path.Homotopy.transReflReparamAux_zero + - Path.Homotopy.trans_assoc_reparam + - Path.Homotopy.transAssocReparamAux_one + - path: Mathlib/Topology/ShrinkingLemma.lean + theorems: + - ShrinkingLemma.PartialRefinement.exists_gt + - exists_subset_iUnion_closure_subset + - ShrinkingLemma.PartialRefinement.find_mem + - ShrinkingLemma.PartialRefinement.mem_find_carrier_iff + - path: Mathlib/Data/List/FinRange.lean + theorems: + - Equiv.Perm.map_finRange_perm + - List.nodup_ofFn_ofInjective + - List.ofFn_eq_pmap + - List.map_coe_finRange + - List.finRange_succ + - List.finRange_succ_eq_map + - Equiv.Perm.ofFn_comp_perm + - List.ofFn_eq_map + - List.nodup_ofFn + - path: Mathlib/Topology/Algebra/Module/LinearPMap.lean + theorems: + - LinearPMap.le_closure + - LinearPMap.closure_def + - LinearPMap.closure_inverse_graph + - LinearPMap.IsClosable.graph_closure_eq_closure_graph + - LinearPMap.IsClosable.leIsClosable + - LinearPMap.IsClosable.closure_mono + - LinearPMap.closureHasCore + - LinearPMap.inverse_isClosable_iff + - LinearPMap.inverse_closure + - LinearPMap.IsClosable.closure_isClosed + - LinearPMap.closure_def' + - LinearPMap.IsClosable.existsUnique + - path: Mathlib/Data/Fintype/Lattice.lean + theorems: + - Finite.exists_min + - Finite.exists_max + - Finset.mem_inf + - path: Mathlib/Algebra/AddTorsor.lean + theorems: + - vsub_left_cancel + - Equiv.right_vsub_pointReflection + - AddTorsor.subsingleton_iff + - eq_of_vsub_eq_zero + - neg_vsub_eq_vsub_rev + - vadd_vsub_vadd_cancel_left + - vsub_vadd_eq_vsub_sub + - Equiv.pointReflection_involutive + - vsub_add_vsub_cancel + - vsub_sub_vsub_cancel_right + - vadd_vsub_eq_sub_vsub + - vsub_sub_vsub_comm + - vadd_vsub_assoc + - Equiv.left_vsub_pointReflection + - Equiv.pointReflection_vsub_right + - vadd_right_cancel + - Set.singleton_vsub_self + - vsub_sub_vsub_cancel_left + - vsub_right_cancel + - Equiv.injective_pointReflection_left_of_injective_bit0 + - Equiv.pointReflection_fixed_iff_of_injective_bit0 + - vsub_self + - vadd_eq_vadd_iff_sub_eq_vsub + - vadd_eq_vadd_iff_neg_add_eq_vsub + - vadd_vsub_vadd_cancel_right + - Equiv.pointReflection_symm + - vsub_vadd_comm + - path: Mathlib/Analysis/InnerProductSpace/OfNorm.lean + theorems: + - InnerProductSpaceable.add_left_aux7 + - InnerProductSpaceable.int_prop + - InnerProductSpaceable.add_left_aux6 + - InnerProductSpaceable.I_prop + - InnerProductSpaceable.add_left_aux1 + - InnerProductSpaceable.add_left_aux5 + - InnerProductSpaceable.add_left_aux2' + - InnerProductSpaceable.add_left_aux2 + - InnerProductSpaceable.innerProp + - InnerProductSpaceable.nat_prop + - InnerProductSpaceable.add_left_aux3 + - Continuous.inner_ + - InnerProductSpaceable.real_prop + - InnerProductSpaceable.inner_.norm_sq + - InnerProductSpaceable.add_left_aux4' + - InnerProductSpaceable.rat_prop + - InnerProductSpaceable.nat + - InnerProductSpaceable.add_left_aux8 + - InnerProductSpaceable.add_left + - InnerProductSpaceable.add_left_aux4 + - InnerProductSpaceable.innerProp_neg_one + - InnerProductSpaceable.inner_.conj_symm + - path: Mathlib/Algebra/Order/Group/DenselyOrdered.lean + theorems: + - le_of_forall_one_lt_div_le + - path: Mathlib/CategoryTheory/Limits/Lattice.lean + theorems: + - CategoryTheory.Limits.CompleteLattice.coprod_eq_sup + - CategoryTheory.Limits.CompleteLattice.pullback_eq_inf + - CategoryTheory.Limits.CompleteLattice.pushout_eq_sup + - CategoryTheory.Limits.CompleteLattice.finite_product_eq_finset_inf + - CategoryTheory.Limits.CompleteLattice.finite_coproduct_eq_finset_sup + - CategoryTheory.Limits.CompleteLattice.prod_eq_inf + - path: Mathlib/GroupTheory/SpecificGroups/Coxeter.lean + theorems: + - "CoxeterMatrix.I\u2082\u2098IsCoxeter" + - "CoxeterMatrix.F\u2084IsCoxeter" + - "CoxeterMatrix.E\u2086IsCoxeter" + - CoxeterSystem.ext + - "CoxeterMatrix.G\u2082IsCoxeter" + - "CoxeterMatrix.E\u2088IsCoxeter" + - "CoxeterMatrix.A\u2099IsCoxeter" + - CoxeterSystem.map_relations_eq_reindex_relations + - "CoxeterMatrix.B\u2099IsCoxeter" + - "CoxeterMatrix.E\u2087IsCoxeter" + - "CoxeterMatrix.H\u2084IsCoxeter" + - "CoxeterMatrix.D\u2099IsCoxeter" + - "CoxeterMatrix.H\u2083IsCoxeter" + - path: Mathlib/Algebra/Homology/ComplexShapeSigns.lean + theorems: + - "ComplexShape.\u03B5\u2081_\u03B5\u2082" + - "ComplexShape.\u03B5_zero" + - "ComplexShape.\u03B5_add" + - "ComplexShape.\u03C3_symm" + - "ComplexShape.\u03C0_symm" + - path: Mathlib/Algebra/Star/Pointwise.lean + theorems: + - Set.image_star + - Set.star_add + - Set.star_inv + - Set.star_subset + - Set.star_mul + - Set.star_inv' + - StarMemClass.star_coe_eq + - Set.star_mem_star + - Set.star_singleton + - path: Mathlib/Topology/ExtremallyDisconnected.lean + theorems: + - exists_compact_surjective_zorn_subset + - StoneCech.projective + - CompactT2.ExtremallyDisconnected.projective + - image_subset_closure_compl_image_compl_of_isOpen + - CompactT2.Projective.extremallyDisconnected + - ExtremallyDisconnected.homeoCompactToT2_injective + - path: Mathlib/SetTheory/Game/Impartial.lean + theorems: + - SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero + - SetTheory.PGame.Impartial.lf_zero_iff + - SetTheory.PGame.impartial_def + - SetTheory.PGame.Impartial.exists_right_move_equiv_iff_fuzzy_zero + - SetTheory.PGame.Impartial.nonneg + - SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero' + - SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero + - SetTheory.PGame.Impartial.nonpos + - SetTheory.PGame.Impartial.le_zero_iff + - SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero + - SetTheory.PGame.Impartial.exists_left_move_equiv_iff_fuzzy_zero + - SetTheory.PGame.Impartial.impartial_congr + - SetTheory.PGame.impartialAux_def + - SetTheory.PGame.Impartial.equiv_or_fuzzy_zero + - path: Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean + theorems: + - ModuleCat.ExtendScalars.map'_comp + - ModuleCat.ExtendScalars.map'_id + - path: Mathlib/CategoryTheory/Sites/SheafOfTypes.lean + theorems: + - CategoryTheory.Sieve.yonedaFamily_fromCocone_compatible + - CategoryTheory.Presieve.isSheaf_bot + - CategoryTheory.Sieve.forallYonedaIsSheaf_iff_colimit + - CategoryTheory.Presieve.isSheaf_pretopology + - path: Mathlib/Algebra/GradedMonoid.lean + theorems: + - SetLike.list_prod_ofFn_mem_graded + - SetLike.list_prod_map_mem_graded + - List.dProd_monoid + - List.dProdIndex_eq_map_sum + - GradedMonoid.list_prod_ofFn_eq_dProd + - SetLike.coe_list_dProd + - GradedMonoid.list_prod_map_eq_dProd + - GradedMonoid.mk_list_dProd + - path: Mathlib/Topology/Bornology/Basic.lean + theorems: + - Bornology.isBounded_compl_iff + - Bornology.isBounded_empty + - Bornology.isBounded_union + - Bornology.isBounded_ofBounded_iff + - Bornology.isBounded_iff_forall_mem + - Bornology.comap_cobounded_le_iff + - Bornology.isBounded_sUnion + - Bornology.nonempty_of_not_isBounded + - Bornology.isBounded_iUnion + - Bornology.ext + - Bornology.isBounded_singleton + - Bornology.cobounded_eq_bot_iff + - Bornology.isBounded_biUnion + - path: Mathlib/Analysis/Asymptotics/Theta.lean + theorems: + - Asymptotics.isTheta_const_mul_right + - Asymptotics.isTheta_of_norm_eventuallyEq' + - Asymptotics.IsTheta.div + - Asymptotics.isTheta_norm_right + - Asymptotics.isTheta_zero_left + - Asymptotics.isTheta_const_mul_left + - Asymptotics.isTheta_norm_left + - Asymptotics.IsTheta.zpow + - Asymptotics.isTheta_of_norm_eventuallyEq + - Asymptotics.IsTheta.isBoundedUnder_le_iff + - Asymptotics.IsTheta.tendsto_zero_iff + - Asymptotics.isTheta_const_const_iff + - Asymptotics.IsTheta.tendsto_norm_atTop_iff + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean + theorems: + - Matrix.trace_eq_sum_roots_charpoly_of_splits + - Matrix.det_eq_prod_roots_charpoly_of_splits + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean + theorems: + - ContinuousLinearMap.nnnorm_def + - ContinuousLinearMap.opNNNorm_le_bound' + - ContinuousLinearMap.sSup_unit_ball_eq_norm + - ContinuousLinearMap.exists_mul_lt_apply_of_lt_opNNNorm + - ContinuousLinearMap.sSup_closed_unit_ball_eq_norm + - ContinuousLinearMap.sSup_unit_ball_eq_nnnorm + - ContinuousLinearMap.exists_lt_apply_of_lt_opNorm + - ContinuousLinearMap.exists_mul_lt_of_lt_opNorm + - ContinuousLinearMap.isLeast_opNNNorm + - ContinuousLinearMap.sSup_closed_unit_ball_eq_nnnorm + - ContinuousLinearMap.exists_lt_apply_of_lt_opNNNorm + - ContinuousLinearMap.opNNNorm_le_of_unit_nnnorm + - path: Mathlib/Topology/ContinuousFunction/Ordered.lean + theorems: + - ContinuousMap.coe_inf' + - ContinuousMap.coe_sup' + - path: Mathlib/MeasureTheory/Covering/Besicovitch.lean + theorems: + - Besicovitch.TauPackage.mem_iUnionUpTo_lastStep + - Besicovitch.exists_disjoint_closedBall_covering_ae_aux + - Besicovitch.TauPackage.lastStep_nonempty + - Besicovitch.SatelliteConfig.hlast' + - Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet + - Besicovitch.TauPackage.monotone_iUnionUpTo + - Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux + - Besicovitch.ae_tendsto_rnDeriv + - Besicovitch.exist_disjoint_covering_families + - Besicovitch.exists_closedBall_covering_tsum_measure_le + - Besicovitch.SatelliteConfig.inter' + - Besicovitch.exist_finset_disjoint_balls_large_measure + - Besicovitch.exists_disjoint_closedBall_covering_ae + - Besicovitch.tendsto_filterAt + - Besicovitch.TauPackage.color_lt + - path: Mathlib/RingTheory/Polynomial/Opposites.lean + theorems: + - Polynomial.leadingCoeff_opRingEquiv + - Polynomial.opRingEquiv_op_monomial + - Polynomial.support_opRingEquiv + - Polynomial.opRingEquiv_symm_C_mul_X_pow + - Polynomial.opRingEquiv_op_C_mul_X_pow + - Polynomial.natDegree_opRingEquiv + - Polynomial.opRingEquiv_symm_monomial + - Polynomial.coeff_opRingEquiv + - path: Mathlib/Data/PNat/Find.lean + theorems: + - PNat.find_le_iff + - PNat.find_comp_succ + - PNat.le_find_iff + - PNat.lt_find_iff + - PNat.find_eq_iff + - PNat.one_le_find + - PNat.find_eq_one + - path: Mathlib/CategoryTheory/Localization/Construction.lean + theorems: + - CategoryTheory.Localization.Construction.morphismProperty_is_top + - CategoryTheory.Localization.Construction.uniq + - CategoryTheory.Localization.Construction.NatTransExtension.app_eq + - CategoryTheory.Localization.Construction.natTrans_hcomp_injective + - CategoryTheory.Localization.Construction.fac + - CategoryTheory.Localization.Construction.natTransExtension_hcomp + - path: Mathlib/Control/Monad/Cont.lean + theorems: + - ContT.ext + - ContT.monadLift_bind + - ExceptT.goto_mkLabel + - StateT.goto_mkLabel + - ReaderT.goto_mkLabel + - WriterT.goto_mkLabel' + - WriterT.goto_mkLabel + - path: Mathlib/Order/RelSeries.lean + theorems: + - RelSeries.append_apply_left + - RelSeries.ext + - RelSeries.smash_natAdd + - RelSeries.append_apply_right + - RelSeries.last_tail + - RelSeries.toList_chain' + - RelSeries.smash_succ_natAdd + - RelSeries.smash_succ_castAdd + - RelSeries.smash_castAdd + - RelSeries.last_append + - RelSeries.head_tail + - RelSeries.last_smash + - RelSeries.last_cons + - RelSeries.head_smash + - path: Mathlib/LinearAlgebra/Matrix/Circulant.lean + theorems: + - Matrix.Fin.circulant_injective + - Matrix.transpose_circulant + - Matrix.circulant_injective + - Matrix.Fin.circulant_isSymm_iff + - Matrix.circulant_single_one + - Matrix.circulant_mul + - Matrix.Fin.circulant_mul + - Matrix.circulant_isSymm_iff + - Matrix.conjTranspose_circulant + - Matrix.circulant_single + - Matrix.Fin.conjTranspose_circulant + - Matrix.Fin.circulant_mul_comm + - Matrix.Fin.circulant_ite + - Matrix.Fin.transpose_circulant + - path: Mathlib/Topology/Bornology/Absorbs.lean + theorems: + - Absorbs.of_boundedSpace + - Absorbs.add + - absorbs_neg_neg + - absorbs_union + - Absorbs.sub + - absorbs_iff_eventually_cobounded_mapsTo + - absorbs_zero_iff + - Absorbs.empty + - Set.Finite.absorbs_sUnion + - Set.Finite.absorbs_biUnion + - absorbs_inter + - absorbent_iff_inv_smul + - Absorbent.vadd_absorbs + - Set.Finite.absorbs_biInter + - Absorbent.absorbs_finite + - Absorbs.univ + - Set.Finite.absorbs_sInter + - path: Mathlib/Computability/Ackermann.lean + theorems: + - ack_succ_succ + - sq_le_two_pow_add_one_minus_three + - ack_add_one_sq_lt_ack_add_three + - add_lt_ack + - ack_succ_zero + - ack_one + - ack_strict_mono_left' + - not_nat_primrec_ack_self + - ack_zero + - ack_succ_right_le_ack_succ_left + - ack_three + - ack_strictMono_right + - exists_lt_ack_of_nat_primrec + - ack_pos + - ack_two + - one_lt_ack_succ_right + - one_lt_ack_succ_left + - not_primrec_ack_self + - ack_add_one_sq_lt_ack_add_four + - path: Mathlib/Algebra/Homology/TotalComplexSymmetry.lean + theorems: + - "HomologicalComplex\u2082.totalFlipIsoX_hom_D\u2082" + - "HomologicalComplex\u2082.\u03B9Total_totalFlipIso_f_hom" + - "HomologicalComplex\u2082.\u03B9Total_totalFlipIso_f_inv" + - "HomologicalComplex\u2082.totalFlipIso_hom_f_D\u2081" + - "HomologicalComplex\u2082.flip_totalFlipIso" + - "HomologicalComplex\u2082.totalFlipIsoX_hom_D\u2081" + - path: Mathlib/Algebra/CharP/MixedCharZero.lean + theorems: + - EqualCharZero.PNat.isUnit_natCast + - split_by_characteristic + - split_by_characteristic_domain + - split_equalCharZero_mixedCharZero + - MixedCharZero.reduce_to_p_prime + - EqualCharZero.of_algebraRat + - EqualCharZero.pnatCast_one + - split_by_characteristic_localRing + - isEmpty_algebraRat_iff_mixedCharZero + - EqualCharZero.nonempty_algebraRat_iff + - EqualCharZero.pnatCast_eq_natCast + - EqualCharZero.to_not_mixedCharZero + - EqualCharZero.of_not_mixedCharZero + - path: Mathlib/CategoryTheory/NatIso.lean + theorems: + - CategoryTheory.NatIso.isIso_inv_app + - CategoryTheory.NatIso.cancel_natIso_hom_right_assoc + - CategoryTheory.NatIso.cancel_natIso_inv_left + - CategoryTheory.NatIso.naturality_2' + - CategoryTheory.NatIso.isIso_of_isIso_app + - CategoryTheory.NatIso.naturality_2 + - CategoryTheory.NatIso.ofComponents.app + - CategoryTheory.NatIso.isIso_map_iff + - CategoryTheory.NatIso.cancel_natIso_inv_right + - CategoryTheory.NatIso.naturality_1 + - CategoryTheory.NatIso.cancel_natIso_hom_right + - CategoryTheory.NatIso.cancel_natIso_hom_left + - CategoryTheory.NatIso.inv_inv_app + - CategoryTheory.NatIso.inv_map_inv_app + - CategoryTheory.NatIso.cancel_natIso_inv_right_assoc + - CategoryTheory.NatIso.naturality_1' + - path: Mathlib/MeasureTheory/Measure/EverywherePos.lean + theorems: + - MeasureTheory.Measure.everywherePosSubset_ae_eq_of_measure_ne_top + - MeasureTheory.Measure.IsEverywherePos.of_forall_exists_nhds_eq + - MeasureTheory.Measure.IsEverywherePos.smul_measure_nnreal + - IsOpen.isEverywherePos + - MeasureTheory.Measure.isEverywherePos_iff_of_forall_exists_nhds_eq + - MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant + - MeasurableSet.everywherePosSubset + - MeasureTheory.Measure.isEverywherePos_everywherePosSubset + - MeasureTheory.Measure.everywherePosSubset_ae_eq + - MeasureTheory.Measure.isEverywherePos_everywherePosSubset_of_measure_ne_top + - MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group + - IsClosed.everywherePosSubset + - MeasureTheory.Measure.exists_isOpen_everywherePosSubset_eq_diff + - MeasureTheory.Measure.IsEverywherePos.smul_measure + - IsCompact.everywherePosSubset + - path: Mathlib/Algebra/Order/Field/Power.lean + theorems: + - Even.zpow_nonneg + - zpow_two_pos_of_ne_zero + - zpow_le_of_le + - zpow_bit0_pos_iff + - Nat.zpow_pos_of_pos + - zpow_injective + - Odd.zpow_pos_iff + - Even.zpow_pos_iff + - zpow_two_nonneg + - zpow_bit1_nonpos_iff + - Odd.zpow_nonpos_iff + - Even.zpow_abs + - Odd.zpow_neg_iff + - zpow_bit1_neg_iff + - Odd.zpow_nonneg_iff + - zpow_le_max_iff_min_le + - path: Mathlib/CategoryTheory/Abelian/Basic.lean + theorems: + - CategoryTheory.Abelian.imageIsoImage_inv + - CategoryTheory.Abelian.factorThruImage_comp_coimageIsoImage'_inv + - CategoryTheory.Abelian.mono_inl_of_factor_thru_epi_mono_factorization + - "CategoryTheory.Abelian.epi_of_cokernel_\u03C0_eq_zero" + - CategoryTheory.Abelian.epi_fst_of_isLimit + - CategoryTheory.Abelian.mono_inl_of_isColimit + - CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation_e' + - CategoryTheory.Abelian.epi_snd_of_isLimit + - CategoryTheory.Abelian.mono_inr_of_isColimit + - "CategoryTheory.Abelian.image_\u03B9_comp_eq_zero" + - "CategoryTheory.Abelian.imageIsoImage_hom_comp_image_\u03B9" + - CategoryTheory.Abelian.coimageIsoImage'_hom + - "CategoryTheory.Abelian.comp_coimage_\u03C0_eq_zero" + - CategoryTheory.Abelian.epi_fst_of_factor_thru_epi_mono_factorization + - path: Mathlib/Probability/Kernel/WithDensity.lean + theorems: + - ProbabilityTheory.kernel.withDensity_apply + - ProbabilityTheory.kernel.IsSFiniteKernel.withDensity + - ProbabilityTheory.kernel.withDensity_congr_ae + - ProbabilityTheory.kernel.withDensity_kernel_sum + - ProbabilityTheory.kernel.withDensity_tsum + - ProbabilityTheory.kernel.withDensity_one + - ProbabilityTheory.kernel.integral_withDensity + - ProbabilityTheory.kernel.withDensity_mul + - ProbabilityTheory.kernel.lintegral_withDensity + - ProbabilityTheory.kernel.withDensity_of_not_measurable + - ProbabilityTheory.kernel.isSFiniteKernel_withDensity_of_isFiniteKernel + - ProbabilityTheory.kernel.withDensity_add_right + - ProbabilityTheory.kernel.withDensity_sub_add_cancel + - ProbabilityTheory.kernel.withDensity_add_left + - ProbabilityTheory.kernel.isFiniteKernel_withDensity_of_bounded + - ProbabilityTheory.kernel.withDensity_absolutelyContinuous + - ProbabilityTheory.kernel.withDensity_apply' + - ProbabilityTheory.kernel.withDensity_zero + - path: Mathlib/Data/Int/Cast/Basic.lean + theorems: + - Int.cast_subNatNat + - Int.cast_two + - Nat.cast_sub + - Int.int_cast_ofNat + - Int.cast_add + - Nat.cast_pred + - Int.cast_negOfNat + - Int.cast_one + - Int.cast_neg + - Int.cast_sub + - Int.cast_four + - Int.cast_bit1 + - Int.cast_three + - path: Mathlib/RingTheory/Nullstellensatz.lean + theorems: + - MvPolynomial.vanishingIdeal_zeroLocus_eq_radical + - MvPolynomial.radical_le_vanishingIdeal_zeroLocus + - MvPolynomial.isMaximal_iff_eq_vanishingIdeal_singleton + - path: .lake/packages/std/Std/Data/Nat/Lemmas.lean + theorems: + - Nat.sub_min_sub_left + - Nat.sum_append + - Nat.compare_def_lt + - Nat.mul_max_mul_left + - Nat.compare_ne_gt + - Nat.compare_def_le + - Nat.compare_eq_lt + - Nat.recDiagAux_zero_right + - Nat.recDiag_succ_zero + - Nat.sub_max_sub_left + - Nat.compare_eq_gt + - Nat.compare_swap + - Nat.recDiagAux_zero_left + - Nat.mul_min_mul_left + - Nat.mul_max_mul_right + - Nat.strongRec_eq + - Nat.mul_min_mul_right + - Nat.compare_ne_lt + - Nat.recDiag_zero_succ + - path: Mathlib/Algebra/Algebra/Unitization.lean + theorems: + - Unitization.inl_mul + - Unitization.inr_mul_inl + - Unitization.inr_star + - NonUnitalAlgHom.toAlgHom_zero + - Unitization.algHom_ext + - Unitization.inr_mul + - Unitization.algHom_ext'' + - Unitization.inl_star + - Unitization.inl_mul_inr + - path: Mathlib/Init/Logic.lean + theorems: + - decide_True' + - decide_False' + - let_body_eq + - let_value_heq + - path: Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean + theorems: + - ContDiffBump.normed_convolution_eq_right + - ContDiffBump.dist_normed_convolution_le + - ContDiffBump.convolution_eq_right + - ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable + - path: Mathlib/Data/Multiset/NatAntidiagonal.lean + theorems: + - Multiset.Nat.antidiagonal_succ_succ' + - Multiset.Nat.antidiagonal_succ + - Multiset.Nat.card_antidiagonal + - Multiset.Nat.antidiagonal_succ' + - Multiset.Nat.mem_antidiagonal + - Multiset.Nat.map_swap_antidiagonal + - path: Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean + theorems: + - natDegree_minpolyDiv_lt + - minpolyDiv_eq_zero + - "eval\u2082_minpolyDiv_self" + - minpolyDiv_ne_zero + - minpolyDiv_eq_of_isIntegrallyClosed + - eval_minpolyDiv_of_aeval_eq_zero + - minpolyDiv_monic + - natDegree_minpolyDiv + - eval_minpolyDiv_self + - minpolyDiv_eval_eq_zero_of_ne_of_aeval_eq_zero + - minpolyDiv_spec + - coeff_minpolyDiv_sub_pow_mem_span + - coeff_minpolyDiv_mem_adjoin + - "eval\u2082_minpolyDiv_of_eval\u2082_eq_zero" + - sum_smul_minpolyDiv_eq_X_pow + - span_coeff_minpolyDiv + - coeff_minpolyDiv + - path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean + theorems: + - HasStrictFDerivAt.approximates_deriv_on_nhds + - HasStrictFDerivAt.map_nhds_eq_of_surj + - HasStrictFDerivAt.approximates_deriv_on_open_nhds + - path: Mathlib/Topology/Instances/Real.lean + theorems: + - Real.totallyBounded_ball + - Int.tendsto_zmultiplesHom_cofinite + - Real.cobounded_eq + - Int.tendsto_coe_cofinite + - Real.isBounded_iff_bddBelow_bddAbove + - Real.mem_closure_iff + - Real.uniformContinuous_neg + - closure_of_rat_image_lt + - Real.isTopologicalBasis_Ioo_rat + - Function.Periodic.compact_of_continuous + - path: Mathlib/FieldTheory/Laurent.lean + theorems: + - RatFunc.taylor_mem_nonZeroDivisors + - RatFunc.laurent_at_zero + - RatFunc.laurent_X + - RatFunc.laurent_C + - RatFunc.laurentAux_div + - RatFunc.laurent_injective + - RatFunc.laurent_laurent + - path: Mathlib/NumberTheory/FunctionField.lean + theorems: + - FunctionField.InftyValuation.map_mul' + - FunctionField.inftyValuation.X + - FunctionField.InftyValuation.map_one' + - FunctionField.inftyValuation_of_nonzero + - FunctionField.ringOfIntegers.algebraMap_injective + - functionField_iff + - FunctionField.InftyValuation.map_add_le_max' + - FunctionField.ringOfIntegers.not_isField + - FunctionField.inftyValuation.polynomial + - FunctionField.inftyValuation.C + - algebraMap_injective + - path: Mathlib/Data/Matrix/Reflection.lean + theorems: + - "Matrix.mul\u1D63_eq" + - "Matrix.dotProduct\u1D63_eq" + - "Matrix.mulVec\u1D63_eq" + - Matrix.etaExpand_eq + - Matrix.forall_iff + - "Matrix.vecMul\u1D63_eq" + - "Matrix.transpose\u1D63_eq" + - Matrix.exists_iff + - path: Mathlib/Probability/Cdf.lean + theorems: + - ProbabilityTheory.measure_cdf + - MeasureTheory.Measure.eq_of_cdf + - ProbabilityTheory.cdf_measure_stieltjesFunction + - ProbabilityTheory.cdf_eq_toReal + - ProbabilityTheory.ofReal_cdf + - MeasureTheory.Measure.cdf_eq_iff + - path: Mathlib/GroupTheory/GroupAction/Group.lean + theorems: + - smul_zpow + - IsUnit.smul_sub_iff_sub_inv_smul + - inv_smul_smul + - smul_eq_zero_iff_eq + - Commute.smul_left_iff + - "inv_smul_eq_iff\u2080" + - smul_inv + - smul_inv_smul + - path: Mathlib/LinearAlgebra/TensorPower.lean + theorems: + - "TensorPower.mul_algebraMap\u2080" + - "TensorPower.algebraMap\u2080_mul" + - "TensorPower.algebraMap\u2080_mul_algebraMap\u2080" + - TensorPower.mul_one + - "TensorPower.algebraMap\u2080_eq_smul_one" + - TensorPower.mul_assoc + - TensorPower.one_mul + - PiTensorProduct.gradedMonoid_eq_of_reindex_cast + - TensorPower.tprod_mul_tprod + - TensorPower.cast_eq_cast + - TensorPower.gradedMonoid_eq_of_cast + - path: Mathlib/MeasureTheory/Measure/Sub.lean + theorems: + - MeasureTheory.Measure.sub_add_cancel_of_le + - MeasureTheory.Measure.sub_eq_zero_of_le + - MeasureTheory.Measure.restrict_sub_eq_restrict_sub_restrict + - MeasureTheory.Measure.sub_apply_eq_zero_of_restrict_le_restrict + - path: Mathlib/Computability/Language.lean + theorems: + - Language.reverse_mem_reverse + - Language.map_kstar + - Language.one_add_kstar_mul_self_eq_kstar + - Language.kstar_def_nonempty + - Language.reverse_kstar + - Language.reverse_mul + - Language.mem_kstar_iff_exists_nonempty + - Language.reverse_one + - Language.kstar_eq_iSup_pow + - Language.map_map + - Language.one_add_self_mul_kstar_eq_kstar + - Language.nil_mem_kstar + - Language.mem_pow + - Language.map_id + - Language.le_mul_congr + - Language.mem_one + - path: Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean + theorems: + - mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt + - cyclotomic_comp_X_add_one_isEisensteinAt + - mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt + - dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt + - mem_adjoin_of_dvd_coeff_of_dvd_aeval + - cyclotomic_prime_pow_comp_X_add_one_isEisensteinAt + - path: Mathlib/Data/Nat/Choose/Vandermonde.lean + theorems: + - Nat.add_choose_eq + - path: Mathlib/Topology/MetricSpace/PartitionOfUnity.lean + theorems: + - EMetric.eventually_nhds_zero_forall_closedBall_subset + - EMetric.exists_continuous_nnreal_forall_closedBall_subset + - Metric.exists_continuous_nnreal_forall_closedBall_subset + - "EMetric.exists_forall_closedBall_subset_aux\u2081" + - path: Mathlib/LinearAlgebra/TensorProduct/Matrix.lean + theorems: + - TensorProduct.toMatrix_comm + - Matrix.toLin_kronecker + - TensorProduct.toMatrix_assoc + - TensorProduct.toMatrix_map + - path: Mathlib/Algebra/Lie/Quotient.lean + theorems: + - LieSubmodule.Quotient.map_mk'_eq_bot_le + - LieSubmodule.Quotient.range_mk' + - LieSubmodule.Quotient.mk'_ker + - path: Mathlib/Data/Finset/Antidiagonal.lean + theorems: + - Finset.map_prodComm_antidiagonal + - Finset.filter_snd_eq_antidiagonal + - Finset.hasAntidiagonal_congr + - Finset.antidiagonal_zero + - Finset.antidiagonal_congr' + - Finset.antidiagonal.snd_le + - Finset.swap_mem_antidiagonal + - Finset.antidiagonal.fst_le + - Finset.antidiagonal_congr + - Finset.filter_fst_eq_antidiagonal + - path: Mathlib/CategoryTheory/Monoidal/Preadditive.lean + theorems: + - "CategoryTheory.rightDistributor_ext\u2082_left" + - CategoryTheory.rightDistributor_ext_right + - "CategoryTheory.biproduct_\u03B9_comp_rightDistributor_hom" + - "CategoryTheory.rightDistributor_hom_comp_biproduct_\u03C0" + - CategoryTheory.leftDistributor_hom + - CategoryTheory.MonoidalPreadditive.tensor_zero + - "CategoryTheory.leftDistributor_inv_comp_biproduct_\u03C0" + - "CategoryTheory.leftDistributor_ext\u2082_left" + - "CategoryTheory.biproduct_\u03B9_comp_rightDistributor_inv" + - "CategoryTheory.biproduct_\u03B9_comp_leftDistributor_inv" + - "CategoryTheory.biproduct_\u03B9_comp_leftDistributor_hom" + - CategoryTheory.leftDistributor_inv + - "CategoryTheory.rightDistributor_ext\u2082_right" + - CategoryTheory.rightDistributor_ext_left + - CategoryTheory.rightDistributor_inv + - "CategoryTheory.rightDistributor_inv_comp_biproduct_\u03C0" + - "CategoryTheory.leftDistributor_hom_comp_biproduct_\u03C0" + - CategoryTheory.rightDistributor_hom + - CategoryTheory.MonoidalPreadditive.add_tensor + - CategoryTheory.leftDistributor_ext_left + - CategoryTheory.tensor_sum + - CategoryTheory.rightDistributor_assoc + - CategoryTheory.sum_tensor + - CategoryTheory.monoidalPreadditive_of_faithful + - CategoryTheory.leftDistributor_ext_right + - CategoryTheory.MonoidalPreadditive.tensor_add + - CategoryTheory.MonoidalPreadditive.zero_tensor + - CategoryTheory.leftDistributor_rightDistributor_assoc + - path: Mathlib/Algebra/Homology/ComplexShape.lean + theorems: + - ComplexShape.prev_eq' + - ComplexShape.symm_symm + - ComplexShape.next_eq' + - path: Mathlib/Analysis/Complex/RemovableSingularity.lean + theorems: + - Complex.differentiableOn_update_limUnder_of_isLittleO + - Complex.differentiableOn_compl_singleton_and_continuousAt_iff + - Complex.tendsto_limUnder_of_differentiable_on_punctured_nhds_of_isLittleO + - Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt + - Complex.two_pi_I_inv_smul_circleIntegral_sub_sq_inv_smul_of_differentiable + - path: Mathlib/NumberTheory/Bernoulli.lean + theorems: + - bernoulli'_two + - bernoulli'_odd_eq_zero + - bernoulli'_one + - sum_Ico_pow + - bernoulli'_def + - sum_bernoulli' + - bernoulli'_zero + - sum_range_pow + - bernoulli'_three + - bernoulli'_spec + - bernoulli_eq_bernoulli'_of_ne_one + - sum_bernoulli + - bernoulli_one + - bernoulli_zero + - bernoulli'PowerSeries_mul_exp_sub_one + - bernoulli'_four + - bernoulli_spec' + - bernoulli'_spec' + - bernoulli'_eq_bernoulli + - bernoulliPowerSeries_mul_exp_sub_one + - path: Mathlib/CategoryTheory/Limits/ConcreteCategory.lean + theorems: + - CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit + - CategoryTheory.Limits.Concrete.isColimit_rep_eq_of_exists + - CategoryTheory.Limits.Concrete.isColimit_exists_of_rep_eq + - CategoryTheory.Limits.Concrete.isColimit_exists_rep + - CategoryTheory.Limits.Concrete.to_product_injective_of_isLimit + - path: Mathlib/RingTheory/PowerBasis.lean + theorems: + - PowerBasis.degree_minpoly + - PowerBasis.constr_pow_aeval + - PowerBasis.exists_eq_aeval + - PowerBasis.adjoin_gen_eq_top + - PowerBasis.constr_pow_mul + - PowerBasis.mem_span_pow + - IsIntegral.mem_span_pow + - PowerBasis.aeval_minpolyGen + - PowerBasis.degree_minpolyGen + - PowerBasis.constr_pow_algebraMap + - PowerBasis.finrank + - PowerBasis.dim_le_degree_of_root + - PowerBasis.exists_eq_aeval' + - PowerBasis.constr_pow_gen + - PowerBasis.minpolyGen_eq + - PowerBasis.algHom_ext + - PowerBasis.equivOfRoot_map + - PowerBasis.minpolyGen_monic + - PowerBasis.adjoin_eq_top_of_gen_mem_adjoin + - PowerBasis.minpolyGen_map + - PowerBasis.mem_span_pow' + - PowerBasis.dim_le_natDegree_of_root + - PowerBasis.leftMulMatrix + - linearIndependent_pow + - path: Mathlib/Data/Set/Intervals/Instances.lean + theorems: + - Set.Ioo.one_minus_lt_one + - Set.Icc.one_sub_le_one + - Set.Icc.coe_eq_one + - Set.Ico.coe_eq_zero + - Set.Icc.coe_eq_zero + - Set.Ioo.one_sub_mem + - Set.Icc.one_sub_mem + - Set.Ioo.one_minus_pos + - path: Mathlib/Analysis/Complex/AbelLimit.lean + theorems: + - Complex.stolzCone_subset_stolzSet_aux + - Complex.nhdsWithin_lt_le_nhdsWithin_stolzSet + - Complex.stolzCone_subset_stolzSet_aux' + - Complex.stolzSet_empty + - Complex.abel_aux + - Complex.nhdsWithin_stolzCone_le_nhdsWithin_stolzSet + - path: Mathlib/Algebra/Order/CompleteField.lean + theorems: + - LinearOrderedField.exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self + - LinearOrderedField.inducedMap_inv_self + - ringHom_monotone + - LinearOrderedField.inducedMap_add + - LinearOrderedField.cutMap_add + - LinearOrderedField.lt_inducedMap_iff + - LinearOrderedField.cutMap_self + - LinearOrderedField.coe_lt_inducedMap_iff + - LinearOrderedField.inducedMap_inducedMap + - LinearOrderedField.inducedMap_rat + - LinearOrderedField.cutMap_coe + - path: Mathlib/Probability/Martingale/OptionalStopping.lean + theorems: + - MeasureTheory.smul_le_stoppedValue_hitting + - MeasureTheory.submartingale_of_expected_stoppedValue_mono + - MeasureTheory.Submartingale.stoppedProcess + - MeasureTheory.maximal_ineq + - MeasureTheory.Submartingale.expected_stoppedValue_mono + - path: Mathlib/Analysis/BoxIntegral/Box/Basic.lean + theorems: + - BoxIntegral.Box.coe_inf + - BoxIntegral.Box.exists_seq_mono_tendsto + - BoxIntegral.Box.withBotCoe_subset_iff + - BoxIntegral.Box.not_disjoint_coe_iff_nonempty_inter + - BoxIntegral.Box.distortion_eq_of_sub_eq_div + - BoxIntegral.Box.isSome_iff + - BoxIntegral.Box.injective_coe + - BoxIntegral.Box.le_TFAE + - BoxIntegral.Box.diam_Icc_le_of_distortion_le + - BoxIntegral.Box.coe_mk' + - BoxIntegral.Box.mapsTo_insertNth_face + - BoxIntegral.Box.disjoint_withBotCoe + - BoxIntegral.Box.biUnion_coe_eq_coe + - BoxIntegral.Box.dist_le_distortion_mul + - BoxIntegral.Box.nndist_le_distortion_mul + - BoxIntegral.Box.withBotCoe_inj + - BoxIntegral.Box.mk'_eq_bot + - path: Mathlib/Algebra/Order/Rearrangement.lean + theorems: + - Antivary.sum_smul_eq_sum_comp_perm_smul_iff + - AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff + - MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff + - MonovaryOn.sum_comp_perm_smul_le_sum_smul + - Antivary.sum_smul_lt_sum_comp_perm_smul_iff + - MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff + - Antivary.sum_smul_lt_sum_smul_comp_perm_iff + - MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff + - Monovary.sum_comp_perm_smul_eq_sum_smul_iff + - MonovaryOn.sum_smul_comp_perm_le_sum_smul + - Antivary.sum_smul_eq_sum_smul_comp_perm_iff + - AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff + - MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff + - Monovary.sum_smul_comp_perm_lt_sum_smul_iff + - path: Mathlib/Condensed/Explicit.lean + theorems: + - Profinite.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition' + - CompHaus.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition' + - Stonean.isSheaf_iff_preservesFiniteProducts + - Profinite.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition + - CategoryTheory.isSheaf_coherent_iff_regular_and_extensive + - Stonean.isSheaf_iff_preservesFiniteProducts' + - CompHaus.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition + - path: Mathlib/CategoryTheory/Localization/Equivalence.lean + theorems: + - CategoryTheory.Functor.IsLocalization.of_equivalences + - CategoryTheory.Localization.equivalence_counitIso_app + - path: .lake/packages/std/Std/Classes/Order.lean + theorems: + - Std.OrientedCmp.cmp_eq_eq_symm + - Ordering.swap_inj + - Std.TransCmp.gt_trans + - Std.OrientedCmp.cmp_eq_gt + - Ordering.swap_swap + - Std.TransCmp.ge_trans + - Std.TransCmp.cmp_congr_right + - path: Mathlib/Algebra/BigOperators/NatAntidiagonal.lean + theorems: + - Finset.Nat.prod_antidiagonal_succ' + - Finset.Nat.prod_antidiagonal_succ + - Finset.Nat.prod_antidiagonal_swap + - Finset.Nat.prod_antidiagonal_subst + - path: .lake/packages/std/Std/Data/ByteArray.lean + theorems: + - ByteArray.append_data + - ByteArray.extract_data + - ByteArray.size_extract + - ByteArray.ofFn_eq_ofFnAux + - ByteArray.get_extract + - ByteArray.get_append_right + - ByteArray.set_set + - ByteArray.get_extract_aux + - ByteArray.get_ofFn + - ByteArray.get_append_left + - ByteArray.size_ofFn + - ByteArray.size_append + - path: Mathlib/Order/Closure.lean + theorems: + - ClosureOperator.closure_sup_closure + - LowerAdjoint.mem_iff + - LowerAdjoint.closure_union_closure + - ClosureOperator.IsClosed.closure_le_iff + - ClosureOperator.eq_ofPred_closed + - ClosureOperator.setOf_isClosed_eq_range_closure + - ClosureOperator.closure_sup_closure_right + - LowerAdjoint.ext + - closureOperator_gi_self + - path: Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean + theorems: + - FiniteDimensional.finrank_linearMap + - cardinal_mk_algHom_le_rank + - card_algHom_le_finrank + - FiniteDimensional.rank_linearMap + - FiniteDimensional.rank_linearMap_self + - FiniteDimensional.finrank_linearMap_self + - Matrix.rank_vecMulVec + - path: Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean + theorems: + - periodic_expMapCircle + - Real.Angle.coe_expMapCircle + - arg_expMapCircle + - Real.Angle.expMapCircle_add + - Real.Angle.arg_expMapCircle + - expMapCircle_eq_expMapCircle + - Real.Angle.expMapCircle_neg + - Real.Angle.expMapCircle_zero + - path: Mathlib/Algebra/Order/AbsoluteValue.lean + theorems: + - AbsoluteValue.map_neg + - AbsoluteValue.le_add + - AbsoluteValue.sub_le + - AbsoluteValue.abs_abv_sub_le_abv_sub + - AbsoluteValue.le_sub + - IsAbsoluteValue.abv_sub_le + - AbsoluteValue.sub_le_add + - AbsoluteValue.map_sub + - AbsoluteValue.map_one_of_isLeftRegular + - path: Mathlib/Topology/Algebra/Polynomial.lean + theorems: + - Polynomial.coeff_le_of_roots_le + - Polynomial.coeff_bdd_of_roots_le + - Polynomial.tendsto_abv_atTop + - Polynomial.eq_one_of_roots_le + - Polynomial.exists_forall_norm_le + - "Polynomial.tendsto_abv_eval\u2082_atTop" + - path: Mathlib/Tactic/NormNum/DivMod.lean + theorems: + - Mathlib.Meta.NormNum.isInt_ediv_zero + - Mathlib.Meta.NormNum.isInt_ediv + - Mathlib.Meta.NormNum.isInt_emod + - Mathlib.Meta.NormNum.isInt_emod_zero + - Mathlib.Meta.NormNum.isNat_neg_of_isNegNat + - Mathlib.Meta.NormNum.isInt_emod_neg + - Mathlib.Meta.NormNum.isInt_ediv_neg + - path: Mathlib/Data/List/Perm.lean + theorems: + - List.perm_of_mem_permutationsAux + - List.Perm.foldr_eq + - List.DecEq_eq + - List.permutations_perm_permutations' + - List.perm_comp_perm + - List.Perm.bagInter_left + - List.nodup_permutations'Aux_iff + - List.Perm.take_inter + - List.count_permutations'Aux_self + - List.perm_replicate_append_replicate + - List.count_eq_count_filter_add + - List.perm_option_to_list + - List.injective_permutations'Aux + - List.Perm.dedup + - "List.perm_comp_forall\u2082" + - List.permutations'Aux_nthLe_zero + - List.perm_lookmap + - "List.forall\u2082_comp_perm_eq_perm_comp_forall\u2082" + - List.nodup_permutations'Aux_of_not_mem + - List.subperm_singleton_iff + - List.Perm.erasep + - List.rel_perm_imp + - List.perm_permutations'Aux_comm + - List.nodup_permutations + - List.length_permutationsAux + - List.map_append_bind_perm + - List.mem_permutations_of_perm_lemma + - List.subperm_iff + - List.Perm.bagInter_right + - List.Perm.dropSlice_inter + - List.Perm.prod_eq' + - List.Perm.permutations' + - List.Perm.bind_left + - List.mem_permutationsAux_of_perm + - List.Perm.drop_inter + - List.bind_append_perm + - List.nthLe_permutations'Aux + - List.cons_subperm_of_mem + - List.Perm.inter_append + - path: Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean + theorems: + - MeasureTheory.integral_llr_tilted_right + - MeasureTheory.exp_llr_of_ac + - MeasureTheory.exp_neg_llr' + - MeasureTheory.exp_llr + - MeasureTheory.llr_tilted_left + - MeasureTheory.integrable_llr_tilted_right + - MeasureTheory.integrable_llr_tilted_left + - MeasureTheory.neg_llr + - MeasureTheory.exp_neg_llr + - MeasureTheory.llr_tilted_right + - MeasureTheory.llr_smul_left + - MeasureTheory.exp_llr_of_ac' + - MeasureTheory.integral_llr_tilted_left + - MeasureTheory.llr_smul_right + - path: Mathlib/Analysis/Calculus/Deriv/Inv.lean + theorems: + - deriv_inv + - HasDerivAt.div + - HasDerivWithinAt.inv + - fderiv_inv + - HasDerivAt.inv + - derivWithin_inv + - HasStrictDerivAt.div + - hasStrictDerivAt_inv + - fderivWithin_inv + - path: Mathlib/RingTheory/FinitePresentation.lean + theorems: + - Algebra.FinitePresentation.trans + - Algebra.FinitePresentation.mvPolynomial + - Algebra.FinitePresentation.quotient + - RingHom.FinitePresentation.comp + - Algebra.FinitePresentation.of_restrict_scalars_finitePresentation + - Algebra.FinitePresentation.iff_quotient_mvPolynomial' + - AlgHom.FinitePresentation.of_surjective + - Algebra.FinitePresentation.of_finiteType + - Algebra.FinitePresentation.equiv + - RingHom.FinitePresentation.of_comp_finiteType + - Algebra.FinitePresentation.iff + - Algebra.FinitePresentation.mvPolynomial_of_finitePresentation + - Algebra.FinitePresentation.ker_fg_of_mvPolynomial + - RingHom.FinitePresentation.of_surjective + - Algebra.FinitePresentation.ker_fG_of_surjective + - path: Mathlib/Data/Set/UnionLift.lean + theorems: + - Set.iUnionLift_of_mem + - Set.iUnionLift_binary + - Set.preimage_iUnionLift + - Set.preimage_liftCover + - Set.iUnionLift_unary + - path: Mathlib/Analysis/Calculus/Darboux.lean + theorems: + - hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne + - exists_hasDerivWithinAt_eq_of_gt_of_lt + - Set.OrdConnected.image_hasDerivWithinAt + - path: Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean + theorems: + - uniqueMDiffWithinAt_iff_uniqueDiffWithinAt + - hasMFDerivWithinAt_iff_hasFDerivWithinAt + - mfderiv_eq_fderiv + - mfderivWithin_eq_fderivWithin + - mdifferentiableOn_iff_differentiableOn + - uniqueMDiffOn_iff_uniqueDiffOn + - mdifferentiableWithinAt_iff_differentiableWithinAt + - mdifferentiableAt_iff_differentiableAt + - hasMFDerivAt_iff_hasFDerivAt + - mdifferentiable_iff_differentiable + - path: Mathlib/Computability/Reduce.lean + theorems: + - ManyOneDegree.le_add_right + - manyOneReducible_refl + - OneOneReducible.trans + - OneOneReducible.of_equiv_symm + - ManyOneDegree.le_refl + - ManyOneReducible.trans + - ManyOneDegree.add_le + - manyOneEquiv_toNat + - oneOneReducible_refl + - ManyOneDegree.le_trans + - manyOneReducible_toNat + - ManyOneDegree.le_add_left + - ManyOneDegree.le_antisymm + - disjoin_manyOneReducible + - toNat_manyOneEquiv + - ManyOneDegree.of_eq_of + - path: .lake/packages/std/Std/Data/UnionFind/Lemmas.lean + theorems: + - Std.UnionFind.rank_push + - Std.UnionFind.rankD_push + - Std.UnionFind.equiv_find + - Std.UnionFind.parentD_linkAux + - Std.UnionFind.root_link + - Std.UnionFind.equiv_union + - Std.UnionFind.equiv_rootD + - Std.UnionFind.equiv_rootD_l + - Std.UnionFind.equiv_link + - Std.UnionFind.equiv_empty + - Std.UnionFind.parent_push + - Std.UnionFind.parent_link + - Std.UnionFind.equiv_rootD_r + - Std.UnionFind.rankMax_push + - Std.UnionFind.parentD_push + - Std.UnionFind.equiv_push + - path: Mathlib/Algebra/Homology/Localization.lean + theorems: + - HomotopyCategory.respectsIso_quasiIso + - HomologicalComplex.homologyFunctor_inverts_quasiIso + - HomologicalComplexUpToQuasiIso.Qh_inverts_quasiIso + - ComplexShape.QFactorsThroughHomotopy_of_exists_prev + - ComplexShape.quotient_isLocalization + - HomotopyCategory.quasiIso_eq_quasiIso_map_quotient + - HomotopyCategory.mem_quasiIso_iff + - HomotopyCategory.quotient_map_mem_quasiIso_iff + - HomologicalComplexUpToQuasiIso.isIso_Q_map_iff_mem_quasiIso + - path: Mathlib/MeasureTheory/Measure/AEMeasurable.lean + theorems: + - aemeasurable_of_map_neZero + - MeasureTheory.Measure.map_mono_of_aemeasurable + - MeasurableEmbedding.aemeasurable_map_iff + - aemeasurable_const' + - MeasurableEmbedding.aemeasurable_comp_iff + - AEMeasurable.sum_measure + - AEMeasurable.exists_measurable_nonneg + - "aemeasurable_indicator_iff\u2080" + - MeasureTheory.Measure.restrict_map_of_aemeasurable + - aemeasurable_add_measure_iff + - AEMeasurable.map_map_of_aemeasurable + - aemeasurable_restrict_iff_comap_subtype + - MeasureTheory.NullMeasurable.aemeasurable_of_aerange + - aemeasurable_indicator_iff + - aemeasurable_indicator_const_iff + - MeasureTheory.NullMeasurable.aemeasurable + - aemeasurable_union_iff + - AEMeasurable.subtype_mk + - aemeasurable_Ioi_of_forall_Ioc + - aemeasurable_zero_measure + - AEMeasurable.exists_ae_eq_range_subset + - MeasureTheory.Measure.map_sum + - aemeasurable_uIoc_iff + - path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/FiniteDimensional.lean + theorems: + - ApproximatesLinearOn.exists_homeomorph_extension + - path: Mathlib/Data/Complex/ExponentialBounds.lean + theorems: + - Real.exp_one_lt_d9 + - Real.log_two_near_10 + - Real.log_two_gt_d9 + - Real.exp_neg_one_lt_d9 + - Real.exp_one_near_10 + - Real.log_two_lt_d9 + - Real.exp_one_gt_d9 + - Real.exp_one_near_20 + - path: Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean + theorems: + - UpperHalfPlane.atImInfty_mem + - UpperHalfPlane.bounded_mem + - UpperHalfPlane.zero_at_im_infty + - UpperHalfPlane.IsBoundedAtImInfty.mul + - path: Mathlib/FieldTheory/KrullTopology.lean + theorems: + - krullTopology_totallyDisconnected + - IntermediateField.fixingSubgroup.bot + - IntermediateField.fixingSubgroup_bot + - IntermediateField.fixingSubgroup_isOpen + - IntermediateField.fixingSubgroup_top + - IntermediateField.fixingSubgroup.antimono + - krullTopology_t2 + - path: Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean + theorems: + - Euclidean_space.volume_closedBall + - Complex.volume_ball + - MeasureTheory.volume_sum_rpow_le + - Complex.volume_sum_rpow_lt_one + - MeasureTheory.measure_le_eq_lt + - Complex.volume_closedBall + - Euclidean_space.volume_ball + - MeasureTheory.measure_unitBall_eq_integral_div_gamma + - Complex.volume_sum_rpow_lt + - InnerProductSpace.volume_closedBall + - InnerProductSpace.volume_ball + - MeasureTheory.volume_sum_rpow_lt_one + - MeasureTheory.volume_sum_rpow_lt + - MeasureTheory.measure_lt_one_eq_integral_div_gamma + - Complex.volume_sum_rpow_le + - path: Mathlib/Topology/UniformSpace/AbstractCompletion.lean + theorems: + - AbstractCompletion.extend_map + - "AbstractCompletion.uniformContinuous_extension\u2082" + - AbstractCompletion.map_unique + - AbstractCompletion.extend_unique + - AbstractCompletion.extend_coe + - AbstractCompletion.inverse_compare + - AbstractCompletion.uniformContinuous_extend + - path: Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean + theorems: + - contMDiff_iff_contDiff + - contMDiffWithinAt_iff_contDiffWithinAt + - contMDiffOn_iff_contDiffOn + - contMDiffAt_iff_contDiffAt + - ContMDiffWithinAt.clm_apply + - path: Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean + theorems: + - TensorProduct.directSumRight_symm_lof_tmul + - TensorProduct.directSum_symm_lof_tmul + - TensorProduct.directSum_lof_tmul_lof + - TensorProduct.directSumRight_tmul_lof + - TensorProduct.directSumLeft_symm_lof_tmul + - TensorProduct.directSumLeft_tmul_lof + - path: Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean + theorems: + - "AlgebraicTopology.DoldKan.identity_N\u2082" + - "AlgebraicTopology.DoldKan.\u0393\u2080_obj_termwise_mapMono_comp_PInfty" + - "AlgebraicTopology.DoldKan.\u0393\u2082N\u2082ToKaroubiIso_inv_app" + - "AlgebraicTopology.DoldKan.\u0393\u2082N\u2082ToKaroubiIso_hom_app" + - "AlgebraicTopology.DoldKan.compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082\ + _natTrans" + - "AlgebraicTopology.DoldKan.identity_N\u2082_objectwise" + - "AlgebraicTopology.DoldKan.\u0393\u2082N\u2082.natTrans_app_f_app" + - AlgebraicTopology.DoldKan.PInfty_comp_map_mono_eq_zero + - path: Mathlib/Analysis/Convex/Caratheodory.lean + theorems: + - convexHull_eq_union + - Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull + - Caratheodory.minCardFinsetOfMemConvexHull_nonempty + - Caratheodory.minCardFinsetOfMemConvexHull_card_le_card + - eq_pos_convex_span_of_mem_convexHull + - Caratheodory.mem_convexHull_erase + - path: Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean + theorems: + - CategoryTheory.Functor.isLeftKanExtension_iff_of_iso + - CategoryTheory.Functor.isRightKanExtension_iff_of_iso + - path: Mathlib/Analysis/Convex/Cone/Proper.lean + theorems: + - ProperCone.pointed_zero + - ProperCone.mem_dual + - ProperCone.nonempty + - ProperCone.comap_comap + - ProperCone.toPointedCone_injective + - ProperCone.hyperplane_separation_of_nmem + - ProperCone.map_id + - ProperCone.hyperplane_separation + - path: Mathlib/LinearAlgebra/Matrix/Dual.lean + theorems: + - Matrix.toLin_transpose + - LinearMap.toMatrix_transpose + - path: Mathlib/GroupTheory/GroupAction/SubMulAction/Pointwise.lean + theorems: + - SubMulAction.subset_coe_pow + - SubMulAction.coe_pow + - path: Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean + theorems: + - ContinuousLinearMap.exist_extension_of_finiteDimensional_range + - Real.exists_extension_norm_eq + - exists_extension_norm_eq + - exists_dual_vector' + - exists_dual_vector'' + - exists_dual_vector + - coord_norm' + - path: Mathlib/RingTheory/EisensteinCriterion.lean + theorems: + - Polynomial.EisensteinCriterionAux.isUnit_of_natDegree_eq_zero_of_isPrimitive + - Polynomial.EisensteinCriterionAux.eval_zero_mem_ideal_of_eq_mul_X_pow + - Polynomial.EisensteinCriterionAux.le_natDegree_of_map_eq_mul_X_pow + - Polynomial.EisensteinCriterionAux.map_eq_C_mul_X_pow_of_forall_coeff_mem + - Polynomial.irreducible_of_eisenstein_criterion + - path: Mathlib/Order/Antichain.lean + theorems: + - isAntichain_and_least_iff + - isAntichain_and_greatest_iff + - IsAntichain.image + - IsStrongAntichain.image + - IsAntichain.subsingleton + - IsAntichain.image_relEmbedding + - path: Mathlib/CategoryTheory/Abelian/NonPreadditive.lean + theorems: + - CategoryTheory.NonPreadditiveAbelian.neg_sub + - CategoryTheory.NonPreadditiveAbelian.sub_self + - CategoryTheory.NonPreadditiveAbelian.add_comp + - CategoryTheory.NonPreadditiveAbelian.comp_add + - CategoryTheory.NonPreadditiveAbelian.sub_zero + - CategoryTheory.NonPreadditiveAbelian.sub_comp + - CategoryTheory.NonPreadditiveAbelian.sub_sub_sub + - CategoryTheory.NonPreadditiveAbelian.add_assoc + - CategoryTheory.NonPreadditiveAbelian.comp_sub + - CategoryTheory.NonPreadditiveAbelian.neg_add + - "CategoryTheory.NonPreadditiveAbelian.diag_\u03C3" + - CategoryTheory.NonPreadditiveAbelian.lift_sub_lift + - CategoryTheory.NonPreadditiveAbelian.add_neg_self + - CategoryTheory.NonPreadditiveAbelian.add_comm + - CategoryTheory.NonPreadditiveAbelian.neg_neg + - "CategoryTheory.NonPreadditiveAbelian.lift_\u03C3" + - CategoryTheory.NonPreadditiveAbelian.add_zero + - CategoryTheory.NonPreadditiveAbelian.neg_add_self + - CategoryTheory.NonPreadditiveAbelian.add_neg + - CategoryTheory.NonPreadditiveAbelian.sub_add + - "CategoryTheory.NonPreadditiveAbelian.\u03C3_comp" + - CategoryTheory.NonPreadditiveAbelian.neg_sub' + - CategoryTheory.NonPreadditiveAbelian.lift_map + - path: Mathlib/ModelTheory/Order.lean + theorems: + - FirstOrder.Language.realize_noBotOrder_iff + - FirstOrder.Language.Term.realize_lt + - FirstOrder.Language.Term.realize_le + - FirstOrder.Language.realize_denselyOrdered_iff + - FirstOrder.Language.relMap_leSymb + - FirstOrder.Language.realize_noTopOrder_iff + - path: Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean + theorems: + - MeasureTheory.SignedMeasure.toJordanDecomposition_eq_of_eq_add_withDensity + - MeasureTheory.SignedMeasure.eq_singularPart + - MeasureTheory.SignedMeasure.singularPart_add + - MeasureTheory.SignedMeasure.rnDeriv_neg + - MeasureTheory.SignedMeasure.integrable_rnDeriv + - MeasureTheory.SignedMeasure.eq_singularPart' + - MeasureTheory.ComplexMeasure.singularPart_add_withDensity_rnDeriv_eq + - MeasureTheory.ComplexMeasure.integrable_rnDeriv + - MeasureTheory.SignedMeasure.singularPart_smul_nnreal + - MeasureTheory.SignedMeasure.singularPart_sub + - MeasureTheory.SignedMeasure.measurable_rnDeriv + - MeasureTheory.SignedMeasure.singularPart_neg + - MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq + - MeasureTheory.SignedMeasure.jordanDecomposition_add_withDensity_mutuallySingular + - MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk + - MeasureTheory.SignedMeasure.rnDeriv_add + - MeasureTheory.SignedMeasure.singularPart_mutuallySingular + - MeasureTheory.SignedMeasure.singularPart_smul + - MeasureTheory.SignedMeasure.eq_rnDeriv + - MeasureTheory.SignedMeasure.rnDeriv_sub + - MeasureTheory.SignedMeasure.singularPart_zero + - MeasureTheory.SignedMeasure.singularPart_totalVariation + - MeasureTheory.SignedMeasure.rnDeriv_smul + - MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk' + - MeasureTheory.SignedMeasure.mutuallySingular_singularPart + - path: Mathlib/AlgebraicTopology/SplitSimplicialObject.lean + theorems: + - SimplicialObject.Splitting.IndexSet.eqId_iff_len_eq + - SimplicialObject.Splitting.IndexSet.eqId_iff_len_le + - SimplicialObject.Splitting.IndexSet.ext + - SimplicialObject.Splitting.cofan_inj_id + - SimplicialObject.Splitting.IndexSet.eqId_iff_eq + - SimplicialObject.Split.cofan_inj_naturality_symm + - SimplicialObject.Splitting.cofan_inj_comp_app + - SimplicialObject.Split.Hom.ext + - SimplicialObject.Splitting.cofan_inj_epi_naturality + - SimplicialObject.Splitting.hom_ext + - SimplicialObject.Split.congr_F + - SimplicialObject.Split.congr_f + - "SimplicialObject.Splitting.\u03B9_desc" + - SimplicialObject.Splitting.IndexSet.eqId_iff_mono + - path: Mathlib/Data/Set/Countable.lean + theorems: + - Set.Countable.sUnion_iff + - Set.countable_iUnion + - Set.countable_setOf_finite_subset + - Set.countable_pi + - Set.Countable.biUnion_iff + - Set.countable_setOf_nonempty_of_disjoint + - Set.countable_insert + - Set.subset_range_enumerate + - Set.Countable.image2 + - Set.range_enumerateCountable_subset + - Set.countable_union + - Set.exists_seq_iSup_eq_top_iff_countable + - Set.Countable.exists_eq_range + - Set.enumerateCountable_mem + - Set.Countable.image + - Set.countable_iff_exists_subset_range + - path: Mathlib/CategoryTheory/Limits/Fubini.lean + theorems: + - "CategoryTheory.Limits.colimitIsoColimitCurryCompColim_\u03B9_hom" + - "CategoryTheory.Limits.limitFlipCompLimIsoLimitCompLim_hom_\u03C0_\u03C0" + - "CategoryTheory.Limits.limitFlipCompLimIsoLimitCompLim_inv_\u03C0_\u03C0" + - "CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim_\u03B9\ + _\u03B9_inv" + - "CategoryTheory.Limits.limitIsoLimitCurryCompLim_hom_\u03C0_\u03C0" + - "CategoryTheory.Limits.colimitUncurryIsoColimitCompColim_\u03B9_\u03B9_inv" + - "CategoryTheory.Limits.colimitFlipCompColimIsoColimitCompColim_\u03B9_\u03B9\ + _inv" + - "CategoryTheory.Limits.limitUncurryIsoLimitCompLim_inv_\u03C0" + - "CategoryTheory.Limits.colimitCurrySwapCompColimIsoColimitCurryCompColim_\u03B9\ + _\u03B9_hom" + - "CategoryTheory.Limits.colimitIsoColimitCurryCompColim_\u03B9_\u03B9_inv" + - "CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_hom_\u03C0\ + _\u03C0" + - "CategoryTheory.Limits.limitUncurryIsoLimitCompLim_hom_\u03C0_\u03C0" + - "CategoryTheory.Limits.limitCurrySwapCompLimIsoLimitCurryCompLim_inv_\u03C0\ + _\u03C0" + - "CategoryTheory.Limits.colimitUncurryIsoColimitCompColim_\u03B9_hom" + - "CategoryTheory.Limits.limitIsoLimitCurryCompLim_inv_\u03C0" + - path: Mathlib/Data/Nat/Set.lean + theorems: + - Nat.zero_union_range_succ + - Nat.range_succ + - Nat.range_of_succ + - Nat.range_rec + - path: Mathlib/Algebra/Ring/BooleanRing.lean + theorems: + - neg_eq + - of_boolalg_symmDiff_aux + - add_eq_zero' + - mul_add_mul + - BooleanRing.inf_assoc + - BooleanRing.inf_sup_self + - BooleanRing.inf_comm + - BooleanRing.sup_assoc + - BooleanRing.le_sup_inf_aux + - mul_one_add_self + - sub_eq_add + - add_self + - BooleanRing.sup_comm + - BooleanRing.le_sup_inf + - BooleanRing.sup_inf_self + - path: Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean + theorems: + - MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable + - MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv + - MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable + - MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable_of_le + - "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2081" + - MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le + - "MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux\u2082" + - path: Mathlib/Topology/UniformSpace/CompactConvergence.lean + theorems: + - ContinuousMap.tendsto_iff_tendstoUniformly + - ContinuousMap.tendsto_iff_tendstoLocallyUniformly + - ContinuousMap.tendsto_of_tendstoLocallyUniformly + - ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn + - path: Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean + theorems: + - CategoryTheory.ShortComplex.mapHomologyIso'_hom_naturality + - CategoryTheory.ShortComplex.mapCyclesIso_hom_naturality + - CategoryTheory.ShortComplex.RightHomologyData.mapHomologyIso'_eq + - CategoryTheory.ShortComplex.mapLeftHomologyIso_inv_naturality + - CategoryTheory.ShortComplex.LeftHomologyData.mapLeftHomologyIso_eq + - CategoryTheory.ShortComplex.LeftHomologyData.map_f' + - CategoryTheory.ShortComplex.RightHomologyData.mapOpcyclesIso_eq + - CategoryTheory.ShortComplex.mapHomologyIso_inv_naturality + - CategoryTheory.ShortComplex.mapHomologyIso_hom_naturality + - CategoryTheory.ShortComplex.mapRightHomologyIso_inv_naturality + - CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesRightHomology + - CategoryTheory.ShortComplex.mapRightHomologyIso_hom_naturality + - CategoryTheory.ShortComplex.RightHomologyData.map_opcyclesMap' + - CategoryTheory.ShortComplex.mapLeftHomologyIso_hom_naturality + - CategoryTheory.ShortComplex.mapOpcyclesIso_inv_naturality + - CategoryTheory.ShortComplex.mapHomologyIso'_inv_naturality + - CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesLeftHomology + - CategoryTheory.ShortComplex.RightHomologyData.mapRightHomologyIso_eq + - CategoryTheory.ShortComplex.LeftHomologyData.map_cyclesMap' + - CategoryTheory.ShortComplex.mapCyclesIso_inv_naturality + - CategoryTheory.ShortComplex.mapOpcyclesIso_hom_naturality + - CategoryTheory.ShortComplex.LeftHomologyData.mapCyclesIso_eq + - CategoryTheory.ShortComplex.LeftHomologyData.mapHomologyIso_eq + - CategoryTheory.ShortComplex.LeftHomologyData.map_leftHomologyMap' + - CategoryTheory.NatTrans.app_homology + - CategoryTheory.ShortComplex.mapHomologyIso'_eq_mapHomologyIso + - CategoryTheory.ShortComplex.mapCyclesIso_hom_iCycles + - CategoryTheory.ShortComplex.RightHomologyData.map_rightHomologyMap' + - CategoryTheory.ShortComplex.RightHomologyData.map_g' + - path: Mathlib/CategoryTheory/Limits/VanKampen.lean + theorems: + - CategoryTheory.IsUniversalColimit.whiskerEquivalence_iff + - CategoryTheory.IsVanKampenColimit.mapCocone_iff + - CategoryTheory.IsUniversalColimit.of_iso + - CategoryTheory.IsVanKampenColimit.precompose_isIso_iff + - CategoryTheory.BinaryCofan.isVanKampen_mk + - CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen + - CategoryTheory.IsVanKampenColimit.whiskerEquivalence_iff + - CategoryTheory.NatTrans.equifibered_of_discrete + - CategoryTheory.IsUniversalColimit.of_mapCocone + - CategoryTheory.mono_of_cofan_isVanKampen + - CategoryTheory.IsUniversalColimit.whiskerEquivalence + - CategoryTheory.BinaryCofan.isVanKampen_iff + - CategoryTheory.mapPair_equifibered + - CategoryTheory.IsUniversalColimit.map_reflective + - CategoryTheory.isVanKampenColimit_of_isEmpty + - CategoryTheory.IsUniversalColimit.precompose_isIso + - CategoryTheory.IsVanKampenColimit.map_reflective + - CategoryTheory.isPullback_of_cofan_isVanKampen + - CategoryTheory.IsVanKampenColimit.precompose_isIso + - CategoryTheory.isUniversalColimit_extendCofan + - CategoryTheory.IsVanKampenColimit.of_iso + - CategoryTheory.IsVanKampenColimit.whiskerEquivalence + - CategoryTheory.IsInitial.isVanKampenColimit + - CategoryTheory.isVanKampenColimit_extendCofan + - CategoryTheory.isVanKampenColimit_of_evaluation + - CategoryTheory.isPullback_initial_to_of_cofan_isVanKampen + - CategoryTheory.hasStrictInitial_of_isUniversal + - CategoryTheory.BinaryCofan.mono_inr_of_isVanKampen + - CategoryTheory.IsVanKampenColimit.of_mapCocone + - path: Mathlib/AlgebraicTopology/DoldKan/PInfty.lean + theorems: + - AlgebraicTopology.DoldKan.QInfty_comp_PInfty + - AlgebraicTopology.DoldKan.QInfty_f_0 + - AlgebraicTopology.DoldKan.Q_is_eventually_constant + - AlgebraicTopology.DoldKan.map_PInfty_f + - AlgebraicTopology.DoldKan.PInfty_add_QInfty + - AlgebraicTopology.DoldKan.PInfty_f_idem + - AlgebraicTopology.DoldKan.karoubi_PInfty_f + - AlgebraicTopology.DoldKan.P_is_eventually_constant + - AlgebraicTopology.DoldKan.PInfty_idem + - AlgebraicTopology.DoldKan.PInfty_f_comp_QInfty_f + - path: Mathlib/Analysis/NormedSpace/Star/Basic.lean + theorems: + - CstarRing.norm_coe_unitary_mul + - CstarRing.norm_mul_coe_unitary + - IsSelfAdjoint.nnnorm_pow_two_pow + - CstarRing.star_mul_self_eq_zero_iff + - CstarRing.norm_coe_unitary + - CstarRing.norm_one + - CstarRing.norm_star_mul_self' + - CstarRing.norm_self_mul_star + - CstarRing.mul_star_self_ne_zero_iff + - CstarRing.mul_star_self_eq_zero_iff + - CstarRing.star_mul_self_ne_zero_iff + - path: Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean + theorems: + - measurableSet_lineDifferentiableAt_uncurry + - stronglyMeasurable_lineDeriv + - measurableSet_lineDifferentiableAt + - measurable_lineDeriv + - measurable_lineDeriv_uncurry + - stronglyMeasurable_lineDeriv_uncurry + - path: Mathlib/GroupTheory/SpecificGroups/Quaternion.lean + theorems: + - QuaternionGroup.orderOf_a + - QuaternionGroup.exponent + - QuaternionGroup.a_one_pow_n + - QuaternionGroup.a_one_pow + - QuaternionGroup.xa_pow_four + - QuaternionGroup.orderOf_a_one + - QuaternionGroup.xa_sq + - QuaternionGroup.quaternionGroup_one_isCyclic + - QuaternionGroup.card + - QuaternionGroup.orderOf_xa + - path: Mathlib/LinearAlgebra/Matrix/Block.lean + theorems: + - Matrix.matrixOfPolynomials_blockTriangular + - Matrix.upper_two_blockTriangular + - Matrix.toBlock_inverse_eq_zero + - Matrix.blockTriangular_reindex_iff + - Matrix.twoBlockTriangular_det + - Matrix.blockTriangular_blockDiagonal + - Matrix.det_toBlock + - Matrix.BlockTriangular.det + - Matrix.det_of_lowerTriangular + - Matrix.BlockTriangular.mul + - Matrix.blockTriangular_inv_of_blockTriangular + - Matrix.det_of_upperTriangular + - Matrix.det_matrixOfPolynomials + - Matrix.BlockTriangular.add + - Matrix.twoBlockTriangular_det' + - Matrix.BlockTriangular.det_fintype + - Matrix.equiv_block_det + - Matrix.blockTriangular_blockDiagonal' + - Matrix.BlockTriangular.sub + - Matrix.blockTriangular_stdBasisMatrix' + - Matrix.BlockTriangular.toBlock_inverse_mul_toBlock_eq_one + - path: Mathlib/Order/Filter/ListTraverse.lean + theorems: + - Filter.mem_traverse_iff + - path: Mathlib/Algebra/MonoidAlgebra/Support.lean + theorems: + - MonoidAlgebra.support_single_mul_subset + - AddMonoidAlgebra.mem_span_support' + - MonoidAlgebra.support_mul_single + - MonoidAlgebra.support_single_mul + - MonoidAlgebra.support_mul_single_subset + - AddMonoidAlgebra.mem_span_support + - MonoidAlgebra.support_single_mul_eq_image + - MonoidAlgebra.support_mul_single_eq_image + - path: Mathlib/NumberTheory/FermatPsp.lean + theorems: + - Nat.AB_id_helper + - Nat.probablePrime_iff_modEq + - Nat.b_id_helper + - Nat.coprime_of_fermatPsp + - Nat.fermatPsp_base_one + - Nat.psp_from_prime_gt_p + - Nat.psp_from_prime_psp + - Nat.coprime_of_probablePrime + - Nat.a_id_helper + - Nat.frequently_atTop_fermatPsp + - Nat.bp_helper + - Nat.exists_infinite_pseudoprimes + - path: Mathlib/Topology/Algebra/ValuedField.lean + theorems: + - Valued.extension_extends + - Valued.continuous_valuation + - Valued.closure_coe_completion_v_lt + - Valuation.inversion_estimate + - Valued.continuous_extension + - path: Mathlib/RingTheory/MvPolynomial/Tower.lean + theorems: + - MvPolynomial.aeval_algebraMap_eq_zero_iff + - MvPolynomial.aeval_algebraMap_eq_zero_iff_of_injective + - MvPolynomial.aeval_map_algebraMap + - Subalgebra.mvPolynomial_aeval_coe + - MvPolynomial.aeval_algebraMap_apply + - path: Mathlib/Data/Fin/Tuple/Finset.lean + theorems: + - Fin.snoc_mem_piFinset_snoc + - Fin.mem_piFinset_succ' + - Fin.cons_mem_piFinset_cons + - Fin.mem_piFinset_succ + - path: Mathlib/GroupTheory/Submonoid/Center.lean + theorems: + - Submonoid.mem_center_iff + - path: Mathlib/Data/Nat/PrimeFin.lean + theorems: + - Nat.mem_primeFactors + - Nat.primeFactors_mul + - Nat.primeFactors_pow + - Nat.primeFactors_prime_pow + - Nat.primeFactors_gcd + - Nat.primeFactors_pow_succ + - Nat.nonempty_primeFactors + - Nat.primeFactors_eq_empty + - Nat.disjoint_primeFactors + - Nat.primeFactors_mono + - Nat.mem_primeFactors_iff_mem_factors + - Nat.mem_primeFactors_of_ne_zero + - path: Mathlib/Algebra/AlgebraicCard.lean + theorems: + - Algebraic.cardinal_mk_le_max + - Algebraic.cardinal_mk_le_mul + - Algebraic.cardinal_mk_lift_le_mul + - Algebraic.cardinal_mk_lift_le_max + - Algebraic.countable + - path: Mathlib/Algebra/Lie/Semisimple.lean + theorems: + - LieAlgebra.isSemisimple_iff_no_abelian_ideals + - LieAlgebra.ad_ker_eq_bot_of_semisimple + - LieAlgebra.abelian_radical_iff_solvable_is_abelian + - LieAlgebra.center_eq_bot_of_semisimple + - LieAlgebra.abelian_radical_of_semisimple + - path: Mathlib/Order/Iterate.lean + theorems: + - Function.Commute.iterate_le_of_map_le + - StrictMono.strictMono_iterate_of_lt_map + - Monotone.monotone_iterate_of_le_map + - Function.Commute.iterate_pos_lt_of_map_lt + - Function.Commute.iterate_pos_le_iff_map_le + - Monotone.seq_lt_seq_of_lt_of_le + - Monotone.seq_le_seq + - Function.Commute.iterate_pos_le_iff_map_le' + - Function.monotone_iterate_of_id_le + - Function.Commute.iterate_pos_lt_iff_map_lt + - Monotone.le_iterate_comp_of_le + - Function.Commute.iterate_pos_eq_iff_map_eq + - Function.id_le_iterate_of_id_le + - path: Mathlib/Analysis/NormedSpace/MatrixExponential.lean + theorems: + - Matrix.exp_diagonal + - Matrix.exp_nsmul + - Matrix.exp_blockDiagonal' + - Matrix.exp_sum_of_commute + - Matrix.exp_units_conj + - Matrix.exp_neg + - Matrix.exp_zsmul + - Matrix.exp_blockDiagonal + - Matrix.exp_add_of_commute + - Matrix.exp_transpose + - Matrix.exp_conj + - Matrix.isUnit_exp + - Matrix.exp_conj' + - path: Mathlib/Algebra/Module/Submodule/Bilinear.lean + theorems: + - "Submodule.map\u2082_bot_left" + - "Submodule.map\u2082_iSup_left" + - "Submodule.map\u2082_span_singleton_eq_map" + - "Submodule.map\u2082_iSup_right" + - "Submodule.map\u2082_span_span" + - "Submodule.apply_mem_map\u2082" + - "Submodule.image2_subset_map\u2082" + - LinearMap.ker_restrictBilinear_eq_of_codisjoint + - "Submodule.map\u2082_span_singleton_eq_map_flip" + - "Submodule.map\u2082_bot_right" + - path: Mathlib/Data/List/NatAntidiagonal.lean + theorems: + - List.Nat.antidiagonal_succ_succ' + - List.Nat.antidiagonal_succ + - List.Nat.map_swap_antidiagonal + - List.Nat.length_antidiagonal + - List.Nat.antidiagonal_succ' + - List.Nat.mem_antidiagonal + - path: Mathlib/Algebra/Group/Units/Hom.lean + theorems: + - Units.liftRight_inv_mul + - Units.map_id + - IsUnit.mul_liftRight_inv + - IsUnit.of_leftInverse + - IsUnit.liftRight_inv_mul + - Units.mul_liftRight_inv + - IsUnit.map + - path: Mathlib/Data/Set/Accumulate.lean + theorems: + - Set.mem_accumulate + - Set.biUnion_accumulate + - Set.iUnion_accumulate + - path: Mathlib/GroupTheory/Subsemigroup/Centralizer.lean + theorems: + - Set.zero_mem_centralizer + - Set.div_mem_centralizer + - Set.centralizer_eq_top_iff_subset + - Set.one_mem_centralizer + - Set.neg_mem_centralizer + - Set.inv_mem_centralizer + - "Set.inv_mem_centralizer\u2080" + - Set.mul_mem_centralizer + - Set.add_mem_centralizer + - "Set.div_mem_centralizer\u2080" + - path: .lake/packages/std/Std/Data/Rat/Basic.lean + theorems: + - Rat.sub.aux + - Rat.normalize.reduced + - Rat.add.aux + - path: Mathlib/Analysis/Analytic/IsolatedZeros.lean + theorems: + - AnalyticAt.order_eq_top_iff + - HasFPowerSeriesAt.locally_zero_iff + - AnalyticAt.eventually_eq_or_eventually_ne + - HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope + - AnalyticAt.order_eq_nat_iff + - HasSum.hasSum_at_zero + - AnalyticAt.unique_eventuallyEq_zpow_smul_nonzero + - HasFPowerSeriesAt.locally_ne_zero + - HasSum.exists_hasSum_smul_of_apply_eq_zero + - AnalyticAt.frequently_eq_iff_eventually_eq + - AnalyticAt.exists_eventuallyEq_pow_smul_nonzero_iff + - HasFPowerSeriesAt.has_fpower_series_dslope_fslope + - HasFPowerSeriesAt.has_fpower_series_iterate_dslope_fslope + - AnalyticAt.unique_eventuallyEq_pow_smul_nonzero + - AnalyticAt.eventually_eq_zero_or_eventually_ne_zero + - HasFPowerSeriesAt.iterate_dslope_fslope_ne_zero + - AnalyticOn.eqOn_of_preconnected_of_frequently_eq + - path: Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean + theorems: + - "map_inv\u2080" + - "eq_on_inv\u2080" + - path: Mathlib/RingTheory/WittVector/IsPoly.lean + theorems: + - "WittVector.IsPoly\u2082.map" + - "WittVector.IsPoly\u2082.ext" + - WittVector.IsPoly.map + - WittVector.IsPoly.ext + - WittVector.poly_eq_of_wittPolynomial_bind_eq' + - "WittVector.bind\u2081_onePoly_wittPolynomial" + - WittVector.poly_eq_of_wittPolynomial_bind_eq + - "WittVector.bind\u2081_zero_wittPolynomial" + - path: Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean + theorems: + - Geometry.SimplicialComplex.vertex_mem_convexHull_iff + - Geometry.SimplicialComplex.mem_space_iff + - Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull + - Geometry.SimplicialComplex.vertices_eq + - Geometry.SimplicialComplex.not_facet_iff_subface + - Geometry.SimplicialComplex.convexHull_subset_space + - path: Mathlib/LinearAlgebra/StdBasis.lean + theorems: + - LinearMap.iSup_range_stdBasis_eq_iInf_ker_proj + - Pi.basisFun_repr + - Matrix.stdBasis_eq_stdBasisMatrix + - LinearMap.stdBasis_eq_pi_diag + - Pi.basisFun_apply + - LinearMap.disjoint_stdBasis_stdBasis + - Module.piEquiv_apply_apply + - LinearMap.stdBasis_apply' + - LinearMap.iInf_ker_proj_le_iSup_range_stdBasis + - Module.surjective_piEquiv_apply_iff + - Pi.basis_apply + - LinearMap.iSup_range_stdBasis_le_iInf_ker_proj + - Pi.basis_repr_stdBasis + - LinearMap.proj_comp_stdBasis + - LinearMap.iSup_range_stdBasis + - Pi.linearIndependent_stdBasis + - path: Mathlib/Data/Sum/Interval.lean + theorems: + - Sum.Lex.Icc_inr_inr + - Finset.mem_sumLexLift + - Sum.Lex.Ico_inl_inl + - Sum.Lex.Ioc_inr_inr + - Sum.Ioo_inr_inl + - Sum.Lex.Ioo_inr_inr + - "Finset.mem_sumLift\u2082" + - Finset.sumLexLift_eq_empty + - Sum.Lex.Ico_inr_inr + - Sum.Ioo_inl_inr + - "Finset.sumLift\u2082_eq_empty" + - Sum.Lex.Icc_inl_inl + - Sum.Lex.Ioo_inl_inl + - Sum.Lex.Ioc_inl_inl + - Finset.sumLexLift_mono + - Finset.sumLexLift_nonempty + - "Finset.inl_mem_sumLift\u2082" + - "Finset.inr_mem_sumLift\u2082" + - "Finset.sumLift\u2082_nonempty" + - Finset.inr_mem_sumLexLift + - Finset.inl_mem_sumLexLift + - path: Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean + theorems: + - CategoryTheory.HomOrthogonal.matrixDecomposition_comp + - CategoryTheory.HomOrthogonal.equiv_of_iso + - CategoryTheory.HomOrthogonal.matrixDecomposition_id + - path: Mathlib/Analysis/NormedSpace/Dual.lean + theorems: + - NormedSpace.polar_ball_subset_closedBall_div + - NormedSpace.double_dual_bound + - NormedSpace.isBounded_polar_of_mem_nhds_zero + - NormedSpace.dualPairing_separatingLeft + - NormedSpace.inclusionInDoubleDual_norm_le + - NormedSpace.norm_le_dual_bound + - NormedSpace.eq_iff_forall_dual_eq + - NormedSpace.isClosed_polar + - NormedSpace.polar_closure + - NormedSpace.smul_mem_polar + - NormedSpace.eq_zero_iff_forall_dual_eq_zero + - NormedSpace.polar_closedBall + - NormedSpace.eq_zero_of_forall_dual_eq_zero + - path: Mathlib/GroupTheory/Subgroup/Finite.lean + theorems: + - Subgroup.card_top + - Subgroup.pi_mem_of_mulSingle_mem + - Subgroup.card_le_one_iff_eq_bot + - Subgroup.card_eq_iff_eq_top + - Subgroup.mem_normalizer_fintype + - Subgroup.pi_le_iff + - Subgroup.pi_mem_of_mulSingle_mem_aux + - Subgroup.eq_bot_iff_card + - Subgroup.eq_top_of_card_eq + - path: Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean + theorems: + - "SmoothFiberwiseLinear.locality_aux\u2081" + - FiberwiseLinear.target_trans_partialHomeomorph + - FiberwiseLinear.source_trans_partialHomeomorph + - "SmoothFiberwiseLinear.locality_aux\u2082" + - mem_aux + - path: Mathlib/Control/Traversable/Lemmas.lean + theorems: + - Traversable.traverse_comp + - Traversable.map_traverse + - Traversable.traverse_map' + - Traversable.traverse_id + - Traversable.naturality_pf + - Traversable.id_sequence + - Traversable.pure_traverse + - Traversable.naturality' + - Traversable.traverse_map + - Traversable.map_traverse' + - Traversable.comp_sequence + - Traversable.traverse_eq_map_id' + - path: Mathlib/Algebra/Group/PNatPowAssoc.lean + theorems: + - ppow_mul + - ppow_eq_pow + - ppow_mul_comm + - ppow_mul' + - ppow_mul_assoc + - path: Mathlib/Topology/Algebra/FilterBasis.lean + theorems: + - GroupFilterBasis.N_one + - GroupFilterBasis.nhds_hasBasis + - GroupFilterBasis.mem_nhds_one + - GroupFilterBasis.nhds_one_eq + - GroupFilterBasis.nhds_one_hasBasis + - GroupFilterBasis.nhds_eq + - ContinuousSMul.of_basis_zero + - path: Mathlib/Combinatorics/Quiver/SingleObj.lean + theorems: + - Quiver.SingleObj.listToPath_pathToList + - Quiver.SingleObj.toPrefunctor_symm_comp + - Quiver.SingleObj.pathToList_listToPath + - path: Mathlib/Topology/SeparatedMap.lean + theorems: + - isSeparatedMap_iff_nhds + - discreteTopology_iff_locallyInjective + - isSeparatedMap_iff_isClosed_diagonal + - IsLocallyInjective.isOpen_eqLocus + - isLocallyInjective_iff_nhds + - IsLocallyInjective_iff_openEmbedding + - embedding_toPullbackDiag + - isSeparatedMap_iff_closedEmbedding + - IsSeparatedMap.pullback + - IsSeparatedMap.eqOn_of_comp_eqOn + - isSeparatedMap_iff_disjoint_nhds + - IsLocallyInjective.comp_right + - Continuous.mapPullback + - IsSeparatedMap.eq_of_comp_eq + - IsSeparatedMap.isClosed_eqLocus + - IsSeparatedMap.comp_right + - path: Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean + theorems: + - CategoryTheory.ShortComplex.i_cyclesMk + - CategoryTheory.ShortComplex.ShortExact.surjective_g + - CategoryTheory.Preadditive.mono_iff_injective' + - "CategoryTheory.ShortComplex.SnakeInput.\u03B4_apply" + - "CategoryTheory.ShortComplex.SnakeInput.\u03B4_apply'" + - CategoryTheory.ShortComplex.zero_apply + - CategoryTheory.Preadditive.epi_iff_surjective + - CategoryTheory.ShortComplex.ShortExact.injective_f + - CategoryTheory.Preadditive.mono_iff_injective + - CategoryTheory.ShortComplex.exact_iff_of_concreteCategory + - path: Mathlib/Analysis/NormedSpace/Connected.lean + theorems: + - isPathConnected_sphere + - Set.Countable.isPathConnected_compl_of_one_lt_rank + - isPreconnected_sphere + - path: Mathlib/Analysis/Matrix.lean + theorems: + - Matrix.frobeniusBoundedSMul + - Matrix.frobenius_norm_col + - Matrix.linfty_opNNNorm_mulVec + - Matrix.linfty_opNNNorm_eq_opNNNorm + - Matrix.nnnorm_col + - Matrix.linftyOpBoundedSMul + - Matrix.nnnorm_le_iff + - Matrix.nnnorm_map_eq + - Matrix.frobenius_norm_row + - Matrix.linfty_opNNNorm_toMatrix + - Matrix.linfty_opNorm_row + - Matrix.frobenius_nnnorm_mul + - Matrix.nnnorm_row + - Matrix.frobenius_nnnorm_one + - Matrix.mul_unitOf + - Matrix.linfty_opNNNorm_col + - Matrix.frobenius_nnnorm_transpose + - Matrix.linfty_opNNNorm_mul + - Matrix.frobenius_nnnorm_map_eq + - Matrix.nnnorm_lt_iff + - Matrix.frobenius_nnnorm_def + - Matrix.norm_unitOf + - Matrix.frobenius_nnnorm_diagonal + - Matrix.linfty_opNNNorm_row + - Matrix.norm_lt_iff + - Matrix.nnnorm_diagonal + - Matrix.frobenius_norm_def + - Matrix.norm_le_iff + - Matrix.linfty_opNNNorm_diagonal + - path: Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean + theorems: + - IsConformalMap.smul + - IsConformalMap.injective + - IsConformalMap.comp + - IsConformalMap.ne_zero + - isConformalMap_of_subsingleton + - isConformalMap_id + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Univ.lean + theorems: + - Matrix.charpoly.univ_map_map + - "Matrix.charpoly.univ_coeff_eval\u2082Hom" + - "Matrix.charpoly.univ_map_eval\u2082Hom" + - Matrix.charpoly.univ_coeff_card + - Matrix.charpoly.optionEquivLeft_symm_univ_isHomogeneous + - path: Mathlib/Geometry/Manifold/BumpFunction.lean + theorems: + - SmoothBumpFunction.isClosed_image_of_isClosed + - SmoothBumpFunction.nhdsWithin_range_basis + - SmoothBumpFunction.nhds_basis_tsupport + - SmoothBumpFunction.isOpen_support + - SmoothBumpFunction.tsupport_subset_symm_image_closedBall + - SmoothBumpFunction.eventuallyEq_one + - SmoothBumpFunction.image_eq_inter_preimage_of_subset_support + - SmoothBumpFunction.smooth + - SmoothBumpFunction.smooth_smul + - SmoothBumpFunction.support_subset_source + - SmoothBumpFunction.exists_r_pos_lt_subset_ball + - SmoothBumpFunction.mem_Icc + - SmoothBumpFunction.support_eq_inter_preimage + - SmoothBumpFunction.tsupport_subset_chartAt_source + - SmoothBumpFunction.support_eq_symm_image + - SmoothBumpFunction.support_mem_nhds + - SmoothBumpFunction.eventuallyEq_one_of_dist_lt + - SmoothBumpFunction.support_updateRIn + - path: Mathlib/Algebra/Algebra/Prod.lean + theorems: + - AlgHom.fst_prod + - AlgHom.snd_prod + - path: Mathlib/Data/MvPolynomial/Derivation.lean + theorems: + - MvPolynomial.derivation_C_mul + - "MvPolynomial.mkDerivation\u2097_X" + - MvPolynomial.leibniz_iff_X + - path: Mathlib/Algebra/Homology/DifferentialObject.lean + theorems: + - CategoryTheory.DifferentialObject.objEqToHom_d + - CategoryTheory.DifferentialObject.eqToHom_f' + - HomologicalComplex.d_eqToHom + - path: Mathlib/Geometry/Manifold/PartitionOfUnity.lean + theorems: + - SmoothBumpCovering.mem_extChartAt_source_of_eq_one + - exists_smooth_one_nhds_of_subset_interior + - exists_msmooth_support_eq_eq_one_iff + - SmoothBumpCovering.mem_support_ind + - SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source_of_isClosed + - exists_msmooth_zero_iff_one_iff_of_isClosed + - BumpCovering.smooth_toPartitionOfUnity + - exists_smooth_zero_one_nhds_of_isClosed + - SmoothBumpCovering.mem_chartAt_source_of_eq_one + - SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq + - SmoothPartitionOfUnity.contMDiffAt_finsum + - SmoothPartitionOfUnity.contDiffAt_finsum + - SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source + - Emetric.exists_smooth_forall_closedBall_subset + - SmoothPartitionOfUnity.exists_pos_of_mem + - IsOpen.exists_msmooth_support_eq_aux + - exists_smooth_zero_one_of_isClosed + - Metric.exists_smooth_forall_closedBall_subset + - exists_contMDiffOn_forall_mem_convex_of_local + - SmoothBumpCovering.exists_isSubordinate + - IsOpen.exists_msmooth_support_eq + - SmoothPartitionOfUnity.exists_isSubordinate + - path: Mathlib/Topology/MetricSpace/DilationEquiv.lean + theorems: + - DilationEquiv.ratio_trans + - DilationEquiv.coe_pow + - IsometryEquiv.toDilationEquiv_ratio + - DilationEquiv.map_cobounded + - DilationEquiv.ratio_symm + - path: Mathlib/Analysis/Subadditive.lean + theorems: + - Subadditive.lim_le_div + - Subadditive.eventually_div_lt_of_div_lt + - Subadditive.tendsto_lim + - path: Mathlib/CategoryTheory/Abelian/Homology.lean + theorems: + - homology'.map_eq_desc'_lift_right + - homology'.map_eq_lift_desc'_right + - "homology'.map_\u03B9" + - "homology'.\u03C0'_map" + - "homology'.condition_\u03C0'" + - homology'.map_eq_desc'_lift_left + - "homology'.\u03C0'_\u03B9" + - homology'.hom_to_ext + - "homology'.\u03C0'_eq_\u03C0" + - homology'.hom_from_ext + - "homology'.lift_\u03B9" + - "homology'.condition_\u03B9" + - homology'.map_eq_lift_desc'_left + - "homology'.\u03C0'_desc'" + - path: Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean + theorems: + - ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection + - LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure + - path: Mathlib/Data/String/Lemmas.lean + theorems: + - String.leftpad_prefix + - String.length_eq_list_length + - String.length_append + - String.length_replicate + - path: Mathlib/CategoryTheory/GradedObject/Trifunctor.lean + theorems: + - "CategoryTheory.GradedObject.\u03B9_mapBifunctorComp\u2082\u2083MapObjIso_hom" + - "CategoryTheory.GradedObject.\u03B9_mapBifunctorComp\u2081\u2082MapObjIso_hom" + - "CategoryTheory.GradedObject.mapBifunctorBifunctor\u2082\u2083MapObj_ext" + - CategoryTheory.GradedObject.mapTrifunctorMapObj_ext + - "CategoryTheory.GradedObject.\u03B9_mapTrifunctorMapMap" + - "CategoryTheory.GradedObject.mapBifunctor\u2081\u2082BifunctorMapObj_ext" + - path: Mathlib/Algebra/GCDMonoid/Multiset.lean + theorems: + - Multiset.dvd_gcd + - Multiset.lcm_ndunion + - Multiset.extract_gcd + - Multiset.lcm_dedup + - Multiset.gcd_add + - Multiset.lcm_add + - Multiset.gcd_union + - Multiset.lcm_eq_zero_iff + - Multiset.gcd_ndunion + - Multiset.normalize_lcm + - Multiset.normalize_gcd + - Multiset.gcd_map_mul + - Multiset.lcm_ndinsert + - Multiset.lcm_union + - Multiset.gcd_ndinsert + - Multiset.gcd_dedup + - Multiset.lcm_dvd + - Multiset.extract_gcd' + - path: Mathlib/LinearAlgebra/AffineSpace/Ordered.lean + theorems: + - lineMap_lt_lineMap_iff_of_lt + - map_le_lineMap_iff_slope_le_slope + - lineMap_lt_right_iff_lt + - map_le_lineMap_iff_slope_le_slope_right + - lineMap_strict_mono_left + - lineMap_le_right_iff_le + - lineMap_strict_mono_endpoints + - lineMap_mono_left + - map_le_lineMap_iff_slope_le_slope_left + - lineMap_mono_right + - left_le_lineMap_iff_le + - lineMap_strict_mono_right + - lineMap_le_lineMap_iff_of_lt + - path: Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean + theorems: + - CategoryTheory.Limits.pullback_diagonal_map_snd_snd_fst + - CategoryTheory.Limits.pullback_diagonal_map_snd_fst_fst + - CategoryTheory.Limits.pullback_map_diagonal_isPullback + - CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_fst + - CategoryTheory.Limits.pullback_fst_map_snd_isPullback + - CategoryTheory.Limits.pullback_lift_map_isPullback + - CategoryTheory.Limits.pullbackDiagonalMapIso_hom_snd + - CategoryTheory.Limits.diagonalObjPullbackFstIso_hom_fst_fst + - CategoryTheory.Limits.pullbackDiagonalMapIso_inv_fst + - CategoryTheory.Limits.pullbackDiagonalMapIso_inv_snd_snd + - CategoryTheory.Limits.pullbackDiagonalMapIdIso_hom_fst + - CategoryTheory.Limits.pullback.diagonal_comp + - CategoryTheory.Limits.diagonalObjPullbackFstIso_inv_fst_snd + - CategoryTheory.Limits.diagonalObjPullbackFstIso_hom_fst_snd + - CategoryTheory.Limits.diagonalObjPullbackFstIso_inv_snd_fst + - CategoryTheory.Limits.diagonal_pullback_fst + - CategoryTheory.Limits.pullbackDiagonalMapIdIso_hom_snd + - CategoryTheory.Limits.pullback_map_eq_pullbackFstFstIso_inv + - CategoryTheory.Limits.diagonalObjPullbackFstIso_inv_snd_snd + - CategoryTheory.Limits.pullbackDiagonalMapIso_inv_snd_fst + - CategoryTheory.Limits.diagonalObjPullbackFstIso_inv_fst_fst + - CategoryTheory.Limits.pullbackDiagonalMapIso_hom_fst + - CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_snd_fst + - path: Mathlib/Combinatorics/Hall/Finite.lean + theorems: + - HallMarriageTheorem.hall_cond_of_erase + - Finset.all_card_le_biUnion_card_iff_existsInjective' + - HallMarriageTheorem.hall_hard_inductive + - HallMarriageTheorem.hall_cond_of_compl + - HallMarriageTheorem.hall_hard_inductive_step_B + - HallMarriageTheorem.hall_hard_inductive_step_A + - HallMarriageTheorem.hall_cond_of_restrict + - path: Mathlib/Order/Filter/EventuallyConst.lean + theorems: + - Filter.EventuallyConst.mulIndicator_const_iff + - Filter.EventuallyEq.eventuallyConst_iff + - Filter.eventuallyConst_atTop_nat + - Filter.HasBasis.eventuallyConst_iff + - Filter.eventuallyConst_pred + - Filter.eventuallyConst_atTop + - Filter.eventuallyConst_pred' + - path: Mathlib/Algebra/Category/GroupCat/Images.lean + theorems: + - AddCommGroupCat.image.lift_fac + - AddCommGroupCat.image.fac + - path: Mathlib/Combinatorics/SimpleGraph/Partition.lean + theorems: + - SimpleGraph.Partition.mem_partOfVertex + - SimpleGraph.Partition.partOfVertex_ne_of_adj + - SimpleGraph.partitionable_iff_colorable + - SimpleGraph.Partition.partOfVertex_mem + - path: Mathlib/Order/CompleteLatticeIntervals.lean + theorems: + - Set.Iic.coe_biInf + - subset_sInf_of_not_bddBelow + - subset_sInf_emptyset + - sInf_within_of_ordConnected + - subset_sSup_emptyset + - Set.Iic.coe_biSup + - sSup_within_of_ordConnected + - Set.Iic.coe_iSup + - subset_sSup_of_within + - subset_sInf_of_within + - subset_sSup_of_not_bddAbove + - path: Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean + theorems: + - SemiNormedGroupCat.explicitCokernelDesc_unique + - "SemiNormedGroupCat.comp_explicitCokernel\u03C0" + - "SemiNormedGroupCat.explicitCokernelIso_hom_\u03C0" + - "SemiNormedGroupCat.explicitCokernelIso_inv_\u03C0" + - "SemiNormedGroupCat.explicitCokernel\u03C0_desc_apply" + - SemiNormedGroupCat.explicitCokernelDesc_comp_eq_zero + - SemiNormedGroupCat.explicitCokernel_hom_ext + - SemiNormedGroupCat.explicitCokernelDesc_comp_eq_desc + - path: Mathlib/Data/Int/LeastGreatest.lean + theorems: + - Int.exists_least_of_bdd + - Int.coe_leastOfBdd_eq + - Int.coe_greatestOfBdd_eq + - Int.exists_greatest_of_bdd + - path: Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean + theorems: + - Module.End.hasEigenvalue_of_isRoot + - Module.End.finite_hasEigenvalue + - Module.End.ker_aeval_ring_hom'_unit_polynomial + - Module.End.eigenspace_aeval_polynomial_degree_1 + - Module.End.aeval_apply_of_hasEigenvector + - path: Mathlib/Algebra/Order/Module/OrderedSMul.lean + theorems: + - inf_eq_half_smul_add_sub_abs_sub' + - sup_eq_half_smul_add_add_abs_sub + - OrderedSMul.mk' + - inf_eq_half_smul_add_sub_abs_sub + - sup_eq_half_smul_add_add_abs_sub' + - path: Mathlib/Topology/Order/UpperLowerSetTopology.lean + theorems: + - Topology.IsUpperSet.closure_singleton + - Topology.IsLowerSet.isClosed_iff_isUpper + - Topology.isUpperSet_orderDual + - Topology.WithUpperSet.toUpperSet_specializes_toUpperSet + - Topology.IsUpperSet.upperSet_le_upper + - Topology.IsUpperSet.isClosed_iff_isLower + - Topology.IsUpperSet.closure_eq_lowerClosure + - Topology.IsLowerSet.closure_singleton + - Topology.IsUpperSet.monotone_to_upperTopology_continuous + - Topology.IsUpperSet.monotone_iff_continuous + - Topology.IsLowerSet.isOpen_iff_isLowerSet + - Topology.IsLowerSet.lowerSet_le_lower + - Topology.IsLowerSet.monotone_iff_continuous + - Topology.WithLowerSet.toLowerSet_specializes_toLowerSet + - Topology.IsUpperSet.isOpen_iff_isUpperSet + - path: Mathlib/MeasureTheory/Covering/Vitali.lean + theorems: + - Vitali.exists_disjoint_covering_ae + - Vitali.exists_disjoint_subfamily_covering_enlargment + - path: Mathlib/Analysis/Calculus/Monotone.lean + theorems: + - Monotone.ae_hasDerivAt + - tendsto_apply_add_mul_sq_div_sub + - MonotoneOn.ae_differentiableWithinAt_of_mem + - StieltjesFunction.ae_hasDerivAt + - Monotone.ae_differentiableAt + - MonotoneOn.ae_differentiableWithinAt + - path: Mathlib/Data/PNat/Prime.lean + theorems: + - PNat.gcd_eq_left_iff_dvd + - PNat.Coprime.coprime_dvd_left + - PNat.gcd_eq_left + - PNat.Coprime.gcd_mul_right_cancel + - PNat.Coprime.gcd_mul_right_cancel_right + - PNat.Coprime.factor_eq_gcd_right_right + - PNat.gcd_one + - PNat.Coprime.gcd_mul + - PNat.Coprime.mul + - PNat.Coprime.gcd_mul_left_cancel + - PNat.gcd_comm + - PNat.Coprime.gcd_mul_left_cancel_right + - PNat.Coprime.mul_right + - PNat.Coprime.symm + - PNat.Coprime.pow + - PNat.one_gcd + - Nat.Primes.coe_pnat_injective + - PNat.Coprime.factor_eq_gcd_left_right + - PNat.dvd_prime + - PNat.Prime.ne_one + - PNat.Prime.not_dvd_one + - PNat.coprime_coe + - PNat.Coprime.factor_eq_gcd_right + - PNat.eq_one_of_lt_two + - PNat.Coprime.factor_eq_gcd_left + - PNat.exists_prime_and_dvd + - path: Mathlib/AlgebraicGeometry/FunctionField.lean + theorems: + - AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint + - AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen + - AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion + - AlgebraicGeometry.germ_injective_of_isIntegral + - AlgebraicGeometry.genericPoint_eq_bot_of_affine + - path: Mathlib/Analysis/Convex/StrictConvexBetween.lean + theorems: + - eq_lineMap_of_dist_eq_mul_of_dist_eq_mul + - eq_midpoint_of_dist_eq_half + - dist_add_dist_eq_iff + - Collinear.sbtw_of_dist_eq_of_dist_lt + - Collinear.wbtw_of_dist_eq_of_dist_le + - path: Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean + theorems: + - GeneralizedContinuedFraction.IntFractPair.succ_nth_stream_b_le_nth_stream_fr_inv + - GeneralizedContinuedFraction.le_of_succ_succ_get?_continuantsAux_b + - GeneralizedContinuedFraction.fib_le_of_continuantsAux_b + - GeneralizedContinuedFraction.zero_le_of_denom + - GeneralizedContinuedFraction.le_of_succ_get?_denom + - GeneralizedContinuedFraction.abs_sub_convergents_le' + - GeneralizedContinuedFraction.IntFractPair.one_le_succ_nth_stream_b + - GeneralizedContinuedFraction.of_part_num_eq_one_and_exists_int_part_denom_eq + - GeneralizedContinuedFraction.of_part_num_eq_one + - GeneralizedContinuedFraction.of_one_le_get?_part_denom + - GeneralizedContinuedFraction.succ_nth_fib_le_of_nth_denom + - GeneralizedContinuedFraction.of_denom_mono + - GeneralizedContinuedFraction.sub_convergents_eq + - GeneralizedContinuedFraction.abs_sub_convergents_le + - GeneralizedContinuedFraction.exists_int_eq_of_part_denom + - GeneralizedContinuedFraction.determinant_aux + - GeneralizedContinuedFraction.zero_le_of_continuantsAux_b + - GeneralizedContinuedFraction.IntFractPair.nth_stream_fr_nonneg_lt_one + - path: Mathlib/Algebra/Divisibility/Prod.lean + theorems: + - pi_dvd_iff + - prod_dvd_iff + - path: Mathlib/LinearAlgebra/Basis/Flag.lean + theorems: + - Basis.flag_mono + - Basis.flag_covBy + - Basis.flag_zero + - Basis.flag_le_ker_coord + - Basis.flag_le_ker_coord_iff + - Basis.flag_last + - Basis.flag_succ + - Basis.flag_strictMono + - Basis.flag_le_ker_dual + - path: Mathlib/Data/Int/Lemmas.lean + theorems: + - Int.toNat_of_nonpos + - Int.natAbs_lt_iff_sq_lt + - Int.natAbs_inj_of_nonpos_of_nonneg + - Int.natAbs_coe_sub_coe_lt_of_lt + - Int.natAbs_inj_of_nonpos_of_nonpos + - Int.natAbs_le_iff_sq_le + - Int.natAbs_eq_iff_sq_eq + - Int.le_coe_nat_sub + - Int.succ_coe_nat_pos + - Int.natAbs_inj_of_nonneg_of_nonpos + - Int.strictAntiOn_natAbs + - Int.natAbs_coe_sub_coe_le_of_le + - Int.div2_bit + - path: Mathlib/CategoryTheory/Adhesive.lean + theorems: + - CategoryTheory.adhesive_of_preserves_and_reflects + - CategoryTheory.IsPushout.isVanKampen_inl + - CategoryTheory.is_coprod_iff_isPushout + - CategoryTheory.IsPushout.IsVanKampen.flip + - CategoryTheory.IsPushout.IsVanKampen.mono_of_mono_right + - CategoryTheory.IsPushout.IsVanKampen.isPullback_of_mono_left + - CategoryTheory.IsPushout.IsVanKampen.isPullback_of_mono_right + - CategoryTheory.adhesive_of_preserves_and_reflects_isomorphism + - CategoryTheory.IsPushout.IsVanKampen.mono_of_mono_left + - CategoryTheory.adhesive_of_reflective + - CategoryTheory.IsPushout.isVanKampen_iff + - path: Mathlib/Algebra/Bounds.lean + theorems: + - ciInf_div + - ciSup_div + - path: Mathlib/Topology/MetricSpace/Completion.lean + theorems: + - UniformSpace.Completion.dist_triangle + - UniformSpace.Completion.uniformity_dist' + - UniformSpace.Completion.uniformity_dist + - LipschitzWith.completion_extension + - UniformSpace.Completion.dist_comm + - Isometry.completion_extension + - UniformSpace.Completion.mem_uniformity_dist + - UniformSpace.Completion.dist_self + - path: Mathlib/Data/Nat/SuccPred.lean + theorems: + - Nat.succ_iterate + - Nat.pred_iterate + - path: Mathlib/Analysis/Calculus/FDeriv/Extend.lean + theorems: + - has_fderiv_at_boundary_of_tendsto_fderiv + - has_deriv_at_interval_right_endpoint_of_tendsto_deriv + - hasDerivAt_of_hasDerivAt_of_ne' + - hasDerivAt_of_hasDerivAt_of_ne + - has_deriv_at_interval_left_endpoint_of_tendsto_deriv + - path: Mathlib/Topology/ClopenBox.lean + theorems: + - TopologicalSpace.Clopens.exists_prod_subset + - TopologicalSpace.Clopens.exists_finset_eq_sup_prod + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean + theorems: + - Polynomial.Chebyshev.algebraMap_eval_U + - Polynomial.Chebyshev.algebraMap_eval_T + - Polynomial.Chebyshev.T_complex_cos + - Polynomial.Chebyshev.aeval_T + - Polynomial.Chebyshev.U_complex_cos + - Polynomial.Chebyshev.aeval_U + - path: Mathlib/LinearAlgebra/Matrix/LDL.lean + theorems: + - LDL.diag_eq_lowerInv_conj + - LDL.lowerInv_eq_gramSchmidtBasis + - LDL.lowerInv_triangular + - LDL.lower_conj_diag + - path: Mathlib/RingTheory/Valuation/ValuationSubring.lean + theorems: + - ValuationSubring.monotone_mapOfLE + - ValuationSubring.valuation_lt_one_iff + - ValuationSubring.ker_unitGroupToResidueFieldUnits + - Valuation.isEquiv_valuation_valuationSubring + - ValuationSubring.principal_units_le_units + - ValuationSubring.nonunits_injective + - ValuationSubring.principalUnitGroup_injective + - ValuationSubring.idealOfLE_le_of_le + - Valuation.isEquiv_iff_valuationSubring + - ValuationSubring.idealOfLE_ofPrime + - ValuationSubring.ofPrime_idealOfLE + - ValuationSubring.unitGroup_le_unitGroup + - ValuationSubring.ofPrime_valuation_eq_one_iff_mem_primeCompl + - ValuationSubring.coe_mem_principalUnitGroup_iff + - ValuationSubring.valuationSubring_valuation + - ValuationSubring.valuation_eq_one_iff + - ValuationSubring.unitGroup_injective + - ValuationSubring.mapOfLE_comp_valuation + - ValuationSubring.toSubring_injective + - ValuationSubring.valuation_unit + - ValuationSubring.image_maximalIdeal + - ValuationSubring.principalUnitGroup_le_principalUnitGroup + - ValuationSubring.nonunits_le_nonunits + - path: Mathlib/Algebra/Module/GradedModule.lean + theorems: + - DirectSum.Gmodule.mul_smul' + - DirectSum.Gmodule.smulAddMonoidHom_apply_of_of + - DirectSum.Gmodule.one_smul' + - path: Mathlib/Topology/CompactOpen.lean + theorems: + - QuotientMap.continuous_lift_prod_right + - ContinuousMap.tendsto_compactOpen_iff_forall + - ContinuousMap.continuous_comp' + - ContinuousMap.continuous_coev + - QuotientMap.continuous_lift_prod_left + - ContinuousMap.continuous_curry + - ContinuousMap.nhds_compactOpen_eq_iInf_nhds_induced + - ContinuousMap.image_coev + - ContinuousMap.continuous_eval_const + - ContinuousMap.continuous_eval + - ContinuousMap.tendsto_nhds_compactOpen + - ContinuousMap.exists_tendsto_compactOpen_iff_forall + - ContinuousMap.inducing_comp + - ContinuousMap.continuous_uncurry + - ContinuousMap.continuous_comp_left + - ContinuousMap.nhds_compactOpen + - path: Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean + theorems: + - ModuleCat.MonoidalCategory.braiding_naturality_right + - ModuleCat.MonoidalCategory.hexagon_forward + - ModuleCat.MonoidalCategory.braiding_naturality + - ModuleCat.MonoidalCategory.hexagon_reverse + - ModuleCat.MonoidalCategory.braiding_naturality_left + - path: Mathlib/Analysis/ODE/Gronwall.lean + theorems: + - gronwallBound_x0 + - "gronwallBound_continuous_\u03B5" + - "gronwallBound_\u03B50" + - ODE_solution_unique_of_mem_Icc + - dist_le_of_trajectories_ODE_of_mem + - hasDerivAt_gronwallBound_shift + - hasDerivAt_gronwallBound + - "gronwallBound_\u03B50_\u03B40" + - ODE_solution_unique_of_mem_Icc_left + - ODE_solution_unique_of_mem_Icc_right + - ODE_solution_unique_of_mem_Ioo + - dist_le_of_approx_trajectories_ODE_of_mem + - path: Mathlib/Combinatorics/SimpleGraph/Ends/Properties.lean + theorems: + - SimpleGraph.end_componentCompl_infinite + - SimpleGraph.nonempty_ends_of_infinite + - path: Mathlib/Data/Tree.lean + theorems: + - Tree.numLeaves_pos + - Tree.numLeaves_eq_numNodes_succ + - Tree.left_node_right_eq_self + - path: Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean + theorems: + - Affine.Simplex.span_eq_top + - collinear_singleton + - collinear_pair + - finrank_vectorSpan_insert_le_set + - finite_of_fin_dim_affineIndependent + - collinear_iff_of_mem + - Collinear.coplanar_insert + - collinear_insert_of_mem_affineSpan_pair + - "ne\u2081\u2083_of_not_collinear" + - Coplanar.finiteDimensional_vectorSpan + - finrank_vectorSpan_range_le + - AffineIndependent.card_lt_card_of_affineSpan_lt_affineSpan + - "ne\u2081\u2082_of_not_collinear" + - finrank_vectorSpan_image_finset_le + - AffineIndependent.affineSpan_eq_of_le_of_card_eq_finrank_add_one + - AffineIndependent.finrank_vectorSpan_image_finset + - AffineIndependent.card_le_card_of_subset_affineSpan + - affineIndependent_iff_not_finrank_vectorSpan_le + - finrank_vectorSpan_insert_le + - affineIndependent_iff_le_finrank_vectorSpan + - affineIndependent_iff_finrank_vectorSpan_eq + - collinear_insert_insert_of_mem_affineSpan_pair + - affineIndependent_iff_not_collinear_set + - Collinear.mem_affineSpan_of_mem_of_ne + - Collinear.collinear_insert_iff_of_ne + - collinear_empty + - collinear_insert_iff_of_mem_affineSpan + - collinear_iff_not_affineIndependent + - AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one + - AffineIndependent.card_le_finrank_succ + - AffineIndependent.finrank_vectorSpan + - AffineIndependent.finrank_vectorSpan_add_one + - collinear_insert_insert_insert_of_mem_affineSpan_pair + - AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one + - "ne\u2082\u2083_of_not_collinear" + - affineIndependent_iff_not_collinear_of_ne + - coplanar_of_finrank_eq_two + - coplanar_insert_iff_of_mem_affineSpan + - collinear_iff_finrank_le_one + - collinear_insert_insert_insert_left_of_mem_affineSpan_pair + - collinear_triple_of_mem_affineSpan_pair + - affineIndependent_iff_not_collinear + - coplanar_iff_finrank_le_two + - collinear_iff_exists_forall_eq_smul_vadd + - AffineBasis.exists_affineBasis_of_finiteDimensional + - path: Mathlib/Data/Finset/NatDivisors.lean + theorems: + - Nat.Prime.divisors_sq + - Nat.divisors_mul + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean + theorems: + - MeasureTheory.condexpIndSMul_add + - MeasureTheory.lintegral_nnnorm_condexpL2_le + - MeasureTheory.set_integral_condexpIndSMul + - MeasureTheory.integrable_condexpIndSMul + - MeasureTheory.condexpIndSMul_nonneg + - MeasureTheory.condexpL2_ae_eq_zero_of_ae_eq_zero + - MeasureTheory.condexpIndSMul_empty + - MeasureTheory.condexpL2_indicator_of_measurable + - MeasureTheory.condexpIndSMul_smul' + - MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le_real + - MeasureTheory.set_lintegral_nnnorm_condexpL2_indicator_le + - MeasureTheory.condexpL2_const_inner + - MeasureTheory.condexpL2_comp_continuousLinearMap + - MeasureTheory.condexpL2_indicator_eq_toSpanSingleton_comp + - MeasureTheory.lintegral_nnnorm_condexpL2_indicator_le + - MeasureTheory.condexpIndSMul_smul + - MeasureTheory.condexpL2_indicator_ae_eq_smul + - MeasureTheory.condexpL2_indicator_nonneg + - MeasureTheory.set_integral_condexpL2_indicator + - MeasureTheory.aeStronglyMeasurable'_condexpIndSMul + - MeasureTheory.set_lintegral_nnnorm_condexpIndSMul_le + - MeasureTheory.inner_condexpL2_eq_inner_fun + - MeasureTheory.norm_condexpL2_coe_le + - MeasureTheory.integral_condexpL2_eq_of_fin_meas_real + - MeasureTheory.lintegral_nnnorm_condexpIndSMul_le + - MeasureTheory.integrable_condexpL2_indicator + - MeasureTheory.integral_condexpL2_eq + - path: Mathlib/CategoryTheory/Sites/CompatiblePlus.lean + theorems: + - CategoryTheory.GrothendieckTopology.whiskerRight_toPlus_comp_plusCompIso_hom + - CategoryTheory.GrothendieckTopology.plusCompIso_whiskerRight + - "CategoryTheory.GrothendieckTopology.diagramCompIso_hom_\u03B9" + - "CategoryTheory.GrothendieckTopology.\u03B9_plusCompIso_hom" + - CategoryTheory.GrothendieckTopology.plusCompIso_whiskerLeft + - CategoryTheory.GrothendieckTopology.toPlus_comp_plusCompIso_inv + - CategoryTheory.GrothendieckTopology.plusCompIso_inv_eq_plusLift + - path: Mathlib/Analysis/SpecialFunctions/Stirling.lean + theorems: + - Stirling.stirlingSeq_has_pos_limit_a + - Stirling.log_stirlingSeq_diff_hasSum + - Stirling.stirlingSeq'_bounded_by_pos_constant + - Stirling.stirlingSeq'_pos + - Stirling.tendsto_self_div_two_mul_self_add_one + - Stirling.log_stirlingSeq_sub_log_stirlingSeq_succ + - Stirling.factorial_isEquivalent_stirling + - Stirling.stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq + - Stirling.log_stirlingSeq_bounded_aux + - Stirling.log_stirlingSeq'_antitone + - Stirling.log_stirlingSeq_formula + - Stirling.tendsto_stirlingSeq_sqrt_pi + - Stirling.stirlingSeq_one + - Stirling.second_wallis_limit + - Stirling.log_stirlingSeq_diff_le_geo_sum + - path: Mathlib/Topology/Bornology/Constructions.lean + theorems: + - boundedSpace_induced_iff + - Bornology.isBounded_prod_self + - Bornology.isBounded_pi + - boundedSpace_subtype_iff + - Bornology.isBounded_prod + - path: Mathlib/NumberTheory/Padics/Hensel.lean + theorems: + - ih_0 + - T_lt_one + - T_def + - newton_seq_succ_dist + - newton_seq_dist + - newton_seq_succ_dist_weak + - soln_dist_to_a_lt_deriv + - newton_seq_dist_aux + - deriv_norm_ne_zero + - calc_deriv_dist + - calc_norm_le_one + - newton_seq_dist_to_a + - T_pos + - soln_unique + - hensels_lemma + - bound' + - a_is_soln + - newton_seq_norm_eq + - a_soln_is_unique + - ncs_tendsto_const + - padic_polynomial_dist + - tendsto_zero_of_norm_tendsto_zero + - bound'_sq + - path: Mathlib/Logic/Function/Iterate.lean + theorems: + - List.foldr_const + - Function.iterate_fixed + - Function.Commute.comp_iterate + - Function.iterate_add_apply + - Function.comp_iterate_pred_of_pos + - Function.iterate_succ' + - Function.iterate_mul + - Function.Commute.iterate_eq_of_map_eq + - Function.iterate_succ_apply' + - Function.iterate_comm + - Function.LeftInverse.iterate + - Function.iterate_add + - Function.Semiconj.iterate_left + - Function.iterate_cancel + - Function.iterate_id + - Function.iterate_add_eq_iterate + - Function.iterate_pred_comp_of_pos + - List.foldl_const + - path: Mathlib/Topology/AlexandrovDiscrete.lean + theorems: + - subset_exterior_iff + - specializes_iff_exterior_subset + - interior_sInter + - exterior_subset_exterior + - exterior_eq_empty + - alexandrovDiscrete_coinduced + - mem_exterior + - isClosed_sUnion + - closure_iUnion + - exterior_singleton_subset_iff_mem_nhds + - exterior_subset_iff_isOpen + - isOpen_exterior + - gc_exterior_interior + - exterior_singleton_eq_ker_nhds + - nhdsSet_exterior + - principal_exterior + - Set.Finite.isCompact_exterior + - Inducing.alexandrovDiscrete + - closure_sUnion + - exterior_minimal + - path: Mathlib/Analysis/Quaternion.lean + theorems: + - Quaternion.hasSum_coe + - Quaternion.coeComplex_add + - Quaternion.continuous_normSq + - Quaternion.tsum_coe + - Quaternion.norm_piLp_equiv_symm_equivTuple + - Quaternion.norm_coe + - Quaternion.coeComplex_mul + - Quaternion.norm_star + - Quaternion.summable_coe + - Quaternion.continuous_im + - Quaternion.coe_real_complex_mul + - Quaternion.normSq_eq_norm_mul_self + - path: Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean + theorems: + - Complex.cpow_sub + - Complex.cpow_neg_one + - Complex.inv_cpow + - Complex.cpow_mul + - Complex.conj_cpow_eq_ite + - Complex.cpow_int_mul + - Complex.cpow_conj + - Complex.zero_cpow_eq_iff + - Complex.cpow_one + - Complex.cpow_nat_inv_pow + - Complex.cpow_int_mul' + - Complex.eq_zero_cpow_iff + - Complex.inv_cpow_eq_ite + - Complex.cpow_nat_cast + - Complex.cpow_mul_int + - Complex.mul_cpow_ofReal_nonneg + - Complex.cpow_zero + - Complex.pow_cpow_nat_inv + - Complex.one_cpow + - Complex.inv_cpow_eq_ite' + - Complex.zero_cpow + - Complex.cpow_eq_zero_iff + - Complex.conj_cpow + - Complex.cpow_int_cast + - Complex.cpow_add + - Complex.cpow_mul_nat + - Complex.natCast_cpow_natCast_mul + - path: Mathlib/Algebra/GroupWithZero/InjSurj.lean + theorems: + - Function.Injective.noZeroDivisors + - Function.Injective.isLeftCancelMulZero + - Function.Injective.isRightCancelMulZero + - path: Mathlib/Analysis/Complex/Schwarz.lean + theorems: + - Complex.dist_le_div_mul_dist_of_mapsTo_ball + - Complex.dist_le_dist_of_mapsTo_ball_self + - Complex.norm_deriv_le_div_of_mapsTo_ball + - Complex.abs_le_abs_of_mapsTo_ball_self + - Complex.abs_deriv_le_one_of_mapsTo_ball + - Complex.norm_dslope_le_div_of_mapsTo_ball + - Complex.schwarz_aux + - Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div + - path: Mathlib/Data/Nat/Hyperoperation.lean + theorems: + - hyperoperation_one + - hyperoperation_zero + - hyperoperation_two_two_eq_four + - hyperoperation_three + - hyperoperation_two + - hyperoperation_ge_four_zero + - hyperoperation_ge_three_eq_one + - hyperoperation_ge_three_one + - hyperoperation_ge_two_eq_self + - hyperoperation_recursion + - path: Mathlib/Algebra/Tropical/BigOperators.lean + theorems: + - Finset.trop_inf + - trop_sum + - List.trop_minimum + - untrop_prod + - List.untrop_prod + - Finset.untrop_sum + - Multiset.trop_sum + - Multiset.trop_inf + - untrop_sum + - untrop_sum_eq_sInf_image + - trop_sInf_image + - Multiset.untrop_sum + - Finset.untrop_sum' + - trop_iInf + - Multiset.untrop_prod + - path: Mathlib/FieldTheory/KummerExtension.lean + theorems: + - adjoinRootXPowSubCEquiv_root + - irreducible_X_pow_sub_C_of_root_adjoin_eq_top + - IntermediateField.adjoin_root_eq_top_of_isSplittingField + - root_X_pow_sub_C_pow + - root_X_pow_sub_C_ne_zero' + - autEquivZmod_symm_apply_natCast + - isSplittingField_AdjoinRoot_X_pow_sub_C + - isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top + - Polynomial.separable_X_pow_sub_C_of_irreducible + - finrank_of_isSplittingField_X_pow_sub_C + - autEquivRootsOfUnity_apply_rootOfSplit + - X_pow_sub_C_irreducible_of_prime + - autEquivZmod_symm_apply_intCast + - isCyclic_tfae + - X_pow_sub_C_irreducible_of_prime_pow + - X_pow_mul_sub_C_irreducible + - X_pow_sub_C_eq_prod + - autEquivRootsOfUnity_smul + - autAdjoinRootXPowSubC_root + - root_X_pow_sub_C_eq_zero_iff + - exists_root_adjoin_eq_top_of_isCyclic + - pow_ne_of_irreducible_X_pow_sub_C + - autAdjoinRootXPowSubCEquiv_symm_smul + - root_X_pow_sub_C_ne_zero + - rootOfSplitsXPowSubC_pow + - ne_zero_of_irreducible_X_pow_sub_C' + - X_pow_sub_C_irreducible_of_odd + - adjoinRootXPowSubCEquiv_symm_eq_root + - X_pow_sub_C_splits_of_isPrimitiveRoot + - Algebra.adjoin_root_eq_top_of_isSplittingField + - ne_zero_of_irreducible_X_pow_sub_C + - path: Mathlib/Algebra/Group/Freiman.lean + theorems: + - MonoidHom.toFreimanHom_injective + - map_prod_eq_map_prod_of_le + - FreimanHom.cancel_left_on + - FreimanHom.toFreimanHom_injective + - FreimanHom.cancel_right_on + - map_mul_map_eq_map_mul_map + - path: Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean + theorems: + - MeasureTheory.tendstoInMeasure_of_tendsto_snorm + - MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae + - MeasureTheory.tendstoInMeasure_iff_norm + - MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable + - MeasureTheory.tendstoInMeasure_of_tendsto_snorm_of_ne_top + - MeasureTheory.TendstoInMeasure.exists_seq_tendstoInMeasure_atTop + - MeasureTheory.ExistsSeqTendstoAe.exists_nat_measure_lt_two_inv + - MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable + - MeasureTheory.tendstoInMeasure_of_tendsto_snorm_top + - MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_spec + - MeasureTheory.TendstoInMeasure.congr' + - MeasureTheory.TendstoInMeasure.aemeasurable + - MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae' + - MeasureTheory.tendstoInMeasure_of_tendsto_ae + - MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono + - MeasureTheory.ExistsSeqTendstoAe.seqTendstoAeSeq_succ + - path: Mathlib/Algebra/Lie/Weights/Cartan.lean + theorems: + - LieAlgebra.rootSpaceProduct_tmul + - LieModule.isWeight_zero_of_nilpotent + - LieAlgebra.rootSpace_zero_eq + - LieAlgebra.coe_rootSpaceWeightSpaceProduct_tmul + - LieAlgebra.is_cartan_of_zeroRootSubalgebra_eq + - LieAlgebra.zeroRootSubalgebra_normalizer_eq_self + - LieAlgebra.toLieSubmodule_le_rootSpace_zero + - LieAlgebra.zeroRootSubalgebra_eq_iff_is_cartan + - LieAlgebra.lie_mem_weightSpace_of_mem_weightSpace + - LieAlgebra.le_zeroRootSubalgebra + - LieAlgebra.mapsTo_toEndomorphism_weightSpace_add_of_mem_rootSpace + - LieAlgebra.mem_zeroRootSubalgebra + - LieAlgebra.mem_range_rootSpaceProductNegSelf + - LieAlgebra.zeroRootSubalgebra_eq_of_is_cartan + - path: Mathlib/SetTheory/Cardinal/SchroederBernstein.lean + theorems: + - Function.Embedding.schroeder_bernstein + - Function.Embedding.min_injective + - path: Mathlib/Tactic/Linarith/Lemmas.lean + theorems: + - Linarith.lt_of_lt_of_eq + - Linarith.mul_eq + - Linarith.mul_neg + - Linarith.lt_of_eq_of_lt + - Linarith.le_of_le_of_eq + - Linarith.eq_of_eq_of_eq + - Linarith.mul_zero_eq + - Linarith.mul_nonpos + - Linarith.zero_mul_eq + - Linarith.le_of_eq_of_le + - path: Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean + theorems: + - Finsupp.toFreeAbelianGroup_comp_toFinsupp + - FreeAbelianGroup.toFinsupp_toFreeAbelianGroup + - FreeAbelianGroup.support_neg + - Finsupp.toFreeAbelianGroup_comp_singleAddHom + - FreeAbelianGroup.support_zero + - FreeAbelianGroup.mem_support_iff + - FreeAbelianGroup.toFinsupp_of + - FreeAbelianGroup.support_zsmul + - FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup + - FreeAbelianGroup.not_mem_support_iff + - Finsupp.toFreeAbelianGroup_toFinsupp + - FreeAbelianGroup.support_of + - FreeAbelianGroup.support_add + - path: Mathlib/CategoryTheory/Core.lean + theorems: + - CategoryTheory.Core.id_hom + - path: Mathlib/RingTheory/PiTensorProduct.lean + theorems: + - SemiconjBy.tprod + - PiTensorProduct.mul_assoc + - PiTensorProduct.mul_tprod_tprod + - PiTensorProduct.algebraMap_apply + - PiTensorProduct.one_mul + - PiTensorProduct.mul_comm + - PiTensorProduct.smul_tprod_mul_smul_tprod + - PiTensorProduct.mul_one + - PiTensorProduct.constantBaseRingEquiv_tprod + - path: Mathlib/Analysis/LocallyConvex/Barrelled.lean + theorems: + - Seminorm.continuous_iSup + - WithSeminorms.banach_steinhaus + - path: Mathlib/GroupTheory/Schreier.lean + theorems: + - Subgroup.closure_mul_image_eq + - Subgroup.exists_finset_card_le_mul + - Subgroup.rank_le_index_mul_rank + - Subgroup.card_commutator_le_of_finite_commutatorSet + - Subgroup.closure_mul_image_mul_eq_top + - Subgroup.closure_mul_image_eq_top + - Subgroup.card_commutator_dvd_index_center_pow + - Subgroup.closure_mul_image_eq_top' + - path: Mathlib/Topology/MetricSpace/Equicontinuity.lean + theorems: + - Metric.equicontinuousAt_iff_pair + - Metric.uniformEquicontinuous_of_continuity_modulus + - Metric.equicontinuousAt_of_continuity_modulus + - path: Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean + theorems: + - "AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_inv_\u03B9\ + _app" + - "AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_\u03C0" + - AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_fac + - AlgebraicGeometry.PresheafedSpace.map_comp_c_app + - AlgebraicGeometry.PresheafedSpace.map_id_c_app + - AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality + - path: Mathlib/Order/Heyting/Regular.lean + theorems: + - Heyting.IsRegular.disjoint_compl_left_iff + - Heyting.IsRegular.inf + - Heyting.isRegular_top + - Heyting.IsRegular.disjoint_compl_right_iff + - Heyting.isRegular_bot + - Heyting.IsRegular.himp + - path: Mathlib/Control/Applicative.lean + theorems: + - Applicative.pure_seq_eq_map' + - Functor.Comp.applicative_comp_id + - Functor.Comp.map_pure + - Applicative.ext + - Applicative.map_seq_map + - Functor.Comp.pure_seq_eq_map + - path: Mathlib/Analysis/Calculus/LHopital.lean + theorems: + - HasDerivAt.lhopital_zero_right_on_Ioo + - HasDerivAt.lhopital_zero_left_on_Ioo + - deriv.lhopital_zero_nhds' + - deriv.lhopital_zero_nhds_left + - HasDerivAt.lhopital_zero_nhds_right + - HasDerivAt.lhopital_zero_right_on_Ico + - deriv.lhopital_zero_atTop_on_Ioi + - HasDerivAt.lhopital_zero_left_on_Ioc + - deriv.lhopital_zero_left_on_Ioo + - HasDerivAt.lhopital_zero_nhds_left + - deriv.lhopital_zero_atTop + - deriv.lhopital_zero_right_on_Ico + - deriv.lhopital_zero_nhds_right + - HasDerivAt.lhopital_zero_atBot + - deriv.lhopital_zero_right_on_Ioo + - HasDerivAt.lhopital_zero_nhds + - HasDerivAt.lhopital_zero_nhds' + - deriv.lhopital_zero_atBot + - HasDerivAt.lhopital_zero_atTop + - HasDerivAt.lhopital_zero_atBot_on_Iio + - path: Mathlib/CategoryTheory/Comma/Arrow.lean + theorems: + - CategoryTheory.Arrow.inv_hom_id_right + - CategoryTheory.Arrow.square_from_iso_invert + - CategoryTheory.Arrow.inv_hom_id_left + - CategoryTheory.Arrow.hom_inv_id_right + - CategoryTheory.Arrow.square_to_iso_invert + - CategoryTheory.Arrow.hom.congr_right + - CategoryTheory.Arrow.hom_inv_id_left + - CategoryTheory.Arrow.mk_injective + - CategoryTheory.Arrow.inv_right + - CategoryTheory.Arrow.inv_left + - CategoryTheory.Arrow.hom.congr_left + - CategoryTheory.Arrow.left_hom_inv_right + - CategoryTheory.Arrow.inv_left_hom_right + - CategoryTheory.Arrow.iso_w + - CategoryTheory.Arrow.mk_eq + - path: Mathlib/Algebra/Lie/Abelian.lean + theorems: + - LieAlgebra.self_module_ker_eq_center + - commutative_ring_iff_abelian_lie_ring + - LieModule.commute_toEndomorphism_of_mem_center_left + - LieModule.isTrivial_iff_max_triv_eq_top + - LieModule.le_max_triv_iff_bracket_eq_bot + - LieModule.coe_linearMap_maxTrivLinearMapEquivLieModuleHom + - LieModule.coe_maxTrivLinearMapEquivLieModuleHom + - LieModule.mem_ker + - LieSubmodule.lie_abelian_iff_lie_self_eq_bot + - LieModule.maxTrivEquiv_of_refl_eq_refl + - LieSubmodule.trivial_lie_oper_zero + - LieModule.ideal_oper_maxTrivSubmodule_eq_bot + - Function.Surjective.isLieAbelian + - path: Mathlib/Topology/Order/T5.lean + theorems: + - Set.compl_section_ordSeparatingSet_mem_nhds + - Set.ordConnectedComponent_mem_nhds + - Set.ordT5Nhd_mem_nhdsSet + - Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Iic + - Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Ici + - path: Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean + theorems: + - MonoidAlgebra.mul_apply_mul_eq_mul_of_uniqueMul + - path: Mathlib/LinearAlgebra/Dimension/DivisionRing.lean + theorems: + - rank_le_one_iff + - linearIndependent_iff_card_le_finrank_span + - rank_dual_eq_card_dual_of_aleph0_le_rank + - exists_linearIndependent_snoc_of_lt_rank + - le_rank_iff_exists_linearIndependent_finset + - le_rank_iff_exists_linearIndependent + - linearIndependent_iff_card_eq_finrank_span + - rank_fun_infinite + - rank_submodule_le_one_iff + - rank_lt_rank_dual + - linearIndependent_of_top_le_span_of_card_eq_finrank + - rank_range_add_rank_ker + - Submodule.rank_add_le_rank_add_rank + - rank_quotient_add_rank + - exists_linearIndependent_cons_of_lt_rank + - rank_add_rank_split + - rank_submodule_le_one_iff' + - lift_rank_lt_rank_dual + - lift_rank_lt_rank_dual' + - max_aleph0_card_le_rank_fun_nat + - exists_linearIndependent_pair_of_one_lt_rank + - Module.rank_le_one_iff_top_isPrincipal + - rank_eq_of_surjective + - Submodule.rank_sup_add_rank_inf_eq + - rank_lt_rank_dual' + - Submodule.rank_le_one_iff_isPrincipal + - path: Mathlib/LinearAlgebra/ExteriorAlgebra/Grading.lean + theorems: + - "ExteriorAlgebra.GradedAlgebra.\u03B9_sq_zero" + - "ExteriorAlgebra.\u03B9Multi_span" + - "ExteriorAlgebra.GradedAlgebra.lift\u03B9_eq" + - "ExteriorAlgebra.GradedAlgebra.\u03B9_apply" + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean + theorems: + - SzemerediRegularity.le_sum_distinctPairs_edgeDensity_sq + - SzemerediRegularity.energy_increment + - SzemerediRegularity.card_increment + - SzemerediRegularity.increment_isEquipartition + - SzemerediRegularity.distinctPairs_increment + - SzemerediRegularity.pairwiseDisjoint_distinctPairs + - path: Mathlib/Algebra/Regular/Pow.lean + theorems: + - IsRightRegular.pow + - IsRightRegular.pow_iff + - IsLeftRegular.pow_iff + - IsLeftRegular.pow + - path: Mathlib/RingTheory/Valuation/ValuationRing.lean + theorems: + - ValuationRing.le_total + - ValuationRing.iff_ideal_total + - ValuationRing.iff_dvd_total + - ValuationRing.unique_irreducible + - Function.Surjective.valuationRing + - ValuationRing.of_integers + - ValuationRing.tFAE + - ValuationRing.range_algebraMap_eq + - ValuationRing.mem_integer_iff + - ValuationRing.iff_isInteger_or_isInteger + - path: Mathlib/Topology/Sheaves/Skyscraper.lean + theorems: + - SkyscraperPresheafFunctor.map'_comp + - StalkSkyscraperPresheafAdjunctionAuxs.fromStalk_to_skyscraper + - skyscraperPresheaf_isSheaf + - skyscraperPresheaf_eq_pushforward + - SkyscraperPresheafFunctor.map'_id + - StalkSkyscraperPresheafAdjunctionAuxs.to_skyscraper_fromStalk + - path: Mathlib/Combinatorics/Hall/Basic.lean + theorems: + - hallMatchingsOn.nonempty + - Fintype.all_card_le_rel_image_card_iff_exists_injective + - Finset.all_card_le_biUnion_card_iff_exists_injective + - path: Mathlib/Data/Int/CardIntervalMod.lean + theorems: + - Int.Ioc_filter_dvd_eq + - Int.Ico_filter_dvd_eq + - Nat.Ioc_filter_modEq_cast + - Nat.Ico_filter_modEq_card + - Nat.count_modEq_card_eq_ceil + - Int.Ico_filter_dvd_card + - Nat.Ico_filter_modEq_cast + - Int.Ioc_filter_dvd_card + - Int.Ico_filter_modEq_eq + - Nat.Ioc_filter_modEq_card + - Int.Ioc_filter_modEq_eq + - Int.Ico_filter_modEq_card + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean + theorems: + - SimpleGraph.left_nonuniformWitnesses_subset + - SimpleGraph.le_card_nonuniformWitness + - SimpleGraph.isUniform_one + - SimpleGraph.not_isUniform_zero + - SimpleGraph.right_nonuniformWitnesses_card + - SimpleGraph.right_nonuniformWitnesses_subset + - SimpleGraph.nonuniformWitness_subset + - Finpartition.botIsUniform + - SimpleGraph.IsUniform.mono + - Finpartition.mk_mem_nonUniforms_iff + - Finpartition.IsUniform.mono + - Finpartition.nonUniforms_bot + - SimpleGraph.isUniform_singleton + - Finpartition.isUniformOne + - SimpleGraph.not_isUniform_iff + - SimpleGraph.IsUniform.symm + - SimpleGraph.left_nonuniformWitnesses_card + - SimpleGraph.nonuniformWitness_spec + - SimpleGraph.nonuniformWitnesses_spec + - Finpartition.isUniformOfEmpty + - path: Mathlib/Topology/Gluing.lean + theorems: + - TopCat.GlueData.open_image_open + - "TopCat.GlueData.\u03B9_eq_iff_rel" + - TopCat.GlueData.range_fromOpenSubsetsGlue + - TopCat.GlueData.preimage_image_eq_image' + - "TopCat.GlueData.\u03B9_injective" + - TopCat.GlueData.fromOpenSubsetsGlue_isOpenMap + - TopCat.GlueData.fromOpenSubsetsGlue_injective + - TopCat.GlueData.preimage_range + - TopCat.GlueData.preimage_image_eq_image + - TopCat.GlueData.image_inter + - TopCat.GlueData.isOpen_iff + - TopCat.GlueData.MkCore.t_inv + - path: Mathlib/Data/Int/Log.lean + theorems: + - Int.clog_mono_right + - Int.log_natCast + - Int.clog_of_left_le_one + - Int.log_one_right + - Int.lt_zpow_succ_log_self + - Int.log_of_left_le_one + - Int.log_of_right_le_one + - Int.self_le_zpow_clog + - Int.neg_log_inv_eq_clog + - Int.clog_of_right_le_one + - Int.log_inv + - Int.zpow_log_le_self + - Int.clog_inv + - Int.log_mono_right + - Int.log_of_right_le_zero + - Int.clog_zpow + - Int.clog_one_right + - Int.clog_natCast + - Int.clog_of_right_le_zero + - Int.log_zpow + - Int.neg_clog_inv_eq_log + - Int.zpow_pred_clog_lt_self + - path: Mathlib/Topology/Algebra/Module/StrongTopology.lean + theorems: + - ContinuousLinearMap.strongUniformity.uniformAddGroup + - ContinuousLinearMap.strongTopology.hasBasis_nhds_zero_of_basis + - ContinuousLinearMap.strongTopology.continuousSMul + - ContinuousLinearMap.strongTopology.t2Space + - ContinuousLinearMap.strongTopology.topologicalAddGroup + - path: Mathlib/Algebra/LinearRecurrence.lean + theorems: + - LinearRecurrence.is_sol_mkSol + - LinearRecurrence.sol_eq_of_eq_init + - LinearRecurrence.eq_mk_of_is_sol_of_eq_init + - LinearRecurrence.mkSol_eq_init + - LinearRecurrence.geom_sol_iff_root_charPoly + - path: Mathlib/Data/DFinsupp/Multiset.lean + theorems: + - DFinsupp.toMultiset_inf + - Multiset.toDFinsupp_union + - DFinsupp.toMultiset_lt_toMultiset + - Multiset.toDFinsupp_singleton + - Multiset.toDFinsupp_inter + - Multiset.toDFinsupp_replicate + - DFinsupp.toMultiset_le_toMultiset + - Multiset.toDFinsupp_le_toDFinsupp + - DFinsupp.toMultiset_sup + - path: Mathlib/Analysis/Normed/Group/Seminorm.lean + theorems: + - NonarchAddGroupSeminorm.smul_sup + - GroupSeminorm.mul_bddBelow_range_add + - AddGroupSeminorm.smul_sup + - map_sub_le_max + - NonarchAddGroupSeminorm.toZeroHom_eq_coe + - GroupSeminorm.smul_sup + - NonarchAddGroupSeminorm.add_bddBelow_range_add + - path: Mathlib/Data/MvPolynomial/PDeriv.lean + theorems: + - MvPolynomial.pderiv_X + - MvPolynomial.pderiv_def + - MvPolynomial.pderiv_X_of_ne + - MvPolynomial.pderiv_monomial_single + - MvPolynomial.pderiv_X_self + - MvPolynomial.pderiv_pow + - MvPolynomial.pderiv_C_mul + - MvPolynomial.pderiv_monomial + - MvPolynomial.pderiv_mul + - path: Mathlib/Algebra/Order/Ring/Abs.lean + theorems: + - exists_abs_lt + - sq_le_one_iff_abs_le_one + - abs_neg_one_pow + - abs_le_iff_mul_self_le + - abs_le_one_iff_mul_self_le_one + - one_le_sq_iff_one_le_abs + - abs_mul + - abs_mul_self + - dvd_abs + - abs_lt_iff_mul_self_lt + - abs_sub_sq + - abs_sq + - sq_lt_one_iff_abs_lt_one + - abs_dvd + - abs_eq_iff_mul_self_eq + - abs_pow_eq_one + - mabs_zpow + - sq_eq_sq_iff_abs_eq_abs + - abs_le_of_sq_le_sq + - sq_abs + - one_lt_sq_iff_one_lt_abs + - sq_le_sq + - sq_lt_sq + - path: Mathlib/Testing/SlimCheck/Functions.lean + theorems: + - SlimCheck.InjectiveFunction.applyId_mem_iff + - SlimCheck.InjectiveFunction.applyId_injective + - SlimCheck.InjectiveFunction.List.applyId_eq_self + - SlimCheck.InjectiveFunction.List.applyId_zip_eq + - SlimCheck.InjectiveFunction.List.applyId_cons + - SlimCheck.InjectiveFunction.injective + - path: Mathlib/RingTheory/Prime.lean + theorems: + - mul_eq_mul_prime_pow + - Prime.abs + - mul_eq_mul_prime_prod + - Prime.neg + - path: Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean + theorems: + - TopCat.Sheaf.interUnionPullbackConeLift_right + - TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections + - TopCat.Sheaf.interUnionPullbackConeLift_left + - TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections + - path: Mathlib/RingTheory/Localization/InvSubmonoid.lean + theorems: + - IsLocalization.toInvSubmonoid_eq_mk' + - IsLocalization.smul_toInvSubmonoid + - IsLocalization.surj'' + - IsLocalization.span_invSubmonoid + - IsLocalization.submonoid_map_le_is_unit + - IsLocalization.mem_invSubmonoid_iff_exists_mk' + - path: Mathlib/NumberTheory/NumberField/FractionalIdeal.lean + theorems: + - NumberField.fractionalIdeal_rank + - NumberField.det_basisOfFractionalIdeal_eq_absNorm + - NumberField.mem_span_basisOfFractionalIdeal + - path: Mathlib/CategoryTheory/GuitartExact.lean + theorems: + - CategoryTheory.TwoSquare.guitartExact_iff_initial + - CategoryTheory.TwoSquare.guitartExact_iff_isConnected_downwards + - path: Mathlib/Data/Matrix/DMatrix.lean + theorems: + - DMatrix.ext_iff + - DMatrix.map_zero + - DMatrix.map_sub + - DMatrix.map_add + - DMatrix.map_map + - path: Mathlib/Geometry/Manifold/LocalDiffeomorph.lean + theorems: + - IsLocalDiffeomorphOn.isLocalHomeomorphOn + - Diffeomorph.isLocalDiffeomorph + - isLocalDiffeomorph_iff + - isLocalDiffeomorphOn_iff + - IsLocalDiffeomorph.isLocalHomeomorph + - path: Mathlib/CategoryTheory/Preadditive/Schur.lean + theorems: + - CategoryTheory.finrank_hom_simple_simple_eq_zero_of_not_iso + - CategoryTheory.finrank_hom_simple_simple_le_one + - CategoryTheory.isIso_iff_nonzero + - CategoryTheory.finrank_hom_simple_simple + - CategoryTheory.finrank_hom_simple_simple_eq_zero_iff + - CategoryTheory.finrank_endomorphism_eq_one + - CategoryTheory.finrank_hom_simple_simple_eq_one_iff + - path: Mathlib/Analysis/Convex/Exposed.lean + theorems: + - IsExposed.isExtreme + - IsExposed.mono + - isExposed_empty + - IsExposed.inter_left + - IsExposed.isClosed + - mem_exposedPoints_iff_exposed_singleton + - IsExposed.inter_right + - IsExposed.eq_inter_halfspace' + - IsExposed.eq_inter_halfspace + - IsExposed.convex + - IsExposed.subset + - IsExposed.inter + - IsExposed.sInter + - path: Mathlib/Data/Fintype/Pi.lean + theorems: + - Fintype.piFinset_of_isEmpty + - Fintype.coe_piFinset + - Fintype.mem_piFinset + - Fintype.piFinset_singleton + - Fintype.piFinset_empty + - Fintype.piFinset_nonempty + - Finset.univ_pi_univ + - Fintype.filter_piFinset_of_not_mem + - Fintype.piFinset_image + - Fintype.eval_image_piFinset + - path: Mathlib/Topology/Sets/Closeds.lean + theorems: + - TopologicalSpace.Closeds.coe_eq_singleton_of_isAtom + - TopologicalSpace.Opens.isCoatom_iff + - TopologicalSpace.Closeds.coe_sSup + - TopologicalSpace.Closeds.coe_iInf + - TopologicalSpace.Closeds.coe_sup + - TopologicalSpace.Closeds.iInf_def + - TopologicalSpace.Closeds.isAtom_coe + - TopologicalSpace.Closeds.isAtom_iff + - TopologicalSpace.Closeds.mem_iInf + - path: Mathlib/CategoryTheory/Category/Basic.lean + theorems: + - CategoryTheory.cancel_epi_id + - CategoryTheory.mono_of_mono_fac + - CategoryTheory.mono_of_mono + - CategoryTheory.id_of_comp_left_id + - CategoryTheory.epi_of_epi_fac + - CategoryTheory.comp_ite + - CategoryTheory.cancel_mono_id + - CategoryTheory.eq_of_comp_right_eq + - CategoryTheory.comp_dite + - CategoryTheory.eq_of_comp_left_eq' + - CategoryTheory.dite_comp + - CategoryTheory.ite_comp + - CategoryTheory.id_of_comp_right_id + - CategoryTheory.epi_comp + - CategoryTheory.eq_of_comp_left_eq + - CategoryTheory.eq_of_comp_right_eq' + - CategoryTheory.epi_of_epi + - CategoryTheory.mono_comp + - path: Mathlib/Analysis/Complex/CauchyIntegral.lean + theorems: + - Complex.circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of_tendsto + - Complex.analyticAt_iff_eventually_differentiableAt + - Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux + - Complex.circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable + - Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable + - Complex.integral_boundary_rect_eq_zero_of_differentiable_on_off_countable + - Complex.integral_boundary_rect_of_differentiableOn_real + - DiffContOnCl.two_pi_i_inv_smul_circleIntegral_sub_inv_smul + - Complex.hasFPowerSeriesOnBall_of_differentiable_off_countable + - DifferentiableOn.analyticAt + - Complex.circleIntegral_eq_zero_of_differentiable_on_off_countable + - Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable + - Complex.analyticOn_univ_iff_differentiable + - path: Mathlib/RingTheory/Adjoin/FG.lean + theorems: + - Subalgebra.fg_top + - Subalgebra.induction_on_adjoin + - Subalgebra.fg_of_submodule_fg + - Algebra.fg_trans + - Subalgebra.FG.prod + - Subalgebra.FG.map + - Subalgebra.fg_of_fg_map + - Subalgebra.fg_of_fg_toSubmodule + - path: Mathlib/Analysis/Calculus/ParametricIntegral.lean + theorems: + - hasFDerivAt_integral_of_dominated_of_fderiv_le + - hasDerivAt_integral_of_dominated_loc_of_lip + - hasDerivAt_integral_of_dominated_loc_of_deriv_le + - hasFDerivAt_integral_of_dominated_of_fderiv_le'' + - hasFDerivAt_integral_of_dominated_loc_of_lip' + - hasFDerivAt_integral_of_dominated_loc_of_lip + - hasFDerivAt_integral_of_dominated_loc_of_lip_interval + - path: Mathlib/Data/Fintype/Quotient.lean + theorems: + - Quotient.finChoiceAux_eq + - Quotient.finChoice_eq + - path: Mathlib/Data/Finset/Preimage.lean + theorems: + - Finset.subset_map_iff + - Finset.preimage_empty + - Finset.preimage_compl + - Finset.sigma_preimage_mk + - Finset.prod_preimage' + - Finset.image_preimage + - Finset.preimage_univ + - Finset.image_preimage_of_bij + - Finset.prod_preimage + - Finset.preimage_inter + - Finset.preimage_map + - Finset.preimage_union + - Finset.image_subset_iff_subset_preimage + - Finset.sigma_preimage_mk_of_subset + - Finset.map_subset_iff_subset_preimage + - path: Mathlib/NumberTheory/Wilson.lean + theorems: + - ZMod.prod_Ico_one_prime + - ZMod.wilsons_lemma + - Nat.prime_of_fac_equiv_neg_one + - Nat.prime_iff_fac_equiv_neg_one + - path: Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean + theorems: + - GaussianInt.prime_of_nat_prime_of_mod_four_eq_three + - GaussianInt.mod_four_eq_three_of_nat_prime_of_prime + - path: Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean + theorems: + - CategoryTheory.hasFiniteProducts_of_has_binary_and_terminal + - CategoryTheory.hasCoproduct_fin + - CategoryTheory.hasProduct_fin + - CategoryTheory.hasFiniteCoproducts_of_has_binary_and_initial + - path: Mathlib/CategoryTheory/Localization/Adjunction.lean + theorems: + - CategoryTheory.Adjunction.localization_unit_app + - CategoryTheory.Adjunction.localization_counit_app + - "CategoryTheory.Adjunction.Localization.\u03B7_app" + - "CategoryTheory.Adjunction.Localization.\u03B5_app" + - path: Mathlib/Topology/Category/Stonean/Limits.lean + theorems: + - Stonean.finiteCoproduct.hom_ext + - Stonean.pullback.hom_ext + - "Stonean.Sigma.openEmbedding_\u03B9" + - Stonean.pullback.condition + - Stonean.pullback.lift_snd + - path: Mathlib/Data/Finset/Sym.lean + theorems: + - Finset.sym2_eq_image + - Finset.sym_inter + - Finset.sym2_toFinset + - Finset.image_diag_union_image_offDiag + - Finset.diag_mem_sym2_mem_iff + - Finset.sym2_mono + - Finset.mk_mem_sym2_iff + - Finset.not_isDiag_mk_of_mem_offDiag + - Finset.mem_sym_iff + - Finset.sym2_nonempty + - Finset.sym_eq_empty + - Finset.diag_mem_sym2_iff + - Finset.mem_sym2_iff + - Finset.sym_nonempty + - Finset.sym2_eq_empty + - Finset.replicate_mem_sym + - Finset.eq_empty_of_sym_eq_empty + - Finset.card_sym2 + - Finset.injective_sym2 + - Finset.sym2_univ + - path: .lake/packages/std/Std/Data/RBMap/Alter.lean + theorems: + - Std.RBNode.Path.Balanced.del + - Std.RBNode.Balanced.alter + - Std.RBSet.ModifyWF.of_eq + - Std.RBNode.Ordered.modify + - Std.RBNode.modify_eq_alter + - "Std.RBNode.Path.zoom_zoomed\u2081" + - Std.RBNode.Path.Ordered.fill + - Std.RBNode.Path.zoom_fill' + - Std.RBNode.Path.zoom_insert + - Std.RBNode.Ordered.alter + - Std.RBNode.Path.zoom_ins + - "Std.RBNode.Path.zoom_zoomed\u2082" + - Std.RBNode.Balanced.zoom + - Std.RBNode.Path.Ordered.ins + - Std.RBNode.find?_eq_zoom + - Std.RBNode.Path.zoom_del + - Std.RBNode.Path.Ordered.del + - Std.RBNode.Path.ins_eq_fill + - Std.RBNode.Path.Balanced.ins + - path: Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean + theorems: + - MeasureTheory.condexp_indicator + - MeasureTheory.condexp_ae_eq_restrict_of_measurableSpace_eq_on + - MeasureTheory.condexp_ae_eq_restrict_zero + - MeasureTheory.condexp_restrict_ae_eq_restrict + - MeasureTheory.condexp_indicator_aux + - path: Mathlib/Data/Multiset/Dedup.lean + theorems: + - Multiset.Nodup.le_dedup_iff_le + - Multiset.le_dedup_self + - Multiset.dedup_map_dedup_eq + - Multiset.dedup_ext + - Multiset.dedup_nsmul + - Multiset.Nodup.le_nsmul_iff_le + - Multiset.dedup_bind_dedup + - Multiset.count_dedup + - path: Mathlib/MeasureTheory/Covering/VitaliFamily.lean + theorems: + - VitaliFamily.eventually_filterAt_measurableSet + - VitaliFamily.FineSubfamilyOn.measure_le_tsum_of_absolutelyContinuous + - VitaliFamily.frequently_filterAt_iff + - VitaliFamily.fineSubfamilyOn_of_frequently + - Filter.HasBasis.vitaliFamily + - path: Mathlib/CategoryTheory/Simple.lean + theorems: + - CategoryTheory.id_nonzero + - CategoryTheory.epi_from_simple_zero_of_not_iso + - CategoryTheory.epi_of_nonzero_to_simple + - CategoryTheory.Simple.not_isZero + - CategoryTheory.Simple.of_iso + - CategoryTheory.kernel_zero_of_nonzero_from_simple + - CategoryTheory.simple_iff_subobject_isSimpleOrder + - CategoryTheory.simple_of_isSimpleOrder_subobject + - CategoryTheory.indecomposable_of_simple + - CategoryTheory.Biprod.isIso_inl_iff_isZero + - CategoryTheory.simple_of_cosimple + - CategoryTheory.mono_to_simple_zero_of_not_iso + - CategoryTheory.zero_not_simple + - CategoryTheory.cokernel_zero_of_nonzero_to_simple + - path: Mathlib/RingTheory/FreeRing.lean + theorems: + - FreeRing.induction_on + - path: Mathlib/RingTheory/RingHomProperties.lean + theorems: + - RingHom.RespectsIso.is_localization_away_iff + - RingHom.RespectsIso.cancel_left_isIso + - RingHom.StableUnderBaseChange.mk + - RingHom.StableUnderBaseChange.pushout_inl + - RingHom.StableUnderComposition.respectsIso + - path: Mathlib/Algebra/Star/CHSH.lean + theorems: + - TsirelsonInequality.tsirelson_inequality_aux + - CHSH_inequality_of_comm + - TsirelsonInequality.sqrt_two_inv_mul_self + - tsirelson_inequality + - CHSH_id + - path: Mathlib/CategoryTheory/Sites/LeftExact.lean + theorems: + - CategoryTheory.GrothendieckTopology.liftToPlusObjLimitObj_fac + - path: Mathlib/SetTheory/Surreal/Dyadic.lean + theorems: + - Surreal.double_powHalf_succ_eq_powHalf + - Surreal.dyadicMap_apply_pow + - Surreal.zsmul_pow_two_powHalf + - Surreal.nsmul_pow_two_powHalf + - SetTheory.PGame.powHalf_moveLeft + - SetTheory.PGame.powHalf_le_one + - Surreal.dyadic_aux + - SetTheory.PGame.powHalf_pos + - SetTheory.PGame.numeric_powHalf + - SetTheory.PGame.birthday_half + - Surreal.dyadicMap_apply + - Surreal.nsmul_pow_two_powHalf' + - SetTheory.PGame.powHalf_leftMoves + - SetTheory.PGame.add_powHalf_succ_self_eq_powHalf + - path: Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean + theorems: + - AlgebraicGeometry.QuasiCompact.affineProperty_toProperty + - AlgebraicGeometry.quasiCompact_iff_affineProperty + - AlgebraicGeometry.compact_open_induction_on + - AlgebraicGeometry.quasiCompact_eq_affineProperty + - AlgebraicGeometry.quasiCompact_iff_spectral + - AlgebraicGeometry.QuasiCompact.affineProperty_isLocal + - AlgebraicGeometry.isCompact_basicOpen + - AlgebraicGeometry.QuasiCompact.affineProperty_stableUnderBaseChange + - AlgebraicGeometry.quasiCompact_iff_forall_affine + - AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact + - AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen + - AlgebraicGeometry.isCompact_open_iff_eq_finset_affine_union + - path: Mathlib/SetTheory/Cardinal/ToNat.lean + theorems: + - Cardinal.toNat_congr + - Cardinal.cast_toNat_of_lt_aleph0 + - Cardinal.toNat_eq_iff + - Cardinal.toNat_lift_add_lift + - Cardinal.toNat_add + - Cardinal.toNat_eq_one + - Cardinal.toNat_eq_zero + - Cardinal.toNat_lift + - Cardinal.toNat_apply_of_aleph0_le + - Cardinal.toNat_apply_of_lt_aleph0 + - Cardinal.toNat_strictMonoOn + - Cardinal.mk_toNat_eq_card + - Cardinal.cast_toNat_of_aleph0_le + - Cardinal.toNat_ne_zero + - path: Mathlib/LinearAlgebra/Matrix/Hermitian.lean + theorems: + - Matrix.isHermitian_inv + - Matrix.isHermitian_mul_conjTranspose_self + - Matrix.IsHermitian.zpow + - Matrix.IsHermitian.fromBlocks + - Matrix.IsHermitian.coe_re_apply_self + - Matrix.IsHermitian.adjugate + - Matrix.IsHermitian.inv + - Matrix.isHermitian_mul_mul_conjTranspose + - Matrix.isHermitian_conjTranspose_mul_mul + - Matrix.IsHermitian.transpose + - Matrix.IsHermitian.ext + - Matrix.isHermitian_transpose_mul_self + - Matrix.isHermitian_submatrix_equiv + - Matrix.isHermitian_iff_isSymmetric + - Matrix.isHermitian_transpose_iff + - Matrix.isHermitian_diagonal_iff + - path: Mathlib/RingTheory/Noetherian.lean + theorems: + - monotone_stabilizes_iff_noetherian + - eventuallyConst_of_isNoetherian + - set_has_maximal_iff_noetherian + - LinearMap.eventually_disjoint_ker_pow_range_pow + - LinearIndependent.finite_of_isNoetherian + - isNoetherian_submodule + - isNoetherian_of_submodule_of_noetherian + - IsNoetherianRing.isNilpotent_nilradical + - isNoetherianRing_of_surjective + - LinearMap.eventually_iSup_ker_pow_eq + - isNoetherian_of_range_eq_ker + - isNoetherian_of_tower + - IsNoetherian.disjoint_partialSups_eventually_bot + - isNoetherian_top_iff + - isNoetherian_iff_fg_wellFounded + - IsNoetherian.injective_of_surjective_endomorphism + - path: Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.lean + theorems: + - charpoly_leftMulMatrix + - path: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Instances.lean + theorems: + - SpectrumRestricts.nnreal_add + - unitary_iff_isStarNormal_and_spectrum_subset_circle + - spectrum_star_mul_self_nonneg + - IsSelfAdjoint.sq_spectrumRestricts + - SpectrumRestricts.eq_zero_of_neg + - SpectrumRestricts.nnreal_iff_nnnorm + - SpectrumRestricts.smul_of_nonneg + - nonneg_iff_isSelfAdjoint_and_spectrumRestricts + - isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts + - path: Mathlib/Data/Set/Intervals/SurjOn.lean + theorems: + - surjOn_Ico_of_monotone_surjective + - surjOn_Ioi_of_monotone_surjective + - surjOn_Icc_of_monotone_surjective + - surjOn_Ici_of_monotone_surjective + - surjOn_Ioo_of_monotone_surjective + - surjOn_Ioc_of_monotone_surjective + - path: Mathlib/Analysis/Convex/Body.lean + theorems: + - ConvexBody.iInter_smul_eq_self + - ConvexBody.zero_mem_of_symmetric + - ConvexBody.hausdorffEdist_coe + - ConvexBody.smul_le_of_le + - path: Mathlib/CategoryTheory/Shift/Pullback.lean + theorems: + - CategoryTheory.pullbackShiftFunctorAdd'_inv_app + - CategoryTheory.pullbackShiftFunctorAdd'_hom_app + - CategoryTheory.pullbackShiftFunctorZero_inv_app + - path: Mathlib/Data/Finset/Pi.lean + theorems: + - Finset.pi_nonempty + - Finset.Pi.cons_injective + - Finset.pi_insert + - Finset.pi_singletons + - path: Mathlib/CategoryTheory/Limits/FunctorCategory.lean + theorems: + - CategoryTheory.Limits.limit_obj_ext + - CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_map + - "CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03B9_inv" + - CategoryTheory.Limits.colimit_map_colimitObjIsoColimitCompEvaluation_hom + - "CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_\u03C0" + - CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom + - "CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03B9_app_hom" + - CategoryTheory.Limits.colimit_obj_ext + - CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map + - "CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_\u03C0_app" + - path: Mathlib/MeasureTheory/Constructions/Polish.lean + theorems: + - MeasureTheory.MeasurablySeparable.iUnion + - MeasurableSet.analyticSet + - MeasureTheory.analyticSet_empty + - MeasureTheory.AnalyticSet.image_of_continuousOn + - MeasureTheory.measurableSet_tendsto_fun + - Measurable.exists_continuous + - MeasureTheory.AnalyticSet.iUnion + - IsClosed.analyticSet + - MeasureTheory.analyticSet_range_of_polishSpace + - MeasureTheory.AnalyticSet.measurableSet_of_compl + - Continuous.map_borel_eq + - MeasurableSet.analyticSet_image + - MeasureTheory.AnalyticSet.measurablySeparable + - MeasureTheory.measurablySeparable_range_of_disjoint + - Continuous.map_eq_borel + - MeasureTheory.measurableEquiv_range_coe_nat_of_infinite_of_countable + - MeasureTheory.borel_eq_borel_of_le + - MeasureTheory.AnalyticSet.iInter + - MeasurableSet.standardBorel + - MeasureTheory.isClopenable_iff_measurableSet + - MeasurableSet.image_of_continuousOn_injOn + - MeasureTheory.exists_measurableEmbedding_real + - Measurable.measurableSet_preimage_iff_inter_range + - MeasureTheory.measurableSet_exists_tendsto + - MeasureTheory.measurableSet_range_of_continuous_injective + - MeasureTheory.AnalyticSet.preimage + - ContinuousOn.measurableEmbedding + - Measurable.map_measurableSpace_eq_borel + - IsClosed.measurableSet_image_of_continuousOn_injOn + - MeasureTheory.exists_subset_real_measurableEquiv + - MeasurableSet.image_of_measurable_injOn + - IsOpen.analyticSet_image + - MeasureTheory.exists_nat_measurableEquiv_range_coe_fin_of_finite + - MeasurableSet.isClopenable' + - Measurable.measurableSet_preimage_iff_of_surjective + - MeasureTheory.analyticSet_iff_exists_polishSpace_range + - path: Mathlib/FieldTheory/Fixed.lean + theorems: + - FixedPoints.minpoly.monic + - "FixedPoints.minpoly.eval\u2082" + - FixedPoints.smul_polynomial + - FixedPoints.minpoly.irreducible_aux + - FixedPoints.rank_le_card + - FixedPoints.toAlgHom_bijective + - "FixedPoints.minpoly.of_eval\u2082" + - FixedPoints.minpoly.ne_one + - FixedPoints.finrank_le_card + - FixedPoints.isIntegral + - FixedPoints.linearIndependent_smul_of_linearIndependent + - path: Mathlib/Topology/Order/Hom/Esakia.lean + theorems: + - PseudoEpimorphism.cancel_left + - EsakiaHom.cancel_left + - path: Mathlib/Data/PFunctor/Multivariate/Basic.lean + theorems: + - MvPFunctor.supp_eq + - MvPFunctor.comp.get_map + - MvPFunctor.liftP_iff + - MvPFunctor.const.mk_get + - MvPFunctor.comp.mk_get + - MvPFunctor.liftR_iff + - MvPFunctor.liftP_iff' + - path: Mathlib/Geometry/RingedSpace/SheafedSpace.lean + theorems: + - AlgebraicGeometry.SheafedSpace.ext + - AlgebraicGeometry.SheafedSpace.congr_app + - AlgebraicGeometry.SheafedSpace.id_c_app + - path: Mathlib/RingTheory/LocalProperties.lean + theorems: + - localizationPreserves_surjective + - isReduced_ofLocalizationMaximal + - Ideal.le_of_localization_maximal + - RingHom.PropertyIsLocal.ofLocalizationSpan + - RingHom.ofLocalizationSpanTarget_iff_finite + - eq_zero_of_localization + - Module.Finite_of_isLocalization + - RingHom.LocalizationPreserves.away + - RingHom.ofLocalizationSpan_iff_finite + - localization_finiteType + - surjective_ofLocalizationSpan + - ideal_eq_bot_of_localization + - IsLocalization.lift_mem_adjoin_finsetIntegerMultiple + - RingHom.OfLocalizationSpanTarget.ofIsLocalization + - finiteType_ofLocalizationSpan + - ideal_eq_bot_of_localization' + - multiple_mem_span_of_mem_localization_span + - finite_ofLocalizationSpan + - IsLocalization.smul_mem_finsetIntegerMultiple_span + - multiple_mem_adjoin_of_mem_localization_adjoin + - IsLocalization.exists_smul_mem_of_mem_adjoin + - localization_finite + - RingHom.PropertyIsLocal.respectsIso + - localization_isReduced + - path: Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean + theorems: + - groupCohomology.Hilbert90.aux_ne_zero + - groupCohomology.isMulOneCoboundary_of_isMulOneCocycle_of_aut_to_units + - path: Mathlib/Data/DFinsupp/WellFounded.lean + theorems: + - DFinsupp.Lex.acc_of_single + - DFinsupp.Lex.wellFounded + - Pi.Lex.wellFounded + - DFinsupp.wellFoundedLT + - DFinsupp.Lex.acc_of_single_erase + - DFinsupp.lex_fibration + - DFinsupp.Lex.acc_single + - path: Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean + theorems: + - WittVector.isUnit_of_coeff_zero_ne_zero + - WittVector.exists_eq_pow_p_mul' + - WittVector.irreducible + - WittVector.exists_eq_pow_p_mul + - WittVector.discreteValuationRing + - path: Mathlib/Order/Booleanisation.lean + theorems: + - Booleanisation.comp_lt_comp + - Booleanisation.lift_le_comp + - Booleanisation.lift_lt_lift + - Booleanisation.lift_lt_comp + - Booleanisation.lift_le_lift + - Booleanisation.comp_le_comp + - path: Mathlib/Condensed/Equivalence.lean + theorems: + - Condensed.StoneanCompHaus.generate_singleton_mem_coherentTopology + - Condensed.StoneanCompHaus.coverPreserving + - Condensed.StoneanProfinite.coherentTopology_is_induced + - Condensed.StoneanProfinite.generate_singleton_mem_coherentTopology + - Condensed.StoneanProfinite.coverPreserving + - Condensed.StoneanCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily + - Condensed.StoneanCompHaus.coherentTopology_is_induced + - path: Mathlib/NumberTheory/NumberField/Discriminant.lean + theorems: + - NumberField.hermiteTheorem.natDegree_le_rankOfDiscrBdd + - NumberField.discr_eq_discr + - NumberField.abs_discr_gt_two + - NumberField.hermiteTheorem.minkowskiBound_lt_boundOfDiscBdd + - NumberField.hermiteTheorem.finite_of_discr_bdd_of_isComplex + - NumberField.finite_of_discr_bdd + - NumberField.exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr + - NumberField.hermiteTheorem.rank_le_rankOfDiscrBdd + - Algebra.discr_eq_discr_of_toMatrix_coeff_isIntegral + - NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis + - NumberField.exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr + - Rat.numberField_discr + - NumberField.abs_discr_ge + - NumberField.hermiteTheorem.finite_of_finite_generating_set + - NumberField.hermiteTheorem.finite_of_discr_bdd_of_isReal + - NumberField.discr_eq_discr_of_algEquiv + - NumberField.discr_ne_zero + - path: Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean + theorems: + - iteratedDerivWithin_const_mul + - iteratedDerivWithin_add + - iteratedDerivWithin_const_add + - iteratedDeriv_const_smul + - iteratedDerivWithin_neg + - iteratedDerivWithin_const_smul + - iteratedDeriv_comp_neg + - iteratedDerivWithin_sub + - iteratedDeriv_neg + - iteratedDerivWithin_const_neg + - iteratedDerivWithin_congr + - iteratedDeriv_const_mul + - path: Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean + theorems: + - BilinForm.le_flip_dualSubmodule + - BilinForm.dualSubmodule_dualSubmodule_of_basis + - BilinForm.dualSubmodule_dualSubmodule_flip_of_basis + - BilinForm.dualSubmodule_span_of_basis + - BilinForm.dualSubmodule_flip_dualSubmodule_of_basis + - BilinForm.dualSubmoduleToDual_injective + - path: Mathlib/Topology/ContinuousFunction/CocompactMap.lean + theorems: + - CocompactMap.isCompact_preimage + - CocompactMap.tendsto_of_forall_preimage + - path: Mathlib/RingTheory/ChainOfDivisors.lean + theorems: + - pow_image_of_prime_by_factor_orderIso_dvd + - multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalizedFactors + - mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors + - DivisorChain.exists_chain_of_prime_pow + - DivisorChain.first_of_chain_isUnit + - factor_orderIso_map_one_eq_bot + - multiplicity_prime_eq_multiplicity_image_by_factor_orderIso + - map_prime_of_factor_orderIso + - mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactors + - DivisorChain.card_subset_divisors_le_length_of_chain + - DivisorChain.eq_second_of_chain_of_prime_dvd + - multiplicity_prime_le_multiplicity_image_by_factor_orderIso + - DivisorChain.eq_pow_second_of_chain_of_has_chain + - Associates.isAtom_iff + - DivisorChain.element_of_chain_eq_pow_second_of_chain + - path: Mathlib/GroupTheory/GroupAction/Quotient.lean + theorems: + - MulAction.injective_ofQuotientStabilizer + - MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group + - QuotientGroup.out'_conj_pow_minimalPeriod_mem + - card_comm_eq_card_conjClasses_mul_card + - MulAction.card_eq_sum_card_group_div_card_stabilizer' + - MulAction.stabilizer_quotient + - MulAction.card_orbit_mul_card_stabilizer_eq_card_group + - ConjClasses.card_carrier + - path: Mathlib/Algebra/Homology/TotalComplexShift.lean + theorems: + - "HomologicalComplex\u2082.D\u2082_totalShift\u2082XIso_hom" + - "HomologicalComplex\u2082.D\u2082_totalShift\u2081XIso_hom" + - "HomologicalComplex\u2082.D\u2081_totalShift\u2081XIso_hom" + - "HomologicalComplex\u2082.D\u2081_totalShift\u2082XIso_hom" + - path: Mathlib/RingTheory/WittVector/InitTail.lean + theorems: + - WittVector.coeff_select + - WittVector.init_neg + - WittVector.coeff_add_of_disjoint + - WittVector.init_pow + - WittVector.init_add + - WittVector.init_mul + - WittVector.init_add_tail + - WittVector.select_add_select_not + - WittVector.init_sub + - WittVector.init_zsmul + - WittVector.init_nsmul + - path: Mathlib/Combinatorics/SimpleGraph/Hasse.lean + theorems: + - SimpleGraph.pathGraph_two_eq_top + - SimpleGraph.hasse_preconnected_of_succ + - SimpleGraph.pathGraph_adj + - SimpleGraph.hasse_prod + - SimpleGraph.hasse_preconnected_of_pred + - path: Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean + theorems: + - legendreSym.at_two + - ZMod.exists_sq_eq_neg_two_iff + - legendreSym.quadratic_reciprocity' + - ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_three + - legendreSym.quadratic_reciprocity_three_mod_four + - legendreSym.quadratic_reciprocity + - ZMod.exists_sq_eq_prime_iff_of_mod_four_eq_one + - legendreSym.quadratic_reciprocity_one_mod_four + - legendreSym.at_neg_two + - ZMod.exists_sq_eq_two_iff + - path: Mathlib/Data/Rat/Cast/Defs.lean + theorems: + - Rat.cast_add_of_ne_zero + - Rat.cast_intCast + - Rat.cast_mk_of_ne_zero + - Rat.cast_sub_of_ne_zero + - MonoidWithZeroHom.ext_rat' + - Rat.cast_neg + - Rat.cast_commute + - Rat.cast_natCast + - Rat.cast_mul_of_ne_zero + - Rat.cast_ofNat + - MonoidWithZeroHom.ext_rat_on_pnat + - Rat.cast_div_of_ne_zero + - Rat.cast_inv_of_ne_zero + - map_ratCast + - eq_ratCast + - path: Mathlib/GroupTheory/EckmannHilton.lean + theorems: + - EckmannHilton.mul_comm + - EckmannHilton.mul_assoc + - EckmannHilton.one + - EckmannHilton.mul + - path: Mathlib/LinearAlgebra/Semisimple.lean + theorems: + - Module.End.isSemisimple_id + - Module.End.eq_zero_of_isNilpotent_isSemisimple + - Module.End.IsSemisimple.minpoly_squarefree + - Module.End.IsSemisimple.aeval + - Module.End.isSemisimple_iff + - Module.End.IsSemisimple.of_mem_adjoin_singleton + - Module.End.isSemisimple_of_squarefree_aeval_eq_zero + - Module.End.IsSemisimple_smul_iff + - Module.End.isSemisimple_zero + - Module.End.IsSemisimple_smul + - Module.End.IsSemisimple.of_mem_adjoin_pair + - path: Mathlib/Topology/Compactness/LocallyCompact.lean + theorems: + - exists_compact_subset + - ClosedEmbedding.locallyCompactSpace + - compact_basis_nhds + - OpenEmbedding.locallyCompactSpace + - exists_compact_superset + - path: Mathlib/Combinatorics/Quiver/Symmetric.lean + theorems: + - Quiver.Symmetrify.lift_reverse + - Quiver.reverse_reverse + - Quiver.Symmetrify.lift_spec + - Quiver.eq_reverse_iff + - Quiver.Path.reverse_reverse + - Quiver.Path.reverse_comp + - Quiver.Symmetrify.lift_unique + - Quiver.reverse_inj + - path: Mathlib/Topology/Algebra/Order/Floor.lean + theorems: + - tendsto_floor_atTop + - tendsto_floor_left_pure_sub_one + - tendsto_ceil_left_pure + - tendsto_floor_left_pure_ceil_sub_one + - tendsto_floor_right_pure + - tendsto_ceil_right_pure_floor_add_one + - tendsto_floor_left + - tendsto_ceil_right + - tendsto_ceil_atBot + - tendsto_ceil_right_pure_add_one + - ContinuousOn.comp_fract' + - tendsto_fract_left' + - path: Mathlib/Analysis/Complex/Polynomial.lean + theorems: + - Complex.exists_root + - Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv + - Polynomial.mul_star_dvd_of_aeval_eq_zero_im_ne_zero + - Irreducible.degree_le_two + - Polynomial.Gal.galActionHom_bijective_of_prime_degree' + - Polynomial.Gal.galActionHom_bijective_of_prime_degree + - Polynomial.quadratic_dvd_of_aeval_eq_zero_im_ne_zero + - path: Mathlib/Topology/Sets/Opens.lean + theorems: + - TopologicalSpace.Opens.mem_sSup + - TopologicalSpace.Opens.not_nonempty_iff_eq_bot + - TopologicalSpace.Opens.isBasis_iff_nbhd + - TopologicalSpace.Opens.eq_bot_or_top + - TopologicalSpace.Opens.coe_iSup + - TopologicalSpace.Opens.isCompactElement_iff + - TopologicalSpace.Opens.isBasis_iff_cover + - TopologicalSpace.Opens.mem_iSup + - TopologicalSpace.Opens.ne_bot_iff_nonempty + - TopologicalSpace.Opens.openEmbedding_of_le + - TopologicalSpace.Opens.IsBasis.isCompact_open_iff_eq_finite_iUnion + - path: Mathlib/Topology/Algebra/UniformConvergence.lean + theorems: + - UniformFun.continuousSMul_induced_of_range_bounded + - UniformOnFun.continuousSMul_induced_of_image_bounded + - UniformOnFun.hasBasis_nhds_one_of_basis + - UniformFun.hasBasis_nhds_one_of_basis + - path: Mathlib/FieldTheory/Galois.lean + theorems: + - IsGalois.IntermediateField.AdjoinSimple.card_aut_eq_finrank + - IsGalois.of_separable_splitting_field_aux + - isGalois_iff_isGalois_bot + - IsGalois.card_fixingSubgroup_eq_finrank + - IsGalois.of_separable_splitting_field + - IsGalois.fixedField_fixingSubgroup + - IntermediateField.fixingSubgroup_fixedField + - IsGalois.of_card_aut_eq_finrank + - IsGalois.of_fixedField_eq_bot + - IsGalois.tfae + - IsGalois.is_separable_splitting_field + - IsGalois.card_aut_eq_finrank + - path: Mathlib/MeasureTheory/Measure/Count.lean + theorems: + - MeasureTheory.Measure.count_injective_image + - MeasureTheory.Measure.count_apply_finset' + - MeasureTheory.Measure.count_apply_eq_top + - MeasureTheory.Measure.count_apply_eq_top' + - MeasureTheory.Measure.count_ne_zero + - MeasureTheory.Measure.count_apply_finite' + - MeasureTheory.Measure.count_singleton' + - MeasureTheory.Measure.count_apply_finite + - MeasureTheory.Measure.count_apply_infinite + - MeasureTheory.Measure.empty_of_count_eq_zero' + - MeasureTheory.Measure.count_ne_zero' + - MeasureTheory.Measure.empty_of_count_eq_zero + - MeasureTheory.Measure.count_empty + - MeasureTheory.Measure.count_injective_image' + - MeasureTheory.Measure.count_apply + - path: Mathlib/NumberTheory/ModularForms/SlashActions.lean + theorems: + - ModularForm.smul_slash + - ModularForm.is_invariant_const + - ModularForm.slash_action_eq'_iff + - ModularForm.slash_mul + - ModularForm.slash_one + - ModularForm.zero_slash + - ModularForm.mul_slash_SL2 + - SlashAction.smul_slash_of_tower + - SlashAction.neg_slash + - ModularForm.add_slash + - ModularForm.mul_slash + - path: Mathlib/Analysis/Analytic/Uniqueness.lean + theorems: + - AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero + - AnalyticOn.eqOn_of_preconnected_of_eventuallyEq + - AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux + - path: Mathlib/Topology/QuasiSeparated.lean + theorems: + - IsQuasiSeparated.image_of_embedding + - isQuasiSeparated_iff_quasiSeparatedSpace + - OpenEmbedding.isQuasiSeparated_iff + - IsQuasiSeparated.of_subset + - isQuasiSeparated_univ_iff + - path: Mathlib/Tactic/Group.lean + theorems: + - Mathlib.Tactic.Group.zpow_trick_one' + - Mathlib.Tactic.Group.zpow_trick_one + - Mathlib.Tactic.Group.zpow_trick + - path: Mathlib/Analysis/Calculus/ContDiff/Bounds.lean + theorems: + - norm_iteratedFDeriv_mul_le + - norm_iteratedFDerivWithin_comp_le + - ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear + - norm_iteratedFDerivWithin_prod_le + - norm_iteratedFDeriv_comp_le + - ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux + - norm_iteratedFDerivWithin_clm_apply_const + - norm_iteratedFDerivWithin_clm_apply + - ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one + - norm_iteratedFDerivWithin_comp_le_aux + - ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear_of_le_one + - ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear + - norm_iteratedFDeriv_prod_le + - path: Mathlib/Data/Int/Associated.lean + theorems: + - Int.natAbs_eq_iff_associated + - path: Mathlib/Data/Real/Pi/Wallis.lean + theorems: + - Real.Wallis.W_eq_factorial_ratio + - Real.Wallis.W_le + - Real.Wallis.tendsto_W_nhds_pi_div_two + - Real.Wallis.le_W + - Real.Wallis.W_eq_integral_sin_pow_div_integral_sin_pow + - Real.Wallis.W_pos + - path: Mathlib/Data/Nat/EvenOddRec.lean + theorems: + - Nat.evenOddRec_odd + - Nat.evenOddRec_even + - path: Mathlib/Topology/UniformSpace/Compact.lean + theorems: + - Continuous.uniformContinuous_of_tendsto_cocompact + - IsCompact.mem_uniformity_of_prod + - CompactSpace.uniformEquicontinuous_of_equicontinuous + - unique_uniformity_of_compact + - IsCompact.uniformContinuousOn_of_continuous + - ContinuousOn.tendstoUniformly + - HasCompactMulSupport.is_one_at_infty + - IsCompact.uniformContinuousAt_of_continuousAt + - CompactSpace.uniformContinuous_of_continuous + - path: Mathlib/Init/Data/List/Instances.lean + theorems: + - List.bind_singleton' + - List.bind_assoc + - List.map_eq_bind + - path: Mathlib/Dynamics/Ergodic/MeasurePreserving.lean + theorems: + - MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le + - MeasureTheory.MeasurePreserving.sigmaFinite + - MeasureTheory.MeasurePreserving.aemeasurable_comp_iff + - MeasureTheory.MeasurePreserving.symm + - MeasureTheory.MeasurePreserving.comp_left_iff + - MeasureTheory.MeasurePreserving.comp + - MeasureTheory.MeasurePreserving.measure_preimage_emb + - MeasureTheory.MeasurePreserving.restrict_preimage + - MeasureTheory.MeasurePreserving.measure_preimage + - MeasureTheory.MeasurePreserving.comp_right_iff + - MeasureTheory.MeasurePreserving.exists_mem_iterate_mem + - MeasureTheory.MeasurePreserving.exists_mem_iterate_mem_of_volume_lt_mul_volume + - path: Mathlib/MeasureTheory/Function/ContinuousMapDense.lean + theorems: + - MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le + - "MeasureTheory.Mem\u2112p.exists_hasCompactSupport_snorm_sub_le" + - MeasureTheory.exists_continuous_snorm_sub_le_of_closed + - "MeasureTheory.Mem\u2112p.exists_boundedContinuous_snorm_sub_le" + - "MeasureTheory.Mem\u2112p.exists_hasCompactSupport_integral_rpow_sub_le" + - ContinuousMap.toLp_denseRange + - "MeasureTheory.Mem\u2112p.exists_boundedContinuous_integral_rpow_sub_le" + - MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le + - MeasureTheory.Integrable.exists_boundedContinuous_integral_sub_le + - MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le + - BoundedContinuousFunction.toLp_denseRange + - path: Mathlib/Analysis/Convex/Quasiconvex.lean + theorems: + - QuasiconvexOn.sup + - quasilinearOn_iff_mem_uIcc + - QuasiconvexOn.convex_lt + - QuasilinearOn.monotoneOn_or_antitoneOn + - path: Mathlib/Probability/ConditionalProbability.lean + theorems: + - ProbabilityTheory.sum_meas_smul_cond_fiber + - ProbabilityTheory.cond_apply + - ProbabilityTheory.cond_univ + - ProbabilityTheory.cond_apply' + - ProbabilityTheory.cond_cond_eq_cond_inter' + - ProbabilityTheory.comap_cond + - ProbabilityTheory.cond_isProbabilityMeasure_of_finite + - ProbabilityTheory.cond_eq_zero_of_meas_eq_zero + - ProbabilityTheory.cond_eq_inv_mul_cond_mul + - ProbabilityTheory.cond_pos_of_inter_ne_zero + - ProbabilityTheory.cond_eq_zero + - ProbabilityTheory.inter_pos_of_cond_ne_zero + - ProbabilityTheory.cond_mul_eq_inter' + - ProbabilityTheory.cond_add_cond_compl_eq + - ProbabilityTheory.cond_empty + - ProbabilityTheory.cond_inter_self + - ProbabilityTheory.cond_toMeasurable_eq + - path: Mathlib/Order/OrderIsoNat.lean + theorems: + - Nat.exists_subseq_of_forall_mem_union + - exists_increasing_or_nonincreasing_subseq' + - Nat.Subtype.orderIsoOfNat_apply + - RelEmbedding.not_acc_of_decreasing_seq + - WellFounded.iSup_eq_monotonicSequenceLimit + - RelEmbedding.wellFounded_iff_no_descending_seq + - RelEmbedding.acc_iff_no_decreasing_seq + - RelEmbedding.not_wellFounded_of_decreasing_seq + - WellFounded.monotone_chain_condition + - WellFounded.monotone_chain_condition' + - RelEmbedding.exists_not_acc_lt_of_not_acc + - exists_increasing_or_nonincreasing_subseq + - path: Mathlib/Data/Polynomial/PartialFractions.lean + theorems: + - div_eq_quo_add_rem_div_add_rem_div + - div_eq_quo_add_sum_rem_div + - path: Mathlib/Order/Max.lean + theorems: + - NoTopOrder.to_noMaxOrder + - not_isMin_iff + - not_isMax_iff + - NoBotOrder.to_noMinOrder + - path: Mathlib/Data/PFunctor/Univariate/Basic.lean + theorems: + - PFunctor.liftp_iff + - PFunctor.supp_eq + - PFunctor.W.dest_mk + - PFunctor.fst_map + - PFunctor.iget_map + - PFunctor.W.mk_dest + - path: Mathlib/Algebra/Lie/Classical.lean + theorems: + - LieAlgebra.Orthogonal.mem_so + - LieAlgebra.Orthogonal.indefiniteDiagonal_transform + - LieAlgebra.SpecialLinear.sl_non_abelian + - LieAlgebra.Orthogonal.soIndefiniteEquiv_apply + - LieAlgebra.Orthogonal.pb_inv + - LieAlgebra.Orthogonal.indefiniteDiagonal_assoc + - LieAlgebra.Orthogonal.pd_inv + - LieAlgebra.Orthogonal.jb_transform + - LieAlgebra.Orthogonal.s_as_blocks + - LieAlgebra.Orthogonal.jd_transform + - LieAlgebra.Orthogonal.pso_inv + - path: Mathlib/CategoryTheory/Functor/Flat.lean + theorems: + - CategoryTheory.PreservesFiniteLimitsOfFlat.uniq + - CategoryTheory.flat_iff_lan_flat + - CategoryTheory.flat_of_preservesFiniteLimits + - CategoryTheory.PreservesFiniteLimitsOfFlat.fac + - path: Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean + theorems: + - GeneralizedContinuedFraction.convergents'Aux_stable_of_terminated + - GeneralizedContinuedFraction.convergents'_stable_of_terminated + - GeneralizedContinuedFraction.continuants_stable_of_terminated + - GeneralizedContinuedFraction.continuantsAux_stable_step_of_terminated + - GeneralizedContinuedFraction.convergents'Aux_stable_step_of_terminated + - GeneralizedContinuedFraction.continuantsAux_stable_of_terminated + - GeneralizedContinuedFraction.numerators_stable_of_terminated + - GeneralizedContinuedFraction.convergents_stable_of_terminated + - GeneralizedContinuedFraction.denominators_stable_of_terminated + - path: Mathlib/Analysis/Convex/Independent.lean + theorems: + - ConvexIndependent.range + - ConvexIndependent.injective + - convexIndependent_iff_finset + - convexIndependent_set_iff_not_mem_convexHull_diff + - convexIndependent_iff_not_mem_convexHull_diff + - ConvexIndependent.comp_embedding + - Subsingleton.convexIndependent + - convexIndependent_set_iff_inter_convexHull_subset + - path: Mathlib/Data/Finset/Interval.lean + theorems: + - Finset.card_Ioc_finset + - Finset.card_Iio_finset + - Finset.monotone_iff_forall_le_insert + - Finset.card_Icc_finset + - Finset.card_Iic_finset + - Finset.card_Ico_finset + - Finset.strictMono_iff_forall_lt_insert + - Finset.card_Ioo_finset + - Finset.monotone_iff_forall_le_cons + - Finset.strictMono_iff_forall_lt_cons + - Finset.Ico_eq_image_ssubsets + - Finset.Icc_eq_image_powerset + - path: Mathlib/Algebra/AddConstMap/Basic.lean + theorems: + - AddConstMapClass.map_add_nsmul + - AddConstMapClass.map_zsmul_add + - AddConstMapClass.map_sub_nat' + - AddConstMapClass.map_add_nat + - AddConstMapClass.map_add_nat' + - AddConstMapClass.map_nsmul_add + - AddConstMapClass.map_nat + - AddConstMapClass.rel_map_of_Icc + - AddConstMapClass.map_sub_int' + - AddConstMapClass.map_sub_int + - AddConstMapClass.map_nsmul_const + - AddConstMapClass.map_add_int + - AddConstMapClass.map_nat' + - AddConstMapClass.map_const + - AddConstMapClass.map_const_add + - AddConstMapClass.map_sub_zsmul + - AddConstMapClass.map_sub_const + - AddConstMapClass.map_zsmul_const + - AddConstMapClass.map_sub_nsmul + - AddConstMapClass.map_nat_add' + - AddConstMapClass.map_int_add + - AddConstMapClass.map_int_add' + - AddConstMapClass.map_nat_add + - AddConstMapClass.map_add_int' + - path: Mathlib/Combinatorics/Quiver/Cast.lean + theorems: + - Quiver.Path.cast_cons + - Quiver.hom_cast_eq_of_cons_eq_cons + - Quiver.Hom.cast_eq_cast + - Quiver.Hom.eq_cast_iff_heq + - Quiver.eq_nil_of_length_zero + - Quiver.Hom.cast_cast + - Quiver.Hom.cast_heq + - Quiver.Path.cast_eq_iff_heq + - Quiver.Path.cast_eq_cast + - Quiver.Path.cast_nil + - Quiver.Path.cast_cast + - Quiver.Path.cast_heq + - Quiver.Hom.cast_eq_iff_heq + - Quiver.cast_eq_of_cons_eq_cons + - path: Mathlib/Topology/Instances/Irrational.lean + theorems: + - dense_irrational + - Irrational.eventually_forall_le_dist_cast_div + - Irrational.eventually_forall_le_dist_cast_rat_of_den_le + - path: Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean + theorems: + - IsLowerSet.memberSubfamily + - IsLowerSet.card_inter_le_finset + - IsUpperSet.le_card_inter_finset + - IsLowerSet.memberSubfamily_subset_nonMemberSubfamily + - IsLowerSet.le_card_inter_finset' + - IsLowerSet.nonMemberSubfamily + - IsUpperSet.card_inter_le_finset + - path: Mathlib/Data/Nat/Choose/Central.lean + theorems: + - Nat.succ_mul_centralBinom_succ + - Nat.two_dvd_centralBinom_succ + - Nat.four_pow_le_two_mul_self_mul_centralBinom + - Nat.four_pow_lt_mul_centralBinom + - Nat.succ_dvd_centralBinom + - Nat.choose_le_centralBinom + - Nat.two_dvd_centralBinom_of_one_le + - path: Mathlib/Algebra/Polynomial/GroupRingAction.lean + theorems: + - Polynomial.smul_eq_map + - prodXSubSMul.eval + - Polynomial.smul_eval_smul + - prodXSubSMul.smul + - prodXSubSMul.coeff + - Polynomial.smul_eval + - Polynomial.eval_smul' + - path: Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean + theorems: + - AlgebraicGeometry.isOpenImmersion_is_local_at_target + - AlgebraicGeometry.isOpenImmersion_respectsIso + - AlgebraicGeometry.isOpenImmersion_stableUnderBaseChange + - AlgebraicGeometry.isOpenImmersion_iff_stalk + - AlgebraicGeometry.isOpenImmersion_stableUnderComposition + - path: Mathlib/Data/Quot.lean + theorems: + - Equivalence.quot_mk_eq_iff + - Quotient.out_equiv_out + - Quotient.induction_on_pi + - Setoid.ext + - Quotient.eq_mk_iff_out + - Quotient.mk_eq_iff_out + - path: Mathlib/Topology/Homotopy/HSpaces.lean + theorems: + - Path.delayReflLeft_one + - Path.delayReflLeft_zero + - unitInterval.qRight_one_left + - unitInterval.qRight_zero_right + - Path.delayReflRight_zero + - Path.delayReflRight_one + - unitInterval.qRight_one_right + - unitInterval.continuous_qRight + - unitInterval.qRight_zero_left + - path: Mathlib/Analysis/Complex/Liouville.lean + theorems: + - Complex.norm_deriv_le_aux + - Complex.deriv_eq_smul_circleIntegral + - Complex.norm_deriv_le_of_forall_mem_sphere_norm_le + - Differentiable.eq_const_of_tendsto_cocompact + - Differentiable.apply_eq_apply_of_bounded + - path: Mathlib/RingTheory/MvPowerSeries/Trunc.lean + theorems: + - MvPowerSeries.trunc_c + - MvPowerSeries.coeff_truncFun + - MvPowerSeries.coeff_trunc + - MvPowerSeries.trunc_one + - path: Mathlib/RingTheory/WittVector/Verschiebung.lean + theorems: + - "WittVector.bind\u2081_verschiebungPoly_wittPolynomial" + - WittVector.aeval_verschiebung_poly' + - WittVector.map_verschiebung + - WittVector.ghostComponent_verschiebungFun + - WittVector.verschiebungFun_coeff_zero + - WittVector.ghostComponent_zero_verschiebungFun + - path: Mathlib/MeasureTheory/Covering/LiminfLimsup.lean + theorems: + - blimsup_thickening_mul_ae_eq + - blimsup_cthickening_ae_eq_blimsup_thickening + - blimsup_cthickening_ae_le_of_eventually_mul_le_aux + - blimsup_cthickening_ae_le_of_eventually_mul_le + - blimsup_thickening_mul_ae_eq_aux + - blimsup_cthickening_mul_ae_eq + - path: Mathlib/SetTheory/Game/Short.lean + theorems: + - SetTheory.PGame.short_birthday + - SetTheory.PGame.subsingleton_short_example + - path: Mathlib/SetTheory/Cardinal/Divisibility.lean + theorems: + - Cardinal.nat_is_prime_iff + - Cardinal.le_of_dvd + - Cardinal.isPrimePow_iff + - Cardinal.is_prime_iff + - Cardinal.prime_of_aleph0_le + - Cardinal.not_irreducible_of_aleph0_le + - Cardinal.isUnit_iff + - Cardinal.nat_coe_dvd_iff + - path: Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean + theorems: + - MeasureTheory.meas_ge_le_mul_pow_snorm + - MeasureTheory.mul_meas_ge_le_pow_snorm + - MeasureTheory.mul_meas_ge_le_pow_snorm' + - MeasureTheory.pow_mul_meas_ge_le_snorm + - path: Mathlib/Data/Matrix/Invertible.lean + theorems: + - Matrix.invOf_mul_self_assoc + - Matrix.isUnit_transpose + - Matrix.mul_invOf_self_assoc + - Matrix.mul_mul_invOf_self_cancel + - Matrix.transpose_invOf + - Matrix.mul_invOf_mul_self_cancel + - path: Mathlib/Topology/Algebra/StarSubalgebra.lean + theorems: + - StarAlgHomClass.ext_topologicalClosure + - StarSubalgebra.closedEmbedding_inclusion + - Subalgebra.topologicalClosure_star_comm + - elementalStarAlgebra.closedEmbedding_coe + - StarAlgHom.ext_topologicalClosure + - elementalStarAlgebra.starAlgHomClass_ext + - path: Mathlib/Topology/Algebra/Ring/Basic.lean + theorems: + - TopologicalRing.of_addGroup_of_nhds_zero + - TopologicalSemiring.continuousNeg_of_mul + - RingTopology.toTopologicalSpace_injective + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean + theorems: + - CliffordAlgebraRing.involute_eq_id + - CliffordAlgebraQuaternion.ofQuaternion_star + - CliffordAlgebraComplex.toComplex_comp_ofComplex + - CliffordAlgebraRing.reverse_apply + - CliffordAlgebraComplex.ofComplex_conj + - CliffordAlgebraQuaternion.ofQuaternion_comp_toQuaternion + - CliffordAlgebraComplex.ofComplex_I + - CliffordAlgebraQuaternion.toQuaternion_comp_ofQuaternion + - "CliffordAlgebraDualNumber.\u03B9_mul_\u03B9" + - CliffordAlgebraComplex.toComplex_involute + - CliffordAlgebraComplex.reverse_apply + - CliffordAlgebraQuaternion.toQuaternion_star + - CliffordAlgebraComplex.ofComplex_comp_toComplex + - path: Mathlib/Algebra/Group/Commute/Basic.lean + theorems: + - Commute.div_mul_div_comm + - Commute.inv_mul_cancel + - Commute.inv_mul_cancel_assoc + - Commute.div_div_div_comm + - path: Mathlib/Data/Multiset/FinsetOps.lean + theorems: + - Multiset.disjoint_ndinsert_right + - Multiset.mem_ndinter + - Multiset.dedup_cons + - Multiset.disjoint_ndinsert_left + - Multiset.cons_ndinter_of_mem + - Multiset.ndunion_le + - Multiset.length_ndinsert_of_not_mem + - Multiset.ndinter_cons_of_not_mem + - Multiset.ndinsert_le + - Multiset.length_ndinsert_of_mem + - Multiset.le_ndinter + - Multiset.attach_ndinsert + - Multiset.ndinter_eq_zero_iff_disjoint + - path: Mathlib/Order/InitialSeg.lean + theorems: + - PrincipalSeg.topLTTop + - InitialSeg.eq_or_principal + - wellFounded_iff_wellFounded_subrel + - RelEmbedding.collapseF.lt + - RelEmbedding.collapseF.not_lt + - InitialSeg.eq + - InitialSeg.init_iff + - PrincipalSeg.irrefl + - InitialSeg.leLT_apply + - PrincipalSeg.top_eq + - path: Mathlib/FieldTheory/Minpoly/Basic.lean + theorems: + - minpoly.aeval + - minpoly.ne_one + - minpoly.unique' + - minpoly.not_isUnit + - minpoly.eq_X_sub_C_of_algebraMap_inj + - minpoly.algHom_eq + - minpoly.two_le_natDegree_iff + - minpoly.irreducible + - minpoly.two_le_natDegree_subalgebra + - minpoly.min + - minpoly.map_ne_one + - minpoly.degree_eq_one_iff + - minpoly.aeval_ne_zero_of_dvdNotUnit_minpoly + - minpoly.monic + - minpoly.mem_range_of_degree_eq_one + - minpoly.subsingleton + - minpoly.natDegree_pos + - minpoly.natDegree_eq_one_iff + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean + theorems: + - "CliffordAlgebra.star_\u03B9" + - CliffordAlgebra.star_algebraMap + - CliffordAlgebra.star_smul + - path: Mathlib/Data/Int/Dvd/Basic.lean + theorems: + - Int.coe_nat_dvd_right + - Int.coe_nat_dvd_left + - Int.coe_nat_dvd + - path: Mathlib/Order/Bounded.lean + theorems: + - Set.unbounded_inter_not + - Set.unbounded_lt_inter_not_lt + - Set.unbounded_lt_iff + - Set.bounded_lt_Iic + - Set.unbounded_le_iff + - Set.bounded_lt_inter_lt + - Set.bounded_le_inter_le + - Set.bounded_le_inter_lt + - Set.unbounded_le_inter_not_le + - Set.bounded_gt_Ici + - Set.bounded_lt_inter_le + - Set.unbounded_lt_inter_le + - Set.unbounded_le_inter_lt + - Set.unbounded_lt_iff_unbounded_le + - Set.unbounded_lt_inter_lt + - Set.unbounded_le_inter_le + - Set.bounded_inter_not + - Set.bounded_le_iff_bounded_lt + - path: Mathlib/Data/Finsupp/Indicator.lean + theorems: + - Finsupp.support_indicator_subset + - Finsupp.indicator_injective + - Finsupp.indicator_apply + - Finsupp.single_eq_indicator + - path: Mathlib/Topology/PartitionOfUnity.lean + theorems: + - PartitionOfUnity.exists_finset_nhd' + - BumpCovering.exists_finset_toPOUFun_eventuallyEq + - BumpCovering.exists_isSubordinate_of_prop + - BumpCovering.sum_toPOUFun_eq + - PartitionOfUnity.finsupport_subset_fintsupport + - BumpCovering.exists_isSubordinate + - PartitionOfUnity.finite_tsupport + - PartitionOfUnity.coe_finsupport + - PartitionOfUnity.exists_pos + - PartitionOfUnity.exists_finset_nhd + - PartitionOfUnity.sum_finsupport_smul_eq_finsum + - PartitionOfUnity.sum_finsupport + - BumpCovering.continuous_toPOUFun + - BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop + - PartitionOfUnity.eventually_fintsupport_subset + - PartitionOfUnity.mem_finsupport + - BumpCovering.toPOUFun_zero_of_zero + - PartitionOfUnity.sum_finsupport' + - BumpCovering.toPOUFun_eq_mul_prod + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean + theorems: + - EulerSine.antideriv_sin_comp_const_mul + - EulerSine.integral_cos_mul_cos_pow_even + - Complex.tendsto_euler_sin_prod + - EulerSine.integral_cos_mul_cos_pow + - EulerSine.integral_cos_pow_eq + - EulerSine.sin_pi_mul_eq + - EulerSine.integral_sin_mul_sin_mul_cos_pow_eq + - EulerSine.antideriv_cos_comp_const_mul + - Real.tendsto_euler_sin_prod + - EulerSine.integral_cos_mul_cos_pow_aux + - path: Mathlib/Algebra/Ring/Commute.lean + theorems: + - mul_self_eq_one_iff + - Commute.mul_self_sub_mul_self_eq' + - Commute.mul_self_eq_mul_self_iff + - Commute.mul_self_sub_mul_self_eq + - mul_self_sub_one + - Units.inv_eq_self_iff + - path: Mathlib/Topology/MetricSpace/ProperSpace.lean + theorems: + - Metric.exists_isLocalMin_mem_ball + - exists_pos_lt_subset_ball + - exists_lt_subset_ball + - path: Mathlib/Topology/DenseEmbedding.lean + theorems: + - DenseInducing.dense_image + - DenseInducing.mk' + - DenseInducing.extend_unique_at + - DenseInducing.continuousAt_extend + - DenseInducing.comap_nhds_neBot + - DenseEmbedding.subtype + - DenseInducing.closure_image_mem_nhds + - Filter.HasBasis.hasBasis_of_denseInducing + - DenseInducing.tendsto_comap_nhds_nhds + - DenseInducing.extend_eq' + - path: Mathlib/RingTheory/GradedAlgebra/Basic.lean + theorems: + - DirectSum.coe_decompose_mul_of_right_mem + - DirectSum.coe_decompose_mul_of_left_mem_of_le + - DirectSum.coe_decompose_mul_of_right_mem_of_le + - DirectSum.coe_decompose_mul_add_of_left_mem + - DirectSum.coe_decompose_mul_of_right_mem_of_not_le + - DirectSum.coe_decompose_mul_of_left_mem + - DirectSum.coe_decompose_mul_of_left_mem_of_not_le + - DirectSum.coe_decompose_mul_add_of_right_mem + - GradedAlgebra.proj_recompose + - GradedRing.proj_recompose + - path: Mathlib/Topology/Algebra/ConstMulAction.lean + theorems: + - isOpenMap_quotient_mk'_mul + - set_smul_mem_nhds_smul + - Dense.smul + - tendsto_const_smul_iff + - "isClosedMap_smul\u2080" + - set_smul_mem_nhds_smul_iff + - set_smul_mem_nhds_zero_iff + - isClosed_setOf_map_smul + - continuous_const_smul_iff + - "closure_smul\u2080" + - path: Mathlib/Topology/ContinuousFunction/Units.lean + theorems: + - ContinuousMap.isUnit_iff_forall_ne_zero + - ContinuousMap.spectrum_eq_range + - ContinuousMap.unitsLift_symm_apply_apply_inv' + - ContinuousMap.continuous_isUnit_unit + - ContinuousMap.isUnit_iff_forall_isUnit + - path: Mathlib/Order/Chain.lean + theorems: + - subset_succChain + - succChain_spec + - isChain_univ_iff + - IsChain.directed + - chainClosure_succ_total_aux + - IsChain.exists3 + - IsChain.succ + - ChainClosure.isChain + - IsChain.superChain_succChain + - chainClosure_empty + - ChainClosure.succ_fixpoint + - Monotone.isChain_range + - chainClosure_succ_total + - path: Mathlib/LinearAlgebra/PID.lean + theorems: + - LinearMap.trace_restrict_eq_of_forall_mem + - path: Mathlib/Algebra/GroupWithZero/Commute.lean + theorems: + - Ring.mul_inverse_rev' + - Commute.div_left + - path: Mathlib/Data/List/MinMax.lean + theorems: + - List.argmax_eq_none + - List.index_of_argmax + - List.argmax_cons + - List.coe_le_maximum_iff + - List.maximum_cons + - List.maximum_of_length_pos_mem + - List.argmax_mem + - List.max_le_of_forall_le + - List.maximum_eq_coe_iff + - List.le_max_of_le + - List.maximum_le_of_forall_le + - List.mem_argmax_iff + - List.argmax_concat + - List.foldl_argAux_mem + - List.le_maximum_of_length_pos_of_mem + - List.maximum_ne_bot_of_length_pos + - List.maximum_ne_bot_of_ne_nil + - List.foldr_max_of_ne_nil + - List.maximum_concat + - List.not_of_mem_foldl_argAux + - List.not_lt_maximum_of_mem' + - List.foldl_argAux_eq_none + - path: Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean + theorems: + - differentiableAt_iff_restrictScalars + - HasFDerivWithinAt.of_restrictScalars + - differentiableWithinAt_iff_restrictScalars + - path: Mathlib/Data/Fintype/Perm.lean + theorems: + - nodup_permsOfList + - length_permsOfList + - mem_of_mem_permsOfList + - card_perms_of_finset + - mem_permsOfList_of_mem + - path: Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean + theorems: + - Polynomial.cardPowDegree_anti_archimedean + - Polynomial.exists_approx_polynomial_aux + - Polynomial.exists_eq_polynomial + - Polynomial.exists_approx_polynomial + - Polynomial.exists_partition_polynomial_aux + - Polynomial.exists_partition_polynomial + - path: Mathlib/Probability/Density.lean + theorems: + - MeasureTheory.pdf_of_not_aemeasurable + - Real.hasPDF_iff_of_aemeasurable + - MeasureTheory.set_lintegral_pdf_le_map + - MeasureTheory.pdf.integral_mul_eq_integral + - MeasureTheory.hasPDF_of_pdf_ne_zero + - MeasureTheory.pdf.indepFun_iff_pdf_prod_eq_pdf_mul_pdf + - MeasureTheory.pdf.lintegral_eq_measure_univ + - Real.hasPDF_iff + - MeasureTheory.pdf.hasFiniteIntegral_mul + - MeasureTheory.map_eq_set_lintegral_pdf + - MeasureTheory.pdf.integrable_pdf_smul_iff + - MeasureTheory.hasPDF_iff_of_aemeasurable + - MeasureTheory.pdf.quasiMeasurePreserving_hasPDF + - MeasureTheory.aemeasurable_of_pdf_ne_zero + - MeasureTheory.measurable_pdf + - MeasureTheory.pdf.eq_of_map_eq_withDensity + - MeasureTheory.hasPDF_of_map_eq_withDensity + - MeasureTheory.map_eq_withDensity_pdf + - MeasureTheory.pdf.lintegral_pdf_mul + - MeasureTheory.pdf.congr + - MeasureTheory.pdf.integral_pdf_smul + - path: Mathlib/CategoryTheory/EffectiveEpi/Comp.lean + theorems: + - CategoryTheory.effectiveEpiFamilyStructIsoComp_aux + - CategoryTheory.effectiveEpiFamilyStructCompIso_aux + - path: Mathlib/LinearAlgebra/Projectivization/Independence.lean + theorems: + - Projectivization.independent_pair_iff_neq + - Projectivization.independent_iff_not_dependent + - Projectivization.dependent_iff + - Projectivization.independent_iff_completeLattice_independent + - Projectivization.dependent_pair_iff_eq + - Projectivization.independent_iff + - Projectivization.dependent_iff_not_independent + - path: Mathlib/CategoryTheory/Monad/Limits.lean + theorems: + - CategoryTheory.hasColimitsOfShape_of_reflective + - path: Mathlib/Algebra/Algebra/Subalgebra/Tower.lean + theorems: + - Subalgebra.restrictScalars_top + - Subalgebra.restrictScalars_injective + - Subalgebra.range_isScalarTower_toAlgHom + - IsScalarTower.adjoin_range_toAlgHom + - path: Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean + theorems: + - CategoryTheory.FreeMonoidalCategory.normalize_naturality + - CategoryTheory.FreeMonoidalCategory.tensorFunc_obj_map + - CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_eq + - CategoryTheory.FreeMonoidalCategory.inclusion_map + - CategoryTheory.FreeMonoidalCategory.normalizeObj_congr + - path: Mathlib/RingTheory/DedekindDomain/PID.lean + theorems: + - IsDedekindDomain.isPrincipalIdealRing_localization_over_prime + - Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne + - IsPrincipalIdealRing.of_finite_primes + - FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top + - FractionalIdeal.isPrincipal.of_finite_maximals_of_inv + - IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime + - path: Mathlib/Order/PFilter.lean + theorems: + - Order.PFilter.sInf_gc + - Order.PFilter.principal_le_principal_iff + - path: Mathlib/Analysis/LocallyConvex/WeakDual.lean + theorems: + - LinearMap.toSeminorm_ball_zero + - LinearMap.toSeminorm_comp + - LinearMap.hasBasis_weakBilin + - path: Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean + theorems: + - Complex.hasSum_sinh + - Complex.hasSum_cosh + - Complex.hasSum_sin' + - Complex.hasSum_cos + - Real.cosh_le_exp_half_sq + - Complex.hasSum_sin + - Complex.hasSum_cos' + - path: Mathlib/Topology/Sheaves/Sheafify.lean + theorems: + - TopCat.Presheaf.stalkToFiber_surjective + - TopCat.Presheaf.stalkToFiber_injective + - path: Mathlib/Combinatorics/SimpleGraph/Prod.lean + theorems: + - SimpleGraph.boxProd_adj_right + - SimpleGraph.Preconnected.boxProd + - SimpleGraph.Walk.ofBoxProdLeft_boxProdRight + - SimpleGraph.Connected.ofBoxProdLeft + - SimpleGraph.boxProd_neighborSet + - SimpleGraph.boxProd_neighborFinset + - SimpleGraph.Preconnected.ofBoxProdLeft + - SimpleGraph.boxProd_adj_left + - SimpleGraph.boxProd_degree + - SimpleGraph.Walk.ofBoxProdLeft_boxProdLeft + - SimpleGraph.Connected.ofBoxProdRight + - SimpleGraph.Preconnected.ofBoxProdRight + - path: Mathlib/Topology/Homotopy/HomotopyGroup.lean + theorems: + - GenLoop.to_from + - HomotopyGroup.symmAt_indep + - Cube.insertAt_boundary + - GenLoop.homotopicTo + - HomotopyGroup.mul_spec + - HomotopyGroup.inv_spec + - GenLoop.copy_eq + - HomotopyGroup.auxGroup_indep + - GenLoop.transAt_distrib + - HomotopyGroup.transAt_indep + - GenLoop.homotopicFrom + - path: Mathlib/Order/Sublattice.lean + theorems: + - Sublattice.coe_iInf + - Sublattice.coe_eq_empty + - Sublattice.map_inf + - Sublattice.mem_iInf + - Sublattice.map_top + - Sublattice.coe_eq_univ + - Sublattice.mem_sInf + - path: Mathlib/Algebra/Order/Sub/WithTop.lean + theorems: + - WithTop.sub_eq_top_iff + - WithTop.sub_top + - WithTop.map_sub + - path: Mathlib/Data/Finsupp/AList.lean + theorems: + - Finsupp.toAList_lookupFinsupp + - AList.empty_lookupFinsupp + - AList.lookupFinsupp_eq_zero_iff + - AList.lookupFinsupp_eq_iff_of_ne_zero + - AList.insert_lookupFinsupp + - AList.singleton_lookupFinsupp + - AList.lookupFinsupp_support + - Finsupp.mem_toAlist + - AList.lookupFinsupp_apply + - Finsupp.toAList_keys_toFinset + - path: Mathlib/NumberTheory/Liouville/Measure.lean + theorems: + - volume_setOf_liouville + - ae_not_liouvilleWith + - ae_not_liouville + - setOf_liouvilleWith_subset_aux + - volume_iUnion_setOf_liouvilleWith + - path: Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean + theorems: + - CategoryTheory.coherentTopology.mem_sieves_of_hasEffectiveEpiFamily + - CategoryTheory.coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily + - CategoryTheory.EffectiveEpiFamily.transitive_of_finite + - path: Mathlib/SetTheory/Lists.lean + theorems: + - Lists.equiv_atom + - Lists'.subset_nil + - Lists'.Subset.refl + - Lists.Equiv.trans + - Lists'.of_toList + - Lists'.mem_cons + - Lists'.subset_def + - Lists.lt_sizeof_cons' + - Lists.to_ofList + - Lists.Equiv.antisymm_iff + - Lists.Equiv.symm + - Lists'.cons_subset + - Lists.sizeof_pos + - Lists'.mem_of_subset' + - Lists'.toList_cons + - Lists.of_toList + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean + theorems: + - "CategoryTheory.Limits.map_\u03C0_preserves_coequalizer_inv_desc" + - "CategoryTheory.Limits.map_\u03C0_preserves_coequalizer_inv_colimMap" + - "CategoryTheory.Limits.map_\u03C0_preserves_coequalizer_inv_colimMap_desc" + - "CategoryTheory.Limits.map_\u03C0_preserves_coequalizer_inv" + - path: Mathlib/Order/Hom/CompleteLattice.lean + theorems: + - "map_iSup\u2082" + - sSupHom.cancel_left + - FrameHom.cancel_left + - "map_iInf\u2082" + - map_iInf + - CompleteLatticeHom.cancel_left + - map_iSup + - sInfHom.cancel_left + - Set.image_sSup + - path: Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean + theorems: + - CategoryTheory.IsKernelPair.comp_of_mono + - CategoryTheory.IsKernelPair.pullback + - CategoryTheory.IsKernelPair.isIso_of_mono + - CategoryTheory.IsKernelPair.cancel_right_of_mono + - CategoryTheory.IsKernelPair.of_isIso_of_mono + - CategoryTheory.IsKernelPair.mono_of_isIso_fst + - CategoryTheory.IsKernelPair.cancel_right + - path: Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean + theorems: + - Submodule.inf_iSup_generalizedEigenspace + - Module.End.exists_eigenvalue + - Module.End.iSup_generalizedEigenspace_restrict_eq_top + - Submodule.eq_iSup_inf_generalizedEigenspace + - Module.End.iSup_generalizedEigenspace_eq_top + - path: Mathlib/Order/Filter/FilterProduct.lean + theorems: + - Filter.Germ.const_abs + - Filter.Germ.min_def + - Filter.Germ.lt_def + - Filter.Germ.const_max + - Filter.Germ.coe_lt + - Filter.Germ.const_min + - Filter.Germ.max_def + - path: Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean + theorems: + - PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain + - PrimeSpectrum.exists_primeSpectrum_prod_le + - path: Mathlib/Order/Filter/Germ.lean + theorems: + - Filter.Germ.isConstant_comp + - Filter.Germ.isConstant_compTendsto + - Filter.Germ.isConstant_coe_const + - Filter.Germ.isConstant_comp_tendsto + - Filter.Germ.map_id + - path: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean + theorems: + - WeierstrassCurve.Jacobian.equation_smul_iff + - WeierstrassCurve.Jacobian.polynomialX_eq + - WeierstrassCurve.Jacobian.eval_polynomialZ + - WeierstrassCurve.Jacobian.nonsingular_iff + - WeierstrassCurve.Jacobian.smul_fin3_ext + - WeierstrassCurve.Jacobian.eval_polynomialY + - WeierstrassCurve.Jacobian.nonsingular_zero + - WeierstrassCurve.Jacobian.polynomialY_eq + - WeierstrassCurve.Jacobian.equation_iff + - WeierstrassCurve.Jacobian.nonsingular_of_equiv + - WeierstrassCurve.Jacobian.fin3_def + - WeierstrassCurve.Jacobian.nonsingular_some + - WeierstrassCurve.Jacobian.eval_polynomialX + - WeierstrassCurve.Jacobian.eval_polynomial + - WeierstrassCurve.Jacobian.equiv_some_of_Z_ne_zero + - WeierstrassCurve.Jacobian.nonsingular_smul_iff + - WeierstrassCurve.Jacobian.nonsingular_zero' + - WeierstrassCurve.Jacobian.equation_zero + - WeierstrassCurve.Jacobian.polynomialZ_eq + - path: Mathlib/Data/Prod/Basic.lean + theorems: + - Prod.map_injective + - Prod.mk.inj_right + - Prod.map_bijective + - Prod.snd_eq_iff + - Prod.fst_eq_iff + - Prod.eq_iff_fst_eq_snd_eq + - Prod.ext_iff + - Prod.map_surjective + - Prod.map_leftInverse + - Prod.mk.inj_left + - Function.LeftInverse.Prod_map + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean + theorems: + - ContinuousLinearMap.tendsto_of_tendsto_pointwise_of_cauchySeq + - ContinuousLinearMap.is_weak_closed_closedBall + - ContinuousLinearMap.opNorm_extend_le + - path: Mathlib/FieldTheory/Finite/Polynomial.lean + theorems: + - "MvPolynomial.ker_eval\u2097" + - "MvPolynomial.map_restrict_dom_eval\u2097" + - MvPolynomial.rank_R + - MvPolynomial.eval_indicator_apply_eq_one + - MvPolynomial.eval_indicator_apply_eq_zero + - "MvPolynomial.range_eval\u1D62" + - MvPolynomial.degrees_indicator + - MvPolynomial.indicator_mem_restrictDegree + - MvPolynomial.frobenius_zmod + - MvPolynomial.eq_zero_of_eval_eq_zero + - path: Mathlib/RingTheory/Localization/NormTrace.lean + theorems: + - Algebra.map_leftMulMatrix_localization + - Algebra.trace_localization + - Algebra.traceMatrix_localizationLocalization + - Algebra.norm_localization + - Algebra.discr_localizationLocalization + - path: Mathlib/Analysis/Calculus/Dslope.lean + theorems: + - dslope_sub_smul_of_ne + - differentiableWithinAt_dslope_of_ne + - continuousAt_dslope_of_ne + - continuousOn_dslope + - DifferentiableWithinAt.of_dslope + - differentiableAt_dslope_of_ne + - continuousAt_dslope_same + - sub_smul_dslope + - continuousWithinAt_dslope_of_ne + - ContinuousLinearMap.dslope_comp + - ContinuousWithinAt.of_dslope + - path: Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean + theorems: + - CategoryTheory.ProjectiveResolution.lift_commutes + - "CategoryTheory.ProjectiveResolution.homotopyEquiv_inv_\u03C0" + - "CategoryTheory.ProjectiveResolution.homotopyEquiv_hom_\u03C0" + - CategoryTheory.ProjectiveResolution.iso_hom_naturality + - CategoryTheory.exact_d_f + - CategoryTheory.ProjectiveResolution.ofComplex_d_1_0 + - CategoryTheory.ProjectiveResolution.lift_commutes_zero + - CategoryTheory.ProjectiveResolution.liftFOne_zero_comm + - CategoryTheory.ProjectiveResolution.ofComplex_exactAt_succ + - path: Mathlib/Algebra/Order/EuclideanAbsoluteValue.lean + theorems: + - AbsoluteValue.abs_isEuclidean + - path: Mathlib/Data/PNat/Basic.lean + theorems: + - PNat.one_add_natPred + - PNat.add_sub_of_lt + - PNat.mod_add_div' + - PNat.le_of_dvd + - PNat.dvd_iff' + - PNat.recOn_succ + - PNat.mul_div_exact + - PNat.div_add_mod' + - PNat.exists_eq_succ_of_ne_one + - PNat.mod_add_div + - PNat.dvd_iff + - PNat.modDivAux_spec + - PNat.mod_le + - PNat.pos_of_div_pos + - path: Mathlib/GroupTheory/GroupAction/Sum.lean + theorems: + - Sum.smul_swap + - path: Mathlib/CategoryTheory/Limits/Cones.lean + theorems: + - CategoryTheory.Limits.Cocone.w + - CategoryTheory.Limits.CoconeMorphism.ext + - CategoryTheory.Limits.Cone.w + - CategoryTheory.Limits.Cocones.cocone_iso_of_hom_iso + - "CategoryTheory.Limits.coconeOfConeLeftOp_\u03B9_app" + - CategoryTheory.Limits.Cones.cone_iso_of_hom_iso + - CategoryTheory.Limits.ConeMorphism.ext + - path: Mathlib/Data/Rat/Sqrt.lean + theorems: + - Rat.sqrt_eq + - Rat.exists_mul_self + - path: Mathlib/RingTheory/WittVector/Identities.lean + theorems: + - WittVector.verschiebung_frobenius + - WittVector.verschiebung_frobenius_comm + - WittVector.coeff_p_zero + - WittVector.frobenius_verschiebung + - WittVector.coeff_p_one + - WittVector.iterate_verschiebung_mul + - WittVector.mul_charP_coeff_zero + - WittVector.coeff_p + - WittVector.iterate_verschiebung_coeff + - WittVector.iterate_verschiebung_mul_left + - WittVector.p_nonzero + - WittVector.mul_charP_coeff_succ + - WittVector.verschiebung_zmod + - WittVector.verschiebung_mul_frobenius + - WittVector.iterate_verschiebung_mul_coeff + - WittVector.FractionRing.p_nonzero + - WittVector.iterate_frobenius_coeff + - path: Mathlib/Analysis/Normed/Group/InfiniteSum.lean + theorems: + - tsum_of_nnnorm_bounded + - summable_iff_vanishing_norm + - tsum_of_norm_bounded + - Summable.of_norm_bounded + - HasSum.norm_le_of_bounded + - cauchySeq_range_of_norm_bounded + - cauchySeq_finset_iff_vanishing_norm + - cauchySeq_finset_of_norm_bounded_eventually + - path: Mathlib/CategoryTheory/Sites/InducedTopology.lean + theorems: + - CategoryTheory.LocallyCoverDense.pushforward_cover_iff_cover_pullback + - CategoryTheory.Functor.locallyCoverDense_of_isCoverDense + - CategoryTheory.over_forget_locallyCoverDense + - path: Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean + theorems: + - intervalIntegral.hasFDerivAt_integral_of_dominated_loc_of_lip + - intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_lip + - intervalIntegral.hasFDerivAt_integral_of_dominated_of_fderiv_le + - intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le + - path: Mathlib/Analysis/Convex/Radon.lean + theorems: + - radon_partition + - path: Mathlib/Data/Finset/Finsupp.lean + theorems: + - Finset.mem_finsupp_iff + - Finset.mem_finsupp_iff_of_support_subset + - Finsupp.card_pi + - path: Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean + theorems: + - AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero + - AlgebraicGeometry.LocallyRingedSpace.preimage_basicOpen + - path: Mathlib/CategoryTheory/Limits/Shapes/Products.lean + theorems: + - "CategoryTheory.Limits.Pi.lift_\u03C0" + - CategoryTheory.Limits.Sigma.map'_eq + - "CategoryTheory.Limits.Sigma.\u03B9_reindex_hom" + - CategoryTheory.Limits.map_lift_piComparison + - CategoryTheory.Limits.Pi.map'_comp_map + - "CategoryTheory.Limits.Sigma.\u03B9_reindex_inv" + - CategoryTheory.Limits.Pi.map_comp_map + - "CategoryTheory.Limits.Sigma.eqToHom_comp_\u03B9" + - CategoryTheory.Limits.Sigma.map'_comp_map + - CategoryTheory.Limits.Sigma.map_id + - "CategoryTheory.Limits.Pi.reindex_hom_\u03C0" + - CategoryTheory.Limits.Pi.map_comp_map' + - CategoryTheory.Limits.Sigma.map'_comp_map' + - CategoryTheory.Limits.Sigma.map_comp_map + - CategoryTheory.Limits.Pi.map'_comp_map' + - CategoryTheory.Limits.Sigma.map_comp_map' + - "CategoryTheory.Limits.Pi.\u03C0_comp_eqToHom" + - "CategoryTheory.Limits.Sigma.\u03B9_desc" + - "CategoryTheory.Limits.Pi.reindex_inv_\u03C0" + - CategoryTheory.Limits.Pi.map'_eq + - CategoryTheory.Limits.Pi.map'_id_id + - CategoryTheory.Limits.Pi.map_id + - CategoryTheory.Limits.sigmaComparison_map_desc + - CategoryTheory.Limits.Sigma.map'_id_id + - path: .lake/packages/std/Std/WF.lean + theorems: + - WellFounded.fixF_eq_fixFC + - Acc.ndrec_eq_ndrecC + - Acc.ndrecOn_eq_ndrecOnC + - Acc.rec_eq_recC + - path: Mathlib/CategoryTheory/Monad/Basic.lean + theorems: + - CategoryTheory.comonadToFunctor_mapIso_comonad_iso_mk + - CategoryTheory.monadToFunctor_mapIso_monad_iso_mk + - path: Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean + theorems: + - Real.hasStrictDerivAt_log_of_pos + - Real.hasSum_log_one_add_inv + - Real.hasSum_log_sub_log_of_abs_lt_one + - Real.hasStrictDerivAt_log + - HasStrictDerivAt.log + - Real.tendsto_mul_log_one_plus_div_atTop + - Real.abs_log_sub_add_sum_range_le + - Real.deriv_log + - Real.contDiffOn_log + - Real.hasSum_pow_div_log_of_abs_lt_one + - HasDerivWithinAt.log + - HasDerivAt.log + - path: Mathlib/Algebra/Lie/DirectSum.lean + theorems: + - DirectSum.lie_of + - DirectSum.lie_of_of_ne + - path: Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean + theorems: + - WittVector.RecursionMain.root_exists + - WittVector.exists_frobenius_solution_fractionRing_aux + - WittVector.RecursionBase.solution_spec' + - WittVector.RecursionBase.solution_nonzero + - WittVector.frobenius_frobeniusRotation + - WittVector.exists_frobenius_solution_fractionRing + - WittVector.RecursionMain.succNthDefiningPoly_degree + - WittVector.RecursionMain.succNthVal_spec' + - WittVector.frobeniusRotation_nonzero + - path: Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean + theorems: + - contDiffOn_succ_iff_fderiv_apply + - contDiffOn_clm_apply + - contDiff_clm_apply_iff + - contDiff_succ_iff_fderiv_apply + - path: Mathlib/LinearAlgebra/Matrix/Polynomial.lean + theorems: + - Polynomial.natDegree_det_X_add_C_le + - Polynomial.leadingCoeff_det_X_one_add_C + - Polynomial.coeff_det_X_add_C_card + - Polynomial.coeff_det_X_add_C_zero + - path: Mathlib/Data/Nat/Cast/Field.lean + theorems: + - Nat.one_div_le_one_div + - Nat.one_div_pos_of_nat + - Nat.cast_div_le + - Nat.cast_inv_le_one + - Nat.cast_div + - Nat.one_div_lt_one_div + - Nat.cast_div_div_div_cancel_right + - path: Mathlib/Probability/Kernel/Invariance.lean + theorems: + - ProbabilityTheory.kernel.const_bind_eq_comp_const + - ProbabilityTheory.kernel.comp_const_apply_eq_bind + - ProbabilityTheory.kernel.Invariant.comp_const + - ProbabilityTheory.kernel.bind_smul + - ProbabilityTheory.kernel.Invariant.comp + - ProbabilityTheory.kernel.bind_add + - path: Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean + theorems: + - IsFreeGroup.unique_lift + - FreeGroupBasis.repr_apply_coe + - path: Mathlib/FieldTheory/AbelRuffini.lean + theorems: + - solvableByRad.isSolvable' + - gal_X_sub_C_isSolvable + - solvableByRad.isIntegral + - solvableByRad.induction2 + - solvableByRad.induction3 + - solvableByRad.isSolvable + - gal_prod_isSolvable + - solvableByRad.induction + - gal_X_isSolvable + - gal_one_isSolvable + - gal_X_pow_sub_C_isSolvable_aux + - gal_zero_isSolvable + - gal_X_pow_sub_one_isSolvable + - gal_X_pow_sub_C_isSolvable + - gal_X_pow_isSolvable + - splits_X_pow_sub_one_of_X_pow_sub_C + - gal_C_isSolvable + - path: Mathlib/Dynamics/BirkhoffSum/Average.lean + theorems: + - birkhoffAverage_apply_sub_birkhoffAverage + - map_birkhoffAverage + - birkhoffAverage_congr_ring' + - Function.IsFixedPt.birkhoffAverage_eq + - birkhoffAverage_zero + - birkhoffAverage_one + - path: Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean + theorems: + - NonarchimedeanRing.mul_subset + - NonarchimedeanGroup.nonarchimedean_of_emb + - NonarchimedeanGroup.prod_self_subset + - NonarchimedeanGroup.prod_subset + - path: Mathlib/NumberTheory/SmoothNumbers.lean + theorems: + - Nat.pow_mul_mem_smoothNumbers + - Nat.eq_prod_primes_mul_sq_of_mem_smoothNumbers + - Nat.smoothNumbers_compl + - Nat.smoothNumbersUpTo_card_le + - Nat.Prime.smoothNumbers_coprime + - Nat.smoothNumbers_succ + - Nat.prod_mem_smoothNumbers + - Nat.mem_smoothNumbers_iff_forall_le + - Nat.roughNumbersUpTo_eq_biUnion + - Nat.mem_smoothNumbersUpTo + - Nat.roughNumbersUpTo_card_le + - Nat.smoothNumbers_zero + - Nat.smoothNumbers_one + - Nat.mem_primesBelow + - Nat.smoothNumbersUpTo_card_add_roughNumbersUpTo_card + - Nat.mem_smoothNumbers_of_dvd + - Nat.primesBelow_succ + - Nat.primesBelow_zero + - Nat.smoothNumbersUpTo_subset_image + - path: Mathlib/Topology/Baire/Lemmas.lean + theorems: + - dense_sInter_of_isOpen + - "Dense.inter_of_G\u03B4" + - eventually_residual + - "IsG\u03B4.dense_sUnion_interior_of_closed" + - nonempty_interior_of_iUnion_of_closed + - "IsG\u03B4.dense_iUnion_interior_of_closed" + - mem_residual + - path: Mathlib/GroupTheory/Archimedean.lean + theorems: + - AddSubgroup.exists_isLeast_pos + - AddSubgroup.cyclic_of_min + - AddSubgroup.cyclic_of_isolated_zero + - Int.subgroup_cyclic + - path: Mathlib/Data/Set/BoolIndicator.lean + theorems: + - Set.mem_iff_boolIndicator + - Set.preimage_boolIndicator + - Set.preimage_boolIndicator_eq_union + - Set.not_mem_iff_boolIndicator + - path: Mathlib/CategoryTheory/Adjunction/Reflective.lean + theorems: + - CategoryTheory.equivEssImageOfReflective_map_counitIso_app_hom + - CategoryTheory.Functor.essImage.unit_isIso + - CategoryTheory.unitCompPartialBijective_natural + - CategoryTheory.unit_obj_eq_map_unit + - CategoryTheory.unitCompPartialBijective_symm_apply + - CategoryTheory.unitCompPartialBijective_symm_natural + - CategoryTheory.unitCompPartialBijectiveAux_symm_apply + - CategoryTheory.mem_essImage_of_unit_isSplitMono + - path: Mathlib/NumberTheory/Harmonic/Defs.lean + theorems: + - harmonic_pos + - harmonic_eq_sum_Icc + - path: Mathlib/LinearAlgebra/Charpoly/Basic.lean + theorems: + - LinearMap.charpoly_natDegree + - LinearMap.minpoly_coeff_zero_of_injective + - LinearMap.pow_eq_aeval_mod_charpoly + - LinearMap.aeval_self_charpoly + - path: Mathlib/Data/Fin/Tuple/Curry.lean + theorems: + - Function.FromTypes.uncurry_curry + - path: Mathlib/RingTheory/Derivation/Basic.lean + theorems: + - Derivation.map_coe_nat + - Derivation.leibniz_inv + - Derivation.map_algebraMap + - Derivation.map_coe_int + - Derivation.leibniz_pow + - path: Mathlib/Algebra/BigOperators/WithTop.lean + theorems: + - WithTop.sum_lt_top_iff + - WithBot.bot_lt_sum_iff + - WithBot.sum_eq_bot_iff + - WithTop.sum_eq_top_iff + - path: Mathlib/Analysis/Calculus/FDeriv/Linear.lean + theorems: + - ContinuousLinearMap.hasStrictFDerivAt + - ContinuousLinearMap.hasFDerivAtFilter + - IsBoundedLinearMap.fderivWithin + - ContinuousLinearMap.fderivWithin + - path: Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean + theorems: + - quadraticChar_odd_prime + - quadraticChar_card_card + - FiniteField.isSquare_two_iff + - FiniteField.isSquare_odd_prime_iff + - FiniteField.isSquare_neg_two_iff + - quadraticChar_neg_two + - path: Mathlib/Data/MvPolynomial/CommRing.lean + theorems: + - MvPolynomial.totalDegree_neg + - MvPolynomial.totalDegree_sub_C_le + - MvPolynomial.degrees_sub + - MvPolynomial.vars_sub_of_disjoint + - MvPolynomial.vars_neg + - MvPolynomial.totalDegree_sub + - "MvPolynomial.eval\u2082Hom_X" + - MvPolynomial.degreeOf_sub_lt + - MvPolynomial.vars_sub_subset + - MvPolynomial.degrees_neg + - path: Mathlib/MeasureTheory/Integral/Asymptotics.lean + theorems: + - Asymptotics.IsBigO.integrable + - Asymptotics.IsBigO.integrableAtFilter + - path: Mathlib/Topology/MetricSpace/Perfect.lean + theorems: + - Perfect.small_diam_splitting + - IsClosed.exists_nat_bool_injection_of_not_countable + - Perfect.exists_nat_bool_injection + - path: Mathlib/Topology/Algebra/ContinuousAffineMap.lean + theorems: + - ContinuousAffineMap.to_continuousMap_injective + - ContinuousAffineMap.mk_coe + - ContinuousLinearMap.toContinuousAffineMap_map_zero + - ContinuousAffineMap.to_affineMap_injective + - path: Mathlib/Analysis/Convex/Uniform.lean + theorems: + - exists_forall_closed_ball_dist_add_le_two_mul_sub + - exists_forall_closed_ball_dist_add_le_two_sub + - path: Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean + theorems: + - Polynomial.cyclotomic_irreducible_pow_of_irreducible_pow + - Polynomial.cyclotomic_mul_prime_dvd_eq_pow + - Polynomial.cyclotomic_expand_eq_cyclotomic + - Polynomial.cyclotomic_expand_eq_cyclotomic_mul + - Polynomial.cyclotomic_mul_prime_eq_pow_of_not_dvd + - Polynomial.cyclotomic_mul_prime_pow_eq + - Polynomial.isRoot_cyclotomic_prime_pow_mul_iff_of_charP + - path: Mathlib/NumberTheory/WellApproximable.lean + theorems: + - approxOrderOf.image_pow_subset_of_coprime + - approxOrderOf.smul_subset_of_coprime + - approxOrderOf.image_pow_subset + - UnitAddCircle.mem_approxAddOrderOf_iff + - UnitAddCircle.mem_addWellApproximable_iff + - mem_approxOrderOf_iff + - AddCircle.exists_norm_nsmul_le + - approxOrderOf.smul_eq_of_mul_dvd + - AddCircle.addWellApproximable_ae_empty_or_univ + - NormedAddCommGroup.exists_norm_nsmul_le + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean + theorems: + - "CliffordAlgebra.reverse_\u03B9" + - CliffordAlgebra.involute_comp_involute + - CliffordAlgebra.involute_eq_of_mem_odd + - CliffordAlgebra.involute_eq_of_mem_even + - "CliffordAlgebra.\u03B9_range_comap_involute" + - "CliffordAlgebra.reverse_prod_map_\u03B9" + - CliffordAlgebra.submodule_map_mul_reverse + - CliffordAlgebra.submodule_map_pow_reverse + - "CliffordAlgebra.involute_prod_map_\u03B9" + - CliffordAlgebra.submodule_comap_mul_reverse + - CliffordAlgebra.evenOdd_comap_involute + - CliffordAlgebra.evenOdd_map_involute + - "CliffordAlgebra.\u03B9_range_comap_reverse" + - CliffordAlgebra.reverse_comp_involute + - CliffordAlgebra.evenOdd_map_reverse + - path: Mathlib/CategoryTheory/Closed/Functor.lean + theorems: + - CategoryTheory.frobeniusMorphism_mate + - CategoryTheory.expComparison_iso_of_frobeniusMorphism_iso + - CategoryTheory.uncurry_expComparison + - CategoryTheory.expComparison_whiskerLeft + - CategoryTheory.frobeniusMorphism_iso_of_expComparison_iso + - CategoryTheory.expComparison_ev + - CategoryTheory.coev_expComparison + - path: Mathlib/RingTheory/Polynomial/GaussLemma.lean + theorems: + - Polynomial.IsPrimitive.dvd_of_fraction_map_dvd_fraction_map + - integralClosure.mem_lifts_of_monic_of_dvd_map + - Polynomial.Monic.dvd_of_fraction_map_dvd_fraction_map + - Polynomial.IsPrimitive.irreducible_iff_irreducible_map_fraction_map + - Polynomial.IsPrimitive.irreducible_of_irreducible_map_of_injective + - Polynomial.IsPrimitive.isUnit_iff_isUnit_map_of_injective + - Polynomial.isIntegrallyClosed_iff' + - Polynomial.isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart + - IsIntegrallyClosed.eq_map_mul_C_of_dvd + - path: Mathlib/MeasureTheory/Integral/DominatedConvergence.lean + theorems: + - MeasureTheory.integral_tsum + - intervalIntegral.continuousAt_parametric_primitive_of_dominated + - intervalIntegral.continuousOn_primitive_interval_left + - MeasureTheory.tendsto_integral_of_dominated_convergence + - intervalIntegral.continuousOn_primitive_Icc + - MeasureTheory.hasSum_integral_of_dominated_convergence + - intervalIntegral.continuousOn_primitive + - intervalIntegral.hasSum_intervalIntegral_of_summable_norm + - intervalIntegral.tendsto_integral_filter_of_dominated_convergence + - intervalIntegral.hasSum_integral_of_dominated_convergence + - intervalIntegral.continuousWithinAt_primitive + - intervalIntegral.continuousOn_primitive_interval' + - MeasureTheory.tendsto_integral_filter_of_dominated_convergence + - intervalIntegral.continuous_primitive + - Antitone.tendsto_set_integral + - path: Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean + theorems: + - Real.tendsto_harmonic_sub_log + - Real.tendsto_eulerMascheroniSeq + - Real.strictAnti_eulerMascheroniSeq' + - Real.eulerMascheroniSeq_lt_eulerMascheroniSeq' + - Real.eulerMascheroniSeq'_six_lt_two_thirds + - Real.eulerMascheroniSeq_lt_eulerMascheroniConstant + - Real.tendsto_eulerMascheroniSeq' + - Real.strictMono_eulerMascheroniSeq + - Real.eulerMascheroniSeq_zero + - Real.eulerMascheroniSeq'_one + - Real.one_half_lt_eulerMascheroniSeq_six + - Real.eulerMascheroniConstant_lt_eulerMascheroniSeq' + - path: Mathlib/Topology/MetricSpace/CauSeqFilter.lean + theorems: + - CauchySeq.isCauSeq + - CauSeq.tendsto_limit + - CauSeq.cauchySeq + - path: Mathlib/RingTheory/Polynomial/Dickson.lean + theorems: + - Polynomial.dickson_one_one_mul + - Polynomial.dickson_one_one_zmod_p + - Polynomial.chebyshev_T_eq_dickson_one_one + - Polynomial.dickson_one_one_charP + - Polynomial.dickson_two + - Polynomial.dickson_two_zero + - Polynomial.dickson_one_one_eq_chebyshev_T + - Polynomial.dickson_of_two_le + - Polynomial.dickson_one_one_comp_comm + - Polynomial.two_mul_C_half_eq_one + - Polynomial.C_half_mul_two_eq_one + - Polynomial.dickson_one_one_eval_add_inv + - path: Mathlib/Analysis/LocallyConvex/StrongTopology.lean + theorems: + - ContinuousLinearMap.strongTopology.locallyConvexSpace + - path: Mathlib/Algebra/Order/Chebyshev.lean + theorems: + - sq_sum_le_card_mul_sum_sq + - Antivary.card_smul_sum_le_sum_smul_sum + - AntivaryOn.card_smul_sum_le_sum_smul_sum + - MonovaryOn.sum_smul_sum_le_card_smul_sum + - AntivaryOn.card_mul_sum_le_sum_mul_sum + - MonovaryOn.sum_mul_sum_le_card_mul_sum + - path: Mathlib/Data/Finsupp/Order.lean + theorems: + - Finsupp.disjoint_iff + - Finsupp.single_le_iff + - Finsupp.subset_support_tsub + - Finsupp.support_inf_union_support_sup + - Finsupp.single_tsub + - Finsupp.le_iff' + - Finsupp.support_monotone + - Finsupp.add_eq_zero_iff + - Finsupp.support_tsub + - Finsupp.support_sup + - Finsupp.support_inf + - path: Mathlib/Algebra/Order/Module/Algebra.lean + theorems: + - algebraMap_le_algebraMap + - algebraMap_nonneg + - algebraMap_lt_algebraMap + - algebraMap_pos + - algebraMap_strictMono + - algebraMap_mono + - path: Mathlib/Order/OrdContinuous.lean + theorems: + - LeftOrdContinuous.map_sup + - LeftOrdContinuous.iterate + - LeftOrdContinuous.comp + - LeftOrdContinuous.mono + - LeftOrdContinuous.lt_iff + - RightOrdContinuous.id + - LeftOrdContinuous.map_sSup + - LeftOrdContinuous.le_iff + - LeftOrdContinuous.map_ciSup + - LeftOrdContinuous.map_iSup + - LeftOrdContinuous.id + - path: Mathlib/Analysis/Convex/Cone/Basic.lean + theorems: + - ConvexCone.Flat.pointed + - Convex.subset_toCone + - ConvexCone.Blunt.salient + - ConvexCone.pointed_zero + - ConvexCone.to_orderedSMul + - ConvexCone.mem_iInf + - Convex.mem_toCone' + - ConvexCone.salient_iff_not_flat + - convexHull_toCone_isLeast + - Convex.mem_toCone + - Convex.toCone_isLeast + - ConvexCone.blunt_iff_not_pointed + - ConvexCone.coe_iInf + - path: Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean + theorems: + - SimpleGraph.IsSRGWith.compl_is_regular + - SimpleGraph.bot_strongly_regular + - SimpleGraph.IsSRGWith.card_commonNeighbors_eq_of_adj_compl + - SimpleGraph.IsSRGWith.top + - SimpleGraph.sdiff_compl_neighborFinset_inter_eq + - SimpleGraph.IsSRGWith.card_neighborFinset_union_of_adj + - SimpleGraph.IsSRGWith.card_commonNeighbors_eq_of_not_adj_compl + - SimpleGraph.IsSRGWith.card_neighborFinset_union_eq + - SimpleGraph.IsSRGWith.matrix_eq + - SimpleGraph.compl_neighborFinset_sdiff_inter_eq + - SimpleGraph.IsSRGWith.param_eq + - SimpleGraph.IsSRGWith.card_neighborFinset_union_of_not_adj + - path: Mathlib/Algebra/Order/Nonneg/Ring.lean + theorems: + - Nonneg.toNonneg_of_nonneg + - Nonneg.toNonneg_lt + - Nonneg.toNonneg_le + - path: Mathlib/Data/Pi/Interval.lean + theorems: + - Pi.card_Ioi + - Pi.card_Ioo + - Pi.card_Ioc + - Pi.card_Iio + - Pi.card_Ico + - path: Mathlib/CategoryTheory/Elements.lean + theorems: + - CategoryTheory.Functor.Elements.ext + - CategoryTheory.CategoryOfElements.to_fromCostructuredArrow_eq + - CategoryTheory.CategoryOfElements.from_toCostructuredArrow_eq + - CategoryTheory.CategoryOfElements.costructuredArrow_yoneda_equivalence_naturality + - path: .lake/packages/std/Std/Data/Int/Gcd.lean + theorems: + - Int.gcd_dvd_right + - Int.neg_gcd + - Int.gcd_dvd_left + - Int.gcd_one + - Int.lcm_ne_zero + - Int.one_gcd + - path: Mathlib/Algebra/Group/Ext.lean + theorems: + - CancelMonoid.toLeftCancelMonoid_injective + - CommMonoid.toMonoid_injective + - Monoid.ext + - CancelCommMonoid.toCommMonoid_injective + - RightCancelMonoid.toMonoid_injective + - Group.toDivInvMonoid_injective + - LeftCancelMonoid.toMonoid_injective + - Group.ext + - DivInvMonoid.ext + - path: Mathlib/Data/Sigma/Order.lean + theorems: + - Sigma.lt_def + - Sigma.le_def + - path: Mathlib/Topology/Order/MonotoneContinuity.lean + theorems: + - Monotone.continuous_of_denseRange + - OrderIso.continuous + - continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin + - StrictMonoOn.continuousWithinAt_right_of_exists_between + - continuousWithinAt_right_of_monotoneOn_of_exists_between + - path: Mathlib/Data/Fintype/List.lean + theorems: + - Multiset.mem_lists_iff + - path: Mathlib/Data/Polynomial/Monomial.lean + theorems: + - Polynomial.monomial_one_eq_iff + - Polynomial.ringHom_ext + - Polynomial.card_support_le_one_iff_monomial + - path: Mathlib/Algebra/Lie/UniversalEnveloping.lean + theorems: + - "UniversalEnvelopingAlgebra.lift_\u03B9_apply" + - UniversalEnvelopingAlgebra.hom_ext + - "UniversalEnvelopingAlgebra.lift_\u03B9_apply'" + - UniversalEnvelopingAlgebra.lift_unique + - path: Mathlib/Data/Matrix/Hadamard.lean + theorems: + - Matrix.one_hadamard + - Matrix.hadamard_one + - Matrix.dotProduct_vecMul_hadamard + - path: Mathlib/CategoryTheory/Preadditive/Mat.lean + theorems: + - CategoryTheory.Mat_.additiveObjIsoBiproduct_naturality' + - CategoryTheory.Mat.id_apply_self + - CategoryTheory.Mat.id_apply_of_ne + - CategoryTheory.Mat_.additiveObjIsoBiproduct_naturality + - "CategoryTheory.Mat_.\u03B9_additiveObjIsoBiproduct_inv" + - "CategoryTheory.Mat_.additiveObjIsoBiproduct_hom_\u03C0" + - CategoryTheory.Mat_.id_apply_self + - CategoryTheory.Mat_.id_apply_of_ne + - path: Mathlib/Testing/SlimCheck/Gen.lean + theorems: + - SlimCheck.Gen.chooseNatLt_aux + - path: Mathlib/LinearAlgebra/Multilinear/Basis.lean + theorems: + - Basis.ext_multilinear + - Basis.ext_multilinear_fin + - path: Mathlib/Data/Polynomial/SpecificDegree.lean + theorems: + - Polynomial.Monic.irreducible_iff_roots_eq_zero_of_degree_le_three + - Polynomial.irreducible_iff_roots_eq_zero_of_degree_le_three + - path: Mathlib/Topology/MetricSpace/Antilipschitz.lean + theorems: + - AntilipschitzWith.to_rightInvOn' + - antilipschitzWith_iff_le_mul_dist + - AntilipschitzWith.to_rightInverse + - AntilipschitzWith.id + - AntilipschitzWith.isBounded_of_image2_left + - LipschitzWith.to_rightInverse + - AntilipschitzWith.properSpace + - AntilipschitzWith.injective + - AntilipschitzWith.comap_uniformity_le + - AntilipschitzWith.mul_le_nndist + - AntilipschitzWith.mul_le_edist + - AntilipschitzWith.of_subsingleton + - path: Mathlib/LinearAlgebra/Matrix/DotProduct.lean + theorems: + - Matrix.conjTranspose_mul_self_mulVec_eq_zero + - Matrix.dotProduct_eq + - Matrix.dotProduct_stdBasis_eq_mul + - Matrix.dotProduct_self_star_eq_zero + - Matrix.mul_conjTranspose_mul_self_eq_zero + - Matrix.dotProduct_star_self_eq_zero + - Matrix.self_mul_conjTranspose_mul_eq_zero + - Matrix.self_mul_conjTranspose_mulVec_eq_zero + - Matrix.conjTranspose_mul_self_mul_eq_zero + - Matrix.dotProduct_self_eq_zero + - Matrix.vecMul_self_mul_conjTranspose_eq_zero + - Matrix.mul_self_mul_conjTranspose_eq_zero + - Matrix.vecMul_conjTranspose_mul_self_eq_zero + - Matrix.dotProduct_stdBasis_one + - path: Mathlib/LinearAlgebra/TensorProduct/Tower.lean + theorems: + - TensorProduct.AlgebraTensorModule.map_add_right + - Submodule.baseChange_top + - Submodule.baseChange_bot + - TensorProduct.AlgebraTensorModule.map_smul_right + - TensorProduct.AlgebraTensorModule.map_add_left + - Submodule.tmul_mem_baseChange_of_mem + - TensorProduct.AlgebraTensorModule.map_smul_left + - Submodule.baseChange_span + - path: Mathlib/NumberTheory/LSeries/Convergence.lean + theorems: + - LSeries.abscissaOfAbsConv_le_one_of_isBigO_one + - LSeries.abscissaOfAbsConv_le_of_le_const + - LSeries.abscissaOfAbsConv_le_of_isBigO_rpow + - LSeriesSummable.abscissaOfAbsConv_le + - LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable + - LSeriesSummable_of_abscissaOfAbsConv_lt_re + - LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re + - LSeries.abscissaOfAbsConv_le_of_le_const_mul_rpow + - path: Mathlib/Algebra/Group/Commute/Defs.lean + theorems: + - Commute.mul_inv_cancel + - Commute.mul_inv_cancel_assoc + - pow_sub_one_mul + - Commute.left_comm + - Commute.mul_mul_mul_comm + - Commute.right_comm + - Commute.mul_inv + - Commute.inv + - path: Mathlib/Data/Fin/Tuple/BubbleSortInduction.lean + theorems: + - Tuple.bubble_sort_induction' + - path: Mathlib/Algebra/Algebra/RestrictScalars.lean + theorems: + - RestrictScalars.addEquiv_symm_map_smul_smul + - path: Mathlib/Control/Fix.lean + theorems: + - Part.fix_def + - Part.fix_def' + - path: Mathlib/CategoryTheory/Subobject/Comma.lean + theorems: + - CategoryTheory.CostructuredArrow.lift_projectQuotient + - CategoryTheory.StructuredArrow.projectSubobject_factors + - CategoryTheory.StructuredArrow.lift_projectSubobject + - CategoryTheory.CostructuredArrow.projectQuotient_factors + - CategoryTheory.CostructuredArrow.unop_left_comp_ofMkLEMk_unop + - path: Mathlib/NumberTheory/FLT/Three.lean + theorems: + - fermatLastTheoremThree_case_1 + - cube_of_not_dvd + - cube_of_castHom_ne_zero + - path: Mathlib/Analysis/NormedSpace/BallAction.lean + theorems: + - ne_neg_of_mem_sphere + - path: Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean + theorems: + - GeneralizedContinuedFraction.convergents_succ + - GeneralizedContinuedFraction.of_convergence + - GeneralizedContinuedFraction.of_convergence_epsilon + - path: Mathlib/Data/Nat/Factorization/Root.lean + theorems: + - Nat.ceilRoot_zero_right + - Nat.ceilRoot_one_right + - Nat.floorRoot_one_left + - Nat.floorRoot_zero_right + - Nat.ceilRoot_pow_self + - Nat.floorRoot_eq_zero + - Nat.floorRoot_one_right + - Nat.floorRoot_pow_self + - Nat.floorRoot_zero_left + - Nat.ceilRoot_one_left + - Nat.dvd_pow_iff_ceilRoot_dvd + - Nat.ceilRoot_zero_left + - Nat.factorization_ceilRoot + - Nat.ceilRoot_ne_zero + - Nat.factorization_floorRoot + - Nat.floorRoot_def + - Nat.pow_dvd_iff_dvd_floorRoot + - Nat.ceilRoot_eq_zero + - Nat.ceilRoot_def + - path: Mathlib/Algebra/Divisibility/Basic.lean + theorems: + - dvd_mul_of_dvd_right + - pow_dvd_pow_of_dvd + - dvd_iff_exists_eq_mul_left + - pow_dvd_pow + - dvd_of_mul_left_dvd + - mul_dvd_mul + - dvd_of_eq + - dvd_mul + - Dvd.intro_left + - dvd_pow + - path: Mathlib/CategoryTheory/GradedObject.lean + theorems: + - CategoryTheory.GradedObject.mapMap_comp + - CategoryTheory.GradedObject.hom_ext + - CategoryTheory.GradedObject.congr_mapMap + - CategoryTheory.GradedObject.comapEq_trans + - CategoryTheory.GradedObject.mapMap_id + - CategoryTheory.GradedObject.eqToHom_proj + - "CategoryTheory.GradedObject.\u03B9_mapMap" + - CategoryTheory.GradedObject.eqToHom_apply + - CategoryTheory.GradedObject.CofanMapObjFun.inj_iso_hom + - "CategoryTheory.GradedObject.\u03B9MapObjOrZero_mapMap" + - "CategoryTheory.GradedObject.\u03B9_descMapObj" + - CategoryTheory.GradedObject.comapEq_symm + - CategoryTheory.GradedObject.isIso_of_isIso_apply + - "CategoryTheory.GradedObject.CofanMapObjFun.\u03B9MapObj_iso_inv" + - path: Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean + theorems: + - Mathlib.Tactic.BicategoryCoherence.bicategoricalComp_refl + - path: Mathlib/Data/Set/Intervals/Monotone.lean + theorems: + - StrictMonoOn.Iic_id_le + - strictMonoOn_Iic_of_lt_succ + - path: Mathlib/Data/Set/Functor.lean + theorems: + - Set.mem_of_mem_coe + - Set.mem_of_mem_image_val + - Set.coe_subset + - Set.image_image_val_eq_restrict_image + - Set.coe_eq_image_val + - Set.image_val_subset + - Set.image2_def + - path: Mathlib/Data/Int/Sqrt.lean + theorems: + - Int.sqrt_eq + - Int.exists_mul_self + - path: Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean + theorems: + - CategoryTheory.Limits.hasFiniteColimits_of_hasFiniteColimits_of_size + - CategoryTheory.Limits.hasFiniteWidePushouts_of_has_finite_limits + - CategoryTheory.Limits.hasFiniteLimits_of_hasFiniteLimits_of_size + - CategoryTheory.Limits.hasFiniteWidePullbacks_of_hasFiniteLimits + - path: Mathlib/Order/SuccPred/Relation.lean + theorems: + - transGen_of_succ_of_reflexive + - reflTransGen_of_succ_of_ge + - reflTransGen_of_succ_of_le + - path: Mathlib/Topology/Instances/Matrix.lean + theorems: + - Continuous.matrix_det + - Continuous.matrix_fromBlocks + - Matrix.blockDiagonal_tsum + - Matrix.conjTranspose_tsum + - Continuous.matrix_blockDiagonal' + - Matrix.diagonal_tsum + - Matrix.transpose_tsum + - Matrix.blockDiagonal'_tsum + - path: Mathlib/Algebra/Order/Group/MinMax.lean + theorems: + - min_div_div_left' + - abs_max_sub_max_le_abs + - max_div_div_left' + - max_inv_one + - max_div_div_right' + - min_div_div_right' + - abs_max_sub_max_le_max + - abs_min_sub_min_le_max + - max_sub_max_le_max + - path: Mathlib/GroupTheory/FreeAbelianGroup.lean + theorems: + - FreeAbelianGroup.map_id_apply + - FreeAbelianGroup.of_mul_of + - FreeAbelianGroup.lift_comp + - FreeAbelianGroup.map_comp + - FreeAbelianGroup.map_comp_apply + - FreeAbelianGroup.lift.add' + - FreeAbelianGroup.lift.of + - path: Mathlib/Data/Set/Pointwise/BigOperators.lean + theorems: + - Set.mem_fintype_prod + - Set.mem_finset_prod + - Set.image_finset_prod_pi + - Set.multiset_prod_mem_multiset_prod + - Set.list_prod_mem_list_prod + - Set.image_fintype_prod_pi + - Set.image_multiset_prod + - Set.list_prod_subset_list_prod + - Set.image_list_prod + - path: Mathlib/CategoryTheory/Preadditive/Generator.lean + theorems: + - CategoryTheory.Preadditive.isCoseparating_iff + - CategoryTheory.isSeparator_iff_faithful_preadditiveCoyonedaObj + - CategoryTheory.isCoseparator_iff_faithful_preadditiveYonedaObj + - CategoryTheory.Preadditive.isSeparator_iff + - CategoryTheory.isSeparator_iff_faithful_preadditiveCoyoneda + - CategoryTheory.Preadditive.isCoseparator_iff + - CategoryTheory.Preadditive.isSeparating_iff + - path: Mathlib/GroupTheory/Perm/Finite.lean + theorems: + - Equiv.Perm.Disjoint.isConj_mul + - Equiv.Perm.Disjoint.extendDomain + - Equiv.Perm.Disjoint.orderOf + - Equiv.Perm.perm_mapsTo_inl_iff_mapsTo_inr + - Equiv.Perm.isConj_of_support_equiv + - Equiv.Perm.support_pow_coprime + - Equiv.Perm.mem_sumCongrHom_range_of_perm_mapsTo_inl + - Equiv.Perm.perm_inv_mapsTo_of_mapsTo + - Equiv.Perm.perm_inv_on_of_perm_on_finset + - path: Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean + theorems: + - QuadraticForm.comp_tensorLId_eq + - QuadraticForm.comp_tensorRId_eq + - QuadraticForm.tmul_comp_tensorComm + - QuadraticForm.tmul_comp_tensorAssoc + - QuadraticForm.tmul_comp_tensorMap + - path: Mathlib/Topology/UniformSpace/Pi.lean + theorems: + - cauchy_pi_iff + - uniformContinuous_pi + - CompleteSpace.iInf + - Cauchy.pi + - cauchy_pi_iff' + - Pi.uniformSpace_comap_precomp' + - Pi.uniformSpace_eq + - Pi.uniformSpace_comap_restrict + - path: Mathlib/AlgebraicTopology/MooreComplex.lean + theorems: + - AlgebraicTopology.NormalizedMooreComplex.d_squared + - path: Mathlib/Deprecated/Submonoid.lean + theorems: + - Monoid.exists_list_of_mem_closure + - IsSubmonoid.pow_mem + - Monoid.image_closure + - Monoid.mem_closure_union_iff + - IsSubmonoid.multiset_prod_mem + - powers.mul_mem + - IsSubmonoid.finset_prod_mem + - Range.isSubmonoid + - Monoid.closure_subset + - Univ.isSubmonoid + - IsSubmonoid.list_prod_mem + - IsSubmonoid.image + - IsSubmonoid.preimage + - path: Mathlib/CategoryTheory/Monad/Algebra.lean + theorems: + - CategoryTheory.Monad.algebra_iso_of_iso + - CategoryTheory.Comonad.coalgebra_iso_of_iso + - path: Mathlib/Order/Filter/CountableInter.lean + theorems: + - EventuallyLE.countable_bUnion + - countable_bInter_mem + - EventuallyLE.countable_bInter + - Filter.countableGenerate_isGreatest + - Filter.mem_countableGenerate_iff + - eventually_countable_forall + - eventually_countable_ball + - Filter.le_countableGenerate_iff_of_countableInterFilter + - path: Mathlib/Data/FinEnum.lean + theorems: + - FinEnum.pi.mem_enum + - FinEnum.mem_toList + - FinEnum.nodup_toList + - FinEnum.mem_pi + - FinEnum.Finset.mem_enum + - path: Mathlib/CategoryTheory/Yoneda.lean + theorems: + - CategoryTheory.Functor.coreprW_app_hom + - CategoryTheory.Yoneda.obj_map_id + - CategoryTheory.yonedaEquiv_symm_map + - CategoryTheory.yonedaEquiv_yoneda_map + - CategoryTheory.yonedaEquiv_naturality + - CategoryTheory.Functor.reprW_app_hom + - path: Mathlib/Algebra/Order/Monoid/Defs.lean + theorems: + - one_le_mul_self_iff + - one_lt_mul_self_iff + - mul_self_lt_one_iff + - mul_self_le_one_iff + - path: Mathlib/Data/Finsupp/Fin.lean + theorems: + - Finsupp.tail_cons + - Finsupp.cons_tail + - Finsupp.cons_ne_zero_of_right + - Finsupp.cons_ne_zero_of_left + - Finsupp.cons_ne_zero_iff + - Finsupp.cons_support + - Finsupp.cons_zero_zero + - path: Mathlib/Analysis/Calculus/AffineMap.lean + theorems: + - ContinuousAffineMap.contDiff + - path: Mathlib/Data/Nat/Cast/NeZero.lean + theorems: + - NeZero.of_neZero_natCast + - path: Mathlib/Algebra/Order/Group/OrderIso.lean + theorems: + - OrderIso.mulLeft_symm + - OrderIso.mulRight_symm + - path: Mathlib/Data/Sym/Card.lean + theorems: + - Sym.card_sym_eq_multichoose + - Sym2.card_subtype_diag + - Sym2.card_subtype_not_diag + - Sym2.two_mul_card_image_offDiag + - Sym.card_sym_fin_eq_multichoose + - Sym2.card_image_offDiag + - Sym2.card_image_diag + - Sym.card_sym_eq_choose + - path: Mathlib/LinearAlgebra/Matrix/Reindex.lean + theorems: + - Matrix.reindexLinearEquiv_comp + - Matrix.reindexLinearEquiv_trans + - path: Mathlib/GroupTheory/GroupAction/FixedPoints.lean + theorems: + - MulAction.fixedBy_mem_fixedBy_of_commute + - MulAction.smul_fixedBy + - MulAction.smul_mem_fixedBy_iff_mem_fixedBy + - MulAction.fixedBy_mul + - MulAction.smul_inv_mem_fixedBy_iff_mem_fixedBy + - MulAction.movedBy_mem_fixedBy_of_commute + - MulAction.set_mem_fixedBy_of_movedBy_subset + - MulAction.fixedBy_eq_univ_iff_eq_one + - MulAction.set_mem_fixedBy_of_subset_fixedBy + - MulAction.not_commute_of_disjoint_movedBy_preimage + - MulAction.set_mem_fixedBy_iff + - MulAction.fixedBy_subset_fixedBy_zpow + - MulAction.fixedBy_inv + - path: Mathlib/Algebra/Category/ModuleCat/Projective.lean + theorems: + - IsProjective.iff_projective + - path: Mathlib/Analysis/Fourier/FourierTransformDeriv.lean + theorems: + - VectorFourier.InnerProductSpace.hasFDerivAt_fourier + - VectorFourier.norm_fderiv_fourier_transform_integrand_right + - VectorFourier.hasFDerivAt_fourier_transform_integrand_right + - hasDerivAt_fourierIntegral + - VectorFourier.hasFDerivAt_fourier + - VectorFourier.norm_fderiv_fourier_transform_integrand_right_le + - path: Mathlib/Topology/Category/Stonean/EffectiveEpi.lean + theorems: + - CategoryTheory.EffectiveEpiFamily.toCompHaus + - Stonean.effectiveEpiFamily_tfae + - Stonean.effectiveEpi_tfae + - path: Mathlib/CategoryTheory/FintypeCat.lean + theorems: + - FintypeCat.hom_ext + - FintypeCat.Skeleton.is_skeletal + - FintypeCat.Skeleton.incl_mk_nat_card + - path: Mathlib/Algebra/DirectSum/Ring.lean + theorems: + - DirectSum.ofList_dProd + - DirectSum.one_mul + - DirectSum.mul_eq_dfinsupp_sum + - DirectSum.ofPow + - DirectSum.of_zero_pow + - DirectSum.mul_comm + - DirectSum.mulHom_of_of + - DirectSum.mul_one + - DirectSum.mul_eq_sum_support_ghas_mul + - DirectSum.of_intCast + - DirectSum.mul_assoc + - DirectSum.list_prod_ofFn_of_eq_dProd + - path: Mathlib/Computability/Encoding.lean + theorems: + - Computability.Encoding.encode_injective + - Computability.decode_encodePosNum + - Computability.decode_encodeNum + - Computability.decode_encodeNat + - path: Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean + theorems: + - GeneralizedContinuedFraction.of_correctness_atTop_of_terminates + - GeneralizedContinuedFraction.of_correctness_of_nth_stream_eq_none + - GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some + - GeneralizedContinuedFraction.compExactValue_correctness_of_stream_eq_some_aux_comp + - path: Mathlib/Analysis/LocallyConvex/Polar.lean + theorems: + - LinearMap.polar_zero + - LinearMap.subset_bipolar + - LinearMap.polar_weak_closed + - LinearMap.zero_mem_polar + - LinearMap.polar_univ + - LinearMap.polar_eq_iInter + - path: Mathlib/Analysis/Convex/Measure.lean + theorems: + - Convex.addHaar_frontier + - path: Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean + theorems: + - CategoryTheory.CommSq.left_adjoint + - CategoryTheory.CommSq.left_adjoint_hasLift_iff + - CategoryTheory.CommSq.right_adjoint + - CategoryTheory.Adjunction.hasLiftingProperty_iff + - CategoryTheory.CommSq.right_adjoint_hasLift_iff + - path: Mathlib/Analysis/InnerProductSpace/Symmetric.lean + theorems: + - LinearMap.IsSymmetric.inner_map_self_eq_zero + - LinearMap.IsSymmetric.coe_reApplyInnerSelf_apply + - LinearMap.IsSymmetric.continuous + - LinearMap.IsSymmetric.add + - LinearMap.IsSymmetric.restrictScalars + - LinearMap.IsSymmetric.inner_map_polarization + - LinearMap.isSymmetric_iff_inner_map_self_real + - path: Mathlib/RingTheory/Polynomial/RationalRoot.lean + theorems: + - den_dvd_of_is_root + - num_isRoot_scaleRoots_of_aeval_eq_zero + - num_dvd_of_is_root + - scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero + - path: Mathlib/Control/Bitraversable/Lemmas.lean + theorems: + - Bitraversable.comp_tsnd + - Bitraversable.tfst_tsnd + - Bitraversable.tsnd_eq_snd_id + - Bitraversable.tfst_eq_fst_id + - Bitraversable.comp_tfst + - Bitraversable.tsnd_tfst + - path: Mathlib/LinearAlgebra/CrossProduct.lean + theorems: + - cross_anticomm + - cross_anticomm' + - dot_self_cross + - leibniz_cross + - cross_self + - dot_cross_self + - triple_product_permutation + - cross_dot_cross + - path: Mathlib/LinearAlgebra/InvariantBasisNumber.lean + theorems: + - card_le_of_injective' + - strongRankCondition_iff_succ + - card_le_of_surjective' + - card_le_of_injective + - card_le_of_surjective + - nontrivial_of_invariantBasisNumber + - path: Mathlib/Algebra/Category/ModuleCat/EpiMono.lean + theorems: + - ModuleCat.mono_iff_injective + - ModuleCat.epi_iff_surjective + - ModuleCat.mono_iff_ker_eq_bot + - path: Mathlib/Order/ConditionallyCompleteLattice/Finset.lean + theorems: + - Set.Nonempty.csSup_mem + - Finset.sup'_eq_csSup_image + - Finset.Nonempty.csSup_mem + - Finset.sup'_univ_eq_ciSup + - Finset.sup'_id_eq_csSup + - Finset.inf'_univ_eq_ciInf + - path: Mathlib/Topology/Sets/Compacts.lean + theorems: + - TopologicalSpace.Compacts.coe_finset_sup + - path: Mathlib/Data/Rat/Cast/Lemmas.lean + theorems: + - Rat.cast_inv_nat + - Rat.cast_inv_int + - path: Mathlib/Topology/MetricSpace/ShrinkingLemma.lean + theorems: + - exists_subset_iUnion_ball_radius_pos_lt + - exists_locallyFinite_subset_iUnion_ball_radius_lt + - exists_subset_iUnion_ball_radius_lt + - path: Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean + theorems: + - CliffordAlgebra.ofBaseChange_comp_toBaseChange + - CliffordAlgebra.toBaseChange_comp_involute + - CliffordAlgebra.ofBaseChange_tmul_one + - "CliffordAlgebra.ofBaseChange_tmul_\u03B9" + - CliffordAlgebra.toBaseChange_comp_reverseOp + - CliffordAlgebra.toBaseChange_comp_ofBaseChange + - CliffordAlgebra.toBaseChange_reverse + - path: Mathlib/NumberTheory/SumFourSquares.lean + theorems: + - Nat.Prime.sum_four_squares + - Int.lt_of_sum_four_squares_eq_mul + - euler_four_squares + - Nat.sum_four_squares + - Nat.euler_four_squares + - Nat.sum_four_squares_of_two_mul_sum_four_squares + - Int.sq_add_sq_of_two_mul_sq_add_sq + - path: Mathlib/Analysis/NormedSpace/Basic.lean + theorems: + - NormedSpace.noncompactSpace + - norm_zsmul + - NormedSpace.cobounded_neBot + - norm_algebraMap' + - algebraMap_isometry + - eventually_nhds_norm_smul_sub_lt + - norm_algebraMap + - NormedSpace.exists_lt_norm + - path: Mathlib/Algebra/Group/Opposite.lean + theorems: + - MulOpposite.semiconjBy_op + - MulOpposite.op_div + - MulOpposite.semiconjBy_unop + - path: Mathlib/NumberTheory/Basic.lean + theorems: + - dvd_sub_pow_of_dvd_sub + - path: Mathlib/Topology/Algebra/Module/CharacterSpace.lean + theorems: + - WeakDual.CharacterSpace.union_zero + - WeakDual.CharacterSpace.isClosed + - WeakDual.CharacterSpace.eq_set_map_one_map_mul + - WeakDual.CharacterSpace.ext_ker + - WeakDual.CharacterSpace.union_zero_isClosed + - path: Mathlib/RingTheory/Polynomial/Tower.lean + theorems: + - Polynomial.eval_map_algebraMap + - Polynomial.aeval_map_algebraMap + - Polynomial.aeval_algebraMap_apply + - Polynomial.aeval_algebraMap_eq_zero_iff_of_injective + - Polynomial.aeval_algebraMap_eq_zero_iff + - path: Mathlib/LinearAlgebra/UnitaryGroup.lean + theorems: + - Matrix.mem_orthogonalGroup_iff' + - Matrix.mem_unitaryGroup_iff' + - Matrix.mem_unitaryGroup_iff + - Matrix.UnitaryGroup.toGL_mul + - Matrix.det_of_mem_unitary + - Matrix.UnitaryGroup.toGL_one + - Matrix.mem_orthogonalGroup_iff + - path: Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean + theorems: + - AlgebraicClosureAux.ofStep_succ + - AlgebraicClosure.toStepSucc.exists_root + - AlgebraicClosure.toStepOfLE'.succ + - AlgebraicClosure.AdjoinMonic.isIntegral + - AlgebraicClosure.toSplittingField_evalXSelf + - AlgebraicClosure.AdjoinMonic.exists_root + - AlgebraicClosure.spanEval_ne_top + - AlgebraicClosureAux.exists_root + - AlgebraicClosure.toStepOfLE.succ + - path: Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean + theorems: + - MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure + - MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure + - MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd + - path: Mathlib/Order/WellFounded.lean + theorems: + - WellFounded.lt_succ_iff + - WellFounded.wellFounded_iff_has_min + - WellFounded.self_le_of_strictMono + - WellFounded.eq_strictMono_iff_eq_range + - WellFounded.lt_succ + - path: Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean + theorems: + - CategoryTheory.effectiveEpiFamilyStructOfEffectiveEpiDesc_aux + - path: Mathlib/Data/Set/Pointwise/Support.lean + theorems: + - support_comp_inv_smul + - "support_comp_inv_smul\u2080" + - mulSupport_comp_inv_smul + - "mulSupport_comp_inv_smul\u2080" + - path: Mathlib/Tactic/NormNum/Result.lean + theorems: + - Mathlib.Meta.NormNum.IsRat.of_raw + - Mathlib.Meta.NormNum.IsRat.neg_to_eq + - Mathlib.Meta.NormNum.IsNat.to_isInt + - Mathlib.Meta.NormNum.IsInt.to_isNat + - Mathlib.Meta.NormNum.IsRat.to_isInt + - Mathlib.Meta.NormNum.IsInt.to_isRat + - Mathlib.Meta.NormNum.IsRat.nonneg_to_eq + - Mathlib.Meta.NormNum.IsRat.to_raw_eq + - Mathlib.Meta.NormNum.IsInt.neg_to_eq + - Mathlib.Meta.NormNum.IsNat.to_isRat + - path: Mathlib/Order/Circular.lean + theorems: + - Set.compl_cIcc + - Set.compl_cIoo + - sbtw_iff_not_btw + - path: Mathlib/Algebra/CharP/Algebra.lean + theorems: + - RingHom.charP + - charZero_of_injective_ringHom + - charP_of_injective_ringHom + - Algebra.ringChar_eq + - path: Mathlib/Topology/Category/Profinite/Product.lean + theorems: + - "Profinite.IndexFunctor.eq_of_forall_\u03C0_app_eq" + - "Profinite.IndexFunctor.surjective_\u03C0_app" + - path: Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean + theorems: + - AbsoluteValue.exists_partition_int + - path: Mathlib/Probability/ProbabilityMassFunction/Basic.lean + theorems: + - PMF.toMeasure_toPMF + - PMF.toOuterMeasure_apply_inter_support + - PMF.toOuterMeasure_apply_singleton + - PMF.toOuterMeasure_apply_finset + - PMF.apply_eq_one_iff + - PMF.toOuterMeasure_caratheodory + - MeasureTheory.Measure.toPMF_toMeasure + - PMF.toMeasure_apply_eq_of_inter_support_eq + - PMF.restrict_toMeasure_support + - PMF.coe_le_one + - PMF.toOuterMeasure_apply_eq_one_iff + - PMF.toMeasure_mono + - PMF.toPMF_eq_iff_toMeasure_eq + - PMF.toMeasure_apply_inter_support + - PMF.toMeasure_eq_iff_eq_toPMF + - PMF.apply_eq_zero_iff + - PMF.toMeasure_apply_singleton + - PMF.toMeasure_apply_eq_zero_iff + - PMF.toOuterMeasure_apply_eq_zero_iff + - path: Mathlib/Order/Filter/ModEq.lean + theorems: + - Nat.frequently_even + - Nat.frequently_odd + - Nat.frequently_mod_eq + - path: Mathlib/Algebra/MonoidAlgebra/Grading.lean + theorems: + - AddMonoidAlgebra.decomposeAux_single + - AddMonoidAlgebra.decomposeAux_coe + - AddMonoidAlgebra.mem_gradeBy_iff + - AddMonoidAlgebra.mem_grade_iff + - AddMonoidAlgebra.mem_grade_iff' + - AddMonoidAlgebra.single_mem_gradeBy + - path: Mathlib/Analysis/NormedSpace/Star/Matrix.lean + theorems: + - Matrix.l2_opNorm_conjTranspose + - Matrix.l2_opNorm_conjTranspose_mul_self + - entry_norm_bound_of_unitary + - Matrix.l2_opNorm_mul + - entrywise_sup_norm_bound_of_unitary + - path: Mathlib/Data/Nat/Periodic.lean + theorems: + - Nat.periodic_coprime + - Nat.periodic_gcd + - Nat.filter_multiset_Ico_card_eq_of_periodic + - Nat.periodic_mod + - Function.Periodic.map_mod_nat + - path: Mathlib/CategoryTheory/PathCategory.lean + theorems: + - CategoryTheory.Paths.lift_toPath + - CategoryTheory.composePath_comp + - CategoryTheory.Paths.ext_functor + - CategoryTheory.Paths.lift_unique + - CategoryTheory.Paths.lift_spec + - path: Mathlib/Combinatorics/SimpleGraph/Girth.lean + theorems: + - SimpleGraph.girth_bot + - SimpleGraph.exists_girth_eq_length + - SimpleGraph.girth_anti + - SimpleGraph.le_girth + - SimpleGraph.girth_eq_top + - path: Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean + theorems: + - "CategoryTheory.InjectiveResolution.\u03B9_f_zero_comp_complex_d" + - CategoryTheory.InjectiveResolution.complex_d_comp + - CategoryTheory.InjectiveResolution.cocomplex_exactAt_succ + - CategoryTheory.InjectiveResolution.exact_succ + - "CategoryTheory.InjectiveResolution.\u03B9_f_succ" + - path: Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean + theorems: + - IsBoundedBilinearMap.hasStrictFDerivAt + - IsBoundedBilinearMap.fderivWithin + - path: Mathlib/NumberTheory/Primorial.lean + theorems: + - primorial_le_4_pow + - primorial_succ + - primorial_add_dvd + - primorial_add + - path: Mathlib/CategoryTheory/Shift/Opposite.lean + theorems: + - CategoryTheory.oppositeShiftFunctorAdd'_hom_app + - CategoryTheory.oppositeShiftFunctorAdd'_inv_app + - CategoryTheory.oppositeShiftFunctorAdd_hom_app + - CategoryTheory.oppositeShiftFunctorZero_hom_app + - path: Mathlib/Algebra/Homology/HomologicalBicomplex.lean + theorems: + - "HomologicalComplex\u2082.d_f_comp_d_f" + - "HomologicalComplex\u2082.shape_f" + - path: Mathlib/SetTheory/Game/Domineering.lean + theorems: + - SetTheory.PGame.Domineering.snd_pred_mem_erase_of_mem_left + - SetTheory.PGame.Domineering.moveLeft_smaller + - SetTheory.PGame.Domineering.moveRight_card + - SetTheory.PGame.Domineering.card_of_mem_right + - SetTheory.PGame.Domineering.moveLeft_card + - SetTheory.PGame.Domineering.moveRight_smaller + - SetTheory.PGame.Domineering.card_of_mem_left + - SetTheory.PGame.Domineering.fst_pred_mem_erase_of_mem_right + - path: Mathlib/Logic/Godel/GodelBetaFunction.lean + theorems: + - Nat.coprimes_lt + - Nat.coprime_mul_succ + - Nat.pairwise_coprime_coprimes + - Nat.beta_unbeta_coe + - path: .lake/packages/std/Std/Data/HashMap/Lemmas.lean + theorems: + - Std.HashMap.Imp.empty_buckets + - Std.HashMap.empty_find? + - path: Mathlib/MeasureTheory/Function/Floor.lean + theorems: + - Nat.measurable_ceil + - Int.measurable_floor + - MeasurableSet.image_fract + - measurable_fract + - Nat.measurable_floor + - Int.measurable_ceil + - path: Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean + theorems: + - DedekindDomain.ProdAdicCompletions.IsFiniteAdele.add + - DedekindDomain.ProdAdicCompletions.IsFiniteAdele.one + - DedekindDomain.ProdAdicCompletions.IsFiniteAdele.mul + - DedekindDomain.ProdAdicCompletions.IsFiniteAdele.neg + - DedekindDomain.ProdAdicCompletions.IsFiniteAdele.zero + - path: Mathlib/Analysis/NormedSpace/FunctionSeries.lean + theorems: + - continuousOn_tsum + - continuous_tsum + - tendstoUniformly_tsum + - tendstoUniformlyOn_tsum + - path: Mathlib/Algebra/Category/MonCat/Colimits.lean + theorems: + - MonCat.Colimits.cocone_naturality + - MonCat.Colimits.cocone_naturality_components + - path: Mathlib/Topology/Order/Priestley.lean + theorems: + - exists_isClopen_upper_or_lower_of_ne + - path: Mathlib/Probability/Process/PartitionFiltration.lean + theorems: + - ProbabilityTheory.measurable_memPartitionSet_subtype + - ProbabilityTheory.iSup_partitionFiltration + - ProbabilityTheory.iSup_partitionFiltration_eq_generateFrom_range + - ProbabilityTheory.iSup_countableFiltration + - path: Mathlib/CategoryTheory/Localization/Resolution.lean + theorems: + - CategoryTheory.LocalizerMorphism.isIso_iff_of_hasRightResolutions + - CategoryTheory.LocalizerMorphism.essSurj_of_hasRightResolutions + - path: Mathlib/LinearAlgebra/DirectSum/Finsupp.lean + theorems: + - finsuppTensorFinsupp'_single_tmul_single + - finsuppTensorFinsupp'_apply_apply + - finsuppTensorFinsupp_apply + - finsuppTensorFinsupp_single + - path: Mathlib/Data/Fintype/Sum.lean + theorems: + - Finset.exists_equiv_extend_of_card_eq + - infinite_sum + - Fintype.card_subtype_or + - image_subtype_ne_univ_eq_image_erase + - Set.MapsTo.exists_equiv_extend_of_card_eq + - Fintype.card_subtype_or_disjoint + - image_subtype_univ_ssubset_image_univ + - path: Mathlib/RingTheory/Ideal/LocalRing.lean + theorems: + - LocalRing.of_unique_nonzero_prime + - LocalRing.local_hom_TFAE + - LocalRing.le_maximalIdeal + - LocalRing.of_surjective' + - LocalRing.jacobson_eq_maximalIdeal + - LocalRing.of_surjective + - RingHom.domain_localRing + - LocalRing.isUnit_or_isUnit_of_isUnit_add + - LocalRing.surjective_units_map_of_local_ringHom + - path: Mathlib/Data/Char.lean + theorems: + - Char.utf8Size_pos + - Char.ofNat_toNat + - path: Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean + theorems: + - AlgebraicTopology.AlternatingCofaceMapComplex.d_eq_unop_d + - AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq + - AlgebraicTopology.karoubi_alternatingFaceMapComplex_d + - AlgebraicTopology.AlternatingCofaceMapComplex.d_squared + - AlgebraicTopology.AlternatingFaceMapComplex.d_squared + - AlgebraicTopology.map_alternatingFaceMapComplex + - AlgebraicTopology.inclusionOfMooreComplexMap_f + - path: Mathlib/RepresentationTheory/Invariants.lean + theorems: + - Representation.averageMap_invariant + - Representation.mem_invariants + - GroupAlgebra.mul_average_right + - Representation.invariants_eq_inter + - Representation.averageMap_id + - GroupAlgebra.mul_average_left + - Representation.linHom.mem_invariants_iff_comm + - path: Mathlib/MeasureTheory/Function/Egorov.lean + theorems: + - MeasureTheory.Egorov.iUnionNotConvergentSeq_subset + - MeasureTheory.Egorov.tendstoUniformlyOn_diff_iUnionNotConvergentSeq + - MeasureTheory.Egorov.mem_notConvergentSeq_iff + - MeasureTheory.tendstoUniformlyOn_of_ae_tendsto' + - MeasureTheory.Egorov.measure_iUnionNotConvergentSeq + - MeasureTheory.Egorov.exists_notConvergentSeq_lt + - MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero + - MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero + - path: Mathlib/CategoryTheory/Limits/Filtered.lean + theorems: + - CategoryTheory.IsFiltered.iff_nonempty_limit + - CategoryTheory.IsCofiltered.iff_nonempty_limit + - path: Mathlib/Probability/Process/Adapted.lean + theorems: + - MeasureTheory.Filtration.adapted_natural + - MeasureTheory.ProgMeasurable.comp + - MeasureTheory.progMeasurable_of_tendsto' + - MeasureTheory.ProgMeasurable.adapted + - MeasureTheory.ProgMeasurable.finset_prod + - path: Mathlib/NumberTheory/FLT/Four.lean + theorems: + - fermatLastTheoremFour + - Fermat42.exists_minimal + - not_fermat_42 + - Fermat42.mul + - Fermat42.ne_zero + - Fermat42.exists_odd_minimal + - Fermat42.neg_of_minimal + - Fermat42.coprime_of_minimal + - Fermat42.not_minimal + - FermatLastTheorem.of_odd_primes + - Int.coprime_of_sq_sum + - Int.coprime_of_sq_sum' + - path: Mathlib/Analysis/LocallyConvex/AbsConvex.lean + theorems: + - with_gaugeSeminormFamily + - nhds_basis_abs_convex_open + - nhds_basis_abs_convex + - path: Mathlib/Data/Polynomial/DenomsClearable.lean + theorems: + - one_le_pow_mul_abs_eval_div + - DenomsClearable.add + - denomsClearable_zero + - denomsClearable_C_mul_X_pow + - path: Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean + theorems: + - TopCat.Presheaf.SheafConditionEqualizerProducts.w + - path: Mathlib/Data/PFunctor/Multivariate/M.lean + theorems: + - MvPFunctor.M.bisim_lemma + - MvPFunctor.M.bisim' + - MvPFunctor.M.dest_map + - "MvPFunctor.M.bisim\u2080" + - MvPFunctor.M.dest_corec + - MvPFunctor.M.map_dest + - MvPFunctor.M.bisim + - path: Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean + theorems: + - ConformalAt.comp + - conformalAt_iff_isConformalMap_fderiv + - path: Mathlib/Order/Antisymmetrization.lean + theorems: + - antisymmetrization_fibration + - path: Mathlib/Analysis/SpecificLimits/IsROrC.lean + theorems: + - IsROrC.tendsto_inverse_atTop_nhds_zero_nat + - path: Mathlib/Data/Bool/AllAny.lean + theorems: + - List.any_iff_exists + - List.all_iff_forall + - List.all_iff_forall_prop + - List.any_iff_exists_prop + - path: Mathlib/Data/DFinsupp/Order.lean + theorems: + - DFinsupp.subset_support_tsub + - DFinsupp.disjoint_iff + - DFinsupp.add_eq_zero_iff + - DFinsupp.single_le_iff + - DFinsupp.support_monotone + - DFinsupp.single_tsub + - DFinsupp.coe_tsub + - DFinsupp.support_inf_union_support_sup + - DFinsupp.support_inf + - DFinsupp.support_sup + - path: Mathlib/Topology/MetricSpace/Algebra.lean + theorems: + - lipschitz_with_lipschitz_const_mul + - path: Mathlib/Data/Matroid/IndepAxioms.lean + theorems: + - Matroid.existsMaximalSubsetProperty_of_bdd + - IndepMatroid.ofFinset_indep + - IndepMatroid.ofFinset_indep' + - IndepMatroid.matroid_indep_iff + - path: Mathlib/Analysis/NormedSpace/CompactOperator.lean + theorems: + - IsCompactOperator.continuous_comp + - isCompactOperator_iff_exists_mem_nhds_isCompact_closure_image + - IsCompactOperator.image_subset_compact_of_isVonNBounded + - IsCompactOperator.image_subset_compact_of_bounded + - IsCompactOperator.isCompact_closure_image_of_bounded + - isClosed_setOf_isCompactOperator + - IsCompactOperator.comp_clm + - IsCompactOperator.sub + - path: Mathlib/Algebra/Jordan/Basic.lean + theorems: + - aux2 + - aux0 + - aux3 + - aux1 + - two_nsmul_lie_lmul_lmul_add_add_eq_zero + - two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add + - path: Mathlib/NumberTheory/LegendreSymbol/GaussSum.lean + theorems: + - MulChar.IsQuadratic.gaussSum_frob_iter + - Char.card_pow_char_pow + - gaussSum_mul_gaussSum_eq_card + - gaussSum_mulShift + - gaussSum_sq + - MulChar.IsQuadratic.gaussSum_frob + - gaussSum_frob + - gaussSum_mul_aux + - FiniteField.two_pow_card + - Char.card_pow_card + - path: Mathlib/Computability/NFA.lean + theorems: + - NFA.mem_stepSet + - DFA.toNFA_correct + - NFA.stepSet_empty + - NFA.mem_accepts + - NFA.evalFrom_append_singleton + - NFA.toDFA_correct + - DFA.toNFA_evalFrom_match + - NFA.pumping_lemma + - path: Mathlib/Data/Multiset/Range.lean + theorems: + - Multiset.range_add_eq_union + - Multiset.range_disjoint_map_add + - Multiset.range_succ + - path: Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean + theorems: + - CategoryTheory.Sieve.effectiveEpimorphic_singleton + - CategoryTheory.Sieve.effectiveEpimorphic_family + - CategoryTheory.Sieve.generateSingleton_eq + - CategoryTheory.Sieve.generateFamily_eq + - path: Mathlib/AlgebraicGeometry/Sites/BigZariski.lean + theorems: + - AlgebraicGeometry.Scheme.zariskiTopology_openCover + - path: Mathlib/Algebra/Lie/Character.lean + theorems: + - LieAlgebra.lieCharacter_apply_lie + - LieAlgebra.lieCharacter_apply_of_mem_derived + - LieAlgebra.lieCharacter_apply_lie' + - path: Mathlib/Deprecated/Subring.lean + theorems: + - Ring.exists_list_of_mem_closure + - Ring.InClosure.recOn + - Ring.image_closure + - path: Mathlib/Combinatorics/Derangements/Basic.lean + theorems: + - derangements.Equiv.RemoveNone.fiber_some + - derangements.Equiv.RemoveNone.mem_fiber + - path: Mathlib/Algebra/Group/AddChar.lean + theorems: + - AddChar.pow_mulShift + - AddChar.map_neg_inv + - AddChar.mulShift_zero + - AddChar.inv_apply' + - AddChar.coe_mk + - AddChar.mulShift_spec' + - AddChar.inv_mulShift + - AddChar.mulShift_mul + - path: Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean + theorems: + - ContMDiffSection.coe_zsmul + - ContMDiffSection.coe_nsmul + - path: Mathlib/CategoryTheory/Functor/Currying.lean + theorems: + - CategoryTheory.Functor.uncurry_obj_injective + - CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip' + - CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip + - CategoryTheory.Functor.flip_injective + - CategoryTheory.Functor.uncurry_obj_curry_obj + - CategoryTheory.Functor.curry_obj_injective + - CategoryTheory.Functor.curry_obj_uncurry_obj + - path: Mathlib/CategoryTheory/SingleObj.lean + theorems: + - CategoryTheory.SingleObj.inv_as_inv + - path: Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean + theorems: + - "AlgebraCat.forget\u2082_map_associator_inv" + - "AlgebraCat.forget\u2082_map_associator_hom" + - path: Mathlib/Data/Vector.lean + theorems: + - Vector.cons_head_tail + - Vector.toList_append + - Vector.toList_take + - Vector.toList_cons + - path: Mathlib/Topology/Metrizable/Uniformity.lean + theorems: + - UniformSpace.metrizable_uniformity + - PseudoMetricSpace.dist_ofPreNNDist_le + - PseudoMetricSpace.le_two_mul_dist_ofPreNNDist + - UniformSpace.metrizableSpace + - path: Mathlib/Algebra/Homology/Single.lean + theorems: + - HomologicalComplex.from_single_hom_ext + - HomologicalComplex.mkHomFromSingle_f + - "CochainComplex.toSingle\u2080Equiv_symm_apply_f_succ" + - HomologicalComplex.isZero_single_obj_X + - HomologicalComplex.to_single_hom_ext + - "CochainComplex.toSingle\u2080Equiv_symm_apply_f_zero" + - HomologicalComplex.mkHomToSingle_f + - "ChainComplex.fromSingle\u2080Equiv_symm_apply_f_zero" + - "CochainComplex.single\u2080_map_f_zero" + - "ChainComplex.single\u2080_map_f_zero" + - "CochainComplex.fromSingle\u2080Equiv_symm_apply_f_zero" + - HomologicalComplex.single_map_f_self + - path: Mathlib/Algebra/QuaternionBasis.lean + theorems: + - QuaternionAlgebra.Basis.j_mul_k + - QuaternionAlgebra.Basis.lift_smul + - QuaternionAlgebra.Basis.k_mul_j + - QuaternionAlgebra.Basis.i_mul_k + - QuaternionAlgebra.Basis.k_mul_i + - QuaternionAlgebra.Basis.lift_add + - QuaternionAlgebra.Basis.lift_one + - QuaternionAlgebra.Basis.lift_zero + - QuaternionAlgebra.Basis.ext + - QuaternionAlgebra.Basis.lift_mul + - QuaternionAlgebra.Basis.k_mul_k + - path: Mathlib/Data/ZMod/Units.lean + theorems: + - ZMod.unitsMap_self + - ZMod.unitsMap_comp + - ZMod.unitsMap_surjective + - ZMod.not_isUnit_of_mem_primeFactors + - path: Mathlib/Data/Rat/Cast/Order.lean + theorems: + - Rat.cast_strictMono + - Rat.cast_nonneg + - Rat.cast_pos + - Rat.cast_lt_zero + - Rat.cast_nonpos + - path: Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean + theorems: + - CategoryTheory.Idempotents.KaroubiKaroubi.idem_f + - CategoryTheory.Idempotents.KaroubiKaroubi.p_comm_f + - path: Mathlib/FieldTheory/ChevalleyWarning.lean + theorems: + - char_dvd_card_solutions + - MvPolynomial.sum_eval_eq_zero + - char_dvd_card_solutions_of_add_lt + - char_dvd_card_solutions_of_sum_lt + - path: Mathlib/Combinatorics/Derangements/Exponential.lean + theorems: + - numDerangements_tendsto_inv_e + - path: Mathlib/Data/Rat/Floor.lean + theorems: + - Rat.ceil_cast + - Rat.floor_int_div_nat_eq_div + - Rat.fract_inv_num_lt_num_of_pos + - Rat.num_lt_succ_floor_mul_den + - Int.mod_nat_eq_sub_mul_floor_rat_div + - Rat.cast_fract + - Nat.coprime_sub_mul_floor_rat_div_of_coprime + - Rat.le_floor + - Rat.round_cast + - path: Mathlib/Algebra/Order/Support.lean + theorems: + - Set.mulIndicator_iUnion_apply + - Set.mulIndicator_apply_le' + - Function.mulSupport_iSup + - Set.mulIndicator_iInter_apply + - Set.indicator_le_indicator_nonneg + - path: Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean + theorems: + - ContinuousMapZero.toContinuousMap_injective + - path: Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean + theorems: + - RingSubgroupsBasis.of_comm + - SubmodulesBasis.nonarchimedean + - SubmodulesRingBasis.toRing_subgroups_basis + - RingSubgroupsBasis.hasBasis_nhds + - RingFilterBasis.submodulesBasisIsBasis + - RingSubgroupsBasis.hasBasis_nhds_zero + - RingSubgroupsBasis.nonarchimedean + - path: Mathlib/CategoryTheory/Monad/Monadicity.lean + theorems: + - CategoryTheory.Monad.MonadicityInternal.comparisonAdjunction_counit_app + - CategoryTheory.Monad.MonadicityInternal.comparisonAdjunction_unit_f + - path: Mathlib/Topology/Algebra/WithZeroTopology.lean + theorems: + - WithZeroTopology.hasBasis_nhds_zero + - WithZeroTopology.isClosed_iff + - WithZeroTopology.singleton_mem_nhds_of_ne_zero + - WithZeroTopology.singleton_mem_nhds_of_units + - WithZeroTopology.tendsto_of_ne_zero + - WithZeroTopology.hasBasis_nhds_of_ne_zero + - WithZeroTopology.Iio_mem_nhds + - WithZeroTopology.isOpen_iff + - WithZeroTopology.nhds_zero + - WithZeroTopology.nhds_eq_update + - WithZeroTopology.tendsto_zero + - path: Mathlib/Logic/Hydra.lean + theorems: + - Relation.cutExpand_singleton_singleton + - Relation.not_cutExpand_zero + - Relation.cutExpand_iff + - Relation.acc_of_singleton + - Relation.cutExpand_le_invImage_lex + - Acc.cutExpand + - path: Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean + theorems: + - MeasureTheory.snorm_add_lt_top + - "MeasureTheory.mem\u2112p_finset_sum'" + - MeasureTheory.snorm_add_le' + - MeasureTheory.snormEssSup_add_le + - MeasureTheory.exists_Lp_half + - MeasureTheory.LpAddConst_zero + - "MeasureTheory.mem\u2112p_finset_sum" + - MeasureTheory.LpAddConst_lt_top + - MeasureTheory.snorm_add_le + - MeasureTheory.snorm_sub_le + - MeasureTheory.snorm_sub_le' + - "MeasureTheory.Mem\u2112p.sub" + - MeasureTheory.snorm'_add_le + - MeasureTheory.snorm'_add_le_of_le_one + - path: Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean + theorems: + - LinearMap.exists_map_addHaar_eq_smul_addHaar' + - LinearMap.exists_map_addHaar_eq_smul_addHaar + - MeasureTheory.ae_comp_linearMap_mem_iff + - path: Mathlib/Tactic/Congr!.lean + theorems: + - implies_congr' + - heq_imp_of_eq_imp + - eq_imp_of_iff_imp + - path: Mathlib/CategoryTheory/Subterminal.lean + theorems: + - CategoryTheory.isSubterminal_of_isIso_diag + - CategoryTheory.isSubterminal_of_mono_terminal_from + - CategoryTheory.IsSubterminal.isIso_diag + - CategoryTheory.isSubterminal_of_mono_isTerminal_from + - path: Mathlib/Algebra/Order/Field/Pi.lean + theorems: + - Pi.exists_forall_pos_add_lt + - path: Mathlib/ModelTheory/Ultraproducts.lean + theorems: + - FirstOrder.Language.Ultraproduct.realize_formula_cast + - FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast + - FirstOrder.Language.Ultraproduct.sentence_realize + - FirstOrder.Language.Ultraproduct.term_realize_cast + - FirstOrder.Language.Ultraproduct.funMap_cast + - path: Mathlib/Algebra/Category/Ring/Basic.lean + theorems: + - CategoryTheory.Iso.commRingIsoToRingEquiv_toRingHom + - CategoryTheory.Iso.commRingIsoToRingEquiv_symm_toRingHom + - path: Mathlib/Algebra/Order/Ring/Canonical.lean + theorems: + - CanonicallyOrderedCommSemiring.mul_pos + - CanonicallyOrderedCommSemiring.mul_lt_mul_of_lt_of_lt + - AddLECancellable.mul_tsub + - AddLECancellable.tsub_mul + - path: Mathlib/Data/Real/Pi/Bounds.lean + theorems: + - Real.pi_gt_31415 + - Real.pi_gt_3141592 + - Real.pi_lower_bound_start + - Real.sqrtTwoAddSeries_step_down + - Real.pi_lt_sqrtTwoAddSeries + - Real.pi_gt_sqrtTwoAddSeries + - Real.pi_gt_three + - Real.sqrtTwoAddSeries_step_up + - Real.pi_lt_3141593 + - Real.pi_lt_315 + - Real.pi_gt_314 + - Real.pi_lt_31416 + - Real.pi_upper_bound_start + - path: Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean + theorems: + - CategoryTheory.preserves_mono_of_preservesLimit + - CategoryTheory.preserves_epi_of_preservesColimit + - CategoryTheory.reflects_mono_of_reflectsLimit + - CategoryTheory.reflects_epi_of_reflectsColimit + - path: Mathlib/Topology/Instances/Int.lean + theorems: + - Int.ball_eq_Ioo + - Int.cobounded_eq + - Int.pairwise_one_le_dist + - Int.cofinite_eq + - Int.closedBall_eq_Icc + - Int.dist_eq' + - path: .lake/packages/std/Std/Data/Sum/Basic.lean + theorems: + - Sum.lex_inl_inl + - Sum.liftRel_inl_inl + - Sum.liftRel_inr_inr + - Sum.lex_inr_inr + - path: Mathlib/RingTheory/Ideal/IdempotentFG.lean + theorems: + - Ideal.isIdempotentElem_iff_eq_bot_or_top + - Ideal.isIdempotentElem_iff_of_fg + - path: Mathlib/AlgebraicTopology/ExtraDegeneracy.lean + theorems: + - "CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_\u03C0_succ" + - SSet.Augmented.StandardSimplex.shiftFun_succ + - CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_base + - "CategoryTheory.Arrow.AugmentedCechNerve.ExtraDegeneracy.s_comp_\u03C0_0" + - path: Mathlib/Algebra/Category/GroupCat/Basic.lean + theorems: + - AddCommGroupCat.asHom_injective + - AddCommGroupCat.injective_of_mono + - path: Mathlib/Topology/Order/LeftRight.lean + theorems: + - nhds_left_sup_nhds_right' + - continuousAt_iff_continuous_left_right + - nhds_left_sup_nhds_right + - continuousAt_iff_continuous_left'_right' + - nhds_left'_sup_nhds_right + - continuousWithinAt_Ioi_iff_Ici + - nhds_left'_sup_nhds_right' + - IsAntichain.interior_eq_empty + - path: Mathlib/CategoryTheory/Closed/Monoidal.lean + theorems: + - CategoryTheory.MonoidalClosed.pre_id + - CategoryTheory.MonoidalClosed.pre_map + - CategoryTheory.MonoidalClosed.uncurry_id_eq_ev + - CategoryTheory.MonoidalClosed.curry_id_eq_coev + - CategoryTheory.MonoidalClosed.uncurry_pre + - CategoryTheory.MonoidalClosed.pre_comm_ihom_map + - path: Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean + theorems: + - "CategoryTheory.Limits.limit_\u03C0_isIso_of_is_strict_terminal" + - CategoryTheory.Limits.hasStrictInitialObjects_of_initial_is_strict + - CategoryTheory.Limits.hasStrictTerminalObjects_of_terminal_is_strict + - CategoryTheory.Limits.IsTerminal.strict_hom_ext + - CategoryTheory.Limits.IsInitial.strict_hom_ext + - path: Mathlib/Algebra/Order/Field/Defs.lean + theorems: + - inv_nonpos + - zpow_pos_of_pos + - inv_pos + - div_nonpos_of_nonneg_of_nonpos + - div_nonneg + - inv_nonneg + - div_nonpos_of_nonpos_of_nonneg + - inv_lt_zero + - div_pos + - zpow_nonneg + - path: Mathlib/Combinatorics/SetFamily/CauchyDavenport.lean + theorems: + - ZMod.min_le_card_add + - Finset.card_add_card_sub_one_le_card_mul + - wellFoundedOn_devosMulRel + - devosMulRel_iff + - Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul + - Finset.min_le_card_mul + - path: Mathlib/GroupTheory/Subgroup/Simple.lean + theorems: + - IsSimpleGroup.isSimpleGroup_of_surjective + - path: Mathlib/Data/Erased.lean + theorems: + - Erased.map_out + - Erased.out_inj + - Erased.mk_out + - path: Mathlib/AlgebraicTopology/CechNerve.lean + theorems: + - CategoryTheory.CechNerveTerminalFrom.wideCospan.limitIsoPi_hom_comp_pi + - path: Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean + theorems: + - AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.negMem' + - AlgebraicGeometry.HomogeneousLocalization.mem_basicOpen + - AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.mulMem' + - path: Mathlib/CategoryTheory/Monoidal/Limits.lean + theorems: + - "CategoryTheory.Limits.limLax_\u03BC" + - path: Mathlib/CategoryTheory/Abelian/RightDerived.lean + theorems: + - CategoryTheory.Functor.rightDerived_map_eq + - CategoryTheory.NatTrans.rightDerived_comp + - CategoryTheory.InjectiveResolution.rightDerived_app_eq + - CategoryTheory.InjectiveResolution.rightDerivedToHomotopyCategory_app_eq + - CategoryTheory.InjectiveResolution.toRightDerivedZero'_naturality + - CategoryTheory.InjectiveResolution.toRightDerivedZero_eq + - CategoryTheory.InjectiveResolution.isoRightDerivedObj_inv_naturality + - CategoryTheory.NatTrans.rightDerived_id + - CategoryTheory.InjectiveResolution.isoRightDerivedObj_hom_naturality + - CategoryTheory.Functor.isZero_rightDerived_obj_injective_succ + - CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_inv_naturality + - CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean + theorems: + - CategoryTheory.Functor.preservesZeroMorphisms_of_map_zero_object + - CategoryTheory.Functor.map_isZero + - CategoryTheory.Functor.map_eq_zero_iff + - path: Mathlib/Algebra/Homology/Homology.lean + theorems: + - HomologicalComplex.boundaries_eq_bot + - boundariesToCycles'_naturality + - cycles'Map_comp + - HomologicalComplex.cycles_eq_top + - cycles'Map_arrow + - cycles'Map_id + - path: Mathlib/Analysis/NormedSpace/MazurUlam.lean + theorems: + - IsometryEquiv.midpoint_fixed + - IsometryEquiv.coe_toRealAffineIsometryEquiv + - IsometryEquiv.map_midpoint + - path: Mathlib/Dynamics/Flow.lean + theorems: + - isInvariant_iff_image + - Flow.continuous_toFun + - Flow.ext + - Flow.isInvariant_iff_image_eq + - path: Mathlib/Data/Int/AbsoluteValue.lean + theorems: + - AbsoluteValue.map_units_int + - AbsoluteValue.map_units_int_cast + - AbsoluteValue.map_units_int_smul + - path: Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean + theorems: + - Polynomial.hermite_eq_deriv_gaussian + - Polynomial.hermite_eq_deriv_gaussian' + - Polynomial.deriv_gaussian_eq_hermite_mul_gaussian + - path: Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean + theorems: + - ContinuousMultilinearMap.norm_map_init_le + - ContinuousMultilinearMap.norm_map_snoc_le + - ContinuousLinearMap.norm_map_tail_le + - ContinuousMultilinearMap.norm_domDomCongr + - ContinuousMultilinearMap.apply_zero_curry0 + - ContinuousMultilinearMap.fin0_apply_norm + - ContinuousLinearMap.curry_uncurryLeft + - ContinuousMultilinearMap.uncurry0_norm + - ContinuousMultilinearMap.norm_map_cons_le + - ContinuousMultilinearMap.curryFinFinset_apply_const + - ContinuousMultilinearMap.uncurry_curryRight + - ContinuousMultilinearMap.curry_uncurryRight + - path: Mathlib/Probability/Martingale/OptionalSampling.lean + theorems: + - MeasureTheory.Martingale.condexp_stopping_time_ae_eq_restrict_eq_const + - MeasureTheory.Martingale.stoppedValue_min_ae_eq_condexp + - MeasureTheory.Martingale.condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const + - MeasureTheory.Martingale.condexp_stoppedValue_stopping_time_ae_eq_restrict_le + - MeasureTheory.Martingale.stoppedValue_ae_eq_restrict_eq + - MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_of_countable_range + - MeasureTheory.Martingale.stoppedValue_ae_eq_condexp_of_le_const_of_countable_range + - path: Mathlib/MeasureTheory/Measure/Doubling.lean + theorems: + - IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul' + - IsUnifLocDoublingMeasure.eventually_measure_mul_le_scalingConstantOf_mul + - IsUnifLocDoublingMeasure.exists_eventually_forall_measure_closedBall_le_mul + - IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul + - path: Mathlib/Order/CompleteSublattice.lean + theorems: + - CompleteSublattice.coe_sSup' + - CompleteSublattice.codisjoint_iff + - CompleteSublattice.isCompl_iff + - CompleteSublattice.disjoint_iff + - CompleteSublattice.isComplemented_iff + - path: Mathlib/Topology/Clopen.lean + theorems: + - continuousOn_boolIndicator_iff_isClopen + - continuous_boolIndicator_iff_isClopen + - isClopen_inter_of_disjoint_cover_clopen + - isClopen_iff_frontier_eq_empty + - path: Mathlib/Topology/Category/UniformSpace.lean + theorems: + - UniformSpaceCat.extension_comp_coe + - path: Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean + theorems: + - CategoryTheory.isSheaf_iff_preservesFiniteProducts + - CategoryTheory.isSheafFor_extensive_of_preservesFiniteProducts + - path: Mathlib/Data/Multiset/Antidiagonal.lean + theorems: + - Multiset.antidiagonal_cons + - Multiset.prod_map_add + - Multiset.antidiagonal_eq_map_powerset + - Multiset.antidiagonal_map_fst + - Multiset.card_antidiagonal + - Multiset.antidiagonal_map_snd + - Multiset.mem_antidiagonal + - path: Mathlib/GroupTheory/GroupAction/SubMulAction.lean + theorems: + - SetLike.forall_smul_mem_iff + - SubMulAction.stabilizer_of_subMul.submonoid + - SubMulAction.smul_of_tower_mem + - SubMulAction.val_preimage_orbit + - SubMulAction.mem_orbit_subMul_iff + - SubMulAction.stabilizer_of_subMul + - SubMulAction.neg_mem + - SubMulAction.neg_mem_iff + - SubMulAction.orbitRel_of_subMul + - path: Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean + theorems: + - ModuleCat.FilteredColimits.colimitSMulAux_eq_of_rel + - ModuleCat.FilteredColimits.colimitModule.one_smul + - ModuleCat.FilteredColimits.colimitModule.add_smul + - path: Mathlib/Data/Subtype.lean + theorems: + - Subtype.surjective_restrict + - Subtype.coind_injective + - Subtype.heq_iff_coe_heq + - Subtype.restrict_apply + - exists_subtype_mk_eq_iff + - path: Mathlib/CategoryTheory/Category/Cat/Limit.lean + theorems: + - "CategoryTheory.Cat.HasLimits.limit_\u03C0_homDiagram_eqToHom" + - path: Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean + theorems: + - Polynomial.tendsto_div_exp_atTop + - path: Mathlib/Data/Int/Defs.lean + theorems: + - Int.coe_pred_of_pos + - Int.succ_neg_succ + - Int.induction_on + - Int.pred_neg_pred + - Int.ediv_of_neg_of_pos + - Int.neg_pred + - Int.two_mul + - path: Mathlib/RingTheory/Binomial.lean + theorems: + - Ring.choose_nat_cast + - Ring.descPochhammer_eq_factorial_smul_choose + - path: Mathlib/CategoryTheory/Grothendieck.lean + theorems: + - CategoryTheory.Grothendieck.id_fiber' + - CategoryTheory.Grothendieck.ext + - CategoryTheory.Grothendieck.congr + - path: Mathlib/MeasureTheory/Measure/AEDisjoint.lean + theorems: + - MeasureTheory.AEDisjoint.exists_disjoint_diff + - MeasureTheory.AEDisjoint.iUnion_left_iff + - MeasureTheory.AEDisjoint.iUnion_right_iff + - MeasureTheory.AEDisjoint.symm + - MeasureTheory.exists_null_pairwise_disjoint_diff + - MeasureTheory.AEDisjoint.union_right_iff + - MeasureTheory.AEDisjoint.union_left_iff + - path: Mathlib/Algebra/Ring/Divisibility/Basic.lean + theorems: + - dvd_mul_sub_mul + - MulEquiv.decompositionMonoid + - dvd_neg + - dvd_add_right + - dvd_sub_right + - dvd_add_left + - dvd_sub_comm + - dvd_iff_dvd_of_dvd_sub + - dvd_add + - dvd_sub_left + - map_dvd_iff + - dvd_sub + - neg_dvd + - path: Mathlib/Algebra/Homology/ShortComplex/Ab.lean + theorems: + - CategoryTheory.ShortComplex.abCyclesIso_inv_apply_iCycles + - CategoryTheory.ShortComplex.ab_exact_iff_range_eq_ker + - CategoryTheory.ShortComplex.ab_exact_iff + - CategoryTheory.ShortComplex.exact_iff_surjective_abToCycles + - CategoryTheory.ShortComplex.ab_zero_apply + - path: Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean + theorems: + - Complex.iter_deriv_exp + - Complex.contDiff_exp + - iteratedDeriv_cexp_const_mul + - iteratedDeriv_exp_const_mul + - Real.iter_deriv_exp + - Complex.hasDerivAt_exp + - path: Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean + theorems: + - AlgebraicTopology.DoldKan.homotopyPToId_eventually_constant + - path: Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean + theorems: + - CochainComplex.MappingConeCompHomotopyEquiv.hom_inv_id + - "CochainComplex.mappingConeCompHomotopyEquiv_comm\u2082" + - "CochainComplex.mappingConeCompHomotopyEquiv_comm\u2081" + - CochainComplex.mappingConeCompHomotopyEquiv_hom_inv_id + - "CochainComplex.mappingConeCompTriangle_mor\u2083_naturality" + - HomotopyCategory.mappingConeCompTriangleh_distinguished + - path: Mathlib/FieldTheory/Extension.lean + theorems: + - Algebra.IsAlgebraic.range_eval_eq_rootSet_minpoly_of_splits + - IntermediateField.Lifts.exists_lift_of_splits + - IntermediateField.exists_algHom_adjoin_of_splits'' + - IntermediateField.exists_algHom_adjoin_of_splits' + - IntermediateField.exists_algHom_adjoin_of_splits + - IntermediateField.exists_algHom_adjoin_of_splits_of_aeval + - IntermediateField.Lifts.exists_lift_of_splits' + - IntermediateField.Lifts.exists_upper_bound + - path: Mathlib/Analysis/Calculus/Deriv/Shift.lean + theorems: + - HasDerivAt.comp_add_const + - HasDerivAt.comp_const_add + - deriv_comp_neg + - path: Mathlib/NumberTheory/Bertrand.lean + theorems: + - Bertrand.real_main_inequality + - centralBinom_factorization_small + - Nat.exists_prime_lt_and_le_two_mul_succ + - Nat.exists_prime_lt_and_le_two_mul + - Nat.exists_prime_lt_and_le_two_mul_eventually + - bertrand_main_inequality + - centralBinom_le_of_no_bertrand_prime + - path: Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean + theorems: + - "CategoryTheory.Idempotents.DoldKan.N\u2082_map_iso\u0393\u2080_hom_app_f" + - "CategoryTheory.Idempotents.DoldKan.h\u03B7" + - "CategoryTheory.Idempotents.DoldKan.h\u03B5" + - path: Mathlib/Topology/MetricSpace/Kuratowski.lean + theorems: + - KuratowskiEmbedding.embeddingOfSubset_isometry + - KuratowskiEmbedding.embeddingOfSubset_dist_le + - LipschitzOnWith.extend_lp_infty + - KuratowskiEmbedding.exists_isometric_embedding + - path: Mathlib/RingTheory/Localization/AtPrime.lean + theorems: + - Localization.localRingHom_comp + - IsLocalization.AtPrime.localRing + - IsLocalization.AtPrime.Nontrivial + - Localization.le_comap_primeCompl_iff + - IsLocalization.AtPrime.mk'_mem_maximal_iff + - Localization.AtPrime.map_eq_maximalIdeal + - IsLocalization.AtPrime.to_map_mem_maximal_iff + - IsLocalization.AtPrime.comap_maximalIdeal + - path: Mathlib/Topology/Category/Profinite/CofilteredLimit.lean + theorems: + - Profinite.exists_locallyConstant_finite_aux + - Profinite.exists_isClopen_of_cofiltered + - Profinite.exists_locallyConstant_fin_two + - Profinite.exists_locallyConstant_finite_nonempty + - Profinite.exists_locallyConstant + - path: Mathlib/Analysis/Calculus/Deriv/Pow.lean + theorems: + - HasDerivAt.pow + - hasStrictDerivAt_pow + - path: Mathlib/Data/ZMod/Defs.lean + theorems: + - ZMod.card + - Fin.left_distrib_aux + - path: Mathlib/Data/Fintype/BigOperators.lean + theorems: + - Fin.prod_univ_eq_prod_range + - Fintype.card_fun + - Finset.prod_fin_eq_prod_range + - card_vector + - Fintype.prod_bool + - Fintype.prod_extend_by_one + - Fintype.card_piFinset + - Fintype.prod_eq_mul + - Fintype.prod_dite + - path: Mathlib/Order/Atoms/Finite.lean + theorems: + - Fintype.IsSimpleOrder.univ + - path: Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean + theorems: + - tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support + - tendsto_integral_exp_smul_cocompact + - fourier_integrand_integrable + - fourier_integral_eq_half_sub_half_period_translate + - fourier_integral_half_period_translate + - tendsto_integral_exp_smul_cocompact_of_inner_product + - tendsto_integral_exp_inner_smul_cocompact + - path: Mathlib/Algebra/GCDMonoid/Div.lean + theorems: + - Finset.Polynomial.gcd_div_eq_one + - Finset.Int.gcd_div_eq_one + - Finset.Nat.gcd_div_eq_one + - path: Mathlib/Algebra/Algebra/Equiv.lean + theorems: + - AlgEquiv.toLinearEquiv_ofLinearEquiv + - AlgEquiv.autCongr_refl + - AlgEquiv.arrowCongr_symm + - AlgEquiv.arrowCongr_refl + - AlgEquiv.arrowCongr_comp + - AlgEquiv.algebraMap_eq_apply + - AlgEquiv.arrowCongr_trans + - AlgEquiv.comp_symm + - AlgEquiv.map_smul + - AlgEquiv.ofLinearEquiv_toLinearEquiv + - AlgEquiv.equivCongr_refl + - AlgEquiv.symm_symm + - AlgEquiv.symm_comp + - path: Mathlib/Data/Nat/Multiplicity.lean + theorems: + - Nat.multiplicity_eq_card_pow_dvd + - Nat.Prime.multiplicity_factorial_mul + - Nat.multiplicity_two_factorial_lt + - Nat.Prime.dvd_choose_pow + - Nat.Prime.pow_dvd_factorial_iff + - Nat.Prime.multiplicity_choose_aux + - Nat.Prime.multiplicity_choose' + - Nat.Prime.multiplicity_choose + - Nat.Prime.multiplicity_factorial_mul_succ + - Nat.Prime.sub_one_mul_multiplicity_factorial + - Nat.Prime.multiplicity_factorial_le_div_pred + - Nat.Prime.multiplicity_factorial + - Nat.Prime.multiplicity_choose_prime_pow_add_multiplicity + - Nat.Prime.multiplicity_le_multiplicity_choose_add + - Nat.Prime.dvd_choose_pow_iff + - path: Mathlib/Algebra/Star/Center.lean + theorems: + - Set.star_mem_center + - Set.star_mem_centralizer' + - path: Mathlib/Combinatorics/Young/SemistandardTableau.lean + theorems: + - SemistandardYoungTableau.row_weak_of_le + - SemistandardYoungTableau.col_weak + - path: Mathlib/CategoryTheory/Limits/Shapes/SingleObj.lean + theorems: + - CategoryTheory.Limits.SingleObj.Types.Quot.Rel.iff_orbitRel + - path: Mathlib/CategoryTheory/Monoidal/Linear.lean + theorems: + - CategoryTheory.monoidalLinearOfFaithful + - path: Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean + theorems: + - CategoryTheory.Limits.Types.descFun_spec + - CategoryTheory.Limits.Types.iUnion_descSet_singleton + - CategoryTheory.Limits.Types.descSet_univ + - CategoryTheory.Limits.Types.descSet_empty + - CategoryTheory.Limits.Types.descSet_inter_of_ne + - CategoryTheory.Limits.Types.descSet_spec + - path: Mathlib/Data/Set/Constructions.lean + theorems: + - FiniteInter.finiteInter_mem + - FiniteInter.finiteInterClosure_insert + - path: Mathlib/RingTheory/FreeCommRing.lean + theorems: + - FreeRing.coe_eq + - FreeCommRing.exists_finset_support + - FreeCommRing.map_subtype_val_restriction + - FreeCommRing.of_cons + - FreeRing.coe_neg + - FreeCommRing.lift_comp_of + - FreeCommRing.isSupported_int + - FreeRing.coe_surjective + - FreeCommRing.induction_on + - FreeRing.coe_sub + - path: Mathlib/Topology/Algebra/InfiniteSum/Order.lean + theorems: + - tsum_eq_zero_iff + - tsum_nonneg + - hasSum_lt + - hasSum_le_inj + - summable_abs_iff + - tsum_ne_zero_iff + - le_hasSum + - hasSum_zero_iff_of_nonneg + - tsum_le_of_sum_le' + - Finite.of_summable_const + - tsum_nonpos + - tsum_pos + - path: Mathlib/Topology/ContinuousFunction/Polynomial.lean + theorems: + - polynomialFunctions_coe + - polynomialFunctions.eq_adjoin_X + - polynomialFunctions.starClosure_eq_adjoin_X + - polynomialFunctions.comap_compRightAlgHom_iccHomeoI + - polynomialFunctions.le_equalizer + - Polynomial.aeval_continuousMap_apply + - polynomialFunctions_separatesPoints + - polynomialFunctions.starClosure_le_equalizer + - path: Mathlib/Algebra/Category/ModuleCat/Kernels.lean + theorems: + - "ModuleCat.cokernel_\u03C0_ext" + - path: Mathlib/Dynamics/Minimal.lean + theorems: + - isMinimal_iff_closed_smul_invariant + - path: Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean + theorems: + - IsCoercive.isClosed_range + - IsCoercive.range_eq_top + - IsCoercive.ker_eq_bot + - path: Mathlib/Data/Nat/Choose/Bounds.lean + theorems: + - Nat.pow_le_choose + - Nat.choose_le_pow + - path: Mathlib/NumberTheory/Cyclotomic/Embeddings.lean + theorems: + - IsCyclotomicExtension.Rat.nrRealPlaces_eq_zero + - IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two + - path: Mathlib/Data/Int/NatPrime.lean + theorems: + - Int.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul + - Int.Prime.dvd_natAbs_of_coe_dvd_sq + - path: Mathlib/Analysis/Complex/OperatorNorm.lean + theorems: + - Complex.reCLM_norm + - Complex.imCLM_norm + - path: Mathlib/Data/List/Count.lean + theorems: + - List.count_cons' + - List.countP_attach + - List.count_bind + - List.count_attach + - List.countP_join + - List.prod_eq_pow_single + - List.length_filter_lt_length_iff_exists + - List.prod_map_eq_pow_single + - path: Mathlib/Control/Bifunctor.lean + theorems: + - Bifunctor.comp_fst + - Bifunctor.fst_snd + - Bifunctor.snd_fst + - Bifunctor.comp_snd + - path: .lake/packages/proofwidgets/ProofWidgets/Demos/Jsx.lean + theorems: + - ghjk + - path: Mathlib/CategoryTheory/Sites/Limits.lean + theorems: + - CategoryTheory.Sheaf.isSheaf_of_isLimit + - path: Mathlib/Data/BinaryHeap.lean + theorems: + - BinaryHeap.size_insert + - BinaryHeap.size_pos_of_max + - path: Mathlib/Data/Int/Basic.lean + theorems: + - Int.natAbs_pow + - Int.cast_mul + - Int.cast_pow + - path: Mathlib/Algebra/Ring/Basic.lean + theorems: + - isCancelMulZero_iff_isDomain_or_subsingleton + - IsRightCancelMulZero.to_noZeroDivisors + - noZeroDivisors_iff_isDomain_or_subsingleton + - isDomain_iff_noZeroDivisors_and_nontrivial + - vieta_formula_quadratic + - IsLeftCancelMulZero.to_noZeroDivisors + - pred_ne_self + - inv_neg' + - succ_ne_self + - path: Mathlib/Data/Set/MemPartition.lean + theorems: + - disjoint_memPartition + - finite_memPartition + - memPartitionSet_succ + - memPartitionSet_mem + - sUnion_memPartition + - mem_memPartitionSet + - memPartitionSet_zero + - path: Mathlib/Topology/Sober.lean + theorems: + - IsGenericPoint.image + - OpenEmbedding.quasiSober + - isGenericPoint_iff_forall_closed + - IsGenericPoint.mem_closed_set_iff + - ClosedEmbedding.quasiSober + - genericPoint_spec + - quasiSober_of_open_cover + - IsGenericPoint.disjoint_iff + - isGenericPoint_iff_specializes + - path: Mathlib/Analysis/Normed/Ring/Seminorm.lean + theorems: + - RingSeminorm.seminorm_one_eq_one_iff_ne_zero + - RingSeminorm.ne_zero_iff + - path: Mathlib/Algebra/Category/Ring/Instances.lean + theorems: + - isLocalRingHom_of_iso + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean + theorems: + - Finpartition.energy_nonneg + - Finpartition.coe_energy + - Finpartition.energy_le_one + - path: Mathlib/Analysis/Convex/Hull.lean + theorems: + - convexHull_eq_iInter + - convexHull_empty_iff + - AffineMap.image_convexHull + - mem_convexHull_iff + - convexHull_nonempty_iff + - affineSpan_convexHull + - Convex.convex_remove_iff_not_mem_convexHull_remove + - convexHull_neg + - convexHull_pair + - path: Mathlib/Logic/Nontrivial/Basic.lean + theorems: + - Subtype.nontrivial_iff_exists_ne + - Pi.nontrivial_at + - exists_pair_lt + - path: Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean + theorems: + - AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr + - AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open + - path: Mathlib/Data/ZMod/IntUnitsPower.lean + theorems: + - uzpow_intCast + - uzpow_natCast + - path: Mathlib/LinearAlgebra/TensorAlgebra/Grading.lean + theorems: + - "TensorAlgebra.GradedAlgebra.\u03B9_apply" + - path: Mathlib/SetTheory/Cardinal/PartENat.lean + theorems: + - Cardinal.toPartENat_eq_top + - Cardinal.toPartENat_natCast + - Cardinal.toPartENat_le_iff_of_le_aleph0 + - Cardinal.toPartENat_lift + - Cardinal.toPartENat_congr + - Cardinal.mk_toPartENat_eq_coe_card + - Cardinal.toPartENat_apply_of_lt_aleph0 + - Cardinal.toPartENat_le_iff_of_lt_aleph0 + - path: Mathlib/Topology/Instances/Nat.lean + theorems: + - Nat.closedBall_eq_Icc + - path: Mathlib/Topology/Algebra/MvPolynomial.lean + theorems: + - MvPolynomial.continuous_eval + - path: Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean + theorems: + - LinearMap.continuousAt_zero_of_locally_bounded + - path: Mathlib/Data/ZMod/Parity.lean + theorems: + - ZMod.eq_one_iff_odd + - ZMod.ne_zero_iff_odd + - path: Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean + theorems: + - CategoryTheory.MonoidalOfChosenFiniteProducts.symmetry + - CategoryTheory.MonoidalOfChosenFiniteProducts.hexagon_reverse + - CategoryTheory.MonoidalOfChosenFiniteProducts.braiding_naturality + - CategoryTheory.MonoidalOfChosenFiniteProducts.hexagon_forward + - path: Mathlib/Logic/Equiv/List.lean + theorems: + - Denumerable.lower_raise + - Denumerable.denumerable_list_aux + - Denumerable.lower_raise' + - Denumerable.raise_lower + - Encodable.decode_list_zero + - Denumerable.list_ofNat_zero + - Encodable.decode_list_succ + - Denumerable.raise_lower' + - path: Mathlib/Init/Control/Lawful.lean + theorems: + - OptionT.run_map + - path: Mathlib/CategoryTheory/Sites/Preserves.lean + theorems: + - CategoryTheory.Presieve.firstMap_eq_secondMap + - CategoryTheory.Presieve.isSheafFor_iff_preservesProduct + - CategoryTheory.Presieve.isSheafFor_of_preservesProduct + - CategoryTheory.Presieve.piComparison_fac + - path: Mathlib/Algebra/GroupWithZero/NeZero.lean + theorems: + - inv_ne_zero + - pullback_nonzero + - path: Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean + theorems: + - Matrix.eq_zero_of_vecMul_eq_zero + - Matrix.eq_zero_of_mulVec_eq_zero + - path: Mathlib/LinearAlgebra/BilinearMap.lean + theorems: + - "LinearMap.compl\u2081\u2082_inj" + - LinearMap.flip_inj + - "LinearMap.compl\u2081\u2082_id_id" + - "LinearMap.compl\u2082_id" + - path: Mathlib/RingTheory/Flat/Stability.lean + theorems: + - Module.Flat.comp + - Module.Flat.auxLTensor_eq + - path: Mathlib/Algebra/Group/Hom/Basic.lean + theorems: + - MonoidHom.comp_div + - MulHom.comp_mul + - MonoidHom.comp_mul + - MonoidHom.comp_inv + - injective_iff_map_eq_one + - path: Mathlib/Combinatorics/HalesJewett.lean + theorems: + - Combinatorics.Line.vertical_apply + - Combinatorics.exists_mono_homothetic_copy + - Combinatorics.Line.apply_none + - Combinatorics.Line.prod_apply + - Combinatorics.Line.map_apply + - Combinatorics.Line.diagonal_apply + - Combinatorics.Line.exists_mono_in_high_dimension + - Combinatorics.Line.exists_mono_in_high_dimension' + - Combinatorics.Line.horizontal_apply + - Combinatorics.Line.apply_of_ne_none + - path: Mathlib/LinearAlgebra/Reflection.lean + theorems: + - Module.Dual.eq_of_preReflection_mapsTo' + - Module.preReflection_apply + - Module.Dual.eq_of_preReflection_mapsTo + - Module.involutive_preReflection + - Module.preReflection_apply_self + - path: Mathlib/CategoryTheory/Preadditive/Basic.lean + theorems: + - CategoryTheory.Preadditive.epi_of_isZero_cokernel' + - CategoryTheory.Preadditive.mono_of_kernel_zero + - CategoryTheory.Preadditive.IsIso.comp_left_eq_zero + - CategoryTheory.Preadditive.epi_of_cokernel_zero + - CategoryTheory.Preadditive.IsIso.comp_right_eq_zero + - "CategoryTheory.Preadditive.kernelForkOfFork_of\u03B9" + - "CategoryTheory.Preadditive.cokernelCoforkOfCofork_of\u03C0" + - CategoryTheory.Preadditive.mono_of_isZero_kernel' + - CategoryTheory.Preadditive.neg_comp_neg + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/Prod.lean + theorems: + - ContinuousLinearMap.opNorm_prod + - ContinuousLinearMap.norm_fst_le + - ContinuousLinearMap.norm_fst + - ContinuousLinearMap.norm_snd_le + - ContinuousLinearMap.norm_snd + - path: Mathlib/Topology/Category/TopCat/Basic.lean + theorems: + - TopCat.of_homeoOfIso + - TopCat.openEmbedding_iff_isIso_comp + - TopCat.of_isoOfHomeo + - TopCat.openEmbedding_iff_isIso_comp' + - TopCat.openEmbedding_iff_comp_isIso' + - path: Mathlib/Topology/ContinuousFunction/Weierstrass.lean + theorems: + - exists_polynomial_near_continuousMap + - polynomialFunctions_closure_eq_top + - exists_polynomial_near_of_continuousOn + - polynomialFunctions_closure_eq_top' + - path: Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean + theorems: + - Subalgebra.mul_self + - Subalgebra.mul_toSubmodule_le + - Subalgebra.mul_toSubmodule + - path: Mathlib/SetTheory/Cardinal/CountableCover.lean + theorems: + - Cardinal.mk_subtype_le_of_countable_eventually_mem_aux + - Cardinal.mk_of_countable_eventually_mem + - Cardinal.mk_subtype_le_of_countable_eventually_mem + - Cardinal.mk_le_of_countable_eventually_mem + - path: Mathlib/Control/Fold.lean + theorems: + - Traversable.toList_eq_self + - Traversable.foldlm_toList + - Traversable.foldr_toList + - Traversable.foldlm.ofFreeMonoid_comp_of + - Traversable.foldrm_toList + - Traversable.foldrm.ofFreeMonoid_comp_of + - Traversable.foldl_toList + - Traversable.length_toList + - Traversable.foldMap_map + - path: Mathlib/Dynamics/Ergodic/Function.lean + theorems: + - PreErgodic.ae_eq_const_of_ae_eq_comp + - "QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range\u2080" + - QuasiErgodic.ae_eq_const_of_ae_eq_comp_ae + - path: Mathlib/Topology/Homotopy/Product.lean + theorems: + - Path.Homotopic.projRight_prod + - Path.Homotopic.projLeft_prod + - Path.Homotopic.pi_proj + - Path.Homotopic.comp_prod_eq_prod_comp + - Path.Homotopic.proj_pi + - Path.Homotopic.pi_lift + - path: Mathlib/MeasureTheory/Function/LpOrder.lean + theorems: + - MeasureTheory.Lp.coeFn_nonneg + - MeasureTheory.Lp.coeFn_le + - path: Mathlib/NumberTheory/LucasPrimality.lean + theorems: + - lucas_primality + - path: Mathlib/Data/Int/Star.lean + theorems: + - Int.addSubmonoid_closure_range_mul_self + - Int.addSubmonoid_closure_range_pow + - path: Mathlib/Data/Set/Enumerate.lean + theorems: + - Set.enumerate_eq_none_of_sel + - Set.enumerate_eq_none + - Set.enumerate_inj + - Set.enumerate_mem + - path: Mathlib/CategoryTheory/Action.lean + theorems: + - CategoryTheory.ActionCategory.cases' + - CategoryTheory.ActionCategory.back_coe + - path: Mathlib/LinearAlgebra/FiniteSpan.lean + theorems: + - LinearEquiv.isOfFinOrder_of_finite_of_span_eq_top_of_mapsTo + - path: Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean + theorems: + - CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono' + - CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono + - CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono + - CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono' + - CategoryTheory.Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono + - path: Mathlib/Topology/Category/Profinite/EffectiveEpi.lean + theorems: + - Profinite.effectiveEpi_tfae + - Profinite.effectiveEpiFamily_tfae + - path: Mathlib/CategoryTheory/Noetherian.lean + theorems: + - CategoryTheory.exists_simple_subobject + - path: Mathlib/ModelTheory/Algebra/Field/CharP.lean + theorems: + - FirstOrder.Field.charP_iff_model_fieldOfChar + - path: Mathlib/Combinatorics/Derangements/Finite.lean + theorems: + - card_derangements_eq_numDerangements + - card_derangements_fin_eq_numDerangements + - numDerangements_succ + - card_derangements_fin_add_two + - numDerangements_sum + - path: Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean + theorems: + - GeneralizedContinuedFraction.squashGCF_nth_of_lt + - GeneralizedContinuedFraction.continuantsAux_eq_continuantsAux_squashGCF_of_le + - GeneralizedContinuedFraction.squashSeq_nth_of_not_terminated + - GeneralizedContinuedFraction.succ_nth_convergent'_eq_squashGCF_nth_convergent' + - GeneralizedContinuedFraction.succ_nth_convergent_eq_squashGCF_nth_convergent + - GeneralizedContinuedFraction.squashSeq_eq_self_of_terminated + - GeneralizedContinuedFraction.squashSeq_nth_of_lt + - GeneralizedContinuedFraction.succ_succ_nth_convergent'_aux_eq_succ_nth_convergent'_aux_squashSeq + - GeneralizedContinuedFraction.squashSeq_succ_n_tail_eq_squashSeq_tail_n + - GeneralizedContinuedFraction.convergents_eq_convergents' + - GeneralizedContinuedFraction.squashGCF_eq_self_of_terminated + - ContinuedFraction.convergents_eq_convergents' + - path: Mathlib/CategoryTheory/Preadditive/Projective.lean + theorems: + - CategoryTheory.Exact.lift_comp + - CategoryTheory.Adjunction.projective_of_map_projective + - CategoryTheory.Projective.of_iso + - CategoryTheory.Equivalence.enoughProjectives_iff + - CategoryTheory.Adjunction.map_projective + - path: Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean + theorems: + - "CategoryTheory.ProjectiveResolution.\u03C0_f_succ" + - CategoryTheory.ProjectiveResolution.exact_succ + - CategoryTheory.ProjectiveResolution.complex_d_succ_comp + - CategoryTheory.ProjectiveResolution.complex_exactAt_succ + - "CategoryTheory.ProjectiveResolution.complex_d_comp_\u03C0_f_zero" + - path: Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean + theorems: + - Real.log_div_self_antitoneOn + - Real.log_mul_self_monotoneOn + - Real.log_div_self_rpow_antitoneOn + - Real.log_div_sqrt_antitoneOn + - path: Mathlib/Topology/Order/ProjIcc.lean + theorems: + - quotientMap_projIcc + - path: Mathlib/CategoryTheory/Bicategory/Strict.lean + theorems: + - CategoryTheory.Bicategory.whiskerLeft_eqToHom + - CategoryTheory.Bicategory.eqToHom_whiskerRight + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean + theorems: + - "CliffordAlgebra.invOf_\u03B9_mul_\u03B9_mul_\u03B9" + - "CliffordAlgebra.\u03B9_mul_\u03B9_mul_invOf_\u03B9" + - "CliffordAlgebra.isUnit_\u03B9_of_isUnit" + - "CliffordAlgebra.invOf_\u03B9" + - "CliffordAlgebra.isUnit_of_isUnit_\u03B9" + - path: Mathlib/FieldTheory/MvPolynomial.lean + theorems: + - MvPolynomial.rank_mvPolynomial + - MvPolynomial.quotient_mk_comp_C_injective + - path: Mathlib/Data/Finset/Sum.lean + theorems: + - Finset.disjSum_eq_empty + - Finset.disjoint_map_inl_map_inr + - path: Mathlib/LinearAlgebra/FinsuppVectorSpace.lean + theorems: + - Finsupp.coe_basis + - Finset.sum_single_ite + - Basis.equivFun_symm_stdBasis + - Finsupp.linearIndependent_single + - path: Mathlib/Algebra/Homology/ShortComplex/Abelian.lean + theorems: + - "CategoryTheory.ShortComplex.kernel_\u03B9_comp_cokernel_\u03C0_comp_cokernelToAbelianCoimage" + - "CategoryTheory.ShortComplex.abelianImageToKernel_comp_kernel_\u03B9_comp_cokernel_\u03C0" + - path: Mathlib/Tactic/NormNum/NatSqrt.lean + theorems: + - Tactic.NormNum.nat_sqrt_helper + - path: Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean + theorems: + - TopologicalSpace.eq_induced_by_maps_to_sierpinski + - TopologicalSpace.productOfMemOpens_inducing + - TopologicalSpace.productOfMemOpens_injective + - path: Mathlib/NumberTheory/ModularForms/SlashInvariantForms.lean + theorems: + - SlashInvariantForm.slash_action_eqn' + - path: Mathlib/CategoryTheory/Sites/CoverLifting.lean + theorems: + - CategoryTheory.Functor.toSheafify_pullbackSheafificationCompatibility + - CategoryTheory.ran_isSheaf_of_isCocontinuous + - CategoryTheory.Functor.pushforwardContinuousSheafificationCompatibility_hom_app_val + - CategoryTheory.RanIsSheafOfIsCocontinuous.gluedSection_isAmalgamation + - CategoryTheory.RanIsSheafOfIsCocontinuous.helper + - CategoryTheory.RanIsSheafOfIsCocontinuous.gluedSection_is_unique + - CategoryTheory.RanIsSheafOfIsCocontinuous.getSection_commute + - CategoryTheory.RanIsSheafOfIsCocontinuous.getSection_is_unique + - path: Mathlib/RingTheory/PowerSeries/Inverse.lean + theorems: + - PowerSeries.coeff_inv_aux + - PowerSeries.coeff_inv + - PowerSeries.constantCoeff_invOfUnit + - path: Mathlib/Data/MvPolynomial/Division.lean + theorems: + - MvPolynomial.X_dvd_X + - MvPolynomial.monomial_dvd_monomial + - MvPolynomial.coeff_modMonomial_of_not_le + - MvPolynomial.monomial_one_dvd_monomial_one + - MvPolynomial.X_dvd_monomial + - path: Mathlib/Order/Filter/IndicatorFunction.lean + theorems: + - Filter.EventuallyEq.mulIndicator_one + - mulIndicator_eventuallyEq + - Antitone.mulIndicator_eventuallyEq_iInter + - Monotone.mulIndicator_eventuallyEq_iUnion + - Filter.EventuallyEq.of_mulIndicator + - mulIndicator_biUnion_finset_eventuallyEq + - path: Mathlib/Analysis/Normed/Field/InfiniteSum.lean + theorems: + - summable_norm_sum_mul_range_of_summable_norm + - summable_norm_sum_mul_antidiagonal_of_summable_norm + - Summable.mul_of_nonneg + - tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm + - path: Mathlib/GroupTheory/Submonoid/Units.lean + theorems: + - Subgroup.mem_ofUnits_iff_toUnits_mem + - Subgroup.mem_iff_toUnits_mem_units + - Subgroup.mem_units_iff_val_mem + - Subgroup.val_mem_ofUnits_iff_mem + - path: Mathlib/Data/IsROrC/Lemmas.lean + theorems: + - FiniteDimensional.proper_isROrC + - IsROrC.reCLM_norm + - path: Mathlib/Data/Polynomial/Derivation.lean + theorems: + - Derivation.compAEval_eq + - Polynomial.mkDerivation_one_eq_derivative + - Polynomial.mkDerivation_one_eq_derivative' + - Polynomial.mkDerivation_X + - Polynomial.C_smul_derivation_apply + - Polynomial.derivation_ext + - Polynomial.mkDerivationEquiv_apply + - Polynomial.mkDerivation_apply + - Derivation.comp_aeval_eq + - path: Mathlib/Algebra/Star/Pi.lean + theorems: + - Function.star_sum_elim + - path: Mathlib/Algebra/GroupWithZero/Bitwise.lean + theorems: + - zpow_bit1' + - "zpow_bit1\u2080" + - path: Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.lean + theorems: + - QuadraticForm.equivalent_weightedSumSquares_units_of_nondegenerate' + - path: Mathlib/Algebra/Order/Invertible.lean + theorems: + - invOf_pos + - invOf_lt_zero + - invOf_nonpos + - invOf_nonneg + - path: Mathlib/Data/Fin/Tuple/Reflection.lean + theorems: + - FinVec.exists_iff + - FinVec.seq_eq + - FinVec.forall_iff + - FinVec.sum_eq + - path: Mathlib/Topology/Sheaves/PUnit.lean + theorems: + - TopCat.Presheaf.isSheaf_of_isTerminal_of_indiscrete + - path: Mathlib/MeasureTheory/Integral/IntervalAverage.lean + theorems: + - interval_average_eq_div + - interval_average_eq + - path: Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean + theorems: + - CliffordAlgebra.even.lift.fFold_fFold + - CliffordAlgebra.even.algHom_ext + - "CliffordAlgebra.even.lift.aux_\u03B9" + - CliffordAlgebra.even.lift.aux_mul + - path: Mathlib/Data/Int/SuccPred.lean + theorems: + - Int.sub_one_covBy + - Int.pred_iterate + - Int.succ_iterate + - Nat.cast_int_covBy_iff + - path: Mathlib/Topology/Homotopy/Contractible.lean + theorems: + - contractible_iff_id_nullhomotopic + - ContinuousMap.Nullhomotopic.comp_right + - ContinuousMap.Nullhomotopic.comp_left + - ContractibleSpace.hequiv + - id_nullhomotopic + - path: Mathlib/Analysis/Calculus/Deriv/Support.lean + theorems: + - support_deriv_subset + - path: Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean + theorems: + - AddChar.IsNontrivial.isPrimitive + - AddChar.zmodChar_apply' + - AddChar.IsPrimitive.zmod_char_eq_one_iff + - AddChar.sum_mulShift + - AddChar.sum_eq_card_of_is_trivial + - AddChar.zmod_char_primitive_of_eq_one_only_at_zero + - AddChar.sum_eq_zero_of_isNontrivial + - AddChar.zmodChar_primitive_of_primitive_root + - AddChar.zmod_char_isNontrivial_iff + - AddChar.to_mulShift_inj_of_isPrimitive + - path: Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean + theorems: + - Real.measurable_of_measurable_exp + - Real.aemeasurable_of_aemeasurable_exp + - path: .lake/packages/std/Std/Control/ForInStep/Lemmas.lean + theorems: + - ForInStep.bindList_append + - ForInStep.bind_bindList_assoc + - ForInStep.bindList_cons' + - ForInStep.done_bindList + - ForInStep.bind_yield_bindList + - path: Mathlib/Analysis/Normed/Group/Completeness.lean + theorems: + - Metric.exists_subseq_summable_dist_of_cauchySeq + - NormedAddCommGroup.completeSpace_of_summable_imp_tendsto + - NormedAddCommGroup.summable_imp_tendsto_of_complete + - path: Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean + theorems: + - IsHomeomorphicTrivialFiberBundle.continuous_proj + - IsHomeomorphicTrivialFiberBundle.isOpenMap_proj + - IsHomeomorphicTrivialFiberBundle.surjective_proj + - path: Mathlib/Tactic/NormNum/NatFib.lean + theorems: + - Mathlib.Meta.NormNum.isFibAux_two_mul_add_one + - Mathlib.Meta.NormNum.isFibAux_two_mul + - path: Mathlib/Data/Num/Prime.lean + theorems: + - PosNum.minFac_to_nat + - PosNum.minFacAux_to_nat + - path: Mathlib/Analysis/NormedSpace/Star/Unitization.lean + theorems: + - Unitization.norm_splitMul_snd_sq + - ContinuousLinearMap.opNorm_mul_flip_apply + - path: Mathlib/Order/Extension/Linear.lean + theorems: + - extend_partialOrder + - path: Mathlib/Algebra/Homology/ConcreteCategory.lean + theorems: + - "CategoryTheory.ShortComplex.ShortExact.\u03B4_apply" + - HomologicalComplex.i_cyclesMk + - path: Mathlib/ModelTheory/ElementaryMaps.lean + theorems: + - FirstOrder.Language.Embedding.isElementary_of_exists + - FirstOrder.Language.ElementaryEmbedding.map_boundedFormula + - FirstOrder.Language.ElementaryEmbedding.theory_model_iff + - FirstOrder.Language.ElementaryEmbedding.injective + - FirstOrder.Language.ElementaryEmbedding.map_fun + - FirstOrder.Language.ElementaryEmbedding.map_sentence + - FirstOrder.Language.ElementaryEmbedding.map_formula + - path: Mathlib/Analysis/Calculus/Deriv/ZPow.lean + theorems: + - deriv_zpow + - iter_deriv_inv + - iter_deriv_zpow' + - iter_deriv_pow + - hasStrictDerivAt_zpow + - path: Mathlib/Topology/Category/TopCat/Limits/Konig.lean + theorems: + - TopCat.partialSections.directed + - TopCat.partialSections.nonempty + - TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system + - TopCat.partialSections.closed + - path: Mathlib/Topology/Algebra/GroupCompletion.lean + theorems: + - AddMonoidHom.completion_zero + - path: Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean + theorems: + - Ideal.adic_basis + - Ideal.hasBasis_nhds_zero_adic + - Ideal.adic_module_basis + - isAdic_iff + - Ideal.hasBasis_nhds_adic + - is_bot_adic_iff + - is_ideal_adic_pow + - path: Mathlib/RingTheory/Localization/Ideal.lean + theorems: + - IsLocalization.isPrime_iff_isPrime_disjoint + - IsLocalization.bot_lt_comap_prime + - IsLocalization.isPrime_of_isPrime_disjoint + - IsLocalization.surjective_quotientMap_of_maximal_of_localization + - IsLocalization.map_comap + - IsLocalization.mem_map_algebraMap_iff + - IsLocalization.comap_map_of_isPrime_disjoint + - path: Mathlib/Tactic/NormNum/GCD.lean + theorems: + - Tactic.NormNum.int_gcd_helper' + - Tactic.NormNum.int_gcd_helper + - Tactic.NormNum.nat_gcd_helper_2 + - Tactic.NormNum.nat_lcm_helper + - Tactic.NormNum.int_lcm_helper + - path: Mathlib/Analysis/BoxIntegral/Partition/Additive.lean + theorems: + - BoxIntegral.BoxAdditiveMap.map_split_add + - BoxIntegral.BoxAdditiveMap.sum_boxes_congr + - path: Mathlib/Combinatorics/Hindman.lean + theorems: + - Hindman.FP.mul + - Hindman.FP_drop_subset_FP + - Hindman.FP.finset_prod + - Hindman.exists_idempotent_ultrafilter_le_FP + - Hindman.FP.singleton + - Hindman.exists_FP_of_large + - Hindman.FP.mul_two + - path: Mathlib/Analysis/Convex/Cone/Extension.lean + theorems: + - exists_extension_of_le_sublinear + - riesz_extension + - RieszExtension.exists_top + - RieszExtension.step + - path: Mathlib/Algebra/Category/MonCat/FilteredColimits.lean + theorems: + - MonCat.FilteredColimits.colimit_one_eq + - MonCat.FilteredColimits.colimitMulAux_eq_of_rel_right + - MonCat.FilteredColimits.colimit_mul_mk_eq + - MonCat.FilteredColimits.colimitMulAux_eq_of_rel_left + - path: Mathlib/Algebra/Lie/Weights/Linear.lean + theorems: + - LieModule.shiftedWeightSpace.toEndomorphism_eq + - LieModule.trace_comp_toEndomorphism_weight_space_eq + - LieModule.exists_forall_lie_eq_smul_of_weightSpace_ne_bot + - LieModule.zero_lt_finrank_weightSpace + - path: Mathlib/Algebra/Group/WithOne/Basic.lean + theorems: + - WithOne.map_map + - WithOne.map_id + - path: Mathlib/Algebra/Order/Interval.lean + theorems: + - NonemptyInterval.mul_eq_one_iff + - Interval.length_add_le + - Interval.mul_eq_one_iff + - NonemptyInterval.length_sub + - Interval.length_sum_le + - Interval.length_sub_le + - path: Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean + theorems: + - LinearMap.charpoly_toMatrix + - LinearMap.charpoly_prodMap + - path: Mathlib/GroupTheory/PresentedGroup.lean + theorems: + - PresentedGroup.toGroup.unique + - PresentedGroup.closure_range_of + - PresentedGroup.ext + - path: Mathlib/Topology/Algebra/Order/Rolle.lean + theorems: + - exists_Ioo_extr_on_Icc + - exists_isExtrOn_Ioo_of_tendsto + - path: Mathlib/Data/Nat/MaxPowDiv.lean + theorems: + - Nat.maxPowDiv.le_of_dvd + - Nat.maxPowDiv.pow_dvd + - Nat.maxPowDiv.zero_base + - Nat.maxPowDiv.go_succ + - Nat.maxPowDiv.zero + - Nat.maxPowDiv.base_mul_eq_succ + - Nat.maxPowDiv.base_pow_mul + - path: Mathlib/Probability/Process/Filtration.lean + theorems: + - MeasureTheory.Filtration.filtrationOfSet_eq_natural + - "MeasureTheory.Filtration.mem\u2112p_limitProcess_of_snorm_bdd" + - MeasureTheory.Filtration.ext + - MeasureTheory.Filtration.stronglyMeasurable_limitProcess + - path: Mathlib/GroupTheory/GroupAction/Prod.lean + theorems: + - Prod.smul_mk_zero + - Prod.smul_zero_mk + - path: Mathlib/Topology/Sheaves/Limits.lean + theorems: + - TopCat.isSheaf_of_isLimit + - path: Mathlib/LinearAlgebra/Coevaluation.lean + theorems: + - coevaluation_apply_one + - contractLeft_assoc_coevaluation' + - contractLeft_assoc_coevaluation + - path: Mathlib/Deprecated/Subfield.lean + theorems: + - Field.closure.isSubfield + - IsSubfield.div_mem + - Range.isSubfield + - Preimage.isSubfield + - Field.closure_subset + - Field.closure.isSubmonoid + - IsSubfield.pow_mem + - path: Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean + theorems: + - QuadraticForm.associated_tmul + - QuadraticForm.polarBilin_baseChange + - QuadraticForm.polarBilin_tmul + - path: Mathlib/Algebra/Module/DedekindDomain.lean + theorems: + - Submodule.isInternal_prime_power_torsion + - Submodule.isInternal_prime_power_torsion_of_is_torsion_by_ideal + - path: Mathlib/Algebra/Category/GroupCat/Zero.lean + theorems: + - GroupCat.isZero_of_subsingleton + - CommGroupCat.isZero_of_subsingleton + - path: Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean + theorems: + - CategoryTheory.NormalEpiCategory.mono_of_zero_kernel + - CategoryTheory.NormalMonoCategory.epi_of_zero_cokernel + - path: Mathlib/NumberTheory/LSeries/Deriv.lean + theorems: + - LSeries.LSeriesSummable_logMul_and_hasDerivAt + - LSeries.abscissaOfAbsConv_logMul + - LSeries.hasDerivAt_term + - LSeries.absicssaOfAbsConv_logPowMul + - LSeries_iteratedDeriv + - path: Mathlib/RingTheory/Adjoin/Tower.lean + theorems: + - Algebra.adjoin_res_eq_adjoin_res + - Algebra.adjoin_restrictScalars + - exists_subalgebra_of_fg + - Algebra.fg_trans' + - path: Mathlib/RepresentationTheory/FdRep.lean + theorems: + - "FdRep.Iso.conj_\u03C1" + - "FdRep.forget\u2082_\u03C1" + - path: Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean + theorems: + - MultilinearMap.free_and_finite_fin + - MultilinearMap.free_and_finite + - path: Mathlib/Analysis/NormedSpace/Span.lean + theorems: + - LinearIsometryEquiv.toSpanUnitSingleton_apply + - LinearMap.toSpanSingleton_homothety + - path: Mathlib/CategoryTheory/Localization/Prod.lean + theorems: + - CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.prod_fac + - "CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.prod_fac\u2082" + - CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.prod_uniq + - path: Mathlib/Geometry/Manifold/MFDeriv/Defs.lean + theorems: + - mdifferentiableWithinAt_iff_liftPropWithinAt + - differentiable_within_at_localInvariantProp + - mdifferentiableAt_iff_liftPropAt + - path: .lake/packages/std/Std/Data/List/Init/Attach.lean + theorems: + - List.pmap_eq_pmapImpl + - path: Mathlib/Data/List/Lex.lean + theorems: + - List.Lex.ne_iff + - List.head!_le_of_lt + - List.head_le_of_lt + - List.Lex.cons_iff + - List.cons_le_cons + - Decidable.List.Lex.ne_iff + - List.lt_iff_lex_lt + - path: Mathlib/RingTheory/WittVector/MulP.lean + theorems: + - WittVector.mulN_coeff + - WittVector.mulN_isPoly + - "WittVector.bind\u2081_wittMulN_wittPolynomial" + - path: Mathlib/Data/SetLike/Basic.lean + theorems: + - SetLike.lt_iff_le_and_exists + - path: Mathlib/LinearAlgebra/Matrix/Diagonal.lean + theorems: + - Matrix.range_diagonal + - LinearMap.rank_diagonal + - Matrix.ker_diagonal_toLin' + - path: Mathlib/Data/Multiset/Sym.lean + theorems: + - Multiset.mk_mem_sym2_iff + - Multiset.sym2_mono + - Multiset.sym2_eq_zero_iff + - Multiset.mem_sym2_iff + - Multiset.card_sym2 + - path: Mathlib/Data/Rat/Cast/CharZero.lean + theorems: + - Rat.cast_bit1 + - Rat.cast_mk + - Rat.cast_eq_zero + - Rat.cast_inj + - path: Mathlib/Data/Finset/Functor.lean + theorems: + - "Finset.image\u2082_def" + - Finset.map_traverse + - Finset.id_traverse + - path: Mathlib/Algebra/Lie/Matrix.lean + theorems: + - Matrix.lieConj_apply + - path: Mathlib/RingTheory/LittleWedderburn.lean + theorems: + - LittleWedderburn.center_eq_top + - Finite.isDomain_to_isField + - LittleWedderburn.InductionHyp.center_eq_top + - path: Mathlib/Analysis/Normed/Group/Lemmas.lean + theorems: + - eventually_norm_sub_lt + - eventually_nnnorm_sub_lt + - path: Mathlib/Data/LazyList/Basic.lean + theorems: + - LazyList.append_bind + - LazyList.mem_cons + - LazyList.append_assoc + - LazyList.append_nil + - LazyList.forall_mem_cons + - path: Mathlib/LinearAlgebra/Projectivization/Basic.lean + theorems: + - Projectivization.map_injective + - Projectivization.finrank_submodule + - Projectivization.submodule_eq + - Projectivization.map_comp + - Projectivization.map_id + - Projectivization.mk_eq_mk_iff' + - Projectivization.submodule_injective + - path: Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean + theorems: + - CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.of_equivalence + - CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.mk' + - CategoryTheory.LocalizerMorphism.nonempty_isEquivalence_iff + - CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.of_isLocalization_of_isLocalization + - path: Mathlib/Analysis/Normed/Group/Tannery.lean + theorems: + - tendsto_tsum_of_dominated_convergence + - path: Mathlib/Topology/Instances/Sign.lean + theorems: + - continuousAt_sign_of_ne_zero + - continuousAt_sign_of_neg + - path: Mathlib/MeasureTheory/Function/SpecialFunctions/IsROrC.lean + theorems: + - aemeasurable_of_re_im + - measurable_of_re_im + - path: Mathlib/Probability/ProbabilityMassFunction/Binomial.lean + theorems: + - PMF.binomial_apply_self + - PMF.binomial_one_eq_bernoulli + - PMF.binomial_apply_zero + - PMF.binomial_apply_last + - path: Mathlib/Algebra/BigOperators/Module.lean + theorems: + - Finset.sum_Ico_by_parts + - path: Mathlib/Geometry/Manifold/WhitneyEmbedding.lean + theorems: + - SmoothBumpCovering.embeddingPiTangent_injOn + - SmoothBumpCovering.exists_immersion_euclidean + - exists_embedding_euclidean_of_compact + - SmoothBumpCovering.embeddingPiTangent_ker_mfderiv + - SmoothBumpCovering.comp_embeddingPiTangent_mfderiv + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean + theorems: + - ContinuousLinearMap.opNorm_ext + - ContinuousLinearMap.map_add_add + - "LinearMap.norm_mkContinuous\u2082_aux" + - ContinuousLinearMap.opNorm_flip + - ContinuousLinearMap.flip_flip + - ContinuousLinearMap.le_norm_flip + - ContinuousLinearMap.norm_precompL_le + - ContinuousLinearMap.norm_precompR_le + - path: Mathlib/Algebra/Homology/Opposite.lean + theorems: + - imageToKernel_unop + - imageToKernel_op + - path: Mathlib/CategoryTheory/DiscreteCategory.lean + theorems: + - CategoryTheory.Discrete.natIso_app + - CategoryTheory.Discrete.functor_map_id + - CategoryTheory.Discrete.mk_as + - CategoryTheory.Discrete.functor_map + - path: Mathlib/Algebra/Lie/TensorProduct.lean + theorems: + - TensorProduct.LieModule.coe_liftLie_eq_lift_coe + - TensorProduct.LieModule.liftLie_apply + - LieModule.toModuleHom_apply + - TensorProduct.LieModule.lie_tmul_right + - LieSubmodule.lieIdeal_oper_eq_tensor_map_range + - path: Mathlib/Probability/ProbabilityMassFunction/Integrals.lean + theorems: + - PMF.integral_eq_sum + - PMF.integral_eq_tsum + - PMF.bernoulli_expectation + - path: Mathlib/Algebra/Lie/CartanSubalgebra.lean + theorems: + - LieIdeal.normalizer_eq_top + - LieSubalgebra.normalizer_eq_self_of_isCartanSubalgebra + - LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra + - LieSubalgebra.isCartanSubalgebra_iff_isUcsLimit + - path: Mathlib/Analysis/Calculus/ContDiff/IsROrC.lean + theorems: + - ContDiff.locallyLipschitz + - HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt + - ContDiffWithinAt.exists_lipschitzOnWith + - ContDiff.lipschitzWith_of_hasCompactSupport + - ContDiffAt.hasStrictFDerivAt' + - path: Mathlib/Tactic/Ring/RingNF.lean + theorems: + - Mathlib.Tactic.RingNF.nat_rawCast_1 + - Mathlib.Tactic.RingNF.int_rawCast_neg + - Mathlib.Tactic.RingNF.rat_rawCast_neg + - Mathlib.Tactic.RingNF.rat_rawCast_pos + - Mathlib.Tactic.RingNF.nat_rawCast_0 + - path: Mathlib/Combinatorics/Optimization/ValuedCSP.lean + theorems: + - Function.HasMaxCutPropertyAt.rows_lt_aux + - Function.HasMaxCutProperty.forbids_commutativeFractionalPolymorphism + - path: Mathlib/CategoryTheory/EpiMono.lean + theorems: + - CategoryTheory.SplitMono.mono + - CategoryTheory.SplitEpi.epi + - CategoryTheory.isIso_of_mono_of_isSplitEpi + - CategoryTheory.IsIso.of_mono_retraction' + - CategoryTheory.isIso_of_epi_of_isSplitMono + - CategoryTheory.IsIso.of_epi_section' + - path: Mathlib/LinearAlgebra/AdicCompletion.lean + theorems: + - Hausdorffification.lift_eq + - IsAdicComplete.le_jacobson_bot + - IsHausdorff.subsingleton + - path: Mathlib/CategoryTheory/Endomorphism.lean + theorems: + - CategoryTheory.isUnit_iff_isIso + - path: Mathlib/Data/List/Palindrome.lean + theorems: + - List.Palindrome.map + - List.Palindrome.of_reverse_eq + - List.Palindrome.append_reverse + - List.Palindrome.reverse_eq + - path: Mathlib/ModelTheory/Skolem.lean + theorems: + - "FirstOrder.Language.Substructure.skolem\u2081_reduct_isElementary" + - "FirstOrder.Language.card_functions_sum_skolem\u2081_le" + - "FirstOrder.Language.card_functions_sum_skolem\u2081" + - FirstOrder.Language.exists_elementarySubstructure_card_eq + - path: Mathlib/Topology/Category/LightProfinite/Basic.lean + theorems: + - LightProfinite.ext + - path: Mathlib/Algebra/Lie/BaseChange.lean + theorems: + - LieAlgebra.ExtendScalars.bracket_leibniz_lie + - LieSubmodule.baseChange_bot + - LieSubmodule.baseChange_top + - LieModule.toEndomorphism_baseChange + - LieSubmodule.lie_baseChange + - LieAlgebra.ExtendScalars.bracket_lie_self + - path: Mathlib/Data/PNat/Defs.lean + theorems: + - PNat.mod_coe + - PNat.div_coe + - Nat.toPNat'_coe + - path: Mathlib/Data/Nat/Factorial/BigOperators.lean + theorems: + - Nat.prod_factorial_dvd_factorial_sum + - Nat.descFactorial_eq_prod_range + - path: Mathlib/Topology/Category/TopCat/OpenNhds.lean + theorems: + - TopologicalSpace.OpenNhds.op_map_id_obj + - TopologicalSpace.OpenNhds.map_id_obj_unop + - path: Mathlib/Algebra/BigOperators/Option.lean + theorems: + - Finset.prod_insertNone + - Finset.prod_eraseNone + - path: Mathlib/Data/Nat/Upto.lean + theorems: + - Nat.Upto.wf + - path: Mathlib/Data/MvPolynomial/Expand.lean + theorems: + - MvPolynomial.map_expand + - MvPolynomial.expand_one_apply + - MvPolynomial.rename_expand + - MvPolynomial.rename_comp_expand + - "MvPolynomial.expand_comp_bind\u2081" + - MvPolynomial.expand_one + - "MvPolynomial.expand_bind\u2081" + - path: Mathlib/CategoryTheory/ConcreteCategory/Basic.lean + theorems: + - CategoryTheory.ConcreteCategory.hom_ext + - "CategoryTheory.forget\u2082_comp_apply" + - CategoryTheory.ConcreteCategory.bijective_of_isIso + - CategoryTheory.ConcreteCategory.isIso_iff_bijective + - path: Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean + theorems: + - CategoryTheory.SemiadditiveOfBinaryBiproducts.add_comp + - CategoryTheory.SemiadditiveOfBinaryBiproducts.comp_add + - CategoryTheory.SemiadditiveOfBinaryBiproducts.isUnital_rightAdd + - CategoryTheory.SemiadditiveOfBinaryBiproducts.distrib + - path: Mathlib/CategoryTheory/Groupoid/Basic.lean + theorems: + - CategoryTheory.Groupoid.isThin_iff + - path: Mathlib/Data/Finsupp/ToDFinsupp.lean + theorems: + - toDFinsupp_support + - sigmaFinsuppEquivDFinsupp_single + - DFinsupp.toFinsupp_support + - DFinsupp.toFinsupp_single + - sigmaFinsuppEquivDFinsupp_add + - sigmaFinsuppEquivDFinsupp_smul + - sigmaFinsuppEquivDFinsupp_support + - Finsupp.toDFinsupp_single + - path: Mathlib/Combinatorics/Quiver/Basic.lean + theorems: + - Prefunctor.ext + - path: Mathlib/Data/Nat/ChineseRemainder.lean + theorems: + - Nat.chineseRemainderOfList_modEq_unique + - Nat.chineseRemainderOfList_lt_prod + - Nat.chineseRemainderOfList_perm + - Nat.modEq_list_prod_iff' + - Nat.modEq_list_prod_iff + - Nat.chineseRemainderOfFinset_lt_prod + - Nat.chineseRemainderOfMultiset_lt_prod + - path: Mathlib/RingTheory/RingHom/FiniteType.lean + theorems: + - RingHom.finiteType_ofLocalizationSpanTarget + - RingHom.finiteType_stableUnderComposition + - RingHom.finiteType_holdsForLocalizationAway + - path: Mathlib/Analysis/Normed/Group/CocompactMap.lean + theorems: + - CocompactMapClass.norm_le + - Filter.tendsto_cocompact_cocompact_of_norm + - path: Mathlib/Topology/Algebra/ContinuousMonoidHom.lean + theorems: + - ContinuousMonoidHom.range_toContinuousMap + - ContinuousMonoidHom.closedEmbedding_toContinuousMap + - ContinuousMonoidHom.toContinuousMap_injective + - path: Mathlib/Algebra/Order/Monoid/WithZero/Defs.lean + theorems: + - WithZero.min_le_iff + - WithZero.covariantClass_add_le + - WithZero.le_max_iff + - path: Mathlib/Analysis/Calculus/Conformal/InnerProduct.lean + theorems: + - conformalAt_iff' + - conformalAt_iff + - path: Mathlib/Deprecated/Ring.lean + theorems: + - IsRingHom.map_sub + - IsSemiringHom.comp + - IsRingHom.map_neg + - IsRingHom.comp + - IsRingHom.id + - IsRingHom.map_zero + - IsSemiringHom.id + - IsSemiringHom.to_isAddMonoidHom + - path: Mathlib/NumberTheory/PrimeCounting.lean + theorems: + - Nat.primeCounting'_add_le + - Nat.primesBelow_card_eq_primeCounting' + - path: Mathlib/Geometry/Manifold/Diffeomorph.lean + theorems: + - Diffeomorph.contMDiffWithinAt_comp_diffeomorph_iff + - Diffeomorph.range_comp + - Diffeomorph.uniqueDiffOn_image + - Diffeomorph.uniqueMDiffOn_image_aux + - Diffeomorph.contMDiffOn_comp_diffeomorph_iff + - ModelWithCorners.extChartAt_transDiffeomorph_target + - path: Mathlib/MeasureTheory/Function/SimpleFuncDense.lean + theorems: + - MeasureTheory.SimpleFunc.nearestPtInd_le + - HasCompactSupport.exists_simpleFunc_approx_of_prod + - HasCompactSupport.measurable_of_prod + - MeasureTheory.SimpleFunc.tendsto_approxOn + - MeasureTheory.SimpleFunc.edist_approxOn_mono + - MeasureTheory.SimpleFunc.approxOn_mem + - MeasureTheory.SimpleFunc.edist_nearestPt_le + - path: Mathlib/Analysis/Convex/Cone/Pointed.lean + theorems: + - PointedCone.mem_dual + - PointedCone.toConvexCone_injective + - PointedCone.toConvexCone_pointed + - path: Mathlib/Data/MvPolynomial/Polynomial.lean + theorems: + - "MvPolynomial.polynomial_eval_eval\u2082" + - MvPolynomial.eval_polynomial_eval_finSuccEquiv + - path: Mathlib/Algebra/GroupWithZero/Hom.lean + theorems: + - MonoidWithZeroHom.cancel_left + - NeZero.of_injective + - path: Mathlib/Order/PrimeIdeal.lean + theorems: + - Order.Ideal.IsPrime.mem_or_mem + - Order.Ideal.PrimePair.F_isPrime + - Order.Ideal.PrimePair.I_isProper + - Order.Ideal.isPrime_of_mem_or_compl_mem + - Order.Ideal.PrimePair.I_isPrime + - Order.Ideal.IsPrime.mem_or_compl_mem + - Order.Ideal.IsPrime.of_mem_or_mem + - path: .lake/packages/std/Std/Data/Array/Monadic.lean + theorems: + - Array.size_modifyM + - Array.SatisfiesM_mapIdxM + - Array.SatisfiesM_foldlM + - Array.SatisfiesM_mapM + - Array.SatisfiesM_foldrM + - Array.SatisfiesM_anyM_iff_exists + - path: Mathlib/Analysis/NormedSpace/IndicatorFunction.lean + theorems: + - norm_indicator_le_of_subset + - norm_indicator_le_norm_self + - path: Mathlib/RingTheory/RootsOfUnity/Complex.lean + theorems: + - Complex.mem_rootsOfUnity + - Complex.isPrimitiveRoot_exp_of_coprime + - Complex.card_primitiveRoots + - IsPrimitiveRoot.arg + - Complex.isPrimitiveRoot_iff + - path: Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean + theorems: + - "AlgebraicGeometry.PresheafedSpace.GlueData.\u03B9_openEmbedding" + - AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app' + - AlgebraicGeometry.PresheafedSpace.GlueData.pullback_base + - "AlgebraicGeometry.PresheafedSpace.GlueData.\u03B9_image_preimage_eq" + - AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app + - "AlgebraicGeometry.PresheafedSpace.GlueData.\u03B9InvApp_\u03C0" + - "AlgebraicGeometry.PresheafedSpace.GlueData.\u03C0_\u03B9InvApp_\u03C0" + - AlgebraicGeometry.PresheafedSpace.GlueData.f_invApp_f_app + - "AlgebraicGeometry.PresheafedSpace.GlueData.\u03C0_\u03B9InvApp_eq_id" + - AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app' + - path: Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean + theorems: + - CategoryTheory.GrothendieckTopology.toSheafify_comp_sheafifyCompIso_inv + - CategoryTheory.GrothendieckTopology.sheafificationWhiskerLeftIso_hom_app + - CategoryTheory.GrothendieckTopology.sheafificationWhiskerLeftIso_inv_app + - CategoryTheory.GrothendieckTopology.sheafificationWhiskerRightIso_inv_app + - CategoryTheory.GrothendieckTopology.sheafificationWhiskerRightIso_hom_app + - CategoryTheory.GrothendieckTopology.whiskerRight_toSheafify_sheafifyCompIso_hom + - CategoryTheory.GrothendieckTopology.sheafifyCompIso_inv_eq_sheafifyLift + - path: Mathlib/Topology/Covering.lean + theorems: + - IsCoveringMapOn.isLocalHomeomorphOn + - IsCoveringMap.isSeparatedMap + - IsEvenlyCovered.toTrivialization_apply + - path: Mathlib/CategoryTheory/ChosenFiniteProducts.lean + theorems: + - CategoryTheory.ChosenFiniteProducts.lift_fst + - CategoryTheory.ChosenFiniteProducts.lift_snd + - path: Mathlib/CategoryTheory/Limits/Constructions/WeaklyInitial.lean + theorems: + - CategoryTheory.hasInitial_of_weakly_initial_and_hasWideEqualizers + - path: Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean + theorems: + - szemeredi_regularity + - path: Mathlib/Topology/Category/CompHaus/Basic.lean + theorems: + - CompHaus.epi_iff_surjective + - CompHaus.isIso_of_bijective + - CompHaus.mono_iff_injective + - path: Mathlib/LinearAlgebra/FreeModule/Determinant.lean + theorems: + - LinearMap.det_zero'' + - path: Mathlib/FieldTheory/IsSepClosed.lean + theorems: + - IsSepClosed.exists_eq_mul_self + - IsSepClosed.roots_eq_zero_iff + - IsSepClosed.splits_codomain + - IsSepClosed.exists_pow_nat_eq + - "IsSepClosed.exists_eval\u2082_eq_zero" + - IsSepClosed.of_exists_root + - IntermediateField.eq_bot_of_isSepClosed_of_isSeparable + - IsSepClosed.algebraMap_surjective + - path: Mathlib/GroupTheory/Commensurable.lean + theorems: + - Commensurable.refl + - Commensurable.commensurable_inv + - path: Mathlib/Combinatorics/DoubleCounting.lean + theorems: + - Finset.sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow + - Finset.card_le_card_of_forall_subsingleton + - Fintype.card_le_card_of_leftTotal_unique + - Fintype.card_le_card_of_rightTotal_unique + - path: Mathlib/Logic/Unique.lean + theorems: + - Unique.forall_iff + - heq_const_of_unique + - Unique.subsingleton_unique' + - Unique.bijective + - unique_iff_subsingleton_and_nonempty + - path: Mathlib/Data/Set/List.lean + theorems: + - Set.range_list_nthLe + - Set.range_list_get? + - Set.range_list_map_coe + - Set.range_list_map + - Set.range_list_getD + - path: Mathlib/Data/Nat/Factorial/Cast.lean + theorems: + - Nat.cast_factorial + - Nat.cast_ascFactorial + - path: Mathlib/Algebra/Order/Group/Lattice.lean + theorems: + - inf_mul_sup + - pow_two_semiclosed + - path: Mathlib/ModelTheory/Quotients.lean + theorems: + - FirstOrder.Language.funMap_quotient_mk' + - FirstOrder.Language.Term.realize_quotient_mk' + - FirstOrder.Language.relMap_quotient_mk' + - path: Mathlib/Algebra/Star/Free.lean + theorems: + - "FreeAlgebra.star_\u03B9" + - FreeAlgebra.star_algebraMap + - path: Mathlib/RingTheory/ZMod.lean + theorems: + - isReduced_zmod + - path: Mathlib/Topology/Algebra/Group/OpenMapping.lean + theorems: + - smul_singleton_mem_nhds_of_sigmaCompact + - isOpenMap_smul_of_sigmaCompact + - MonoidHom.isOpenMap_of_sigmaCompact + - path: Mathlib/Tactic/ReduceModChar.lean + theorems: + - Tactic.ReduceModChar.CharP.isInt_of_mod + - Tactic.ReduceModChar.CharP.cast_int_eq_mod + - Tactic.ReduceModChar.CharP.neg_eq_sub_one_mul + - Tactic.ReduceModChar.CharP.neg_mul_eq_sub_one_mul + - path: Mathlib/Tactic/CategoryTheory/Coherence.lean + theorems: + - Mathlib.Tactic.Coherence.insert_id_lhs + - Mathlib.Tactic.Coherence.insert_id_rhs + - path: Mathlib/CategoryTheory/Linear/Basic.lean + theorems: + - CategoryTheory.Linear.units_smul_comp + - CategoryTheory.Linear.comp_units_smul + - path: Mathlib/NumberTheory/Harmonic/Bounds.lean + theorems: + - log_le_harmonic_floor + - log_add_one_le_harmonic + - harmonic_le_one_add_log + - harmonic_floor_le_one_add_log + - path: Mathlib/Topology/Algebra/Affine.lean + theorems: + - AffineMap.continuous_iff + - AffineMap.homothety_isOpenMap + - AffineMap.homothety_continuous + - path: Mathlib/Topology/Algebra/Equicontinuity.lean + theorems: + - equicontinuous_of_equicontinuousAt_one + - uniformEquicontinuous_of_equicontinuousAt_one + - path: Mathlib/CategoryTheory/GradedObject/Single.lean + theorems: + - CategoryTheory.GradedObject.singleObjApplyIsoOfEq_inv_single_map + - CategoryTheory.GradedObject.single_map_singleObjApplyIso_hom + - CategoryTheory.GradedObject.singleObjApplyIso_inv_single_map + - CategoryTheory.GradedObject.single_map_singleObjApplyIsoOfEq_hom + - path: Mathlib/Analysis/Normed/Group/ControlledClosure.lean + theorems: + - controlled_closure_of_complete + - controlled_closure_range_of_complete + - path: Mathlib/Analysis/Complex/Conformal.lean + theorems: + - isConformalMap_complex_linear + - isConformalMap_iff_is_complex_or_conj_linear + - IsConformalMap.is_complex_or_conj_linear + - path: Mathlib/Data/Analysis/Filter.lean + theorems: + - Filter.Realizer.mem_sets + - Filter.Realizer.ne_bot_iff + - CFilter.ofEquiv_val + - path: Mathlib/GroupTheory/Perm/Closure.lean + theorems: + - Equiv.Perm.closure_isCycle + - Equiv.Perm.closure_prime_cycle_swap + - Equiv.Perm.closure_cycle_coprime_swap + - Equiv.Perm.closure_cycle_adjacent_swap + - path: Mathlib/Topology/VectorBundle/Constructions.lean + theorems: + - Trivialization.continuousLinearEquivAt_prod + - Trivialization.coordChangeL_prod + - Bundle.Trivial.trivialization.coordChangeL + - path: Mathlib/Analysis/Convex/StoneSeparation.lean + theorems: + - not_disjoint_segment_convexHull_triple + - path: Mathlib/CategoryTheory/Galois/GaloisObjects.lean + theorems: + - CategoryTheory.PreGaloisCategory.isGalois_iff_pretransitive + - path: Mathlib/Algebra/MonoidAlgebra/Ideal.lean + theorems: + - MonoidAlgebra.mem_ideal_span_of_image + - path: Mathlib/CategoryTheory/Sites/Equivalence.lean + theorems: + - CategoryTheory.Equivalence.coverPreserving + - CategoryTheory.Equivalence.locallyCoverDense + - CategoryTheory.Equivalence.hasSheafCompose + - path: Mathlib/Order/Category/NonemptyFinLinOrd.lean + theorems: + - NonemptyFinLinOrd.epi_iff_surjective + - NonemptyFinLinOrd.mono_iff_injective + - path: Mathlib/Algebra/Algebra/Hom.lean + theorems: + - AlgHom.toLinearMap_ofLinearMap + - AlgHom.ofLinearMap_toLinearMap + - path: Mathlib/CategoryTheory/Limits/Presheaf.lean + theorems: + - CategoryTheory.coconeOfRepresentable_naturality + - CategoryTheory.ColimitAdj.extendAlongYoneda_map + - CategoryTheory.ColimitAdj.restrictYonedaHomEquiv_natural + - path: Mathlib/Tactic/CategoryTheory/MonoidalComp.lean + theorems: + - CategoryTheory.monoidalComp_refl + - path: Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean + theorems: + - AlgebraicGeometry.locallyOfFiniteTypeOfComp + - path: Mathlib/Control/Functor.lean + theorems: + - Functor.Comp.id_map + - Functor.ext + - Functor.Comp.comp_map + - path: Mathlib/Topology/Algebra/Order/UpperLower.lean + theorems: + - Set.OrdConnected.interior + - IsUpperSet.interior + - path: Mathlib/Control/Traversable/Basic.lean + theorems: + - ApplicativeTransformation.preserves_map' + - ApplicativeTransformation.coe_inj + - ApplicativeTransformation.preserves_map + - ApplicativeTransformation.ext + - path: .lake/packages/std/Std/Data/RBMap/Basic.lean + theorems: + - Std.RBNode.all_iff + - Std.RBNode.any_iff + - path: Mathlib/Data/ZMod/Coprime.lean + theorems: + - ZMod.eq_zero_iff_gcd_ne_one + - path: Mathlib/CategoryTheory/Abelian/Images.lean + theorems: + - CategoryTheory.Abelian.coimageImageComparison_eq_coimageImageComparison' + - CategoryTheory.Abelian.coimage_image_factorisation + - path: Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean + theorems: + - SimpleGraph.chromaticNumber_pathGraph + - path: Mathlib/CategoryTheory/Category/Grpd.lean + theorems: + - "CategoryTheory.Grpd.piIsoPi_hom_\u03C0" + - path: Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean + theorems: + - rieszContentAux_sup_le + - rieszContentAux_image_nonempty + - exists_lt_rieszContentAux_add_pos + - path: Mathlib/RingTheory/RingHom/Finite.lean + theorems: + - RingHom.finite_stableUnderBaseChange + - RingHom.finite_stableUnderComposition + - RingHom.finite_respectsIso + - path: Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean + theorems: + - FirstOrder.Ring.realize_termOfFreeCommRing + - FirstOrder.Ring.exists_term_realize_eq_freeCommRing + - path: Mathlib/Data/Int/Div.lean + theorems: + - Int.eq_mul_div_of_mul_eq_mul_of_dvd_left + - Int.eq_zero_of_dvd_of_natAbs_lt_natAbs + - path: Mathlib/Algebra/Ring/Regular.lean + theorems: + - isLeftRegular_of_non_zero_divisor + - isRightRegular_of_non_zero_divisor + - isRegular_iff_ne_zero' + - path: Mathlib/Tactic/NormNum/Ineq.lean + theorems: + - Mathlib.Meta.NormNum.isRat_lt_true + - Mathlib.Meta.NormNum.isRat_le_true + - path: Mathlib/CategoryTheory/DifferentialObject.lean + theorems: + - CategoryTheory.DifferentialObject.eqToHom_f + - path: Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean + theorems: + - Filter.ZeroAtFilter.boundedAtFilter + - Filter.ZeroAtFilter.smul + - Filter.BoundedAtFilter.add + - Filter.ZeroAtFilter.neg + - Filter.ZeroAtFilter.add + - Filter.BoundedAtFilter.mul + - path: Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean + theorems: + - IsDedekindDomain.selmerGroup.valuation_ker_eq + - IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_mod_eq + - IsDedekindDomain.selmerGroup.fromUnit_ker + - IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroToFun_eq + - IsDedekindDomain.selmerGroup.fromUnitLift_injective + - path: Mathlib/Topology/MetricSpace/Closeds.lean + theorems: + - EMetric.isClosed_subsets_of_isClosed + - EMetric.continuous_infEdist_hausdorffEdist + - EMetric.NonemptyCompacts.isClosed_in_closeds + - Metric.lipschitz_infDist_set + - Metric.lipschitz_infDist + - path: Mathlib/Geometry/Euclidean/Angle/Unoriented/Conformal.lean + theorems: + - InnerProductGeometry.IsConformalMap.preserves_angle + - path: Mathlib/Algebra/Module/LinearMap/Pointwise.lean + theorems: + - "preimage_smul_set\u209B\u2097" + - "image_smul_set\u209B\u2097" + - path: Mathlib/Geometry/Manifold/Metrizable.lean + theorems: + - ManifoldWithCorners.metrizableSpace + - path: Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean + theorems: + - AlgebraicTopology.DoldKan.degeneracy_comp_PInfty + - "AlgebraicTopology.DoldKan.HigherFacesVanish.comp_\u03C3" + - "AlgebraicTopology.DoldKan.\u03C3_comp_P_eq_zero" + - path: Mathlib/Topology/Algebra/UniformField.lean + theorems: + - UniformSpace.Completion.mul_hatInv_cancel + - UniformSpace.Completion.coe_inv + - UniformSpace.Completion.continuous_hatInv + - path: Mathlib/Data/Nat/ForSqrt.lean + theorems: + - Nat.sqrt.iter_sq_le + - Nat.iter_fp_bound + - Nat.div_mul_div_le + - Nat.mul_le_of_le_div + - Nat.AM_GM + - Nat.sqrt.lt_iter_succ_sq + - path: Mathlib/CategoryTheory/Abelian/Injective.lean + theorems: + - CategoryTheory.injective_of_preservesFiniteColimits_preadditiveYonedaObj + - path: Mathlib/CategoryTheory/Closed/Ideal.lean + theorems: + - CategoryTheory.prodComparison_iso + - CategoryTheory.bijection_symm_apply_id + - CategoryTheory.ExponentialIdeal.mk' + - CategoryTheory.ExponentialIdeal.mk_of_iso + - path: Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean + theorems: + - CategoryTheory.Limits.colimitLimitToLimitColimit_surjective + - CategoryTheory.Limits.colimitLimitToLimitColimit_injective + - path: Mathlib/RingTheory/Localization/Module.lean + theorems: + - Basis.ofIsLocalizedModule_span + - Basis.ofIsLocalizedModule_apply + - LinearIndependent.localization + - Basis.ofIsLocalizedModule_repr_apply + - span_eq_top_of_isLocalizedModule + - path: Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean + theorems: + - CategoryTheory.Limits.IsZero.of_iso + - CategoryTheory.Limits.IsZero.obj + - CategoryTheory.Functor.isZero + - path: Mathlib/CategoryTheory/Pi/Basic.lean + theorems: + - CategoryTheory.Functor.pi'_eval + - CategoryTheory.Functor.eqToHom_proj + - CategoryTheory.Functor.pi_ext + - path: Mathlib/CategoryTheory/Galois/Examples.lean + theorems: + - CategoryTheory.FintypeCat.Action.pretransitive_of_isConnected + - CategoryTheory.FintypeCat.Action.isConnected_of_transitive + - path: Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean + theorems: + - "CategoryTheory.PreservesImage.hom_comp_map_image_\u03B9" + - CategoryTheory.PreservesImage.factorThruImage_comp_hom + - "CategoryTheory.PreservesImage.inv_comp_image_\u03B9_map" + - path: Mathlib/Topology/FiberBundle/Constructions.lean + theorems: + - Trivialization.Prod.continuous_to_fun + - inducing_pullbackTotalSpaceEmbedding + - Bundle.Trivial.inducing_toProd + - Pullback.continuous_lift + - Trivialization.Prod.right_inv + - Pullback.continuous_totalSpaceMk + - Pullback.continuous_proj + - Trivialization.Prod.left_inv + - Trivialization.Prod.continuous_inv_fun + - path: Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean + theorems: + - AlgebraicGeometry.Polynomial.isOpenMap_comap_C + - AlgebraicGeometry.Polynomial.isOpen_imageOfDf + - AlgebraicGeometry.Polynomial.imageOfDf_eq_comap_C_compl_zeroLocus + - path: Mathlib/RingTheory/Derivation/ToSquareZero.lean + theorems: + - liftOfDerivationToSquareZero_mk_apply + - liftOfDerivationToSquareZero_mk_apply' + - path: Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean + theorems: + - CategoryTheory.Projective.projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj + - CategoryTheory.Projective.projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj' + - path: Mathlib/Topology/Metrizable/Urysohn.lean + theorems: + - TopologicalSpace.exists_inducing_l_infty + - path: Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean + theorems: + - orbit_fixingSubgroup_compl_subset + - mem_fixingSubgroup_iff_subset_fixedBy + - mem_fixingSubgroup_compl_iff_movedBy_subset + - path: Mathlib/RingTheory/RingHom/Surjective.lean + theorems: + - RingHom.surjective_stableUnderComposition + - RingHom.surjective_ofLocalizationSpan + - RingHom.surjective_stableUnderBaseChange + - RingHom.surjective_respectsIso + - path: Mathlib/CategoryTheory/Shift/CommShift.lean + theorems: + - CategoryTheory.Functor.map_shiftFunctorComm_hom_app + - CategoryTheory.Functor.CommShift.isoAdd_hom_app + - CategoryTheory.Functor.commShiftIso_add' + - CategoryTheory.Functor.CommShift.isoAdd_inv_app + - path: Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean + theorems: + - Module.Finite.of_basis + - path: Mathlib/Algebra/Group/Embedding.lean + theorems: + - mulLeftEmbedding_eq_mulRightEmbedding + - path: Mathlib/Topology/Instances/RatLemmas.lean + theorems: + - Rat.not_firstCountableTopology_opc + - Rat.not_countably_generated_cocompact + - Rat.not_countably_generated_nhds_infty_opc + - Rat.not_secondCountableTopology_opc + - path: Mathlib/RingTheory/WittVector/Isocrystal.lean + theorems: + - WittVector.isocrystal_classification + - path: Mathlib/Data/Int/Order/Lemmas.lean + theorems: + - Int.dvd_div_of_mul_dvd + - Int.eq_zero_of_abs_lt_dvd + - Int.natAbs_eq_iff_mul_self_eq + - Int.pow_right_injective + - Int.natAbs_le_iff_mul_self_le + - Int.natAbs_lt_iff_mul_self_lt + - path: Mathlib/CategoryTheory/GradedObject/Associator.lean + theorems: + - "CategoryTheory.GradedObject.\u03B9_mapBifunctorAssociator_inv" + - "CategoryTheory.GradedObject.\u03B9_mapBifunctorAssociator_hom" + - path: Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean + theorems: + - CategoryTheory.isSheaf_coherent + - CategoryTheory.coherentTopology.isSheaf_yoneda_obj + - path: Mathlib/Algebra/Lie/CartanMatrix.lean + theorems: + - CartanMatrix.Relations.adF_of_eq_eq_zero + - CartanMatrix.Relations.adE_of_eq_eq_zero + - path: Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean + theorems: + - EisensteinSeries.pairwise_disjoint_gammaSet + - EisensteinSeries.vecMul_SL2_mem_gammaSet + - EisensteinSeries.eisensteinSeries_slash_apply + - EisensteinSeries.eisSummand_SL2_apply + - path: Mathlib/NumberTheory/NumberField/Basic.lean + theorems: + - NumberField.isIntegral_of_mem_ringOfIntegers + - Int.not_isField + - NumberField.RingOfIntegers.not_isField + - NumberField.mem_span_integralBasis + - path: Mathlib/Topology/Homotopy/Basic.lean + theorems: + - ContinuousMap.Homotopy.extend_apply_of_one_le + - ContinuousMap.Homotopy.trans_apply + - ContinuousMap.HomotopyRel.eq_snd + - ContinuousMap.Homotopy.extend_apply_of_le_zero + - ContinuousMap.Homotopy.symm_symm + - ContinuousMap.Homotopy.symm_trans + - ContinuousMap.Homotopic.equivalence + - ContinuousMap.HomotopicRel.equivalence + - path: Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean + theorems: + - CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.hom_ext + - path: Mathlib/Algebra/CharP/CharAndCard.lean + theorems: + - isUnit_iff_not_dvd_char_of_ringChar_ne_zero + - prime_dvd_char_iff_dvd_card + - path: Mathlib/Analysis/Calculus/Deriv/Inverse.lean + theorems: + - HasDerivAt.eventually_ne + - not_differentiableWithinAt_of_local_left_inverse_hasDerivWithinAt_zero + - not_differentiableAt_of_local_left_inverse_hasDerivAt_zero + - path: Mathlib/GroupTheory/NoncommCoprod.lean + theorems: + - MonoidHom.noncommCoprod_comp_inl + - MonoidHom.noncommCoprod_comp_inr + - MonoidHom.noncommCoprod_unique + - MonoidHom.comp_noncommCoprod + - path: Mathlib/RingTheory/Bezout.lean + theorems: + - Function.Surjective.isBezout + - IsBezout.TFAE + - IsBezout.iff_span_pair_isPrincipal + - path: Mathlib/MeasureTheory/Covering/DensityTheorem.lean + theorems: + - IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt + - IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub + - IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div + - IsUnifLocDoublingMeasure.ae_tendsto_average + - IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mul + - path: Mathlib/Data/NNRat/Lemmas.lean + theorems: + - NNRat.num_div_den + - Rat.toNNRat_div' + - path: Mathlib/CategoryTheory/EffectiveEpi/Basic.lean + theorems: + - CategoryTheory.EffectiveEpiFamily.hom_ext + - path: Mathlib/Data/Analysis/Topology.lean + theorems: + - Ctop.Realizer.mem_nhds + - Ctop.Realizer.ext' + - Ctop.Realizer.isOpen + - Ctop.Realizer.is_basis + - Ctop.Realizer.ofEquiv_F + - Ctop.Realizer.isClosed_iff + - path: Mathlib/Topology/Algebra/Field.lean + theorems: + - IsPreconnected.eq_one_or_eq_neg_one_of_sq_eq + - IsPreconnected.eq_of_sq_eq + - IsPreconnected.eq_or_eq_neg_of_sq_eq + - IsLocalMin.inv + - path: Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean + theorems: + - CategoryTheory.Limits.hasCoproducts_of_finite_and_filtered + - path: Mathlib/Logic/Embedding/Set.lean + theorems: + - Equiv.asEmbedding_range + - path: Mathlib/Order/Monotone/Odd.lean + theorems: + - strictMono_of_odd_strictMonoOn_nonneg + - path: Mathlib/Data/Polynomial/Cardinal.lean + theorems: + - Polynomial.cardinal_mk_le_max + - Polynomial.cardinal_mk_eq_max + - path: Mathlib/Tactic/PushNeg.lean + theorems: + - Mathlib.Tactic.PushNeg.empty_ne_eq_nonempty + - Mathlib.Tactic.PushNeg.not_iff + - Mathlib.Tactic.PushNeg.ne_empty_eq_nonempty + - path: Mathlib/CategoryTheory/Abelian/FunctorCategory.lean + theorems: + - CategoryTheory.Abelian.FunctorCategory.coimageImageComparison_app + - path: Mathlib/Order/SuccPred/IntervalSucc.lean + theorems: + - Monotone.biUnion_Ico_Ioc_map_succ + - path: Mathlib/Topology/UniformSpace/CompleteSeparated.lean + theorems: + - IsComplete.isClosed + - path: Mathlib/Algebra/CharP/Quotient.lean + theorems: + - CharP.quotient_iff + - CharP.quotient' + - CharP.quotient_iff_le_ker_natCast + - Ideal.Quotient.index_eq_zero + - path: Mathlib/Order/Directed.lean + theorems: + - constant_of_monotoneOn_antitoneOn + - directedOn_pair' + - directedOn_range + - DirectedOn.insert + - exists_lt_of_directed_ge + - constant_of_monotone_antitone + - Directed.extend_bot + - directedOn_iff_directed + - path: Mathlib/Algebra/Category/GroupCat/Limits.lean + theorems: + - AddCommGroupCat.kernelIsoKer_hom_comp_subtype + - "AddCommGroupCat.kernelIsoKer_inv_comp_\u03B9" + - path: Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean + theorems: + - exists_deriv_eq_zero' + - path: Mathlib/Tactic/Positivity/Core.lean + theorems: + - Mathlib.Meta.Positivity.nz_of_isNegNat + - Mathlib.Meta.Positivity.pos_of_isRat + - Mathlib.Meta.Positivity.nz_of_isRat + - Mathlib.Meta.Positivity.pos_of_isNat + - Mathlib.Meta.Positivity.nonneg_of_isRat + - Mathlib.Meta.Positivity.nonneg_of_isNat + - path: .lake/packages/std/Std/Data/BinomialHeap/Lemmas.lean + theorems: + - Std.BinomialHeap.Imp.HeapNode.WF.realSize_eq + - Std.BinomialHeap.Imp.Heap.WF.size_eq + - Std.BinomialHeap.Imp.Heap.findMin_val + - Std.BinomialHeap.Imp.Heap.deleteMin_fst + - path: Mathlib/CategoryTheory/Abelian/Projective.lean + theorems: + - CategoryTheory.projective_of_preservesFiniteColimits_preadditiveCoyonedaObj + - path: Mathlib/Algebra/BigOperators/Pi.lean + theorems: + - Finset.univ_prod_mulSingle + - pi_eq_sum_univ + - MonoidHom.functions_ext + - prod_mk_prod + - path: Mathlib/NumberTheory/MaricaSchoenheim.lean + theorems: + - Nat.grahamConjecture_of_squarefree + - path: Mathlib/Algebra/Ring/CentroidHom.lean + theorems: + - NonUnitalNonAssocCommSemiring.mem_center_iff + - CentroidHom.comp_mul_comm + - NonUnitalNonAssocSemiring.mem_center_iff + - CentroidHom.cancel_left + - path: Mathlib/NumberTheory/NumberField/Norm.lean + theorems: + - RingOfIntegers.isUnit_norm + - RingOfIntegers.isUnit_norm_of_isGalois + - RingOfIntegers.norm_algebraMap + - RingOfIntegers.dvd_norm + - RingOfIntegers.norm_norm + - path: Mathlib/Algebra/Ring/Hom/Basic.lean + theorems: + - Function.Injective.isDomain + - RingHom.codomain_trivial_iff_range_eq_singleton_zero + - path: Mathlib/CategoryTheory/Abelian/Transfer.lean + theorems: + - CategoryTheory.AbelianOfAdjunction.hasKernels + - CategoryTheory.AbelianOfAdjunction.hasCokernels + - CategoryTheory.AbelianOfAdjunction.coimageIsoImage_hom + - path: Mathlib/Init/Meta/WellFoundedTactics.lean + theorems: + - Nat.zero_lt_one_add + - Nat.lt_add_of_zero_lt_left + - path: Mathlib/GroupTheory/ClassEquation.lean + theorems: + - Group.card_center_add_sum_card_noncenter_eq_card + - Group.nat_card_center_add_sum_card_noncenter_eq_card + - Group.sum_card_conj_classes_eq_card + - sum_conjClasses_card_eq_card + - path: Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean + theorems: + - CategoryTheory.R_faithful_of_counit_isIso + - CategoryTheory.L_faithful_of_unit_isIso + - path: Mathlib/Init/Data/Int/Order.lean + theorems: + - Int.eq_zero_or_eq_zero_of_mul_eq_zero + - path: Mathlib/Data/Finsupp/Antidiagonal.lean + theorems: + - Finsupp.antidiagonal_single + - Finsupp.prod_antidiagonal_swap + - path: Mathlib/Analysis/Analytic/Linear.lean + theorems: + - ContinuousLinearMap.hasFPowerSeriesOnBall_bilinear + - ContinuousLinearMap.hasFPowerSeriesOnBall + - path: Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean + theorems: + - ContinuousLinearMap.opNorm_lsmul + - ContinuousLinearMap.opNorm_lsmul_le + - ContinuousLinearMap.norm_toSpanSingleton + - path: Mathlib/Algebra/Order/Algebra.lean + theorems: + - algebraMap_monotone + - path: Mathlib/Data/Set/Pointwise/ListOfFn.lean + theorems: + - Set.mem_list_prod + - Set.mem_prod_list_ofFn + - path: Mathlib/Geometry/Euclidean/Sphere/Ptolemy.lean + theorems: + - EuclideanGeometry.mul_dist_add_mul_dist_eq_mul_dist_of_cospherical + - path: Mathlib/Logic/Equiv/Functor.lean + theorems: + - Bifunctor.mapEquiv_refl_refl + - path: Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean + theorems: + - "SemiNormedGroupCat\u2081.iso_isometry" + - SemiNormedGroupCat.iso_isometry_of_normNoninc + - "SemiNormedGroupCat\u2081.isZero_of_subsingleton" + - SemiNormedGroupCat.isZero_of_subsingleton + - path: Mathlib/RingTheory/EuclideanDomain.lean + theorems: + - left_div_gcd_ne_zero + - right_div_gcd_ne_zero + - path: Mathlib/Geometry/Manifold/Sheaf/Smooth.lean + theorems: + - smoothSheaf.eval_surjective + - smoothSheafCommRing.forgetStalk_inv_comp_eval + - "smoothSheafCommRing.\u03B9_forgetStalk_inv" + - smoothSheafCommRing.eval_surjective + - smoothSheafCommRing.forgetStalk_hom_comp_evalHom + - path: Mathlib/Data/Sum/Order.lean + theorems: + - Sum.noMaxOrder_iff + - Sum.denselyOrdered_iff + - Sum.noMinOrder_iff + - path: Mathlib/Topology/Order/Hom/Basic.lean + theorems: + - ContinuousOrderHom.cancel_left + - path: Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean + theorems: + - TopCat.isTopologicalBasis_cofiltered_limit + - path: Mathlib/Topology/Category/Profinite/Basic.lean + theorems: + - Profinite.epi_iff_surjective + - Profinite.mono_iff_injective + - path: Mathlib/GroupTheory/Subgroup/MulOpposite.lean + theorems: + - Subgroup.unop_closure + - Subgroup.op_closure + - path: Mathlib/Order/CompactlyGenerated/Intervals.lean + theorems: + - complementedLattice_of_complementedLattice_Iic + - Set.Iic.isCompactElement + - path: Mathlib/LinearAlgebra/Basis/Bilinear.lean + theorems: + - LinearMap.sum_repr_mul_repr_mul + - "LinearMap.sum_repr_mul_repr_mul\u209B\u2097" + - path: Mathlib/Data/ZMod/Factorial.lean + theorems: + - ZMod.cast_descFactorial + - path: Mathlib/Analysis/Analytic/RadiusLiminf.lean + theorems: + - FormalMultilinearSeries.radius_eq_liminf + - path: Mathlib/LinearAlgebra/FreeAlgebra.lean + theorems: + - FreeAlgebra.rank_eq + - path: Mathlib/RingTheory/HahnSeries/PowerSeries.lean + theorems: + - HahnSeries.ofPowerSeries_C + - HahnSeries.ofPowerSeries_X + - HahnSeries.ofPowerSeries_apply_coeff + - HahnSeries.ofPowerSeries_apply + - HahnSeries.ofPowerSeries_X_pow + - path: Mathlib/SetTheory/Cardinal/UnivLE.lean + theorems: + - univLE_total + - univLE_iff_cardinal_le + - path: Mathlib/Analysis/Calculus/FDeriv/Pi.lean + theorems: + - hasFDerivAt_update + - path: Mathlib/Tactic/NoncommRing.lean + theorems: + - Mathlib.Tactic.NoncommRing.nat_lit_mul_eq_nsmul + - Mathlib.Tactic.NoncommRing.mul_nat_lit_eq_nsmul + - path: Mathlib/CategoryTheory/Limits/KanExtension.lean + theorems: + - CategoryTheory.Ran.reflective + - CategoryTheory.Lan.coreflective + - path: Mathlib/Data/Int/Cast/Field.lean + theorems: + - Int.cast_neg_natCast + - Int.cast_div + - path: Mathlib/Control/EquivFunctor.lean + theorems: + - EquivFunctor.mapEquiv_refl + - EquivFunctor.mapEquiv_trans + - path: Mathlib/LinearAlgebra/TensorProduct/DirectLimit.lean + theorems: + - TensorProduct.directLimitRight_tmul_of + - TensorProduct.directLimitRight_symm_of_tmul + - TensorProduct.toDirectLimit_tmul_of + - path: Mathlib/CategoryTheory/Triangulated/TriangleShift.lean + theorems: + - CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_eq + - path: Mathlib/CategoryTheory/Monoidal/Free/Basic.lean + theorems: + - CategoryTheory.FreeMonoidalCategory.Hom.inductionOn + - path: Mathlib/Algebra/Module/BigOperators.lean + theorems: + - Finset.cast_card + - Multiset.sum_smul_sum + - Finset.sum_smul_sum + - path: Mathlib/Data/Vector/Zip.lean + theorems: + - Vector.zipWith_tail + - Vector.zipWith_get + - path: Mathlib/Algebra/CharP/LocalRing.lean + theorems: + - charP_zero_or_prime_power + - path: Mathlib/Data/Nat/Cast/Commute.lean + theorems: + - Nat.cast_commute + - path: Mathlib/Data/Set/Intervals/Image.lean + theorems: + - Set.image_subtype_val_Ixx_Iix + - MonotoneOn.mapsTo_Ici + - Set.image_subtype_val_Ixx_Ixi + - AntitoneOn.mapsTo_Iic + - AntitoneOn.mapsTo_Ici + - MonotoneOn.mapsTo_Iic + - path: Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean + theorems: + - analyticAt_clog + - AnalyticAt.cpow + - analyticOn_cexp + - path: Mathlib/Algebra/Order/Ring/Pow.lean + theorems: + - one_add_mul_le_pow' + - one_add_mul_sub_le_pow + - path: Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean + theorems: + - Complex.volume_preserving_equiv_pi + - path: Mathlib/Analysis/NormedSpace/Star/Exponential.lean + theorems: + - Commute.expUnitary + - Commute.expUnitary_add + - path: Mathlib/SetTheory/Game/State.lean + theorems: + - SetTheory.PGame.turnBound_ne_zero_of_right_move + - SetTheory.PGame.turnBound_ne_zero_of_left_move + - path: Mathlib/Analysis/NormedSpace/Complemented.lean + theorems: + - Submodule.ClosedComplemented.of_quotient_finiteDimensional + - ContinuousLinearMap.ker_closedComplemented_of_finiteDimensional_range + - path: Mathlib/Algebra/Group/Units/Equiv.lean + theorems: + - toUnits_val_apply + - path: Mathlib/Data/Multiset/Sections.lean + theorems: + - Multiset.prod_map_sum + - Multiset.coe_sections + - Multiset.sections_add + - Multiset.mem_sections + - Multiset.card_sections + - path: Mathlib/Data/Polynomial/Degree/CardPowDegree.lean + theorems: + - Polynomial.cardPowDegree_isEuclidean + - Polynomial.cardPowDegree_apply + - path: Mathlib/CategoryTheory/PUnit.lean + theorems: + - CategoryTheory.equiv_punit_iff_unique + - CategoryTheory.Functor.punit_ext' + - path: Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean + theorems: + - QuadraticModuleCat.Hom.toIsometry_injective + - path: Mathlib/Topology/Category/TopCat/Yoneda.lean + theorems: + - ContinuousMap.piComparison_fac + - path: Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean + theorems: + - lineMap_one_half + - homothety_one_half + - path: Mathlib/CategoryTheory/NatTrans.lean + theorems: + - CategoryTheory.congr_app + - path: Mathlib/RingTheory/MvPolynomial/Ideal.lean + theorems: + - MvPolynomial.mem_ideal_span_X_image + - MvPolynomial.mem_ideal_span_monomial_image + - MvPolynomial.mem_ideal_span_monomial_image_iff_dvd + - path: Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean + theorems: + - TensorAlgebra.rank_eq + - "TensorAlgebra.equivFreeAlgebra_\u03B9_apply" + - path: Mathlib/AlgebraicTopology/Quasicategory.lean + theorems: + - SSet.Quasicategory.hornFilling + - SSet.quasicategory_of_filler + - path: Mathlib/RingTheory/Complex.lean + theorems: + - Algebra.trace_complex_apply + - Algebra.norm_complex_apply + - Algebra.leftMulMatrix_complex + - path: Mathlib/CategoryTheory/Bicategory/Free.lean + theorems: + - "CategoryTheory.FreeBicategory.liftHom\u2082_congr" + - path: Mathlib/Algebra/Category/GroupCat/Abelian.lean + theorems: + - AddCommGroupCat.exact_iff + - path: Mathlib/RingTheory/Localization/Cardinality.lean + theorems: + - IsLocalization.card_le + - path: Mathlib/Tactic/LinearCombination.lean + theorems: + - Mathlib.Tactic.LinearCombination.eq_of_add + - path: Mathlib/AlgebraicGeometry/Limits.lean + theorems: + - AlgebraicGeometry.bot_isAffineOpen + - AlgebraicGeometry.Scheme.empty_ext + - path: Mathlib/CategoryTheory/Limits/Constructions/Pullbacks.lean + theorems: + - CategoryTheory.Limits.hasColimit_span_of_hasColimit_pair_of_hasColimit_parallelPair + - CategoryTheory.Limits.hasLimit_cospan_of_hasLimit_pair_of_hasLimit_parallelPair + - path: Mathlib/Data/Complex/Determinant.lean + theorems: + - Complex.det_conjAe + - Complex.linearEquiv_det_conjAe + - path: Mathlib/Algebra/Invertible/GroupWithZero.lean + theorems: + - nonzero_of_invertible + - path: Mathlib/Algebra/Field/Defs.lean + theorems: + - Rat.smul_one_eq_coe + - path: Mathlib/Tactic/Peel.lean + theorems: + - Mathlib.Tactic.Peel.eventually_congr + - Mathlib.Tactic.Peel.and_imp_left_of_imp_imp + - Mathlib.Tactic.Peel.frequently_congr + - path: Mathlib/Data/MvPolynomial/Cardinal.lean + theorems: + - MvPolynomial.cardinal_mk_eq_max + - MvPolynomial.cardinal_mk_eq_max_lift + - MvPolynomial.cardinal_mk_le_max + - MvPolynomial.cardinal_lift_mk_le_max + - path: Mathlib/CategoryTheory/WithTerminal.lean + theorems: + - CategoryTheory.WithTerminal.lift_map_liftStar + - CategoryTheory.WithInitial.liftStar_lift_map + - path: Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean + theorems: + - IsLocalization.surj_of_gcd_domain + - path: Mathlib/Topology/Algebra/Module/WeakDual.lean + theorems: + - WeakBilin.tendsto_iff_forall_eval_tendsto + - path: Mathlib/CategoryTheory/Limits/Constructions/Over/Products.lean + theorems: + - CategoryTheory.Over.over_hasTerminal + - path: Mathlib/Condensed/TopComparison.lean + theorems: + - equalizerCondition_yonedaPresheaf + - factorsThrough_of_pullbackCondition + - path: Mathlib/CategoryTheory/Filtered/Small.lean + theorems: + - CategoryTheory.IsCofiltered.small_fullSubcategory_cofilteredClosure + - CategoryTheory.IsFiltered.small_fullSubcategory_filteredClosure + - path: Mathlib/RingTheory/Localization/BaseChange.lean + theorems: + - isLocalizedModule_iff_isBaseChange + - IsLocalizedModule.isBaseChange + - path: Mathlib/Analysis/NormedSpace/RieszLemma.lean + theorems: + - Metric.closedBall_infDist_compl_subset_closure + - riesz_lemma_of_norm_lt + - riesz_lemma + - path: Mathlib/Order/FixedPoints.lean + theorems: + - OrderHom.lfp_lfp + - OrderHom.lfp_induction + - path: Mathlib/Topology/Sheaves/SheafOfFunctions.lean + theorems: + - TopCat.Presheaf.toTypes_isSheaf + - path: Mathlib/CategoryTheory/Idempotents/FunctorCategories.lean + theorems: + - CategoryTheory.Idempotents.toKaroubi_comp_karoubiFunctorCategoryEmbedding + - path: Mathlib/Order/ZornAtoms.lean + theorems: + - IsCoatomic.of_isChain_bounded + - path: Mathlib/Algebra/Symmetrized.lean + theorems: + - SymAlg.sym_mul_self + - SymAlg.mul_comm + - SymAlg.unsym_mul_self + - path: Mathlib/Analysis/Convex/Contractible.lean + theorems: + - StarConvex.contractibleSpace + - path: .lake/packages/std/Std/Classes/BEq.lean + theorems: + - beq_eq_false_iff_ne + - path: Mathlib/Data/Nat/Choose/Dvd.lean + theorems: + - Nat.Prime.dvd_choose_add + - path: Mathlib/Data/Nat/Cast/Prod.lean + theorems: + - Prod.snd_natCast + - Prod.fst_natCast + - path: Mathlib/FieldTheory/Tower.lean + theorems: + - FiniteDimensional.Subalgebra.isSimpleOrder_of_finrank_prime + - FiniteDimensional.right + - path: Mathlib/Data/List/NodupEquivFin.lean + theorems: + - List.sublist_iff_exists_orderEmbedding_get?_eq + - List.duplicate_iff_exists_distinct_get + - path: Mathlib/Analysis/SpecialFunctions/Sqrt.lean + theorems: + - Real.deriv_sqrt_aux + - path: Mathlib/CategoryTheory/Comma/Basic.lean + theorems: + - CategoryTheory.Comma.eqToHom_left + - CategoryTheory.Comma.eqToHom_right + - path: Mathlib/Topology/Hom/Open.lean + theorems: + - ContinuousOpenMap.cancel_left + - path: Mathlib/Data/Multiset/Sort.lean + theorems: + - Multiset.mem_sort + - path: Mathlib/Topology/ContinuousFunction/Sigma.lean + theorems: + - ContinuousMap.embedding_sigmaMk_comp + - path: Mathlib/Topology/Instances/Complex.lean + theorems: + - Complex.subfield_eq_of_closed + - Complex.uniformContinuous_ringHom_eq_id_or_conj + - path: Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean + theorems: + - "AlgebraicTopology.DoldKan.MorphComponents.id_\u03C6" + - AlgebraicTopology.DoldKan.decomposition_Q + - "AlgebraicTopology.DoldKan.MorphComponents.postComp_\u03C6" + - path: Mathlib/Algebra/Field/IsField.lean + theorems: + - uniq_inv_of_isField + - path: Mathlib/Data/Prod/Lex.lean + theorems: + - Prod.Lex.toLex_mono + - Prod.Lex.toLex_strictMono + - path: Mathlib/Data/Fin/FlagRange.lean + theorems: + - IsMaxChain.range_fin_of_covBy + - path: Mathlib/CategoryTheory/Shift/Localization.lean + theorems: + - CategoryTheory.MorphismProperty.IsCompatibleWithShift.iff + - CategoryTheory.MorphismProperty.IsCompatibleWithShift.shiftFunctor_comp_inverts + - path: Mathlib/Topology/Instances/TrivSqZeroExt.lean + theorems: + - TrivSqZeroExt.nhds_def + - path: Mathlib/LinearAlgebra/QuotientPi.lean + theorems: + - Submodule.piQuotientLift_mk + - Submodule.quotientPi_aux.left_inv + - Submodule.quotientPi_aux.right_inv + - path: Mathlib/Geometry/Euclidean/Inversion/Calculus.lean + theorems: + - EuclideanGeometry.hasFDerivAt_inversion + - DifferentiableAt.inversion + - path: Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean + theorems: + - CategoryTheory.extensive_regular_generate_coherent + - CategoryTheory.effectiveEpi_desc_iff_effectiveEpiFamily + - path: Mathlib/Analysis/NormedSpace/WeakDual.lean + theorems: + - WeakDual.isClosed_polar + - NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology + - path: Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean + theorems: + - CompHaus.effectiveEpiFamily_tfae + - CompHaus.effectiveEpi_tfae + - path: Mathlib/LinearAlgebra/TensorProduct/Basis.lean + theorems: + - Basis.tensorProduct_apply + - Basis.tensorProduct_apply' + - Basis.tensorProduct_repr_tmul_apply + - path: Mathlib/Algebra/Order/Sub/Basic.lean + theorems: + - AddHom.le_map_tsub + - le_tsub_mul + - OrderIso.map_tsub + - path: Mathlib/Topology/ExtendFrom.lean + theorems: + - continuousOn_extendFrom + - continuous_extendFrom + - path: Mathlib/Data/Finsupp/WellFounded.lean + theorems: + - Finsupp.Lex.acc + - path: Mathlib/Analysis/Hofer.lean + theorems: + - hofer + - path: Mathlib/Data/Polynomial/Identities.lean + theorems: + - Polynomial.poly_binom_aux3 + - Polynomial.poly_binom_aux2 + - path: Mathlib/Algebra/CharP/Invertible.lean + theorems: + - not_ringChar_dvd_of_invertible + - path: Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.lean + theorems: + - "CategoryTheory.reflectsIsomorphisms_forget\u2082" + - path: Mathlib/Logic/Small/Defs.lean + theorems: + - not_small_type + - Shrink.ext + - path: .lake/packages/std/Std/Data/HashMap/Basic.lean + theorems: + - Std.HashMap.Imp.WF.empty + - path: Mathlib/CategoryTheory/Limits/ColimitLimit.lean + theorems: + - "CategoryTheory.Limits.\u03B9_colimitLimitToLimitColimit_\u03C0" + - "CategoryTheory.Limits.\u03B9_colimitLimitToLimitColimit_\u03C0_apply" + - path: Mathlib/Analysis/Complex/UnitDisc/Basic.lean + theorems: + - Complex.UnitDisc.normSq_lt_one + - Complex.UnitDisc.coe_ne_neg_one + - path: Mathlib/CategoryTheory/Abelian/Ext.lean + theorems: + - isZero_Ext_succ_of_projective + - path: Mathlib/Algebra/Category/ModuleCat/Abelian.lean + theorems: + - ModuleCat.exact_iff + - path: Mathlib/Order/Monotone/Extension.lean + theorems: + - MonotoneOn.exists_monotone_extension + - path: Mathlib/Topology/Metrizable/Basic.lean + theorems: + - TopologicalSpace.IsSeparable.secondCountableTopology + - path: Mathlib/AlgebraicTopology/SimplicialSet.lean + theorems: + - SSet.standardSimplex.map_apply + - SSet.horn.hom_ext + - path: Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.lean + theorems: + - Equiv.Perm.exists_with_cycleType_iff + - path: Mathlib/LinearAlgebra/Matrix/Symmetric.lean + theorems: + - Matrix.IsSymm.pow + - path: .lake/packages/aesop/Aesop/Util/Basic.lean + theorems: + - Aesop.Array.size_modify + - path: Mathlib/Lean/Thunk.lean + theorems: + - Thunk.ext + - path: Mathlib/Topology/Sheaves/Sheaf.lean + theorems: + - TopCat.Presheaf.isSheaf_unit + - path: Mathlib/Algebra/Homology/Bifunctor.lean + theorems: + - "HomologicalComplex.\u03B9_mapBifunctorMap" + - path: Mathlib/ModelTheory/Algebra/Field/Basic.lean + theorems: + - FirstOrder.Field.FieldAxiom.realize_toSentence_iff_toProp + - path: Mathlib/Dynamics/BirkhoffSum/Basic.lean + theorems: + - birkhoffSum_add + - Function.IsFixedPt.birkhoffSum_eq + - birkhoffSum_apply_sub_birkhoffSum + - path: Mathlib/Logic/Equiv/Nat.lean + theorems: + - Equiv.natSumNatEquivNat_apply + - path: Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean + theorems: + - MeasureTheory.hahn_decomposition + - path: Mathlib/RingTheory/Coprime/Ideal.lean + theorems: + - Ideal.iSup_iInf_eq_top_iff_pairwise + - path: Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean + theorems: + - CategoryTheory.Limits.HasEqualizersOfHasPullbacksAndBinaryProducts.pullbackFst_eq_pullback_snd + - CategoryTheory.Limits.HasCoequalizersOfHasPushoutsAndBinaryCoproducts.pushoutInl_eq_pushout_inr + - path: Mathlib/Data/Int/Range.lean + theorems: + - Int.mem_range_iff + - path: Mathlib/Data/Finsupp/Pointwise.lean + theorems: + - Finsupp.support_mul + - path: Mathlib/Algebra/Category/GroupCat/Injective.lean + theorems: + - Module.Baer.of_divisible + - path: Mathlib/RingTheory/QuotientNilpotent.lean + theorems: + - Ideal.isRadical_iff_quotient_reduced + - IsNilpotent.isUnit_quotient_mk_iff + - Ideal.IsNilpotent.induction_on + - path: Mathlib/Data/List/Sections.lean + theorems: + - List.mem_sections + - path: Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean + theorems: + - CategoryTheory.NatTrans.app_sum + - path: Mathlib/NumberTheory/Cyclotomic/PID.lean + theorems: + - IsCyclotomicExtension.Rat.three_pid + - IsCyclotomicExtension.Rat.five_pid + - path: Mathlib/CategoryTheory/Idempotents/Biproducts.lean + theorems: + - CategoryTheory.Idempotents.Karoubi.karoubi_hasFiniteBiproducts + - path: Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean + theorems: + - CategoryTheory.regularTopology.mem_sieves_iff_hasEffectiveEpi + - CategoryTheory.regularTopology.mem_sieves_of_hasEffectiveEpi + - path: Mathlib/Data/FunLike/Basic.lean + theorems: + - DFunLike.coe_fn_eq + - path: Mathlib/Algebra/Ring/Prod.lean + theorems: + - NonUnitalRingHom.prod_unique + - false_of_nontrivial_of_product_domain + - path: Mathlib/Data/Rat/Denumerable.lean + theorems: + - Cardinal.mkRat + - path: Mathlib/Algebra/Order/Field/Canonical/Basic.lean + theorems: + - tsub_div + - path: Mathlib/AlgebraicTopology/TopologicalSimplex.lean + theorems: + - SimplexCategory.continuous_toTopMap + - path: Mathlib/RingTheory/Valuation/Integral.lean + theorems: + - Valuation.Integers.mem_of_integral + - path: Mathlib/CategoryTheory/Abelian/Refinements.lean + theorems: + - CategoryTheory.ShortComplex.exact_iff_exact_up_to_refinements + - CategoryTheory.epi_iff_surjective_up_to_refinements + - CategoryTheory.ShortComplex.Exact.exact_up_to_refinements + - path: Mathlib/Geometry/Manifold/Sheaf/LocallyRingedSpace.lean + theorems: + - smoothSheafCommRing.nonunits_stalk + - path: Mathlib/NumberTheory/PrimesCongruentOne.lean + theorems: + - Nat.exists_prime_gt_modEq_one + - Nat.frequently_atTop_modEq_one + - path: Mathlib/Topology/Algebra/Order/Group.lean + theorems: + - tendsto_zero_iff_abs_tendsto_zero + - path: Mathlib/CategoryTheory/Balanced.lean + theorems: + - CategoryTheory.balanced_opposite + - path: Mathlib/Topology/Algebra/UniformFilterBasis.lean + theorems: + - AddGroupFilterBasis.cauchy_iff + - path: Mathlib/Algebra/CharP/Reduced.lean + theorems: + - ExpChar.pow_prime_pow_mul_eq_one_iff + - iterateFrobenius_inj + - isSquare_of_charTwo' + - path: Mathlib/Topology/Spectral/Hom.lean + theorems: + - SpectralMap.cancel_left + - path: Mathlib/Combinatorics/Additive/RuzsaCovering.lean + theorems: + - Set.exists_subset_mul_div + - Finset.exists_subset_mul_div + - path: Mathlib/Data/Opposite.lean + theorems: + - Opposite.unop_injective + - path: Mathlib/CategoryTheory/Limits/Constructions/Over/Connected.lean + theorems: + - CategoryTheory.Over.CreatesConnected.raised_cone_lowers_to_original + - path: Mathlib/Order/Hom/Bounded.lean + theorems: + - BoundedOrderHom.cancel_left + - BotHom.cancel_left + - map_eq_bot_iff + - TopHom.cancel_left + - map_eq_top_iff + - path: Mathlib/Analysis/NormedSpace/LpEquiv.lean + theorems: + - "Mem\u2113p.all" + - equiv_lpPiLp_norm + - path: Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean + theorems: + - CategoryTheory.Injective.injective_iff_preservesEpimorphisms_preadditive_yoneda_obj' + - CategoryTheory.Injective.injective_iff_preservesEpimorphisms_preadditiveYoneda_obj + - path: Mathlib/LinearAlgebra/QuadraticForm/Dual.lean + theorems: + - LinearMap.separatingLeft_dualProd + - path: Mathlib/RingTheory/Polynomial/IrreducibleRing.lean + theorems: + - Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical + - path: Mathlib/Tactic/Linarith/Preprocessing.lean + theorems: + - Linarith.without_one_mul + - path: Mathlib/GroupTheory/Subgroup/Saturated.lean + theorems: + - Subgroup.saturated_iff_zpow + - AddSubgroup.ker_saturated + - path: Mathlib/Data/UInt.lean + theorems: + - UInt8.toChar_aux + - path: Mathlib/Data/ZMod/Module.lean + theorems: + - ZMod.smul_mem + - ZMod.map_smul + - path: Mathlib/CategoryTheory/Linear/LinearFunctor.lean + theorems: + - CategoryTheory.Functor.map_units_smul + - path: Mathlib/Data/Fin/Tuple/Monotone.lean + theorems: + - liftFun_vecCons + - path: Mathlib/Order/Filter/Interval.lean + theorems: + - Filter.tendstoIxxClass_inf + - Filter.tendstoIxxClass_principal + - path: Mathlib/Analysis/InnerProductSpace/ConformalLinearMap.lean + theorems: + - isConformalMap_iff + - path: Mathlib/Data/Complex/Cardinality.lean + theorems: + - not_countable_complex + - path: Mathlib/CategoryTheory/Triangulated/Triangulated.lean + theorems: + - CategoryTheory.IsTriangulated.mk' + - path: Mathlib/Data/Polynomial/CancelLeads.lean + theorems: + - Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm + - path: Mathlib/Analysis/InnerProductSpace/ProdL2.lean + theorems: + - OrthonormalBasis.prod_apply + - path: Mathlib/NumberTheory/DirichletCharacter/Bounds.lean + theorems: + - DirichletCharacter.unit_norm_eq_one + - DirichletCharacter.norm_le_one + - path: Mathlib/ModelTheory/ElementarySubstructures.lean + theorems: + - FirstOrder.Language.ElementarySubstructure.theory_model_iff + - path: Mathlib/CategoryTheory/EssentialImage.lean + theorems: + - CategoryTheory.Functor.essSurj_of_surj + - path: Mathlib/Init/Data/Nat/GCD.lean + theorems: + - Nat.gcd_def + - path: Mathlib/Topology/Algebra/Module/Alternating/Basic.lean + theorems: + - ContinuousAlternatingMap.toAlternatingMap_injective + - ContinuousMultilinearMap.alternatization_apply_toAlternatingMap + - path: Mathlib/FieldTheory/AxGrothendieck.lean + theorems: + - ax_grothendieck_of_locally_finite + - path: Mathlib/CategoryTheory/Limits/Indization/IndObject.lean + theorems: + - CategoryTheory.Limits.isIndObject_of_isFiltered_of_finallySmall + - path: Mathlib/Dynamics/FixedPoints/Topology.lean + theorems: + - isFixedPt_of_tendsto_iterate + - path: Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean + theorems: + - GeneralizedContinuedFraction.IntFractPair.stream_isSeq + - path: Mathlib/Algebra/Order/Ring/Star.lean + theorems: + - StarOrderedRing.mul_le_mul_of_nonneg_left + - path: Mathlib/Topology/Algebra/UniformMulAction.lean + theorems: + - UniformInducing.uniformContinuousConstSMul + - path: Mathlib/Tactic/NormNum/IsCoprime.lean + theorems: + - Tactic.NormNum.int_not_isCoprime_helper + - path: Mathlib/RingTheory/DedekindDomain/SInteger.lean + theorems: + - Set.integer_eq + - path: Mathlib/LinearAlgebra/GeneralLinearGroup.lean + theorems: + - LinearMap.GeneralLinearGroup.generalLinearEquiv_to_linearMap + - path: Mathlib/CategoryTheory/Skeletal.lean + theorems: + - CategoryTheory.skeleton_skeletal + - path: Mathlib/CategoryTheory/Whiskering.lean + theorems: + - CategoryTheory.Functor.triangle + - path: Mathlib/RepresentationTheory/GroupCohomology/Basic.lean + theorems: + - inhomogeneousCochains.d_eq + - path: Mathlib/CategoryTheory/Products/Basic.lean + theorems: + - CategoryTheory.isIso_prod_iff + - path: Mathlib/Analysis/Complex/Convex.lean + theorems: + - Complex.convexHull_reProdIm + - path: Mathlib/Probability/ConditionalExpectation.lean + theorems: + - MeasureTheory.condexp_indep_eq + - path: Mathlib/Combinatorics/Quiver/Push.lean + theorems: + - Quiver.Push.lift_unique + - Quiver.Push.lift_comp + - path: Mathlib/NumberTheory/FrobeniusNumber.lean + theorems: + - frobeniusNumber_pair + - path: Mathlib/Analysis/Analytic/Polynomial.lean + theorems: + - AnalyticAt.aeval_mvPolynomial + - AnalyticAt.aeval_polynomial + - path: Mathlib/Topology/Category/TopCat/EffectiveEpi.lean + theorems: + - TopCat.effectiveEpi_iff_quotientMap + - path: Mathlib/Data/Set/Pointwise/Finite.lean + theorems: + - Group.card_pow_eq_card_pow_card_univ + - path: Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean + theorems: + - AEMeasurable.inner + - path: Mathlib/Data/ENat/Lattice.lean + theorems: + - ENat.iInf_coe_ne_top + - path: Mathlib/Data/Set/Pointwise/Iterate.lean + theorems: + - smul_eq_self_of_preimage_zpow_eq_self + - path: Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean + theorems: + - ContinuousLinearEquiv.homothety_inverse + - path: Mathlib/RingTheory/NormTrace.lean + theorems: + - Algebra.norm_one_add_smul + - path: Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean + theorems: + - CategoryTheory.MonoidalNatIso.ofComponents.inv_app + - path: Mathlib/Algebra/Homology/ShortComplex/Basic.lean + theorems: + - "CategoryTheory.ShortComplex.\u03C0\u2081To\u03C0\u2082_comp_\u03C0\u2082To\u03C0\ + \u2083" + - path: Mathlib/Analysis/Distribution/FourierSchwartz.lean + theorems: + - SchwartzMap.hasFDerivAt_fourier + - path: Mathlib/MeasureTheory/Integral/ExpDecay.lean + theorems: + - integrable_of_isBigO_exp_neg + - exp_neg_integrableOn_Ioi + - path: Mathlib/CategoryTheory/Abelian/Generator.lean + theorems: + - CategoryTheory.Abelian.has_projective_separator + - path: Mathlib/GroupTheory/GroupAction/Support.lean + theorems: + - MulAction.Supports.smul + - path: Mathlib/Data/Finset/PiInduction.lean + theorems: + - Finset.induction_on_pi_of_choice + - path: Mathlib/Init/Data/List/Lemmas.lean + theorems: + - List.bex_cons + - path: Mathlib/Tactic/CategoryTheory/Elementwise.lean + theorems: + - Tactic.Elementwise.hom_elementwise + - path: Mathlib/Data/Bool/Set.lean + theorems: + - Bool.range_eq + - path: Mathlib/Topology/Sheaves/LocallySurjective.lean + theorems: + - TopCat.Presheaf.locally_surjective_iff_surjective_on_stalks + - path: Mathlib/Order/CompletePartialOrder.lean + theorems: + - CompletePartialOrder.scottContinuous + - path: Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean + theorems: + - TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover + - path: Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean + theorems: + - Ideal.finrank_quotient_eq_sum + - path: Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean + theorems: + - Matrix.det_mvPolynomialX_ne_zero + - path: Mathlib/Topology/Separation/NotNormal.lean + theorems: + - IsClosed.mk_lt_continuum + - path: Mathlib/Topology/Algebra/Ring/Ideal.lean + theorems: + - QuotientRing.isOpenMap_coe + - path: Mathlib/CategoryTheory/Limits/FunctorToTypes.lean + theorems: + - CategoryTheory.FunctorToTypes.jointly_surjective + - path: Mathlib/Data/Polynomial/Module/FiniteDimensional.lean + theorems: + - Module.AEval.isTorsion_of_aeval_eq_zero + - path: Mathlib/Analysis/Calculus/InverseFunctionTheorem/ContDiff.lean + theorems: + - ContDiffAt.to_localInverse + - path: Mathlib/Analysis/NormedSpace/DualNumber.lean + theorems: + - DualNumber.exp_smul_eps + - path: Mathlib/CategoryTheory/CatCommSq.lean + theorems: + - CategoryTheory.CatCommSq.hInv_hInv + - path: Mathlib/CategoryTheory/Subobject/WellPowered.lean + theorems: + - CategoryTheory.wellPowered_of_equiv + - path: Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean + theorems: + - "AlgebraicTopology.DoldKan.compatibility_N\u2082_N\u2081_karoubi" + - path: Mathlib/Data/Countable/Basic.lean + theorems: + - uncountable_iff_isEmpty_embedding + - path: Mathlib/Data/Finset/Order.lean + theorems: + - Directed.finset_le diff --git a/lean/train/local.meta.json b/lean/train/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..610f832e235bd4fb00ed8f2812dc41c7c95dbf6e --- /dev/null +++ b/lean/train/local.meta.json @@ -0,0 +1,3 @@ +version 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b/math-comp/eval/math-comp_eval.yaml new file mode 100644 index 0000000000000000000000000000000000000000..aa527ef6e8b1ea54c5dae2c35f98d59982f8e43f --- /dev/null +++ b/math-comp/eval/math-comp_eval.yaml @@ -0,0 +1,880 @@ +name: math-comp_eval +num_files: 70 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 729 +datasets: +- project: /math-comp/ + files: + - path: mathcomp/solvable/frobenius.v + theorems: + - Frobenius_semiregularP + - Frobenius_cent1_ker + - prime_FrobeniusP + - Frobenius_coprime_quotient + - stab_semiprime + - card_support_normedTI + - injm_Frobenius + - path: mathcomp/solvable/abelian.v + theorems: + - pnElemS + - abelemS + - pdiv_p_elt + - exponent_morphim + - rank_witness + - Ohm_cont + - quotient_rank_abelian + - morphim_Mho + - pnElem_prime + - Mho_p_cycle + - abelem_abelian + - Mho0 + - injm_Ohm + - extend_cyclic_Mho + - path: mathcomp/ssreflect/order.v + theorems: + - max_idPl + - max_minr + - eq_meetl + - lexx + - idfun_is_bottom_morphism + - diff_eq0 + - complI + - le_sorted_leq_nth + - anti + - leUr + - joins_sup_seq + - comparable_ltP + - subset_display + - joinIBC + - le_wval + - sort_le_id + - bigminIl + - diffErcompl + - lex1 + - le_path_pairwise + - contra_leN + - enum_rankK + - bigmaxUr + - lt_path_mask + - diffIK + - leUx + - eq_ltLR + - lt_sorted_ltn_nth + - le0x + - le_refl + - compl1 + - lteif_andb + - le_Rank + - comparable_maxC + - opred0 + - incomparable_ltF + - ge_min_id + - ltNge + - enum_set1 + - lt_def + - leBKU + - lteifxx + - lcmnn + - bigmax_sup + - contraTle + - tnth_rcompl + - minAC + - maxxK + - path: mathcomp/algebra/ssralg.v + theorems: + - subrI + - mulIr + - valM + - mulVf + - mulf_div + - and_dnfP + - natr_div + - invr_neq0 + - pair_mul0r + - rpredBr + - ffun_mul_addr + - commrV + - mulrb + - exprAC + - opB + - rpredN + - divIf + - eval_If + - prodr_const_nat + - holds_fsubst + - divfI + - can2_rmorphism + - fieldP + - rmorphMsign + - Frobenius_aut_is_multiplicative + - prodf_div + - Frobenius_aut1 + - mulrBr + - mulr_sumr + - subrKA + - path: mathcomp/algebra/finalg.v + theorems: + - unit_inv_proof + - path: mathcomp/field/algC.v + theorems: + - floorC_itv + - floorCpK + - aut_Cint + - CintEge0 + - algC_invaut_is_rmorphism + - Cint1 + - conjL_K + - rpred_Cint + - mulVf + - aut_Crat + - mul1 + - eqCmodMr + - path: mathcomp/ssreflect/prime.v + theorems: + - partn_part + - logn_div + - max_pdiv_leq + - totient_count_coprime + - divn_count_dvd + - pnatM + - partnX + - eq_in_partn + - dvdn_prime2 + - path: mathcomp/algebra/ssrnum.v + theorems: + - sgr_ge0 + - ger_real + - nneg_addr_closed + - leif_mean_square + - normrX + - pmulrn_rle0 + - ge0_cp + - ltrDr + - a1 + - real_ltr_distlBl + - ReM + - subr_ge0 + - ltrNl + - leN_total + - psumr_eq0 + - ltrn1 + - ler0P + - comparabler0 + - negrE + - sub_ge0 + - divC_Crect + - ltr_nwDr + - realBC + - deg_le2_poly_le0 + - normC_Re_Im + - unitf_gt0 + - ler_norm_sum + - ltr_distlCBl + - ltr_distlC + - monic_Cauchy_bound + - leif_AGM + - invf_nle + - ler_sum_nat + - gerB_real + - lerB_normD + - ltr_pdivrMl + - real_normK + - Im_rootC_ge0 + - ReMil + - realn_mono + - addr_minr + - subr_ge0 + - lt0N + - ltr_sqrt + - ltr_pwDl + - real_oppr_closed + - oppr_max + - poly_ivt + - normfV + - root0C + - mulr_sg_norm + - real_arg_maxP + - oppr_ge0 + - lt_trans + - exprn_odd_lt0 + - Im_i + - real_eqr_norm2 + - deg2_poly_min + - psumr_neq0 + - conjCK + - comparablerE + - realE + - nmulr_rle0 + - path: mathcomp/ssreflect/fintype.v + theorems: + - negb_exists + - eq_liftF + - ordS_inj + - injF_onto + - extremum_inP + - card_bool + - pickP + - eq_proper_r + - subset_catr + - split_subproof + - ltn_unsplit + - proper_sub + - cast_ord_comp + - eq_proper + - canF_eq + - fintype_le1P + - lshift_subproof + - unsplitK + - exists_eqP + - arg_maxnP + - forall_inPn + - exists_inPP + - proper_irrefl + - cardUI + - path: mathcomp/algebra/matrix.v + theorems: + - tr_submxrow + - mxtrace_block + - mul_mxrow_mxdiag + - mxOver_mul_subproof + - map_col' + - block_mxEur + - diag_mx_is_diag + - comm_mx1 + - perm_mxEsub + - drsubmxEsub + - mulmx_rsub + - diag_mx_is_semi_additive + - col_mxEd + - mxcolP + - const_mx_is_semi_additive + - det_scalar + - is_scalar_mxP + - map2_xrow + - submxcolK + - mul_mxblock_mxdiag + - mul_delta_mx_0 + - tr_diag_mx + - scalemx_eq0 + - mulmx1_min + - col_mxrow + - invmx1 + - tr_tperm_mx + - mxrow_recl + - path: mathcomp/ssreflect/ssrnat.v + theorems: + - leq_pmul2r + - ltnS + - ltn_half_double + - ltnW_nhomo + - addBnAC + - subSn + - uphalf_half + - minnE + - double_pred + - leq_mono + - leq_add2r + - eqn_exp2l + - multE + - leqW_nmono + - ex_maxn_subproof + - ltn_ind + - leqW_mono + - leq_psubRL + - addnBC + - addnCB + - halfK + - geq_uphalf_double + - odd_halfK + - maxn_idPl + - contra_not_ltn + - nat_irrelevance + - leqn0 + - ubnPgeq + - leq_subCr + - ltn_sub2lE + - leq_sub2l + - leq_add + - addSn + - path: mathcomp/character/mxrepresentation.v + theorems: + - card_linear_irr + - rfix_mxP + - mx_rsim_refl + - mxtrace_sub_fact_mod + - sum_mxsimple_direct_compl + - irr_mx_sum + - rstabs_group_set + - val_genJmx + - rstab_map + - Clifford_simple + - mx_iso_simple + - gen_satP + - in_submod_eq0 + - rcent_sub + - in_gen0 + - subSocle_semisimple + - rfix_morphim + - morphpre_mx_irr + - mxmodule_quo + - principal_comp_key + - irr_reprK + - dom_hom_invmx + - submod_mx_faithful + - eqg_mx_irr + - gen_mulDr + - val_factmod_module + - mxval_sub + - section_module + - rker_linear + - gen_mx_faithful + - rconj_mxE + - gen_mulA + - mx_rsim_in_submod + - simple_Socle + - path: mathcomp/algebra/poly.v + theorems: + - lreg_size + - map_poly_eq0 + - deg2_poly_factor + - leq_sizeP + - derivnN + - scale_polyE + - size_poly_leq0P + - comm_poly0 + - coef_prod_XsubC + - coefN + - root1 + - size_poly_eq0 + - coefMX + - drop_poly_eq0 + - add_polyC + - map_poly_eq0_id0 + - hornerM + - map_comp_poly + - size_poly_leq0 + - comp_polyA + - deriv_is_linear + - drop_polyD + - coef_sumMXn + - horner_coef + - lead_coefXaddC + - closed_nonrootP + - coef_add_poly + - size_Msign + - closed_field_poly_normal + - polyC_exp + - take_polyMXn + - derivZ + - lead_coef_map_id0 + - polyOverP + - rreg_lead0 + - deg2_poly_root2 + - size_mulX + - path: mathcomp/solvable/gseries.v + theorems: + - quotient_maximal + - subnormalEr + - maximal_eqJ + - minnormal_maxnormal + - simpleP + - sub_setIgr + - minnormal_exists + - morphpre_maximal_eq + - path: mathcomp/character/mxabelem.v + theorems: + - sub_rowg_mx + - rowg_group_set + - rowg1 + - rVabelemJmx + - path: mathcomp/solvable/sylow.v + theorems: + - card_Syl_mod + - nilpotent_pcore_Hall + - Sylow_Jsub + - path: mathcomp/character/character.v + theorems: + - cfMod_lin_char + - cfRepr_map + - cfDprodl_irr + - xcfun_rE + - Cnat_cfdot_char + - eq_sum_nth_irr + - cfker_Ind_irr + - dprod_IirrC + - cfun_irr_sum + - dprodl_Iirr_eq0 + - inv_dprod_IirrK + - cfdotC_char + - max_cfRepr_norm_scalar + - tprod1 + - sdprod_Iirr_inj + - constt_Ind_Res + - cfdot_aut_irr + - irr_repr_lin_char + - xcfun_annihilate + - sum_norm_irr_quo + - pgroup_cyclic_faithful + - path: mathcomp/algebra/polydiv.v + theorems: + - reducible_cubic_root + - dvd0pP + - mup_geq + - eqpW + - lc_expn_scalp_neq0 + - gdcop_recP + - gcd0p + - divp_addl_mul_small + - divp_eq0 + - gdcop_rec_map + - eqp_rtrans + - modp_eq0P + - ltn_rmodp + - eq_rdvdp + - eqp_sym + - gcdp_mulr + - eqp_root + - dvdpNr + - size_gcd1p + - cubic_irreducible + - path: mathcomp/character/inertia.v + theorems: + - constt_Inertia_bijection + - cfConjg_irr + - inertia_mod_pre + - inertia_quo + - cfConjgMod_norm + - prime_invariant_irr_extendible + - inertia_bigdprod + - path: mathcomp/algebra/intdiv.v + theorems: + - eqz_div + - coprimez_pexpr + - gcdzz + - modz0 + - gcdz0 + - zprimitive_eq0 + - eqz_modDl + - dvdz_exp2l + - dvdz_zmod_closed + - dvdz_lcml + - modzMr + - lcmz0 + - ltz_mod + - coprimezXl + - dvdz_contents + - gcd1z + - mulz_divCA_gcd + - path: mathcomp/fingroup/morphism.v + theorems: + - morphpreK + - isog_sym + - morphpreT + - trivial_isog + - morphim0 + - morphicP + - factmE + - morphpre_subcent + - ker_trivg_morphim + - misom_isog + - ker_trivm + - morphimK + - morphpre_gen + - injm_ifactm + - sub_morphpre_im + - eq_homgl + - morphimGI + - path: mathcomp/solvable/burnside_app.v + theorems: + - F_s23 + - Sd1_inj + - r05_inv + - S3_inv + - sh_inv + - S1_inv + - rot_r1 + - path: mathcomp/algebra/ssrint.v + theorems: + - pmulrz_rlt0 + - nexpIrz + - PoszD + - exp0rz + - mulrz_eq0 + - coefMrz + - intP + - leqifD_dist + - mulzS + - mulz2 + - addSnz + - scaler_int + - distnDr + - expN1r + - mulrzBl_nat + - rpredXsign + - path: mathcomp/field/fieldext.v + theorems: + - module_baseVspace + - divp_polyOver + - subfx_scaler1r + - mem_vspaceOver + - path: mathcomp/solvable/commutator.v + theorems: + - derg0 + - der_char + - commMG + - quotient_cents2r + - path: mathcomp/algebra/rat.v + theorems: + - mul1q + - mul_subdefA + - rat_field_theory + - ltrq0 + - QintP + - path: mathcomp/solvable/maximal.v + theorems: + - nilpotent_Fitting + - card_p3group_extraspecial + - Fitting_max + - FittingJ + - Aut_extraspecial_full + - path: mathcomp/field/galois.v + theorems: + - dim_fixed_galois + - kHomP + - normalFieldP + - kHom_inv + - kHom1 + - normalFieldf + - kAHomP + - path: mathcomp/ssreflect/div.v + theorems: + - coprime_sym + - edivnP + - modnXm + - modn_dvdm + - dvdn_sub + - leq_trunc_div + - eqn_mod_dvd + - divnBr + - dvdn_subr + - coprime_pexpl + - modnB + - leq_divRL + - dvdn_lcml + - lcmnC + - path: mathcomp/solvable/primitive_action.v + theorems: + - n_act_is_action + - path: mathcomp/fingroup/automorphism.v + theorems: + - char1 + - autE + - charY + - injm_Aut + - injm_conj + - Aut_group_set + - morphic_aut + - path: mathcomp/algebra/mxpoly.v + theorems: + - integral_horner + - horner_mx_diag + - char_poly_trace + - map_horner_mx + - diagonalizable_forPex + - map_char_poly_mx + - algebraic_inv + - algebraic_add + - comm_horner_mx2 + - conjmxM + - eval_mxvec + - conjumx + - horner_mx_uconjC + - mx_inv_horner0 + - simmxW + - conj0mx + - coef_rVpoly + - split_diagA + - path: mathcomp/field/falgebra.v + theorems: + - sub1_agenv + - cent_centerv + - subX_agenv + - prodvDr + - amull_is_linear + - lfun_mulRVr + - algidr + - centerv_sub + - path: mathcomp/ssreflect/path.v + theorems: + - subseq_path + - iota_sorted + - sorted_ltn_index_in + - rot_to_arc + - eq_fpath + - trajectS + - path_min_sorted + - shortenP + - sort_stable + - perm_merge + - mem2lr_splice + - sorted_map + - homo_sort_map + - cycle_prev + - map_path + - take_path + - sorted_sort + - prev_rotr + - mem2rf + - allrel_merge + - arc_rot + - sorted_subseq_sort_in + - path: mathcomp/ssreflect/bigop.v + theorems: + - big_rec2 + - big_nseq_cond + - big_distrr + - big_nat_recl + - big_orE + - big_nat1_eq + - big_ord1_eq + - sig_big_dep_idem + - big_ord_narrow_cond_leq + - big_enum_rank_cond + - le_big_ord_cond + - reindex_omap + - big_nat + - big_cat + - addm0 + - path: mathcomp/ssreflect/choice.v + theorems: + - tagged_hasChoice + - seq_hasChoice + - path: mathcomp/algebra/mxalgebra.v + theorems: + - mulmx_max_rank + - row_base0 + - rowV0Pn + - row_freePn + - qidmx_eq1 + - addmx_sub + - summx_sub_sums + - proj_mx_proj + - eqmx_rowsub_comp + - genmxE + - genmxP + - binary_mxsum_proof + - kermx_eq0 + - rV_eqP + - stableNmx + - mulmx_sub + - path: mathcomp/ssreflect/fingraph.v + theorems: + - finv_inj_in + - fcycle_undup + - sym_connect_sym + - roots_root + - findex0 + - orbit_rot_cycle + - cycle_orbit_in + - adjunction_n_comp + - connect0 + - eq_froot + - path: mathcomp/solvable/hall.v + theorems: + - coprime_comm_pcore + - Hall_Jsub + - path: mathcomp/solvable/extraspecial.v + theorems: + - card_isog8_extraspecial + - pX1p2_pgroup + - path: mathcomp/solvable/extremal.v + theorems: + - def_q + - generators_2dihedral + - normal_rank1_structure + - odd_not_extremal2 + - path: mathcomp/algebra/vector.v + theorems: + - memv_add + - span_bigcat + - zero_lfunE + - mul_mxof + - comp_lfunDl + - seq1_free + - lker0_compfV + - cat_free + - lfunP + - seq1_basis + - lfun_addN + - vspace_modr + - capvA + - path: mathcomp/algebra/ring_quotient.v + theorems: + - prime_idealrM + - path: mathcomp/solvable/cyclic.v + theorems: + - quotient_generator + - ker_eltm + - prime_cyclic + - path: mathcomp/fingroup/gproduct.v + theorems: + - extprod_mul1g + - cprod_card_dprod + - injm_cprodm + - bigcprod_card_dprod + - pprodmEl + - morphim_coprime_sdprod + - dprodW + - sdprodmEr + - subcent_dprod + - divgr_id + - morphim_bigcprod + - injm_dprodm + - pprod1g + - im_sdprodm1 + - path: mathcomp/field/separable.v + theorems: + - base_separable + - separable_prod_XsubC + - base_inseparable + - separable_inseparable_element + - separable_generator_maximal + - separable_add + - dvdp_separable + - path: mathcomp/ssreflect/generic_quotient.v + theorems: + - enc_mod_rel_is_equiv + - pi_morph2 + - pi_mono1 + - path: mathcomp/fingroup/action.v + theorems: + - dvdn_orbit + - actbyE + - actby_is_groupAction + - morph_astabs + - perm_faithful + - astabsD + - actpermK + - ract_is_groupAction + - actK + - modact_coset_astab + - actsP + - orbit_inv_in + - gacentE + - act1 + - atransR + - reindex_acts + - gactM + - subgacent1E + - card_orbit_in_stab + - gactJ + - atrans_acts_card + - path: mathcomp/character/integral_char.v + theorems: + - irr_gring_center + - path: mathcomp/solvable/alt.v + theorems: + - rfd_morph + - simple_Alt5 + - card_Alt + - path: mathcomp/algebra/fraction.v + theorems: + - equivf_sym + - path: mathcomp/solvable/center.v + theorems: + - center_ncprod + - center_prod + - path: mathcomp/algebra/interval.v + theorems: + - leBSide + - itv_joinKI + - itv_leEmeet + - itv_split1U + - path: mathcomp/ssreflect/binomial.v + theorems: + - fact_split + - ffact_gt0 + - ffactnSr + - bin_factd + - path: mathcomp/algebra/qpoly.v + theorems: + - poly_of_size_mod + - qpoly_scale1l + - qpoly_scaleDl + - irreducibleP + - path: mathcomp/field/algnum.v + theorems: + - Aint_subring_exists + - Qn_aut_exists + - path: mathcomp/solvable/nilpotent.v + theorems: + - lcn_subS + - ucn_id + - TI_center_nil + - meet_center_nil + - injm_nil + - quotient_nil + - path: mathcomp/algebra/zmodp.v + theorems: + - Zp_mulVz + - Zp_group_set + - Fp_Zcast + - path: mathcomp/field/closed_field.v + theorems: + - rmulpT + - path: mathcomp/algebra/polyXY.v + theorems: + - swapXY_X + - path: mathcomp/field/qfpoly.v + theorems: + - qlogp_lt + - path: mathcomp/solvable/finmodule.v + theorems: + - fmodM + - fmod_addrC + - fmvalK + - mulg_exp_card_rcosets + - path: mathcomp/character/vcharacter.v + theorems: + - vcharP + - zcharD1E + - orthogonal_span + - dirr_inj + - path: mathcomp/solvable/jordanholder.v + theorems: + - acompsP + - maxainv_proper + - maxainv_asimple_quo + - path: mathcomp/algebra/archimedean.v + theorems: + - lt_succ_floor + - natr_norm_int + - trunc1 + - ceil_floor + - norm_natr + - floorN + - prod_truncK + - int_num0 + - path: mathcomp/field/finfield.v + theorems: + - primeChar_abelem + - primeChar_scale1 + - primeChar_scaleAl + - path: mathcomp/fingroup/perm.v + theorems: + - porbitP + - porbit_setP + - preim_permV + - path: mathcomp/fingroup/quotient.v + theorems: + - rcoset_kercosetP + - path: mathcomp/ssreflect/ssrbool.v + theorems: + - if_implyb + - path: mathcomp/ssreflect/finfun.v + theorems: + - supportE diff --git a/math-comp/test/local.meta.json b/math-comp/test/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..5655368022c15da6fdd3b92d313734a749208b85 --- /dev/null +++ b/math-comp/test/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2d7872f8b3bf60f3727a6217cb1a7ab6ae1b99cac2082f7695b5d939f5bd468d +size 388 diff --git a/math-comp/test/local_data_0000001378.json b/math-comp/test/local_data_0000001378.json new file mode 100644 index 0000000000000000000000000000000000000000..dc7dd1d07454203c0ea4cf2139df2a8221994deb --- /dev/null +++ b/math-comp/test/local_data_0000001378.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:16594fc4400ebfbfd89950a7649065c5b8aa6b77c6b019c18d543b13839c6e7d +size 2034563 diff --git a/math-comp/test/local_lemma_0000000000.json b/math-comp/test/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/math-comp/test/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/math-comp/test/math-comp_test.yaml b/math-comp/test/math-comp_test.yaml new file mode 100644 index 0000000000000000000000000000000000000000..2700b13d2aeebfe30c62896f9fb37cf327aca85b --- /dev/null +++ b/math-comp/test/math-comp_test.yaml @@ -0,0 +1,677 @@ +name: math-comp_test +num_files: 65 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 536 +datasets: +- project: /math-comp/ + files: + - path: mathcomp/algebra/ssralg.v + theorems: + - telescope_prodf_eq + - opp_fun_is_additive + - scalerK + - iter_mulr_1 + - prodf_seq_eq0 + - invr0 + - exprSr + - mulC_unitP + - invrZ + - mulVKr + - scalerDr + - in_alg_is_rmorphism + - fst_is_multiplicative + - prodrXl + - natrX + - rmorph_sum + - addUC + - opp_fun_is_scalable + - rmorphMNn + - exprDn_comm + - eqr_opp + - expfB_cond + - rmorphN1 + - exprM + - fpredMr + - pair_scaleA + - scalerN + - path: mathcomp/algebra/matrix.v + theorems: + - adj1 + - mxrowEblock + - cormen_lup_detL + - mul_col_mx + - map2_row' + - eq_mx + - thinmxOver + - mxtrace_mxdiag + - invmxK + - map_xrow + - row_dsubmx + - mulmxr_is_linear + - block_mxA + - col_ind + - tr_submxcol + - scalar_mx_is_semi_additive + - row'Ku + - trmx_ursub + - mul_mx_diag + - lift0_mx_is_perm + - row_perm_key + - map_castmx + - mxblockB + - submxcolD + - path: mathcomp/algebra/ssrnum.v + theorems: + - deg2_poly_gt0 + - ltr_pdivrMr + - lt0r + - sqr_norm_eq1 + - lerXn2r + - minr_pMl + - le_total + - Nreal_leF + - real_lteif_distl + - ler_nM2r + - mulr_sign_lt0 + - ler_ndivlMr + - ring_display + - ler_wiXn2l + - lern1 + - sqrtr_sqr + - ltrP + - le_def' + - mulr_ile1 + - aNge0 + - real_exprn_odd_le0 + - gtrDr + - pnatr_eq0 + - ler_pMn2l + - bigmax_real + - sqrtr_eq0 + - real_ltr_normlW + - ltr_nMn2l + - mulr_ilt1 + - psumr_neq0P + - deg2_poly_root2 + - exprn_odd_ge0 + - real_ltr_distl + - real_lteifNE + - ler_rootCl + - real_neqr_lt + - normr_ge0 + - le_def' + - ler01 + - real_ltrNnormlW + - ge0_def + - normrN1 + - sgrN1 + - natr_indexg_gt0 + - sgr_odd + - agt0 + - ieexprIn + - path: mathcomp/ssreflect/ssrnat.v + theorems: + - mul2n + - odd_gt2 + - leqif_geq + - subn2 + - leq_pmulr + - homo_leq_in + - gtn_min + - contra_ltnT + - half_gt0 + - sqrnD + - mulE + - subnA + - doubleMr + - uphalfK + - predn_sub + - iterD + - mul_expE + - mulnSr + - leqif_add + - path: mathcomp/ssreflect/bigop.v + theorems: + - big_tnth + - eq_big_idem + - dvdn_biggcdP + - big_ord1_cond_eq + - addmA + - big_all_cond + - big_uniq + - opm1 + - eq_bigl + - eq_bigmax + - big_rcons + - bigA_distr_bigA + - big_sumType + - eq_big + - big_tuple + - path: mathcomp/fingroup/morphism.v + theorems: + - morphim_normal + - kerE + - morphimEdom + - sgvalmK + - isog_symr + - morphpre_sub + - isogEcard + - path: mathcomp/algebra/polydiv.v + theorems: + - dvdp_XsubCl + - gcd1p + - rdvdpp + - rdvd0p + - divpp + - coprimepP + - dvdp_mull + - polyXsubCP + - leq_trunc_divp + - rmodp_eq0P + - rmodpX + - dvdpp + - dvdpN0 + - gcdp_modr + - coprimepZr + - dvdp_gdco + - rmodp_sum + - coprimep_expr + - rdvdp_eqP + - eqp_scale + - coprimepp + - gcdp_addl_mul + - dvdp_exp + - gcdp_map + - edivpP + - rmodp_id + - edivp_eq + - path: mathcomp/field/galois.v + theorems: + - fixedField_galois + - comp_kHom + - normal_field_splitting + - gal_adjoin_eq + - regular_splittingAxiom + - kAutfE + - galNormM + - splitting_normalField + - comp_AEndA + - kAut_to_gal + - kHomExtend_additive_subproof + - gal_fixedField + - path: mathcomp/character/vcharacter.v + theorems: + - mem_zchar + - vchar_norm2 + - cfnorm_dchi + - zchar_subset + - cfun1_vchar + - path: mathcomp/solvable/cyclic.v + theorems: + - injm_Zp_unitm + - orderXdiv + - path: mathcomp/algebra/ring_quotient.v + theorems: + - mulqA + - equiv_is_equiv + - path: mathcomp/fingroup/quotient.v + theorems: + - cosetpre1 + - index_morphim_ker + - morphpre_quotm + - quotient_normal + - path: mathcomp/solvable/maximal.v + theorems: + - subcent1_extraspecial_maximal + - min_card_extraspecial + - isog_special + - path: mathcomp/algebra/mxalgebra.v + theorems: + - addsmxA + - mxrank_compl + - mulmx_ker + - col_ebase_unit + - rank_leq_col + - LUP_card_GL + - row_leq_rank + - mulmx_base + - pinvmxE + - mxrank_sum_cap + - map_ltmx + - capmxS + - genmx_muls + - mxrank_eq0 + - col_base_full + - path: mathcomp/solvable/extremal.v + theorems: + - extremal_generators_facts + - def2 + - path: mathcomp/fingroup/action.v + theorems: + - actKVin + - astab1R + - afixP + - actperm_Aut + - acts_qact_dom_norm + - astabs_ract + - astab1JG + - afixYin + - gacent1E + - acts_actby + - mem_orbit + - path: mathcomp/algebra/poly.v + theorems: + - prim_order_exists + - size_XaddC + - size_scale + - poly_alg_initial + - map_monic + - coef_comp_poly + - size_addl + - poly_ind + - size_odd_poly_eq + - derivN + - take_poly_is_linear + - map_Poly_id0 + - polyCB + - rootC + - add_poly_key + - polyseq1 + - char_prim_root + - comp_polyCr + - monicMr + - size_comp_poly_leq + - derivX + - horner_evalE + - map_polyXsubC + - coef1 + - scale_polyDl + - size_XnaddC + - path: mathcomp/ssreflect/order.v + theorems: + - comp_is_top_morphism + - lexi_cons + - rcomplPjoin + - meetBI + - joinCA + - comparable_bigr + - meetUr + - enum1 + - incomparable_leF + - ltEsig + - meetEseq + - joinKUC + - complEdiff + - le_refl + - joins_sup + - lteif_orb + - le_refl + - le_enum_rank_in + - contraTlt + - meetCA + - nonincn_inP + - le_nmono_in + - joinC + - lt_path_min + - idfun_is_meet_morphism + - meetC + - comparable_contra_ltn_lt + - minEle + - comparable_gt_max + - lt_le_asym + - ltEprodlexi + - le_def + - opredI + - joinIB + - leBl + - joinACA + - botEseq + - bigmin_mkcondr + - leUidl + - comparable_gt_min + - idfun_is_nondecreasing + - decn_inP + - ltW_homo + - max_idPr + - botEsig + - le_nmono + - meetAC + - ge_max + - anti + - leBx + - neqhead_lexiE + - contraNle + - meetA + - comparable_maxCA + - path: mathcomp/character/mxabelem.v + theorems: + - mx_repr_action_faithful + - astabs_rowg_repr + - path: mathcomp/character/mxrepresentation.v + theorems: + - mx_abs_irr_cent_scalar + - map_reprJ + - bigcapmx_module + - map_gring_row + - component_mx_module + - repr_mxM + - rcenter_normal + - mx_second_rsim + - kquo_repr_coset + - in_factmod_addsK + - mx_Jacobson_density + - card_irr + - genmx_Socle + - gring_opJ + - val_factmodJ + - val_submod_eq0 + - irr_modeV + - gen_is_multiplicative + - mxmodule1 + - mxsimple_isoP + - mxmodule_eigenvector + - mxsemisimpleS + - Wedderburn_ideal + - hom_cyclic_mx + - socle_rsimP + - path: mathcomp/solvable/abelian.v + theorems: + - rank_cycle + - primes_exponent + - exponent_Zgroup + - Ohm1 + - exponent2_abelem + - p_rank_abelem + - TIp1ElemP + - path: mathcomp/field/finfield.v + theorems: + - finRing_nontrivial + - path: mathcomp/ssreflect/prime.v + theorems: + - Euclid_dvd_prod + - trunc_log_ltn + - up_log2_double + - trunc_log_up_log + - totient_prime + - trunc_logP + - lognM + - primesX + - pdiv_id + - pdiv_pfactor + - path: mathcomp/algebra/polyXY.v + theorems: + - sizeYE + - swapXY_is_scalable + - max_size_coefXY + - poly_XaY0 + - path: mathcomp/algebra/vector.v + theorems: + - dimv0 + - lfun_addC + - sumv_sup + - comp_lfun0l + - memvK + - span_basis + - comp_lfunNl + - free_span + - fun_of_lfunK + - memv_capP + - vs2mxP + - capvSl + - hommx1 + - sub_span + - dim_span + - path: mathcomp/fingroup/perm.v + theorems: + - perm_mulP + - out_perm + - cast_perm_inj + - porbit_id + - porbit_perm + - odd_tperm + - tpermKg + - inj_tperm + - path: mathcomp/algebra/ssrint.v + theorems: + - addSz + - ler_pMz2r + - ltr_pMz2l + - ler_wnMz2l + - exprSz + - exprz_out + - sqrn_dist + - mulrN1z + - sgzN1 + - distnEr + - path: mathcomp/algebra/intdiv.v + theorems: + - divzN + - coprimez_dvdr + - gcdzMl + - modzz + - coprimezMl + - path: mathcomp/ssreflect/choice.v + theorems: + - pickle_taggedK + - sig2W + - choose_id + - path: mathcomp/field/fieldext.v + theorems: + - minPoly_dvdp + - vsval_multiplicative + - subfx_inj_is_multiplicative + - dim_refBaseField + - path: mathcomp/ssreflect/fintype.v + theorems: + - val_seq_sub_enum + - predT_subset + - eq_forallb + - nth_enum_ord + - bumpC + - image_pre + - disjointWl + - enum_rank_in_inj + - card_geqP + - forall_inP + - fintype1 + - cardID + - neq_lift + - path: mathcomp/algebra/zmodp.v + theorems: + - unit_Zp_mulgC + - sub_Zp_1 + - Zp_abelian + - Fp_cast + - path: mathcomp/character/character.v + theorems: + - cfRepr_prod + - irr_inj + - xcfunZl + - xcfun_r_is_additive + - cfRepr_muln + - constt_charP + - cfRes_irr_irr + - irr_classK + - cfker_repr + - lin_Res_IirrE + - cfRes_eq0 + - cfAut_char + - lin_charV + - irr_consttE + - char1_gt0 + - path: mathcomp/solvable/sylow.v + theorems: + - Sylow_superset + - ZgroupS + - path: mathcomp/field/falgebra.v + theorems: + - adjoin_cons + - expv0 + - lfun_invE + - expv1n + - agenv_sub_modl + - lfun_mulrRV + - path: mathcomp/field/algC.v + theorems: + - truncC0 + - truncCM + - Cnat_prod_eq1 + - posP + - eqCmod_addl_mul + - eqCmodMl + - eqCmodD + - eqCmod_trans + - CtoL_is_multiplicative + - path: mathcomp/ssreflect/path.v + theorems: + - mem2E + - splitP2r + - rotr_ucycle + - mem_merge + - rotr_cycle + - mem2r_cat + - sorted_eq_in + - cycle_relI + - drop_sorted + - splitPr + - path: mathcomp/algebra/mxpoly.v + theorems: + - map_mx_companion + - map_char_poly + - rVpoly_is_linear + - horner_rVpoly_inj + - path: mathcomp/algebra/fraction.v + theorems: + - tofrac0 + - equivf_trans + - path: mathcomp/ssreflect/div.v + theorems: + - eqn_div + - eqn_modDl + - dvdn_eq + - modnMmr + - dvdn0 + - dvdn_lcmr + - path: mathcomp/solvable/frobenius.v + theorems: + - Frobenius_kerS + - FrobeniusJgroup + - path: mathcomp/solvable/alt.v + theorems: + - simple_Alt_3 + - path: mathcomp/solvable/jordanholder.v + theorems: + - JordanHolderUniqueness + - gactsI + - gastabsP + - gactsP + - path: mathcomp/solvable/burnside_app.v + theorems: + - S23_inv + - S05_inj + - eqperm_map2 + - path: mathcomp/algebra/rat.v + theorems: + - scalqE + - fracq0 + - normqE + - numq_le0 + - natq_div + - addqC + - path: mathcomp/algebra/qpoly.v + theorems: + - size_mk_monic_gt1 + - npolyXE + - qpoly_mulVz + - big_coef_npoly + - path: mathcomp/character/inertia.v + theorems: + - inertia_isom + - im_cfclass_Iirr + - nNG + - path: mathcomp/fingroup/gproduct.v + theorems: + - complP + - bigcprodW + - sdprodg1 + - injm_pairg1 + - cprodP + - perm_bigcprod + - dprodJ + - divgrMid + - astabsEsd + - sdprod_subr + - xsdprodm_dom1 + - bigcprodYP + - cprodWpp + - dprodWC + - path: mathcomp/field/qfpoly.v + theorems: + - plogpD + - path: mathcomp/solvable/gseries.v + theorems: + - morphpre_maximal + - path: mathcomp/ssreflect/generic_quotient.v + theorems: + - right_trans + - path: mathcomp/algebra/archimedean.v + theorems: + - aut_natr + - floor_ge_int + - ceil_def + - path: mathcomp/algebra/interval.v + theorems: + - le_ninfty + - oppr_itvco + - le_bound_trans + - path: mathcomp/solvable/nilpotent.v + theorems: + - solvableS + - ucn1 + - path: mathcomp/ssreflect/fingraph.v + theorems: + - eq_froots + - order_id + - relU_sym + - order_set_finv + - finv_eq_can + - finv_f_cycle + - path: mathcomp/solvable/center.v + theorems: + - center_dprod + - center_sub + - path: mathcomp/field/separable.v + theorems: + - Derivation_mul_poly + - char0_PET + - path: mathcomp/solvable/commutator.v + theorems: + - comm3G1P + - der_subS + - commgX + - isog_der + - path: mathcomp/solvable/gfunctor.v + theorems: + - gFsub + - path: mathcomp/ssreflect/ssrAC.v + theorems: + - count_memE + - path: mathcomp/field/closed_field.v + theorems: + - ex_elim_qf + - path: mathcomp/algebra/finalg.v + theorems: + - zmod_abelian + - path: mathcomp/fingroup/automorphism.v + theorems: + - ker_autm + - path: mathcomp/ssreflect/binomial.v + theorems: + - ffactSS + - path: mathcomp/solvable/hall.v + theorems: + - coprime_quotient_cent diff --git a/math-comp/train/local.meta.json b/math-comp/train/local.meta.json new file mode 100644 index 0000000000000000000000000000000000000000..a578af92e3e17c0c584a73a2d03c60a8b4a74100 --- /dev/null +++ b/math-comp/train/local.meta.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:19ae08589bce8acaad141d581ef1ecb488aedbf73cea8aada9b8c3a6f39c96ee +size 390 diff --git a/math-comp/train/local_data_0000010000.json b/math-comp/train/local_data_0000010000.json new file mode 100644 index 0000000000000000000000000000000000000000..62d86e8d7f3258f6b38e252465384594574a5d79 --- /dev/null +++ b/math-comp/train/local_data_0000010000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b5e0af1a3f5e54b0f0a50f83d93af1cf4644e4fcd73484f6c029ba5e5ef14617 +size 16090910 diff --git a/math-comp/train/local_data_0000020000.json b/math-comp/train/local_data_0000020000.json new file mode 100644 index 0000000000000000000000000000000000000000..358f6de1399d917d4245cbf6ea5ef61a7041aaf8 --- /dev/null +++ b/math-comp/train/local_data_0000020000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a4a0aca28af175c00ff4507003531ad76cf401a3d5d087efa57c65e58c4b46dc +size 38786695 diff --git a/math-comp/train/local_data_0000030000.json b/math-comp/train/local_data_0000030000.json new file mode 100644 index 0000000000000000000000000000000000000000..59af3970556391cdbf16e79980bb11d04f77c42a --- /dev/null +++ b/math-comp/train/local_data_0000030000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b249e5781252a4e805dd460e814ad619be25e90f5b20b7474f6ece7519dca664 +size 29441520 diff --git a/math-comp/train/local_data_0000034196.json b/math-comp/train/local_data_0000034196.json new file mode 100644 index 0000000000000000000000000000000000000000..d55050536631a6f8b203f19e84da7dd7fae14a01 --- /dev/null +++ b/math-comp/train/local_data_0000034196.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:91741c9bedd7d1c9700caaa098a03f3b1ead8aee4dbce1f996eb856e030f2f87 +size 5343781 diff --git a/math-comp/train/local_lemma_0000000000.json b/math-comp/train/local_lemma_0000000000.json new file mode 100644 index 0000000000000000000000000000000000000000..a3e2a68db13cb94b4741a93f2bd8d26245d532e3 --- /dev/null +++ b/math-comp/train/local_lemma_0000000000.json @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d45d1bb11ab18584e9c29bd2b628f2d845f4693962929973a89245afba9ab131 +size 24 diff --git a/math-comp/train/math-comp_train.yaml b/math-comp/train/math-comp_train.yaml new file mode 100644 index 0000000000000000000000000000000000000000..6aa947308a69cff11ed57ced44ba829900f60f97 --- /dev/null +++ b/math-comp/train/math-comp_train.yaml @@ -0,0 +1,11542 @@ +name: math-comp_train +num_files: 75 +language: COQ +few_shot_data_path_for_retrieval: null +few_shot_metadata_filename_for_retrieval: null +dfs_data_path_for_retrieval: null +dfs_metadata_filename_for_retrieval: local.meta.json +theorem_cnt: 11381 +datasets: +- project: /math-comp/ + files: + - path: mathcomp/solvable/abelian.v + theorems: + - trivg_exponent + - abelian_type_dvdn_sorted + - Ohm_leq + - abelem_Ohm1P + - expg_exponent + - TI_Ohm1 + - rankJ + - abelian_splits + - isog_abelem + - morphim_pElem + - nElem1P + - rank_Ohm1 + - pnElem0 + - rankS + - pElemP + - nElem0 + - Ohm1Eexponent + - p_rank_Ohm1 + - rank_pgroup + - pmaxElemS + - is_abelemP + - p_rank_pmaxElem_exists + - exponent_witness + - p_rank_dprod + - abelian_type_dprod_homocyclic + - morphim_rank_abelian + - quotient_p_rank_abelian + - dprod_exponent + - abelem_cyclic + - isog_rank + - cprod_abelem + - Ohm1_id + - abelian_type_gt1 + - morphim_LdivT + - card_pnElem + - isog_homocyclic + - p_rankS + - injm_pnElem + - card_p1Elem + - Ohm1_homocyclicP + - morphim_Ohm + - quotient_pnElem + - eq_abelian_type_isog + - abelian_type_sorted + - p_rank_Hall + - max_card_abelian + - p_rank_p'quotient + - isog_Mho + - abelian_type_pgroup + - LdivT_J + - Ohm1Eprime + - OhmEabelian + - nElemS + - abelem_Ohm1 + - Mho1 + - Mho_leq + - pnElemP + - quotient_LdivT + - p_rank1 + - nElemP + - quotient_grank + - exponent_quotient + - abelemP + - sub_Ldiv + - Ohm1_eq1 + - isog_abelem_card + - rank_gt0 + - fin_lmod_char_abelem + - abelem_order_p + - injm_rank + - p_rank_le_logn + - Ohm_dprod + - pnElemPcard + - pElemJ + - pnat_exponent + - rank_abelian_pgroup + - injm_pElem + - p_rank_le_rank + - cyclic_abelem_prime + - p_rank_gt0 + - dprod_abelem + - Ohm_Mho_homocyclic + - injm_abelem + - OhmS + - pmaxElem_exists + - pi_of_exponent + - card_p1Elem_p2Elem + - quotient_Ldiv + - pmaxElem_LdivP + - abelem_pgroup + - OhmJ + - Ohm0 + - exponent_injm + - grank_abelian + - nElemI + - pmaxElemP + - partn_exponentS + - abelian_type_subproof + - injm_pmaxElem + - abelem1 + - abelem_homocyclic + - Mho_sub + - OhmPredP + - exponent_isog + - Mho_dprod + - isog_Ohm + - p_rank_witness + - quotient_pElem + - injm_nElem + - exponent1 + - exponentS + - exponentP + - trivg_Mho + - Ohm_char + - LdivJ + - p_rankJ + - Mho_normal + - p_rankElem_max + - size_abelian_type + - Ohm_sub + - cyclic_pgroup_dprod_trivg + - quotient_abelem + - pElemI + - rank_abelem + - nt_pnElem + - exponent_Hall + - isog_abelian_type + - card_p1Elem_pnElem + - dprod_homocyclic + - exponent_cyclic + - morphim_grank + - morphim_abelem + - LdivP + - pmaxElemJ + - Ohm_p_cycle + - injm_Ldiv + - cprod_exponent + - p_rank_abelian + - abelian_structure + - abelemE + - morphim_p_rank_abelian + - MhoE + - p1ElemE + - Ohm1_abelem + - fin_Fp_lmod_abelem + - rank1 + - Ohm_normal + - homocyclic1 + - Ohm1_cent_max + - p2Elem_dprodP + - morphim_pnElem + - exponent_gt0 + - group_Ldiv + - p_rank_Sylow + - prime_abelem + - morphim_Ldiv + - sub_LdivT + - MhoJ + - exponent_dvdn + - Mho_cprod + - Mho_char + - Ohm_id + - logn_le_p_rank + - exponent_dprod_homocyclic + - mul_card_Ohm_Mho_abelian + - cycle_abelem + - injm_p_rank + - dvdn_exponent + - pElemS + - pnElemI + - grank_min + - OhmE + - abelemJ + - abelem_splits + - grank_witness + - Mho_p_elt + - pnElemE + - isog_p_rank + - meet_Ohm1 + - abelem_pnElem + - pnElemJ + - MhoEabelian + - Ohm1_cyclic_pgroup_prime + - rank_Sylow + - abelian_type_abelem + - rank_geP + - homocyclic_Ohm_Mho + - p_rank_quotient + - abelian_type_homocyclic + - MhoS + - logn_quotient + - card_homocyclic + - abelian_exponent_gen + - exponentJ + - def_pnElem + - isog_grank + - is_abelem_pgroup + - fin_ring_char_abelem + - exponent_cycle + - count_logn_dprod_cycle + - abelian_rank1_cyclic + - Mho_cont + - piOhm1 + - p_rank_geP + - injm_grank + - path: mathcomp/algebra/mxalgebra.v + theorems: + - mxrank_cap_compl + - eqmxMfull + - rowV0P + - ltmx_irrefl + - adds0mx + - map_capmx_gen + - addsmx_nop0 + - cent_mx_ideal + - mxrank_sum_leqif + - mxrank0 + - stablemxN + - row_free_castmx + - map_row_base + - mulsmxDl + - map_row_ebase + - ltmx1 + - capmx_eq_norm + - mxrank_mul_ker + - mxrankS + - mxrankE + - diffmxE + - map_capmx + - eqmxMfree + - ltmxErank + - eq_row_full + - row_sub + - sub_capmx + - summx_sub + - eqmx_conform + - matrix_modr + - row_fullP + - mxrankM_maxr + - capmx_idPl + - cap0mx + - mxrank_opp + - logn_card_GL_p + - mxrank_adds_leqif + - rank_leq_row + - map_submx + - mulsmx_subP + - stablemx0 + - negb_row_free + - stablemx_sums + - mxring_id_uniq + - sub_daddsmx + - rank_ltmx + - sub_addsmxP + - eqmx_col + - mxrank_fullrowsub + - sub_dsumsmx + - mxrank_ker + - eqmx_eq0 + - capmx_norm_eq + - row_subP + - capmxC + - nary_mxsum_proof + - addsmx_addKl + - genmx_id + - rowsub_comp_sub + - sub_ltmx_trans + - genmx_cap + - addsmx_nop_eq0 + - addsmx_compl_full + - mxdirect_addsP + - maxrowsub_free + - mxrankMfree + - mulsmxS + - sub_rVP + - lt1mx + - eq_maxrowsub + - row_full_unit + - stablemxC + - rank_rV + - proj_mx_sub + - center_mxP + - fullrankfun_inj + - addsmx_addKr + - capmx_diff + - sumsmx_subP + - eqmxMunitP + - mulsmxA + - card_GL + - mulmxKpV + - stablemx_unit + - memmx1 + - mulmxP + - eqmxP + - lt0mx + - center_mx_sub + - inj_row_free + - addsmx_idPr + - eq_genmx + - rowsub_sub + - mxdirect_adds_center + - proj_mx_compl_sub + - row_full_inj + - eq_row_base + - eqmx_stable + - mulmx_coker + - row_subPn + - genmx_sums + - eq_rank_unitmx + - addmx_sub_adds + - submx_full + - genmx_diff + - rank_diag_block_mx + - addsmx_nop_id + - submx_rowsub + - eqmx_sym + - memmx_addsP + - genmx0 + - addsmxE + - submx_refl + - submxElt + - ltmxE + - capmx1 + - capmx_idPr + - submxMl + - addsmxS + - map_genmx + - fullrowsub_full + - lt_eqmx + - mulmx1_min_rank + - eqmx_rank + - row_full_castmx + - capmx_nopP + - stablemxD + - sub_sumsmxP + - memmx0 + - mxdirect_trivial + - row_base_free + - mxrank_tr + - eqmx_scale + - mxrank_disjoint_sum + - mxrank_injP + - maxrowsub_full + - mxrank_leqif_sup + - sub0mx + - row_free_map + - addsmx_diff_cap_eq + - capmxMr + - adds0mx_id + - adds_eqmx + - mem0mx + - map_cokermx + - pinvmx_full + - mxrank_map + - memmx_map + - mulsmxP + - eigenvalueP + - kermx0 + - mulsmx0 + - eigenvectorP + - mxdirect_sumsE + - capmxSr + - mulmx0_rank_max + - add_proj_mx + - submx0 + - mxdirect_sums_center + - nz_row_sub + - complete_unitmx + - scalar_mx_cent + - map_center_mx + - mulmxKp + - eigenvalue_map + - cokermx_eq0 + - eqmx_opp + - map_kermx + - eqmx_sums + - eq_fullrowsub + - col_mx_sub + - card_GL_1 + - capmx0 + - stablemxM + - sub_kermxP + - sumsmx_sup + - muls0mx + - eqmxMr + - eq_row_sub + - mxdirect_sumsP + - mulVpmx + - mxring_idP + - mulsmxDr + - mxrank1 + - Gaussian_elimination_map + - stable0mx + - capmxE + - sub_bigcapmxP + - cap1mx + - eqmx_refl + - mxdirect_sums_recP + - row_full_map + - mxrank_scale_nz + - mxrank_coker + - row_free_inj + - rV_subP + - map_pinvmx + - map_cent_mx + - mxdirect_sum_eigenspace + - cent_mx_fun_is_linear + - mxrank_gen + - cent_mxP + - mulmxKV_ker + - row_freeP + - mxdirect_delta + - rank_col_mx0 + - scalemx_sub + - comm_mx_stable + - mxrank_delta + - mxdirect_addsE + - submx_trans + - map_col_base + - rank_copid_mx + - comm_mx_stable_ker + - capTmx + - capmx_nop_id + - map_eigenspace + - stableCmx + - eqmx0 + - sub_qidmx + - addsmx_sub + - eqmx_sum_nop + - capmx_normP + - stablemx_row_base + - mxdirectEgeq + - memmx_sumsP + - map_mulsmx + - pinvmx_free + - addsmxSr + - rank_mxdiag + - stableDmx + - eqmx_rowsub_comp_perm + - has_non_scalar_mxP + - rank_col_0mx + - capmxSl + - mulmx_ebase + - addsmx0_id + - addsmxC + - row_free_unit + - ltmx_trans + - genmx1 + - genmx_bigcap + - ltmx_sub_trans + - mem_mulsmx + - proj_mx_0 + - maxrankfun_inj + - map_eqmx + - capmxA + - qidmx_cap + - mxrankM_maxl + - eqmx_rowsub + - stablemx_full + - sumsmxMr + - cap_eqmx + - comm_mx_stable_eigenspace + - mxdirectE + - fullrowsub_free + - submxMr + - bigcapmx_inf + - memmx_subP + - mxrank_leqif_eq + - addsmxMr + - mulmxVp + - map_col_ebase + - fullrowsub_unit + - rank_row_mx0 + - submx0null + - eqmx_cast + - matrix_modl + - ltmx0 + - mxdirectP + - sub_sums_genmxP + - mxrank_add + - addsmx_idPl + - card_GL_2 + - ltmxW + - eigenspaceP + - cent_mx_ring + - mxrank_Frobenius + - diffmxSl + - sub_kermx + - submxMfree + - row_ebase_unit + - mxrank_mul_min + - genmx_witnessP + - memmx_eqP + - sumsmxS + - capmx_witnessP + - map_complmx + - addsmx0 + - mxrank_scale + - sub_capmx_gen + - submxE + - muls_eqmx + - sub1mx + - capmx_compl + - proj_mx_id + - cent_rowP + - rank_row_0mx + - eqmx_trans + - mulmx_free_eq0 + - submxP + - addsmxSl + - capmxT + - sumsmxMr_gen + - col_leq_rank + - submx1 + - eqmx0P + - ltmxEneq + - mxrank_unit + - map_diffmx + - genmx_adds + - rank_pid_mx + - map_addsmx + - path: mathcomp/algebra/matrix.v + theorems: + - map_col_mx + - det0 + - scalemx_inj + - mx11_scalar + - mxcol_const + - mxcol_sum + - unitmx1 + - trmxK + - lin1_mx_key + - row_mxEl + - map_mx_key + - mxsize_recl + - eq_mxdiag + - mul_mxcol_mxrow + - mul_row_block + - col_permM + - mxrow0 + - mxrowD + - mxvecE + - colsub_comp + - row_usubmx + - mx_vec_lin + - scalar_mx_sum_delta + - map2_usubmx + - scale1mx + - det_inv + - map2_col_perm + - mxtrace_is_scalar + - map_mxN + - map2_row_mx + - mul_scalar_mx + - add_row_mx + - unitmxE + - mx0_is_diag + - map_xcol + - delta_mx_rshift + - map2_row + - col_mxblock + - mxtrace_mxblock + - col'_eq + - col_mx_key + - map2_mxsub + - tr_scalar_mx + - mxblockEh + - row_mx0 + - determinant_alternate + - mul_rVP + - submxblockB + - trmx_adj + - row_mxKr + - col_rsubmx + - map2_1mx + - mxsub_comp + - mxcolB + - mx_rV_lin + - map_mx_id_in + - trmx_delta + - mulVmx + - scalar_mx_block + - trmx_drsub + - is_perm_mx_tr + - mulmx_is_scalable + - addmx_key + - map_tperm_mx + - det_mulmx + - eq_castmx + - scale_scalar_mx + - map2_lsubmx + - scalar_mx_is_additive + - tr_col + - mulmxN + - mulmx_colsub + - is_diag_block_mx + - map_row_mx + - mul_pid_mx + - mxcol_recu + - scalemxAr + - usubmxEsub + - mxsub_mul + - mxvec_cast + - diag_mx_sum_delta + - cast_col_mx + - mulNmx + - mulmxBl + - rowE + - map2_conform_mx + - trmx_const + - invmx_out + - xcolE + - mulmx1 + - submxblockK + - mxblock_recul + - rowP + - xcol_const + - perm_mxV + - diag_const_mx + - row_mxsub + - mulKVmx + - mul_rV_lin + - colP + - comm_mx_sym + - map_mxsub + - col_mx_eq0 + - tr_row_mx + - xrowEsub + - flatmxOver + - rowsubE + - det_diag + - map_mxZ + - mxblockD + - mul_vec_lin_row + - rsubmxEsub + - tr_row' + - map_ursubmx + - trmx_usub + - row_mx_key + - mul_delta_mx + - diag_mx_is_linear + - invmx_scalar + - mxOver_opp_subproof + - cormen_lup_correct + - mul_block_col + - row_rowsub + - eq_mxsub + - delta_mx_dshift + - pid_mx_row + - scalemxDl + - rowK + - curry_mxvec_bij + - map_lin1_mx + - block_mxEdr + - map_mxB + - comm_mxB + - lift0_perm_eq0 + - castmxKV + - mxOverM + - map2_col_mx + - scale_col_mx + - map2_mxC + - col_mxsub + - idmxE + - mxtrace_scalar + - is_perm_mxMr + - mul_row_col + - tr_submxblock + - submxK + - lift0_perm0 + - col_mxdiag + - map2_mxvec + - trmx_cast + - map_row_perm + - block_mx0 + - eq_mxcol + - mul_diag_mx + - mul_xcol + - mxcolN + - map_drsubmx + - map2_vec_mx + - map2_mx0 + - mxOverZ + - opp_row_mx + - block_mxEul + - const_mx_key + - col_perm_const + - vec_mx_eq0 + - scalemxDr + - map2_mxA + - intro_unitmx + - map2_ursubmx + - tr_mxrow + - trmx1 + - col_colsub + - map2_mx1 + - row1 + - map_mxM + - row'Kd + - mxblock0 + - GL_VxE + - opp_block_mx + - mul_rowsub_mx + - col'_const + - tr_mxcol + - map2_rsubmx + - is_diag_trmx + - mxEmxrow + - scalemxAl + - tr_pid_mx + - submxrow_matrix + - eq_rowsub + - trigmx_ind + - mul_mxrow_mxblock + - block_mxEdl + - submxrowD + - row_permEsub + - adjZ + - map2_mx_left + - expand_det_row + - det_lblock + - map_col_perm + - is_scalar_mx_is_diag + - ursubmxEsub + - col_eq + - row_mxrow + - rowsub_comp + - vsubmxK + - lift0_perm_lift + - mxblockK + - mulmx_sum_row + - mxOverS + - mxOver_diagE + - mulmxDr + - tr_xrow + - col_mxEu + - map_scalar_mx + - mxrowP + - cast_row_mx + - matrix_key + - mx0_is_trig + - det_tr + - rowEsub + - cormen_lup_upper + - diagsqmx_ind + - map2_mx_key + - mul_mx_row + - tr_row + - is_trig_mxblockP + - GL_ME + - mul_mxdiag_mxcol + - GL_unitmx + - block_mx_eq0 + - mul_mxdiag_mxblock + - mxdiagN + - tr_perm_mx + - mul_mxrow + - mxsub_id + - mxtrace_is_semi_additive + - mxsub_ffun + - row_perm1 + - mul_mx_scalar + - is_diag_mxP + - submxcol_sum + - mxcol_mul + - row_mx_eq0 + - mx0_is_scalar + - is_scalar_mx_is_trig + - mxrowN + - tr_col' + - castmx_comp + - is_perm_mxMl + - map_lin_mx + - mx11_is_diag + - eq_block_mx + - lin_mulmx_is_linear + - addNmx + - mxrow_const + - mul_pid_mx_copid + - opp_col_mx + - trmx0 + - trmx_mxsub + - row_mxA + - mxvec_delta + - delta_mx_key + - col_mx0 + - rV0Pn + - mxdiagD + - is_diag_mxEtrig + - unitmxZ + - mxcol0 + - map_mxD + - map_usubmx + - eq_mxblockP + - row_permM + - tr_block_mx + - map2_trmx + - invmx_block_diag + - col_perm_key + - mulmx1_unit + - dsubmxEsub + - comm_mxM + - summxE + - scale_block_mx + - ulsubmx_diag + - mul_delta_mx_cond + - add_block_mx + - eq_in_map_mx + - col0 + - mul_dsub_mx + - map2_xcol + - unitmx_mul + - vec_mxK + - mxtraceD + - trmx_mul + - map2_mxDl + - trmxV + - cV0Pn + - col'Kl + - mxOver_scalarE + - cofactor_map_mx + - mul_mxblock + - mxOver_add_subproof + - mxOver_constE + - pid_mx_col + - mxblockEv + - mxvec_indexP + - all_comm_mx1 + - map_copid_mx + - mul_adj_mx + - map_diag_mx + - block_mxKdl + - mxblock_recu + - mxsub_ffunl + - pid_mx_minh + - unitmx_inv + - conform_castmx + - mxOverP + - mxsub_const + - submxrowN + - mxrowB + - scalemxA + - mxdiagB + - col'_col_mx + - col_lsubmx + - map_pid_mx + - map_mx_id + - submxcol_matrix + - cofactorZ + - col1 + - map2_dlsubmx + - is_trig_mxP + - is_trig_block_mx + - col_permE + - delta_mx_lshift + - eq_row_mx + - xrowE + - mxsubcr + - mxdiag0 + - scalemx_const + - comm_mxN + - copid_mx_id + - trmx_inv + - vec_mx_delta + - mxOver_const + - mx1_sum_delta + - eq_map2_mx + - submxcolN + - row_mx_const + - map_mx_eq0 + - lift0_permK + - col_const + - pid_mx_id + - all_comm_mxP + - pid_mx_block + - row_const + - ulsubmxEsub + - block_mxKul + - scalar_mx_key + - colKr + - comm_mxE + - delta_mx_ushift + - row_mxdiag + - row_diag_mx + - comm1mx + - diagmx_ind + - scalar_mx_is_diag + - row_permE + - det_mx11 + - swizzle_mx_is_semi_additive + - col_mxA + - comm_mx_refl + - ursubmx_trig + - is_diag_mxblock + - trmx_mul_rev + - scalar_mx_is_multiplicative + - map_mx_is_scalar + - mul_mxblock_mxrow + - mxcolK + - xrow_const + - submxcol0 + - castmx_id + - submxrow0 + - eq_map_mx + - map_mx_is_multiplicative + - scalar_mxM + - mxvec_eq0 + - mulmx1C + - usubmx_key + - tr_col_mx + - mxsub_ffunr + - col_permEsub + - mulmx0 + - submxblockN + - map_const_mx + - trmx_dlsub + - trmx_dsub + - mxsub_ind + - map_mx_adj + - submxblockEv + - mulmx_lsub + - col_mxKd + - map2_ulsubmx + - mulmxA + - pid_mx_minv + - colsub_cast + - submxblockEh + - row_perm_const + - unitmx_tr + - row_eq + - all_comm_mx2P + - block_mxEh + - mul_copid_mx_pid + - scalar_mxC + - map_row + - submxrow_sum + - eq_colsub + - castmx_sym + - submxblock_diag + - mul_col_perm + - submxrowB + - map_lsubmx + - col_mxcol + - block_mxEv + - row'_const + - is_perm_mxP + - comm_mxP + - mxcolD + - mulmx_diag + - matrix_eq0 + - map2_0mx + - diag_mx_comm + - mxrowK + - castmxK + - eq_col_mx + - mul_rV_lin1 + - colKl + - drsubmx_diag + - perm_mxM + - row'_eq + - flatmx0 + - map_block_mx + - eq_mxcolP + - all_comm_mx_cons + - trace_map_mx + - map_conform_mx + - ringmx_ind + - mul_submxrow + - map2_const_mx + - comm_mx_scalar + - det_map_mx + - mul_col_row + - eq_mxrow + - trmx_lsub + - map_mx_inv + - detM + - submxblock0 + - expand_det_col + - comm0mx + - card_mx + - rowKd + - tr_xcol + - pid_mx_key + - castmxEsub + - lsubmxEsub + - row'_row_mx + - row0 + - swizzle_mx_is_additive + - rowKu + - submxblock_sum + - oppmx_key + - map_dsubmx + - map2_dsubmx + - col_row_permC + - mxOver0 + - map2_col' + - dlsubmxEsub + - diag_mxrow + - map_unitmx + - map_mx_comp + - hsubmxK + - pid_mx_1 + - scalar_mx_is_trig + - trace_mx11 + - mulmx_block + - const_mx_is_additive + - map2_mx_right_in + - mxsub_eq_id + - scalar_mx_is_scalar + - mxtrace_is_additive + - detZ + - map_mx1 + - ulsubmx_trig + - block_diag_mx_unit + - mul_row_perm + - mul_vec_lin + - mul1mx + - dsubmx_key + - map_dlsubmx + - diag_mx_row + - cormen_lup_lower + - exp_block_diag_mx + - vec_mx_key + - mxdiag_recl + - map_row' + - eq_mxblock + - map_mx_inj + - row_thin_mx + - xcolEsub + - map2_drsubmx + - map_invmx + - mulKmx + - map2_row_perm + - perm_mx1 + - col_mx_const + - map_mx_unit + - add_col_mx + - mul0mx + - det_mx00 + - mx'_cast + - swizzle_mx_is_scalable + - det_scalar1 + - mxblock_sum + - mul_usub_mx + - is_diag_mxblockP + - eq_mxrowP + - colE + - mxtraceZ + - map_trmx + - mxtrace_tr + - trmx_conform + - is_perm_mxV + - col'Esub + - mxblock_recl + - matrixP + - invmxZ + - mxtrace1 + - nz_row_eq0 + - mxsub_eq_colsub + - row_ind + - mxOver_scalar + - mx11_is_trig + - mxEmxblock + - mulmx_key + - tperm_mxEsub + - map2_mx_right + - unitmx_perm + - mulmxK + - eq_in_map2_mx + - comm_mxD + - map_ulsubmx + - mxvecK + - mxdiagZ + - diag_mx_key + - dlsubmx_diag + - GL_MxE + - det_perm + - mxdiag_sum + - mxsub_eq_rowsub + - GL_1E + - row_mxKl + - map2_block_mx + - col_perm1 + - mxE + - mxvec_dotmul + - lin_mul_row_is_linear + - eq_mxdiagP + - GL_det + - mulmxE + - comm_mx_sum + - row_matrixP + - map2_castmx + - mxEmxcol + - tr_row_perm + - col_id + - trmx_rsub + - map_delta_mx + - map2_col + - is_trig_mxblock + - castmxE + - det0P + - map2_mxDr + - unitr_trmx + - det1 + - row_row_mx + - diag_mxP + - rowsub_cast + - map_col + - det_trig + - det_ublock + - cormen_lup_perm + - comm_scalar_mx + - is_diag_mx_is_trig + - GL_unit + - matrix_sum_delta + - col_flat_mx + - cofactor_tr + - mul_mxrow_mxcol + - mxtrace_mulC + - colEsub + - mul_mx_adj + - diag_mx_is_trig + - submxrowK + - mxtrace_diag + - lift0_mx_perm + - diag_mxC + - scalemx1 + - row'Esub + - trigsqmx_ind + - trmx_eq0 + - scalemx_key + - adjugate_key + - lin_mulmxr_is_linear + - row_mul + - map2_mx_left_in + - mulmx_suml + - is_perm_mx1 + - row_id + - trmx_ulsub + - map_mx0 + - submxblockD + - map_perm_mx + - tr_mxdiag + - pid_mxErow + - mxblock_const + - scale_row_mx + - submxcol_mul + - row_mxEr + - sqmx_ind + - map_mxvec + - castmx_const + - pid_mxEcol + - mulmxKV + - mxsubrc + - detV + - mulmxDl + - mxOver_diag + - GL_VE + - mxtrace0 + - expand_cofactor + - mxrow_sum + - mx_ind + - mulmx_sumr + - mulmxBr + - matrix0Pn + - matrix_nonzero1 + - col'Kr + - row_mxcol + - col_mxKu + - map_vec_mx + - determinant_multilinear + - perm_mx_is_perm + - tr_col_perm + - mulmxnE + - drsubmx_trig + - comm_mxN1 + - diag_mx_is_additive + - mxblockN + - col_col_mx + - mxblockP + - submxcolB + - row_mxblock + - row_sum_delta + - lsubmx_key + - comm_mx0 + - det_Vandermonde + - nonconform_mx + - conform_mx_id + - mxcolEblock + - block_mx_const + - map_rsubmx + - block_mxKur + - rsubmx_key + - trmx_key + - tr_mxblock + - block_mxKdr + - thinmx0 + - mxsub_cast + - pid_mx_0 + - trmx_inj + - mulmxV + - path: mathcomp/ssreflect/order.v + theorems: + - lt_Taggedl + - diffE + - leU2E + - bigmin_inf + - sub_bigmax_seq + - joinxB + - gt_min + - le_sorted_filter + - join0x + - min_minKx + - joinUA + - meetUr + - subset_bigmax + - ltxi_tuplePlt + - rcomplPmeet + - refl + - lcomparable_leP + - comparable_lteifNE + - codiffErcompl + - bigmaxD1 + - compl_joins + - setKIC + - arg_maxP + - refl + - comparable_bigl + - lexU + - lt_def + - meetUl + - leIxr + - orEbool + - le0x + - trans + - meetxx + - enumT + - le0s + - size_enum_ord + - nhomo_ltn_lt_in + - ltxx + - join_cons + - disj_diffr + - lcmE + - lt_def + - sig_bij_on + - trans + - join_idPr + - decnP + - ltn_def + - ltgtP + - trans + - incomparable_eqF + - count_lt_nth + - joinKIC + - diffxB + - meet_eq1 + - eq_minr + - comparable_contra_leq_lt + - le_sig + - joinBI + - contra_not_lt + - comparable_contra_leq_le + - sub_in_bigmax + - complEcodiff + - incn_inP + - enum_setT + - meetUl + - comparable_lteif_minr + - lt_asym + - meetUA + - mono_leif + - eq_le + - rcomplPmeet + - compl_inj + - diffxx + - le_path_filter + - count_lt_ge + - le_meetU + - les0 + - sub_bigmin + - enum_val_inj + - opred_joins + - meetKU + - meetIB + - meet1x + - contra_lt_le + - joinKI + - diffKI + - sort_le_sorted + - bigmin_geP + - meetUl + - rcomplPmeet + - max_r + - maxxx + - meetUl + - comparable_maxl + - comparable_contra_lt_le + - maxEge + - ltxi_cons + - opred1 + - bigmax_eq_arg + - comparable_contraTle + - joinA + - contra_leq_le + - leifP + - le_path_min + - lteifN + - sig2K + - lt_sorted_pairwise + - meet_def_le + - comparable_leNgt + - ltxi0s + - lcomparable_ltP + - joinE + - comparable_min_maxr + - leU2l_le + - botEprodlexi + - nmono_in_leif + - count_lt_le_mem + - minEge + - min_l + - bigmin_le_cond + - le_cons + - bigmax_ge_id + - meetxC + - diffBx + - comparable_arg_maxP + - sub_tprod_lexi + - comparable_ge_max + - inj_homo_lt_in + - lteif_minr + - le_bigmax_ord + - joinxx + - compl_meets + - comparable_contraPle + - joinC + - ge_trans + - comparable_minC + - join_r + - comparable_maxEge + - joinUKC + - anti + - comparable_le_max + - joins_disjoint + - subset_bigmin + - le_bigmax2 + - homo_ltn_lt_in + - comparable_minCA + - comparable_max_idPl + - leU2 + - eqTleif + - lex1 + - comparable_ltgtP + - eq_diff + - leI2 + - ltEnat + - leI2l_le + - joinIKC + - gt_comparable + - le_anti + - eq_joinl + - joinEprod + - comparable_contra_not_le + - minEgt + - max_minl + - anti + - eq_leLR + - leC + - total + - enum_val_nth + - gcdE + - nonincnP + - lexi_pair + - leUx + - comparable_contraFle + - nth_count_le + - idfun_is_join_morphism + - joinBx + - diff1x + - bigmin_idl + - filter_le_nth + - meetUKC + - le_lt_asym + - diffIx + - min_idPl + - lt1x + - refl + - lexUr + - maxElt + - enum_uniq + - leW_mono + - ge_comparable + - lteif_imply + - lt_val + - rcomplEprod + - lt_trans + - lexi_tupleP + - le_anti + - maxKx + - lexUl + - leUl + - trans + - enum_ord + - omorph0 + - bigmin_gtP + - meetUl + - le_Taggedl + - le_bigmin_nat_cond + - leEseq + - lex1 + - lexI + - maxCA + - joinA + - join_idPl + - wlog_lt + - lt_Taggedr + - comparable_sym + - contra_leq_lt + - diffKU + - lexI + - minC + - eqRank + - comparable_maxKx + - lt_max + - le_max + - enum_valP + - lt_min + - bigmax_mkcondl + - nth_enum_ord + - enum_valK + - maxAC + - tnth_meet + - comparable_max_minr + - enum_rank_inj + - omorphI + - joinKI + - le_anti + - le_enum_val + - refl + - eq_bigmax + - joinAC + - bigmax_idr + - sub_prod_lexi + - codiffEprod + - joinUC + - leif_le + - le_bigmin_ord_cond + - meets_total + - comparable_minEgt + - join1x + - bigmin_set1 + - le_trans + - sig_bij + - lt_path_sortedE + - leEsig + - diffUx + - leEseqlexi + - meetEprod + - joinxx + - topEdual + - diffErcompl + - comparable_contra_lt + - bigmax_mkcond + - meetA + - lteif_maxl + - lt_eqF + - tnth_join + - joinIl + - dec_inj + - comparable_contraNlt + - codiffErcompl + - botEtprod + - bigmin_eq_arg + - topEord + - sigK + - bigmax_ltP + - ltx1 + - mask_sort_le + - leEdual + - joinEsubset + - lt_def + - joinCx + - tnth_codiff + - meetEsubset + - lt_leif + - subseq_lt_sorted + - diffKI + - bigmax_le + - lteifNE + - meets_setU + - eq_meetr + - le_def + - le_max_id + - rank_bij + - comparable_maxAC + - leI2E + - meetsP_seq + - neqhead_ltxiE + - subseq_lt_path + - leBRL + - lt_sorted_leq_nth + - lteifF + - diffErcompl + - subseq_le_sorted + - lcomparable_ltgtP + - subseq_le_path + - joinx1 + - meets_inf_seq + - meetKUC + - bigmin_mkcondl + - enum_rank_bij + - count_le_gt + - opred_meets + - rcomplEtprod + - codiffErcompl + - ltrW_lteif + - contra_lt_not + - lt_nsym + - neq_lt + - meetUl + - comparable_contra_ltn_le + - lt_neqAle + - joinA + - bigmin_idr + - gtE + - meet_eq0E_diff + - valI + - ltxI + - bigminUl + - lt_sorted_eq + - sort_lt_sorted + - bigminUr + - inj_homo_lt + - maxEgt + - comparable_ge_min + - comparable_contra_le + - comparable_lt_min + - le_eqVlt + - index_enum_ord + - le_meets + - diff0x + - sub_in_bigmin + - eq_joinr + - set_enum + - comparable_maxxK + - le_bigmax_nat_cond + - rankEsum + - comparable_contraPlt + - valD + - ge_anti + - bigmax_eq_id + - botEsubset + - omorphU + - meetIKC + - diffxU + - inj_nhomo_lt + - leif_trans + - meetKI + - contra_ltF + - joinKU + - meetKU + - lt_rank + - sorted_mask_sort_le + - lt_path_pairwise + - nth_enum_rank_in + - gt_max + - nth_count_gt + - enum_rankK_in + - bigmax_idl + - ltEprod + - comparable_contra_le_lt + - joinsP_seq + - ltEseqlexi + - contra_leF + - leI2r_le + - le_bigmin + - joins_setU + - le_le_trans + - prod_display_unit + - sort_lt_id + - lteif_maxr + - lexI + - meetA + - comparable_minl + - maxEle + - bigmin_split + - leIl + - rcomplPmeet + - maxC + - ltn_def + - rcomplKU + - lt_pair + - eq_leRL + - dvdE + - sub_bigmin_seq + - complK + - rcomplPmeet + - ltW_nhomo_in + - inj_nhomo_lt_in + - omorph_lt + - meet0x + - perm_sort_leP + - diffxI + - leEtprod + - rank_inj + - joinKI + - nmono_leif + - le_min + - complEsubset + - comparable_minAC + - le_enum_rank + - comp_is_bottom_morphism + - joinA + - comparable_lteif_maxl + - ord_display + - joins_min_seq + - lex1 + - le_bigmax + - le_sorted_ltn_nth + - le_sorted_eq + - count_le_nth + - leif_refl + - anti + - sdvdE + - complEbool + - meet_idPr + - le_def + - nth_count_ge + - val1 + - meetUC + - lt_def + - le_gtF + - diffEtprod + - meetBx + - anti + - meetsP + - ltW_homo_in + - meetE + - leBUK + - comparable_minEge + - complEdiff + - meetx1 + - complEcodiff + - disj_leC + - lt_def + - mem_enum + - meet_cons + - comparable_le_min + - joinBIC + - meets_inf + - comparable_arg_minP + - sub_bigmax_cond + - minElt + - meetA + - complEtprod + - joinKI + - incnP + - meetxx + - lteifNF + - topEsubset + - bigminD1 + - bigmaxIr + - gt_def + - nat_display + - lteif_trans + - comparable_minACA + - le_trans + - meets_ge + - Rank1K + - joinC + - contra_le_not + - lt_wval + - comparable_contraTlt + - ltW + - cardT + - leW_mono_in + - le_path_mask + - arg_minP + - eq_Rank + - joinxx + - lteif_anti + - complU + - leIx2 + - join_l + - ge_leif + - lt_sorted_mask + - nth_count_lt + - bigmax_leP + - meet_idPl + - eqhead_ltxiE + - nondecnP + - opredU + - min_minxK + - meetxx + - le_bigmin_nat + - le_comparable + - bigminD + - leEord + - comparable_maxr + - bigmaxD + - meetUl + - meets_seq + - min_maxr + - maxEnat + - maxA + - comparable_min_maxl + - bigminIr + - leBr + - leEbool + - contra_ltn_lt + - homo_ltn_lt + - enum_set0 + - enum_val_bij + - complEdiff + - ltEbool + - botEord + - subset_bigmin_cond + - lexI + - meet_eql + - lexis0 + - bigmax_imset + - join_eq0 + - complEprod + - disjoint_lexUr + - codiffErcompl + - le_mono_in + - totalT + - minACA + - compl0 + - setKUC + - rcomplPjoin + - leIr + - complB + - lex1 + - ltx0 + - joinsP + - rect + - comparable_maxA + - leIx + - leEmeet + - le_sig1 + - leUidr + - cover_leIxr + - le_pair + - meetEtotal + - comparable_eq_maxl + - enum_valK_in + - joinUKI + - comparable_lteif_minl + - joinBKC + - comp_is_nondecreasing + - complEcodiff + - meetUK + - leP + - rank_bij_on + - diffUK + - mono_sorted_enum + - le_bigmax_cond + - sorted_filter_le + - leU2r_le + - wlog_le + - posxP + - minxx + - leNgt + - bigmaxU + - contra_ltT + - joins_min + - lexI + - rcomplKI + - bigmax_split + - leEjoin + - complEdiff + - mono_in_leif + - meetxB + - RankEsum + - lt_sorted_filter + - comparable_leP + - lexi_display + - setTDsym + - bigmaxID + - contra_le_leq + - leUx + - leBLR + - meet_r + - lt_sig + - leCx + - bigmax_set1 + - min_idPr + - trans + - filter_sort_le + - comparable_lteif_maxr + - leW_nmono + - le_trans + - lt_total + - mono_unique + - contra_leT + - le0x + - bigmaxIl + - complEdiff + - bigmax_lt + - contra_le_lt + - le_sorted_mask + - comparable_min_idPr + - joinIr + - opredI + - comparable_minKx + - joinxC + - comparable_max_minl + - ltxi_pair + - leB2 + - idfun_is_top_morphism + - inc_inj + - nondecn_inP + - comparable_ltNge + - complErcompl + - min_r + - le_Taggedr + - le_rank + - subEsubset + - le_total + - refl + - complEcodiff + - leEsubset + - lex0 + - bigmin_imset + - topEtlexi + - lexI + - sub_bigmin_cond + - meetKU + - bigminID + - ltxU + - comparable_minxK + - le_sorted_pairwise + - lteif_minl + - diffEprod + - ltW_nhomo + - bigmin_mkcond + - filter_lt_nth + - eq_enum_rank_in + - comparable_eq_minr + - seqprod_display + - diffx0 + - val0 + - bigmaxUl + - leBC + - lcomparableP + - le0x + - le_anti + - leIidr + - sig_inj + - contra_not_le + - le0x + - omorph1 + - topEprod + - meetEdual + - leEjoin + - contraFle + - joinUK + - ltEdual + - tnth_compl + - meetCx + - ltUx + - lt_comparable + - leEprodlexi + - enum0 + - contra_lt_ltn + - bigminU + - eq_bigmin + - comparable_maxACA + - contra_ltN + - nth_enum_rank + - le_mono + - dec_inj_in + - comparableT + - le_bigmax_nat + - total + - orbE + - setIDv + - ltIx + - comparable_contra_not_lt + - lteifW + - ge_total + - ge_min + - leIxl + - enum_rank_in_inj + - rankE + - lt_def + - minEnat + - botEprod + - contraPle + - comparable_lt_max + - joinEdual + - trans + - lt_geF + - contraFlt + - topEprodlexi + - meetKU + - nth_ord_enum + - nat1E + - sub_bigmax + - meetIK + - ge_antiT + - comparableP + - comp_is_join_morphism + - meets_max_seq + - geE + - joinBK + - joinEtotal + - eq_leif + - leEnat + - lt_path_filter + - ltxi_lehead + - card + - minA + - nth_count_eq + - inc_inj_in + - eq_cardT + - ge_anti + - subEbool + - le_lt_trans + - joinC + - contra_lt + - leEmeet + - le0x + - lt_bigmin + - meetUl + - val_enum_ord + - lteifS + - disjoint_lexUl + - lt0B + - diffErcompl + - ltEord + - disj_le + - comparablexx + - le_bigmax_ord_cond + - seqlexi_display + - eq_minl + - meetUKU + - ltNleif + - le1x + - dvd_display + - leUx + - eqhead_lexiE + - meetKI + - meet_l + - enum_val_bij_in + - leUx + - contra_le_ltn + - comp_is_meet_morphism + - meetx0 + - contra_le + - comparable_minA + - botEtlexi + - contraPlt + - total + - contra_lt_leq + - opredU + - joinKI + - bigmin_le_id + - joinEseq + - meetC + - min_maxl + - disj_diffl + - lt_Rank + - comparable_maxEgt + - meetC + - lexi_lehead + - le_trans + - joins_seq + - Rank2K + - sub_seqprod_lexi + - mem2_sort_le + - lt_le_trans + - lt_sorted_uniq_le + - bigmin_eq_id + - botEnat + - le_path_sortedE + - valU + - sorted_subseq_sort_le + - lt_leAnge + - subset_bigmax_cond + - max_maxxK + - le_bigmin_ord + - comparable_contraNle + - lt0x + - leif_eq + - lexU2 + - rankK + - sigE12 + - minKx + - bigmax_mkcondr + - sorted_filter_ge + - meetKIC + - meetEtprod + - meetxx + - subseq_sort_le + - le0x + - leIidl + - totalU + - ltxis0 + - lt_def + - comparable_contraFlt + - tnth_diff + - ltEtprod + - contra_ltn_le + - joinIB + - meets_max + - nhomo_ltn_lt + - andEbool + - leW_nmono_in + - comparable_minr + - lteifT + - minCA + - leEprod + - bool_display + - contraNlt + - sorted_filter_lt + - eq_maxr + - LatticePred.opred_meets + - le_joins + - bigmin_le + - lt_sorted_uniq + - disj_leC + - andbE + - minxK + - omorph_le + - join_def_le + - meetACA + - botEdual + - eq_ltRL + - le_bigmin2 + - max_l + - ltxi_tupleP + - gt_eqF + - rcomplPjoin + - eq_enum + - anti + - lt_val + - meetC + - topEsig + - rcomplPjoin + - joinx0 + - cardE + - topEtprod + - joinEtprod + - refl + - maxACA + - cover_leIxl + - joinIl + - nat0E + - codiffEtprod + - eq_maxl + - max_maxKx + - joins_le + - lexi0s + - lexC + - joinIK + - sorted_filter_gt + - path: mathcomp/solvable/gfunctor.v + theorems: + - idGfun_closed + - gFmod_cont + - gFgroupset + - gFnormal_trans + - trivGfun_cont + - gFmod_hereditary + - gFunctorS + - morphimF + - injmF_sub + - gFnorm_trans + - gFsub_trans + - injmF + - pcontinuous_is_hereditary + - gFisog + - idGfun_cont + - gFcont + - gFnorm + - gFunctorI + - gFmod_closed + - idGfun_monotonic + - gFchar + - gFnorms + - gFcompS + - gFchar_trans + - gFhereditary + - continuous_is_iso_continuous + - gFiso_cont + - gFcomp_cont + - gFnormal + - gFid + - gFcomp_closed + - gF1 + - gFisom + - pmorphimF + - pcontinuous_is_continuous + - path: mathcomp/ssreflect/ssrnat.v + theorems: + - expnS + - addnK + - lt0b + - ltn_subCl + - ltn0 + - eq_binP + - ltn_sqr + - addnn + - contra_leqF + - mono_leqif + - eqSS + - ltn_predRL + - minnSS + - ltnSE + - sqrnD_sub + - leq_nmono + - eq_ex_maxn + - gtn_max + - leq_eqVlt + - ltn_Sdouble + - addnBCA + - ltn_pmul2r + - half_leq + - ltn_exp2l + - maxnSS + - minnMl + - leqW + - anti_leq + - subnBAC + - doubleS + - odd_geq + - add1n + - subnDr + - addn4 + - gtn_half_double + - iter_muln + - leq_mul2l + - doubleE + - half_bit_double + - leq_addr + - addn_eq0 + - leq_total + - subnAC + - ltn_predK + - addn_eq1 + - muln_eq0 + - leqP + - ltn_neqAle + - iter_succn + - mulnAC + - addnE + - expn_eq0 + - subnCBA + - iter_addn_0 + - eqn_add2l + - leq_sqr + - minKn + - ltnNge + - eqnP + - eq_iterop + - doubleD + - addSnnS + - addn_negb + - muln_gt0 + - leq_subCl + - doubleK + - leq_exp2r + - leq_add2l + - nat_of_mul_pos + - contraTltn + - subDnCAC + - ltn_psubLR + - prednK + - muln0 + - eqnE + - ltn_sub2l + - minn_maxl + - decn_inj_in + - expnAC + - addE + - ltn_pfact + - addnAC + - uphalf_leq + - subDnAC + - expIn + - leq_sub + - mulSnr + - oddB + - addn_maxr + - contra_ltn + - maxn_minr + - subSS + - exp1n + - leq_subRL + - ubnPleq + - minn_idPl + - ltnW + - mulnDl + - leq_pmull + - addnBAC + - ltn_add2l + - contra_leqN + - ltnNleqif + - leqif_eq + - leq_exp2l + - posnP + - double_eq0 + - leq_sub2rE + - oddE + - anti_geq + - inj_homo_ltn_in + - maxnAC + - subn1 + - inj_homo_ltn + - ltP + - leqNgt + - nat_of_add_bin + - expnMn + - iter_muln_1 + - addnI + - addBnA + - leq_fact + - ex_maxnP + - leqif_mul + - ltn_subRL + - ltn_min + - odd_uphalfK + - eq_leq + - lt0n + - maxnn + - mulnDr + - ltn_subrR + - addn_minl + - iteropS + - nat_semi_morph + - ltnP + - eqn_pmul2r + - contra_ltnN + - ltn_subLR + - minnK + - leqSpred + - mulnn + - minn_idPr + - addnCA + - eq_ex_minn + - factS + - even_halfK + - leqif_refl + - expnM + - doubleB + - oddD + - subn_minl + - ltn_psubCl + - max0n + - addn_maxl + - subn0 + - geq_minr + - iter_in + - nat_of_mul_bin + - subn_maxl + - contra_leq_ltn + - find_ex_minn + - ltn_addl + - leq_pfact + - halfD + - oddX + - expE + - ltngtP + - leq_min + - ltn_double + - addn1 + - add3n + - leq_uphalf_double + - muln1 + - leq_sub2r + - nat_AGM2 + - minn_maxr + - geq_leqif + - eqb0 + - expnE + - plusE + - double0 + - leq_mul2r + - exp0n + - eq_leqif + - iter_predn + - expnSr + - minnACA + - subnBl_leq + - add2n + - expn0 + - neq0_lt0n + - addnBl_leq + - addnCBA + - eqn_add2r + - eqn_exp2r + - subn_eq0 + - iterX + - muln_eq1 + - subnK + - ltn_fact + - oddS + - iterM + - leqifP + - leqW_mono_in + - ltn_mul2r + - addnBA + - minusE + - minnA + - minnCA + - leq_sub2lE + - factE + - mulnb + - add4n + - contra_not_leq + - iteriS + - mulnACA + - ltn_sub2rE + - eqn_sub2rE + - addnACl + - nat_Cauchy + - maxnMr + - gtn_eqF + - maxn0 + - even_uphalfK + - leq_half_double + - leq_maxl + - lt0n_neq0 + - contraFleq + - contra_ltn_not + - oddN + - leqif_trans + - doubleMl + - oddb + - uphalfE + - fact_geq + - homo_ltn + - eqn_leq + - uphalf_gt0 + - leq_addl + - iterS + - addnBr_leq + - addnS + - ltn_Pmulr + - ltn_expl + - subnDAC + - iter_addn + - nat_of_binK + - mulnC + - incn_inj_in + - addn2 + - ltn_trans + - mulnA + - ltn_addr + - ltn_pexp2l + - eqn0Ngt + - iterSr + - doubleE + - ltn_exp2r + - ltn_pmul2l + - decn_inj + - leq_double + - maxnCA + - incn_inj + - mulnS + - homo_leq + - sqrn_gt0 + - fact_gt0 + - mulnBr + - maxnC + - contra_leq_not + - leq_subrR + - gtn_neqAge + - addnBn + - odd_ltn + - nat_of_exp_bin + - subnDl + - eqb1 + - contraPltn + - iter_fix + - eqn_pmul2l + - homo_ltn_in + - contraFltn + - inj_nhomo_ltn_in + - subnKC + - leq_pexp2l + - subn_if_gt + - lt_irrelevance + - maxnK + - leqnSn + - ltn_add2r + - odd_double + - eqn_sub2lE + - maxKn + - sub1b + - subnSK + - ltnW_homo + - eq_iteri + - ltn_mul + - maxnE + - contraNleq + - leq_b1 + - ltn_predL + - leq_maxr + - geq_half_double + - maxnACA + - mulnBl + - bin_of_natK + - eqn_mul2r + - nat_of_succ_pos + - ltnn + - subSKn + - addn0 + - ubnPeq + - muln2 + - expnD + - expn1 + - eqn_sqr + - odd_double_half + - maxn_idPr + - ltn_eqF + - double_gt0 + - ex_minnP + - maxnA + - subSnn + - leq_mul + - ltn_subrL + - addnABC + - maxnMl + - subDnCA + - subKn + - fact0 + - leq_psubCr + - maxn_minl + - mulnCA + - ltn_uphalf_double + - subnDA + - contra_ltnF + - nat_semi_ring + - leq_nmono_in + - subnBr_leq + - ltn_geF + - contraPleq + - minnMr + - subBnAC + - succnK + - ltnSn + - addKn + - leq_max + - leq_ltn_trans + - mulnbl + - addn3 + - ltn_sub2r + - addnACA + - leq_subr + - mulSn + - geq_minl + - leqW_nmono_in + - uphalf_double + - leq_pmul2l + - addnA + - mulnE + - add0n + - ltn_mul2l + - ltnW_nhomo_in + - geq_min + - addnC + - iter_succn_0 + - gtn_uphalf_double + - subn_sqr + - min0n + - contra_leqT + - minn0 + - contraTleq + - addBnCAC + - ltn_leqif + - leq_pred + - addn_min_max + - leq_gtF + - addn_minr + - subn_gt0 + - sub0n + - oddM + - leqnn + - contra_ltn_leq + - le_irrelevance + - eqTleqif + - minnC + - leq_mono_in + - sqrn_inj + - ltnW_homo_in + - succn_inj + - mulnbr + - subnBA + - add_mulE + - minnn + - leq0n + - mul0n + - leq_subLR + - leP + - ltn0Sn + - neq_ltn + - mul1n + - ltn_subCr + - inj_nhomo_ltn + - subnS + - ubnP + - subnn + - addn_gt0 + - contra_leq + - expnI + - addnCAC + - nat_of_add_pos + - contraNltn + - nat_power_theory + - subnE + - eqn_mul2l + - ltn_Pmull + - minnAC + - sqrnB + - expn_gt0 + - odd_gt0 + - addIn + - leq_trans + - leq_Sdouble + - geq_max + - eq_iter + - leqVgt + - path: mathcomp/solvable/nilpotent.v + theorems: + - nilpotentS + - nilpotent1 + - lcn_normalS + - ucn_char + - ucn_sub + - lcn_norm + - quotient_sol + - injm_sol + - der_bigdprod + - lcn_cont + - lcn1 + - ucn_nilpotent + - abelian_nil + - der_bigcprod + - quotient_center_nil + - nil_comm_properl + - lcn_sub + - ucn0 + - morphim_ucn + - isog_nil_class + - nil_class0 + - morphim_nil + - ucn_norm + - lcnSnS + - nilpotent_proper_norm + - sol_der1_proper + - ucnSn + - cyclic_nilpotent_quo_der1_cyclic + - isog_nil + - der_cprod + - der_dprod + - lcnSn + - lcn_char + - series_sol + - nilpotent_sub_norm + - derivedP + - abelian_sol + - centrals_nil + - ucn_lcnP + - ucn_bigcprod + - morphim_lcn + - ucn_normalS + - ucn_normal + - ucn_group_set + - ucn_central + - lcnE + - metacyclic_sol + - bigdprod_nil + - lcn_bigcprod + - ucn_dprod + - lcn_dprod + - ucn_bigdprod + - ucnP + - morphim_sol + - ucn_subS + - nil_class_quotient_center + - ucn_cprod + - nilpotent_sol + - lcn0 + - ucn_nil_classP + - ucn_pmap + - mulg_nil + - nil_class_injm + - solvable1 + - lcn2 + - lcn_sub_leq + - lcnS + - ucn_comm + - nilpotent_class + - nil_class_morphim + - nil_comm_properr + - ucn_sub_geq + - center_nil_eq1 + - cprod_nil + - nil_class1 + - lcn_normal + - lcn_nil_classP + - injm_ucn + - dprod_nil + - lcnP + - lcn_bigdprod + - quotient_ucn_add + - lcn_central + - lcn_cprod + - ucnE + - nilpotent_subnormal + - nil_class_ucn + - ucnSnR + - lcn_group_set + - isog_sol + - path: mathcomp/algebra/fraction.v + theorems: + - pi_opp + - addN_l + - mulC + - pi_mul + - tofracMn + - tofrac_eq0 + - pi_inv + - equivf_def + - equivf_r + - Ratio_numden + - Ratio0 + - tofrac_is_multiplicative + - tofracMNn + - tofrac1 + - addC + - denom_ratioP + - inv0 + - equivf_l + - tofracB + - mulA + - mul1_l + - Ratio_numden + - add0_l + - tofracD + - tofracM + - RatioP + - mul_addl + - equivf_refl + - pi_add + - numer0 + - tofrac_eq + - equivfE + - denom_Ratio + - numer_Ratio + - addA + - tofracXn + - nonzero1 + - tofrac_is_additive + - tofracN + - mulV_l + - path: mathcomp/fingroup/action.v + theorems: + - astab1Js + - actpermM + - orbit_transl + - atrans_dvd_index_in + - orbit_inv + - is_total_action + - astabR + - gacent1 + - qactE + - porbit_actperm + - afix_cycle + - injm_faithful + - sub_astabQ + - setactVin + - contra_orbit + - astabsR + - orbit_sym + - afix_gen_in + - conjG_is_action + - astabs_Aut_isom + - gact_stable + - acts_joing + - afix_actby + - abelian_classP + - actperm_id + - afixS + - quotient_astabQ + - gactX + - comp_is_groupAction + - afixRs_rcosets + - sub_afixRs_norms + - orbit_in_sym + - astab_setact_in + - subset_faithful + - qact_domE + - acts_quotient + - qact_proof + - restr_permE + - astab_norm + - astabs_act + - aperm_is_action + - conj_astabQ + - astabs_mod + - orbit_conjsg_in + - gacent_ract + - setactE + - trans_subnorm_fixP + - orbitE + - orbit_partition + - card_orbit_stab + - comp_is_action + - acts_sum_card_orbit + - astab_comp + - afixJ + - acts_in_orbit + - sub_afixRs_norm + - acts_subnorm_fix + - gacent_actby + - astabsP + - acts_ract + - astabQ + - atrans_supgroup + - card_orbit1 + - astabsJ + - dom_qactJ + - gacentY + - orbit_in_eqP + - actCJV + - acts_gen + - afix_cycle_in + - atrans_acts + - subgroup_transitiveP + - actXin + - astabU + - astab1_set + - amoveK + - act_reprK + - atrans_dvd_in + - astabs_comp + - afix_comp + - sum_card_class + - acts_subnorm_subgacent + - astab_gen + - astabs_range + - acts_sub_orbit + - afix_ract + - orbitJ + - gacentM + - qactJ + - actX + - astabsU + - afixM + - sub_astab1_in + - im_actm + - gacentIdom + - actsI + - modactEcond + - astabP + - im_restr_perm + - mem_setact + - astab_normal + - autactK + - astabs_setact + - card_setact + - astab_subact + - orbit_conjsg + - fixSH + - Cayley_isom + - act_inj + - orbit_stabilizer + - triv_restr_perm + - Aut_in_isog + - restr_perm_isom + - perm_act1P + - astabM + - sub_astab1 + - Cayley_isog + - faithful_isom + - orbit_in_trans + - gacentC + - ker_actperm + - mactE + - astab1Rs + - rcoset_is_action + - astab_sub + - orbit_lcoset + - morph_gacent + - atransPin + - orbitR + - orbitJs + - conjg_is_groupAction + - atrans_acts_in + - orbit_trans + - setactJ + - gacts_range + - morph_astab + - actMin + - orbit_actr + - afixU + - subgacentE + - ract_is_action + - morph_gastab + - orbit_refl + - astabs_subact + - orbit_eq_mem + - afix_subact + - amove_orbit + - astab_act + - actKin + - orbit_lcoset_in + - morphim_actm + - gacentD1 + - morph_gact_irr + - afixMin + - astab1_act_in + - actsD + - gacent_comp + - astabs_quotient + - astab1_act + - Aut_restr_perm + - index_cent1 + - astabsQ + - modact_faithful + - astabs1 + - acts_act + - orbit_rcoset + - qact_is_groupAction + - gacentQ + - card_orbit + - atransP + - group_set_astab + - atransP2in + - injm_Aut_full + - orbit_in_transl + - gacentJ + - card_conjugates + - afixJG + - gacent_cycle + - actsU + - faithfulR + - perm_mact + - astabs_actby + - gactR + - subgroup_transitivePin + - mact_is_action + - atrans_dvd + - astabCin + - modactE + - gacent_gen + - atrans_orbit + - astabsIdom + - afix_mod + - astab1P + - astabC + - val_subact + - acts_char + - orbit_rcoset_in + - sub_act_proof + - astabRs_rcosets + - isom_restr_perm + - astabs_dom + - modact_is_action + - acts_irr_mod + - afix1P + - qactEcond + - astab_mod + - setact_orbit + - restr_perm_commute + - setact_is_action + - group_set_astabs + - gacentU + - gacts_char + - astabIdom + - classes_partition + - restr_perm_on + - card_classes_abelian + - reindex_astabs + - class_formula + - astabsI + - astab_trans_gcore + - sub_astabQR + - Frobenius_Cauchy + - orbitRs + - gactV + - orbitP + - group_set_gacent + - porbitE + - astabQR + - orbit_eqP + - astab_actby + - gacent_mod + - astab_ract + - amove_act + - actmE + - actM + - atransP2 + - acts_qact_dom + - comp_actE + - restr_perm_Aut + - morph_afix + - astab_setact + - ractE + - orbit_morphim_actperm + - acts_orbit + - gact1 + - faithfulP + - afixD1 + - orbit_act_in + - card_orbit_in + - astabs_set1 + - gacentS + - actKV + - actCJ + - orbit_actr_in + - ker_restr_perm + - actmEfun + - astab_range + - astabsD1 + - injm_actm + - astab1J + - autact_is_groupAction + - orbit1P + - acts_irr_mod_astab + - orbit_act + - actby_is_action + - modact_is_groupAction + - astabsC + - ractpermE + - acts_fix_norm + - orbit_transversalP + - astabJ + - modgactE + - astab1 + - injm_Aut_sub + - astab_dom + - actsRs_rcosets + - actsQ + - Aut_sub_fullP + - actpermE + - afix_gen + - acts_dom + - astabS + - gact_out + - afix1 + - SymE + - actmM + - transRs_rcosets + - morph_gastabs + - gacentIim + - acts_subnorm_gacent + - subact_is_action + - im_actperm_Aut + - qact_subdomE + - path: mathcomp/algebra/qpoly.v + theorems: + - in_qpoly_small + - size_lagrange_ + - qpolyCN + - qpolyC0 + - qpoly_mulz1 + - npolypK + - qpolyC_proof + - lagrange_gen + - qpolyCM + - qpolyC_is_multiplicative + - npoly_is_a_poly_of_size + - lagrange_free + - in_qpolyZ + - rVnpolyK + - qpolyCD + - lagrange_sample + - qpoly_intro_unit + - lagrangeE + - mk_monic_neq0 + - mk_monic_X + - card_monic_qpoly + - monic_mk_monic + - mem_npoly_enum + - qpoly_mulA + - poly_of_qpolyZ + - size_mk_monic_gt0 + - nth_npolyX + - qpolyXE + - in_qpoly0 + - npoly_vect_axiom + - poly_of_qpolyD + - npolyP + - in_qpolyM + - coefn_sum + - npoly_rV_K + - npolyp_key + - in_qpoly_multiplicative + - coef_npolyp + - qpoly_scaleDr + - npolyX_gen + - mk_monic_Xn + - qpoly_nontrivial + - qpoly_scaleAl + - card_npoly + - qpoly_scaleAr + - qpolyC_natr + - npoly_enum_uniq + - npoly_submod_closed + - qpoly_inv_out + - size_mk_monic + - card_qpoly + - lagrange_def_sample + - size_npoly0 + - qpolyC_is_additive + - poly_of_qpoly_sum + - lagrange_full + - nth_lagrange + - irreducible_poly_coprime + - size_npoly + - poly_of_qpolyX + - in_qpoly1 + - polyn_is_linear + - npolyX_coords + - npolyX_full + - poly_of_qpolyM + - char_qpoly + - qpoly_scaleA + - npolyX_free + - qpoly_mul_addl + - qpoly_mul_addr + - size_lagrange_def + - size_lagrange + - qpoly_mul1z + - in_qpoly_is_linear + - in_qpolyD + - dim_polyn + - qpoly_mulC + - qpolyCE + - lagrange_coords + - lagrange_key + - qpoly_mulzV + - path: mathcomp/character/inertia.v + theorems: + - norm_inertia + - cfConjg1 + - cfConjg_eqE + - inertia_dprod + - inertia_morph_im + - cfRes_Ind_invariant + - inertia_opp + - cfclass_transr + - Inertia1 + - inertia_mod_quo + - inertia_valJ + - cfConjgDprodr + - cfConjgMorph + - conj_cfConjg + - cfConjgInd_norm + - cfConjgInd + - sub_inertia + - cfConjgRes + - cfRes_prime_irr_cases + - constt_Ind_ext + - Inertia_sub + - conjg_IirrKV + - cfConjg_is_linear + - cfclassInorm + - card_cfclass_Iirr + - inertia_dprodr + - cfclass_inertia + - cfResInd + - cfConjg_char + - inertia_bigdprod_irr + - cfConjgIsom + - dvdn_constt_Res1_irr1 + - cfConjgMnorm + - cfConjgE + - conjg_Iirr_inj + - inertia_bigdprodi + - inertia_injective + - cfAutConjg + - cfConjgMod + - inertia_add + - normal_inertia + - inertia_id + - irr_induced_Frobenius_ker + - sub_Inertia + - sub_inertia_Res + - cfConjgRes_norm + - cfConjg_iso + - cfclass_sym + - extend_to_cfdet + - cfclass1 + - extend_linear_char_from_Sylow + - extend_solvable_coprime_irr + - cfclass_Ind + - cfConjgK + - inertia_sum + - sNG + - cfclass_IirrE + - inertia_irr0 + - reindex_cfclass + - cfConjg_is_multiplicative + - cfConjgQuo_norm + - cfdot_irr_conjg + - cfConjg_eq1 + - inertia_prod + - cfDetConjg + - Clifford_Res_sum_cfclass + - inertia_irr_prime + - extendible_irr_invariant + - inertia0 + - cent_sub_inertia + - conjg_Iirr_eq0 + - eq_cfclass_IirrE + - cfConjgEJ + - invariant_chief_irr_cases + - cfConjgKV + - sub_inertia_Ind + - cfConjgQuo + - inertia_Frobenius_ker + - cfConjg_subproof + - cfConjg_lin_char + - cfConjgBigdprodi + - cfConjg_id + - inertia_scale_nz + - cfConjg_cfuniJ + - conjg_IirrK + - cfclass_uniq + - cfConjgDprodl + - cfConjgEin + - normal_Inertia + - cfclassP + - cent_sub_Inertia + - conjg_IirrE + - cfker_conjg + - inertia_mul + - inertia_dprodl + - cfclass_invariant + - center_sub_Inertia + - inertia_sdprod + - norm_Inertia + - cfConjg_cfun1 + - inertia_dprod_irr + - cfclass_refl + - inertia_scale + - inertiaJ + - cfConjgEout + - conjg_inertia + - constt_Ind_mul_ext + - cfConjgM + - inertia_morph_pre + - group_set_inertia + - inertia1 + - extend_coprime_linear_char + - cfConjgJ1 + - size_cfclass + - constt0_Res_cfker + - cfConjg_cfuni + - cfConjgSdprod + - cfConjgDprod + - cfdot_Res_conjg + - Frobenius_Ind_irrP + - cfConjgBigdprod + - solvable_irr_extendible_from_det + - conjg_Iirr0 + - path: mathcomp/field/galois.v + theorems: + - comp_kHom_img + - fixedPoly_gal + - inAEndK + - gal_oneP + - aut_mem_eqP + - gal_generated + - galNormV + - galM + - kHom_to_gal + - galTrace_fixedField + - gal_kAut + - kHom_extends + - normalFieldS + - galTrace_is_additive + - galNormX + - gal_matrix + - kHom_lrmorphism + - normalField_kAut + - galNorm_gal + - galois_connection_subset + - gal_kHom + - kAut1E + - galK + - galNorm_fixedField + - mem_fixedFieldP + - fixedFieldS + - gal_eqP + - normalField_galois + - limg_gal + - kHomExtend_scalable_subproof + - normalField_cast_eq + - kHomS + - Hilbert's_theorem_90 + - kHom_eq + - gal_reprK + - memv_gal + - kHom_poly_id + - gal_cap + - galoisS + - normalField_isog + - normalField_normal + - kAEnd_norm + - splittingFieldForS + - galois_connection + - normalField_isom + - kHom_root + - kAut_eq + - normalField_root_minPoly + - gal_independent + - galois_fixedField + - mem_galNorm + - kHomExtend_poly + - kHomExtend_id + - gal_AEnd + - fixedField_bound + - k1HomE + - kHom_to_AEnd + - comp_AEndK + - kHom_horner + - gal_mulP + - galS + - splittingPoly + - kAutS + - gal_sgvalK + - kHomExtendP + - comp_AEnd1l + - kAEnd_group_set + - mem_kAut_coset + - inv_is_ahom + - galois_dim + - galV + - gal_is_morphism + - normalField_img + - gal_repr_inj + - normalField_castM + - kHomSr + - fixedFieldP + - gal_id + - root_minPoly_gal + - normalField_ker + - galNorm_prod + - kHom_kAut_sub + - gal_invP + - kHomSl + - gal_independent_contra + - kHom_is_additive + - enum_AEnd + - galois_factors + - fixed_gal + - fixedField_is_aspace + - kHomExtend_val + - galNorm_eq0 + - mem_galTrace + - splitting_galoisField + - kAutf_lker0 + - k1AHom + - eq_galP + - galois_connection_subv + - kHom_is_multiplicative + - galTrace_gal + - splittingFieldP + - galNorm1 + - kHom_dim + - fieldOver_splitting + - normalField_factors + - kHom_root_id + - kHomExtendE + - kAutE + - splitting_field_normal + - gal_conjg + - dim_fixedField + - inv_kHomf + - galNorm0 + - normal_fixedField_galois + - path: mathcomp/field/qfpoly.v + theorems: + - card_primitive_qpoly + - qlogp0 + - plogp0 + - map_fpoly_div_inj + - qX_expK + - qpoly_mulVp + - coprimep_unit + - sh_gt1 + - card_qfpoly + - qpoly_inv0 + - qX_exp_neq0 + - qX_neq0 + - map_poly_div_inj + - qX_in_unit + - powX_eq_mod + - qlogp_eq0 + - qlogp_qX + - primitive_poly_in_qpoly_eq0 + - card_qfpoly_gt1 + - gX_order + - in_qpoly_comp_horner + - qX_order_dvd + - plogp1 + - pred_card_qT_gt0 + - qlogpD + - plogp_div_eq0 + - mk_monicE + - primitive_polyP + - qlogp1 + - qX_exp_inj + - gX_all + - qX_order_card + - plogp_lt + - primitive_mi + - plogp_X + - dvdp_order + - path: mathcomp/solvable/finmodule.v + theorems: + - fmodV + - actsgHG + - fmodX + - act0r + - actr_is_groupAction + - sgG + - transfer_cycle_expansion + - fmod_addrA + - injHGg + - fmod_addNr + - congr_fmod + - actZr + - sum_index_rcosets_cycle + - injHg + - fmodK + - fmval0 + - rcosets_cycle_transversal + - actNr + - injm_fmod + - fmodP + - coprime_abel_cent_TI + - fmodKcond + - actrKV + - actrM + - rcosets_cycle_partition + - Gaschutz_transitive + - transferM + - fmvalN + - actr1 + - fmod_add0r + - fmvalJ + - fmodJ + - actr_is_action + - transfer_indep + - defHGg + - fmod1 + - transfer_morph_subproof + - actrK + - fmvalA + - fmvalJcond + - fmod_inj + - actAr + - fmvalZ + - sXG + - Gaschutz_split + - path: mathcomp/algebra/intdiv.v + theorems: + - dvdzz + - gcd0z + - Gauss_dvdzr + - zprimitive_irr + - modzMl + - dvdpP_rat_int + - gcdz1 + - mulz_modl + - ltz_divRL + - zchinese_mod + - gcdz_idPl + - divz_small + - divzMpl + - modz_ge0 + - divz_abs + - eisenstein + - zchinese_remainder + - dvdz_Pexp2l + - sgz_contents + - size_zprimitive + - zpolyEprim + - gcdzCA + - map_poly_divzK + - modNz_nat + - modz1 + - coprimeNz + - dvdz_lcm + - dvdz_charf + - dvdp_rat_int + - egcdzP + - zprimitiveZ + - modz_absm + - dvdz_exp2r + - expz_min + - Gauss_dvdz + - divz0 + - dvdz_lcmr + - modzDm + - dvdz_mull + - zcontentsZ + - modzDr + - divzMr + - modz_small + - Gauss_dvdzl + - gcdzN + - dvd1z + - divzMl + - lcmz_neq0 + - ltz_ceil + - modzNm + - gcdzDr + - zprimitive_id + - zcontents_primitive + - dvdz_eq + - coprimez_pexpl + - divz_ge0 + - coprimezE + - gcdz_eq0 + - Qint_dvdz + - sgz_lead_primitive + - dvdz_mul2r + - divNz_nat + - gcdzDl + - Gauss_gcdzl + - dvdz0 + - zprimitive0 + - Qnat_dvd + - modzDmr + - zcontentsM + - dvdz_mul2l + - lez_pdiv2r + - size_rat_int_poly + - divzMA + - modz_abs + - gcdz_idPr + - gcdz_modl + - mulzK + - modzMmr + - lez_divRL + - dvdzE + - dvdz_mulr + - divz_eq + - dvdz_gcd + - eqz_modDr + - coprimezP + - dec_Qint_span + - mulKz + - lcm0z + - lez_divLR + - zcontents_monic + - ltz_pmod + - modzDml + - divzMDl + - divzz + - mod0z + - expzB + - divzK + - gcdzMDl + - zchinese_modl + - coprimezMr + - modz_nat + - dvdz_trans + - dvdz_pexp2r + - mulz_modr + - lcmzC + - coprimez_dvdl + - div0z + - dvdzP + - lcmz_ge0 + - divzDr + - gcdzC + - eqz_mod_dvd + - divzMA_ge0 + - modzMm + - eqz_mul + - dvdz1 + - coprimez_sym + - zprimitive_monic + - dvdz_exp + - gcdNz + - mulz_gcdr + - ltz_divLR + - gcdzMr + - gcdzA + - int_Smith_normal_form + - modzDl + - dvd0z + - dvdz_mod0P + - rat_poly_scale + - dvdz_gcdl + - divz_mulAC + - mulz_divA + - polyOver_dvdzP + - modzMml + - gcdzACA + - gcdzAC + - Bezoutz + - dvdz_gcdr + - modzMDl + - lez_div + - mulz_divCA + - dvdz_mul + - zprimitiveM + - Gauss_gcdzr + - divzA + - divzAC + - dvdpP_int + - modzXm + - zcontents0 + - coprimezXr + - modz_mod + - zchinese_modr + - lez_floor + - divzMpr + - modzN + - mulz_gcdl + - zprimitive_min + - gcdz_modr + - divzDl + - divz1 + - zcontents_eq0 + - divz_nat + - coprimezN + - path: mathcomp/algebra/ssralg.v + theorems: + - rmorph_sign + - oppr_eq0 + - pair_mulA + - valZ + - pair_mulC + - rmorph_alg + - lastr_eq0 + - charf'_nat + - mull_fun_is_semi_additive + - raddfZnat + - unitrV + - mulrDr + - exprB + - natrXE + - charf0P + - Frobenius_autMn + - natrDE + - cat_dnfP + - scale_is_scalable + - divalg_closedZ + - prodfV + - ffun_mul_addl + - linearN + - lregM + - sqrrD + - natr_mod_char + - divalg_closedBdiv + - scalarP + - natn + - semiring_closedM + - exprDn_char + - rpred_nat + - subr_sqrDB + - iter_addr + - expr_sum + - fmorph_eq + - scale0r + - unitrX_pos + - dnf_to_rform + - unitrN1 + - unitrX + - mulr1 + - dnf_to_form_qf + - mulr_signM + - rpred_div + - exprBn_comm + - lregMl + - commr_sym + - sum_ffun + - mulr2n + - mulKr + - foldExistsP + - iter_addr_0 + - mulr_natl + - scalerA' + - rregP + - rpredMNn + - mulrDl + - semiringClosedP + - scaler0 + - solP + - natrD + - unitrPr + - eq_sol + - char_lalg + - fmorphV + - mulrI_eq0 + - pair_mulVl + - ffun_addC + - divrr + - mulr1_eq + - scalerBl + - mul0r + - addrNK + - rpredMsign + - submod_closedB + - val1 + - idfun_is_scalable + - rmorphMn + - unitr0 + - eval_Pick + - rmorphD + - scalerAl + - proj_satP + - exprDn + - commrN + - sub0r + - exprNn_char + - size_sol + - scalarAr + - Frobenius_autE + - rpred_sign + - algMixin + - ffun_scale_addl + - mulr_fun_is_semi_additive + - exprZn + - scalable_linear + - addrCA + - pair_addC + - rmorphN + - rpred_prod + - scalerCA + - scaler_prodr + - rmorph_unit + - lalgMixin + - compN1op + - idfun_is_semi_additive + - addr0 + - raddf0 + - scaler_suml + - natr1E + - addNr + - mulrAC + - telescope_prodf + - sumrMnr + - fmorph_eq1 + - lreg1 + - sqrf_eq1 + - expf_eq0 + - prodrMl + - divr1_eq + - exprNn + - natf_neq0 + - mulrnDl + - subr_sqr + - in_algE + - rpred_sum + - mulrI0_lreg + - pair_mulDl + - addNKr + - ffun_mulA + - divr1 + - exprVn + - ffun_scaleA + - addrACA + - charf_prime + - signr_odd + - mulIf + - addrAC + - mul0rn + - addKr + - Frobenius_autX + - bool_fieldP + - can2_linear + - valB + - addrI + - rpredDr + - prodrM_comm + - scaler_unit + - scalerDl + - signrN + - scalarZ + - pair_add0 + - unitrMr + - eq_sat + - mulKf + - invr_out + - prodrN + - rpred_divl + - lregX + - expr0 + - ffun_mul_0l + - sdivr_closedM + - rmorphV + - rpredV + - rmorph1 + - signrZK + - scaler_prod + - subrX1 + - raddfD + - raddf_sum + - rmorph_eq1 + - mulrnDr + - sumr_const_nat + - mulf_eq0 + - scaler_eq0 + - linearMn + - invr_inj + - imaginary_exists + - addrr_char2 + - pairMnE + - raddfZsign + - rreg_neq0 + - rpredMl + - natrB + - exprBn + - submodClosedP + - scalerKV + - subring_closed_semi + - sub_fun_is_additive + - pair_mulVr + - unitrN + - oner_eq0 + - raddfMn + - pair_unitP + - prodf_neq0 + - eqr_oppLR + - sqrrB + - mulr1n + - rpredN1 + - mulVr + - commrN1 + - quantifier_elim_rformP + - add0U + - val0 + - rpred1M + - commrD + - prodrMn_const + - divrr + - can2_semi_additive + - valD + - fpred_divr + - expr1n + - idfun_is_multiplicative + - sqrf_eq0 + - valD + - subr0 + - unitrM_comm + - lregN + - prodr_const + - linearB + - Frobenius_autB_comm + - expr_mod + - pair_mulr0 + - raddf_eq0 + - natrME + - signr_addb + - rev_unitrP + - rpredBl + - mulrn_char + - prodr_undup_exp_count + - invb_out + - Frobenius_autD_comm + - pair_mul1l + - mulrACA + - mulrN1 + - scalerMnr + - additive_linear + - invr_sign + - pair_invr_out + - signr_eq0 + - addIr + - ffun_mul_0r + - prodrMr_comm + - eqr_div + - linearPZ + - rreg1 + - addUA + - pair_scaleAr + - comRingMixin + - qf_evalP + - comp_is_multiplicative + - mulrC + - invr_out + - null_fun_is_semi_additive + - div1r + - mulrnAC + - sumrN + - expfB + - signrE + - addrN + - natr0E + - mulfVK + - telescope_sumr_eq + - divKr + - sqrr_sign + - divIr + - pair_addA + - expr2 + - mulf_neq0 + - mull_fun_is_scalable + - lreg_neq0 + - subr_sqr_1 + - pair_mulDr + - invr1 + - ffun_mulC + - exprD + - opp_is_additive + - mulrS + - commrM + - add_fun_is_semi_additive + - to_rform_rformula + - mulrSr + - mulVb + - eq_eval + - linearZ + - iter_mulr + - mulr_fun_is_scalable + - raddfB + - expr_dvd + - raddfMnat + - addrK_char2 + - rpred0 + - divKf + - mulNr + - unitrE + - mulrnBr + - rmorph_nat + - mulrBl + - unitfE + - linearZZ + - linearD + - prodrMn + - mulrN + - mulrC + - opprD + - lreg_sign + - rpredZeq + - subIr + - mulrNN + - prodf_seq_neq0 + - null_fun_is_scalable + - mul0r + - mulrVK + - subr_eq0 + - charf_eq + - exprMn_n + - mulfI + - unitr1 + - divrI + - mulr_suml + - commr1 + - prodrXr + - ffunMnE + - raddfMNn + - to_rterm_id + - rregX + - linearZ_LR + - eqr_sum_div + - rpredZsign + - ffun_addN + - divr_closedM + - fmorph_char + - Frobenius_autM_comm + - pair_scaleDl + - If_form_rf + - smulr_closedM + - pair_scaleDr + - can2_scalable + - scaler_sign + - prodrMr + - valD + - addf_div + - foldForallP + - rregM + - IdomainMixin + - unitrMl + - mulVKf + - mulrCA + - divring_closedBM + - rpredX + - mulr_sign + - invr_eq1 + - subr_eq + - scaleN1r + - fst_is_scalable + - rpredZnat + - oppr0 + - submod_closedZ + - fpred_divl + - lregP + - id + - rpredDl + - commr_refl + - Frobenius_aut0 + - inv_out + - sub_fun_is_scalable + - subr0_eq + - fpredMl + - exprD1n + - Frobenius_autN + - raddf_inj + - opp_is_scalable + - linearP + - invrN1 + - sol_subproof + - If_form_qf + - addrK + - same_env_sym + - divff + - valM + - raddfN + - scaler_sumr + - raddf0 + - ffun_mul_1l + - rmorphXn + - subring_closedB + - scale_fun_is_scalable + - invrN + - commrB + - valN + - mulrnAl + - subKr + - ffun_scale_addr + - pair_mul1r + - raddfMsign + - mulrA + - mulrK + - pair_scaleAl + - qf_to_dnf_rterm + - add_fun_is_scalable + - eqf_sqr + - scaler_prodl + - subring_closedM + - subrXX + - pair_addN + - commr_nat + - ffun_scale1 + - intro_unit + - divr_closedV + - mulNrn + - sum_ffunE + - semiring_closedD + - sumr_const + - commrX + - invfM + - revrX + - sumrMnl + - Frobenius_aut_is_additive + - telescope_prodr + - scaler_injl + - expr1 + - pair_one_neq0 + - invr_signM + - expr0n + - rmorphismMP + - oner_neq0 + - mulr_natr + - exprS + - sqrrN + - fmorph_eq0 + - scalerBr + - mulr_algl + - exprMn + - addrKA + - sqrrD1 + - natr1 + - mulr_algr + - scalerMnl + - mulVr + - subalgClosedP + - rmorphB + - unitrM + - divr_signM + - signrMK + - invrM + - to_rformP + - subr_char2 + - rmorph_prod + - invr_eq0 + - smulr_closedN + - invrK + - sqrrB1 + - ffun_addA + - prodrMl_comm + - rmorph_comm + - nat1r + - opprB + - rpred_divr + - scalerI + - mulfK + - commr_sum + - rpredD + - unitrP + - subalg_closedBM + - rmorph_eq_nat + - rmorph_char + - linearMNn + - divrNN + - commr0 + - ffun1_nonzero + - mulrnAr + - comm_alg + - divringClosedP + - expr_div_n + - unitr_sdivr_closed + - oppr_char2 + - zmodClosedP + - rpredMn + - telescope_sumr + - sumrB + - rmorph0 + - rregMr + - fmorph_unit + - opprK + - mul1r + - fst_is_semi_additive + - linear_sum + - bin_lt_charf_0 + - raddfD + - in_alg_is_additive + - Pick_form_qf + - subrXX_comm + - commr_prod + - scaler_nat + - mulr0 + - valB + - Frobenius_aut_nat + - fmorph_div + - linear_closedB + - fmorph_inj + - natrM + - zmod_closedD + - val0 + - rpredXN + - linear0 + - mulIr0_rreg + - telescope_prodr_eq + - scalerAr + - divrN + - comp_is_scalable + - comp_is_semi_additive + - expf_neq0 + - mulr0 + - pair_scale1 + - rmorph_div + - N1op + - rpredB + - addKr_char2 + - addr_eq0 + - snd_is_multiplicative + - mulr0 + - snd_is_scalable + - sdivr_closed_div + - subalg_closedZ + - scaleNr + - rpredMr + - rpredBC + - rregN + - qf_to_dnfP + - can2_additive + - ffun_add0 + - ffun_mul_1r + - mulC_mulrV + - commr_sign + - scale1r + - mulrnA + - zmod_closedN + - oppr_inj + - mulN1r + - mulr0n + - rmorphM + - invf_div + - mulrnBl + - snd_is_semi_additive + - subringClosedP + - natr_prod + - charf0 + - quantifier_elim_wf + - mulIr_eq0 + - char0_natf_div + - exprMn_comm + - prodf_eq0 + - commrMn + - valM1 + - eval_tsubst + - expfS_eq1 + - natf0_char + - dvdn_charf + - eq_holds + - addr0_eq + - mulrI + - divring_closed_div + - path: mathcomp/fingroup/gproduct.v + theorems: + - sdprod_isog + - dprodEsd + - sdprodWY + - divgrM + - pprodP + - isog_set1X + - cprodJ + - sdpair1_morphM + - cprod_normal2 + - ker_pprodm + - setX_prod + - dprodYP + - remgr_id + - dprodWY + - injm_pair1g + - dprodWcp + - cprodWC + - cprodE + - bigcprodEY + - sdpairE + - pairg1_morphM + - morphim_sdprodml + - sdpair_act + - pprodWY + - remgrM + - dprodP + - sdprod_mul_proof + - xsdprodm_act + - morphim_pprodmr + - morphim_coprime_dprod + - quotient_pprod + - sdprod_recr + - dprodE + - sdprodm_sub + - sdprod_normal_complP + - morphim_sdprodm + - dprodWsdC + - morphim_dprodmr + - cprodmEl + - actsEsd + - cprodEY + - divgr_eq + - dprod_normal2 + - gacentEsd + - sdprod_context + - dprodmEl + - setX_gen + - im_xsdprodm + - cprod0g + - injm_bigdprod + - mem_dprod + - dprodm_eqf + - subcent_TImulg + - cprod_modr + - cprod_modl + - ker_sdprodm + - im_cprodm + - im_sdprodm + - bigdprodYP + - trivg0 + - setX_dprod + - sdprodm_norm + - sdpair2_morphM + - sdprodWC + - pprodWC + - mem_sdprod + - injm_sdpair1 + - morphim_cprodm + - im_sdpair + - bigcprodWY + - morphim_pairg1 + - morphim_pprodm + - complgC + - dprod_modl + - sdprod_compl + - cprodW + - sdprodmE + - triv_cprod + - quotient_coprime_dprod + - dprodA + - splitsP + - remgrP + - mul0g + - snd_morphM + - sdprod_recl + - injm_sdpair2 + - group_not0 + - isog_setX1 + - cprodC + - morphim_sdprodmr + - isog_dprod + - sdpair_setact + - divgrMl + - bigdprod_card + - sdprod_mul1g + - dprodg1 + - sdprod_mulVg + - sdprodE + - morphim_cprodml + - quotient_sdprodr_isom + - dprodm_cprod + - sdprod_modl + - pprodmM + - quotient_sdprodr_isog + - quotient_coprime_sdprod + - injm_sdprodm + - index_sdprod + - dprodEY + - dprod_modr + - pair1g_morphM + - sdprodJ + - dprod_card + - cprodg1 + - sdprodm_eqf + - dprod1g + - sdprod_modr + - bigdprodWY + - pprodE + - morphim_cprodmr + - bigdprodW + - cprodm_sub + - remgrMid + - sdprodWpp + - sdprod_inv_proof + - index_sdprodr + - ker_dprodm + - reindex_bigcprod + - imset_mulgm + - extprod_mulVg + - injm_dprod + - cprod_ntriv + - morphim_pprod + - sdprodEY + - morphim_cprod + - bigdprodWcp + - pprodmEr + - sdprodP + - sdprod_mulgA + - group0 + - remgrMl + - pprodg1 + - sdprod_sdpair + - mem_divgr + - xsdprodm_dom2 + - injm_xsdprodm + - dprodmEr + - cprodWY + - morphim_pair1g + - morphim_fstX + - bigcprod_coprime_dprod + - sdprodW + - mulgmP + - cprodA + - cprodmEr + - astabEsd + - mem_bigdprod + - pprodW + - extprod_mulgA + - sdprod_card + - dprodWsd + - pprodmE + - im_sdpair_TI + - remgr1 + - cprodmE + - group_setX + - morphim_dprodm + - morphim_coprime_bigdprod + - pprodEY + - mulg0 + - morphim_sndX + - quotient_cprod + - im_sdprodm2 + - cprodm_actf + - dprodmE + - sdprod_isom + - subcent_sdprod + - pprodJ + - cprod1g + - morphim_pprodml + - dprodC + - im_dprodm + - injm_sdprod + - im_sdpair_norm + - mem_remgr + - cprodm_norm + - fst_morphM + - dprodEcp + - sdprodmEl + - injm_pprodm + - sdprod1g + - morphim_dprodml + - ker_cprodm + - path: mathcomp/character/mxrepresentation.v + theorems: + - linear_mxsimple + - eqg_mx_abs_irr + - Clifford_astab1 + - hom_mxmodule + - mxsimpleP + - mx_rsim_abs_irr + - hom_component_mx + - rstabs_submod + - row_gen_sum_mxval + - Wedderburn_min_ideal + - gen_mul1r + - rfix_mx_rstabC + - mxval_is_multiplicative + - mxval_gen1 + - map_group_ring + - map_section_repr + - rkerP + - Clifford_is_action + - val_submodP + - gring_indexK + - rstabs_act + - rstabs_in_gen + - rker_gen + - gen_addA + - morphim_mx_irr + - mx_iso_refl + - rker_map + - submod_mx_repr + - quo_mx_quotient + - mxvalM + - mxmodule_eqg + - rcent_eqg + - mx_faithful_irr_center_cyclic + - hom_mxsemisimple_iso + - mx_reducibleS + - annihilator_mxP + - mxsemisimple_module + - gring_mxA + - rsim_regular_factmod + - rsim_rcons + - norm_sub_rstabs_rfix_mx + - gring_free + - gring_mxJ + - rstab_norm + - rstab_act + - factmod_mx_faithful + - mx_JordanHolder_max + - envelop_mx1 + - gring_opM + - mx_Schur_onto + - rfix_gen + - val_submod_inj + - primitive_root_splitting_abelian + - rfix_factmod + - classg_base_center + - mxsimple_morphim + - rker_factmod + - in_factmod_eq0 + - rfix_morphpre + - Wedderburn_mulmx0 + - mxval_grootXn + - irr_degree_abelian + - kquo_mxE + - rstabs_quo + - val_submodS + - in_submod_module + - irr1_mode + - mxvalV + - val_gen_row + - gen_addC + - mxsimple_eqg + - gen_dim_ex_proof + - quo_repr_coset + - gring_mxK + - map_mxminpoly_groot + - val_genD + - in_genZ + - gen_is_additive + - gring_projE + - rfix_submod + - irr_mode_unit + - reducible_Socle1 + - quo_mx_coset + - Wedderburn_annihilate + - conj_mx_irr + - Clifford_atrans + - rcent_quo + - mxvalN + - val_factmod_inj + - mxsemisimple0 + - mxrank_in_submod + - regular_op_inj + - sG_f'fG + - classg_base_free + - add_sub_fact_mod + - cyclic_mx_module + - map_regular_subseries + - rowval_gen_stable + - val_factmodE + - mx_butterfly + - mx_irr_abelian_linear + - in_genK + - irr_center_scalar + - mem_sub_gring + - mx_JordanHolder + - center_kquo_cyclic + - mxsimple_module + - subg_mx_abs_irr + - gen_addNr + - rsim_regular_series + - rker_subg + - mxmodule_envelop + - mxsimple_semisimple + - rstabs_morphpre + - component_socle + - gring_op_id + - Socle_module + - mxmodule_conj + - Wedderburn_sum + - Clifford_rank_components + - in_gen_row + - mx_faithful_irr_abelian_cyclic + - repr_mxMr + - rstabs_subg + - rank_irr1 + - Wedderburn_sum_id + - rfix_mx_id + - Clifford_component_basis + - Socle_direct + - rker_morphpre + - component_mx_key + - rsim_last + - subg_mx_repr + - addsmx_module + - ker_irr_comp_op + - factmod_mx_repr + - rstab_eqg + - irr_degree_gt0 + - mem_gring_mx + - rstab_morphim + - gring_op1 + - rker_quo + - rcent_map + - gen_mx_irr + - val_submodE + - rker_morphim + - in_submodK + - map_enveloping_algebra_mx + - mx_iso_module + - socle_mem + - quo_mx_irr + - sumsmx_module + - sums_R + - coset_splitting_field + - rstab_group_set + - component_mx_isoP + - repr_mx1 + - rstabs_conj + - eqmx_rstabs + - mx_series_rcons + - subSocle_direct + - repr_mxX + - mxval_inj + - submod_mx_irr + - rcenter_group_set + - mx_rsim_def + - socle_exists + - mxsimple_exists + - eqmx_module + - in_factmod_module + - Wedderburn_closed + - mx_Maschke + - kquo_mx_faithful + - gring_rowK + - irr1_rfix + - socle_simple + - mxval1 + - mxmoduleP + - map_regular_mx + - Wedderburn_subring_center + - irr_repr'_op0 + - morphpre_mx_abs_irr + - val_factmod_eq0 + - mxsimple_abelian_linear + - extend_group_splitting_field + - val_gen0 + - mxsimple_cyclic + - cent_mx_scalar_abs_irr + - Wedderburn_id_mem + - mx_rsim_scalar + - quotient_splitting_field + - not_rsim_op0 + - capmx_subSocle + - mxtrace_dadd_mod + - component_mx_id + - kermx_centg_module + - rstabs_factmod + - val_genK + - rstab_in_gen + - sG_f'fG + - mxval_groot + - mx_rsim_sym + - map_rfix_mx + - mx_faithful_inj + - val_gen_rV + - rstab_sub + - mx_irrP + - mxsimple_subg + - irr_modeM + - splitting_cyclic_primitive_root + - socleP + - gen_mulC + - val_Clifford_act + - val_factmodP + - rfix_eqg + - mx_iso_trans + - rstabs_rowval_gen + - Wedderburn_direct + - mx_irr_map + - reducible_Socle + - max_size_mx_series + - map_mx_faithful + - degree_irr1 + - sum_mxsimple_direct_sub + - op_Wedderburn_id + - Wedderburn_is_id + - genmx_component + - val_submod1 + - rstabs_morphim + - base_free + - eqmx_semisimple + - regular_mx_repr + - in_factmodJ + - map_gring_op + - Clifford_componentJ + - principal_comp_subproof + - nth_map_rVval + - irr_comp_rsim + - Clifford_basis + - socle_can_subproof + - in_genD + - rker_mx_rsim + - Clifford_rstabs_simple + - eqg_repr_proof + - section_eqmx + - map_mx_repr + - in_factmodE + - irr_reprE + - mx_abs_irrP + - cyclic_mxP + - abelian_abs_irr + - mx_Schreier + - mxnonsimpleP + - regular_module_ideal + - nz_row_mxsimple + - group_splitting_field_exists + - Wedderburn_disjoint + - irr_mx_mult + - mx_Schur_inj + - component_mx_semisimple + - rstab_conj + - irr_comp_envelop + - val_factmodS + - rowval_genK + - socle_irr + - capmx_module + - gring_opG + - eqg_mx_faithful + - hom_mxP + - rcent_subg + - submx_in_gen + - rker_normal + - mxmodule_morphpre + - mxtrace_Socle + - repr_mxK + - max_submod_eqmx + - mx_rsim_faithful + - mxtrace_submod1 + - proj_mx_hom + - gen_dim_gt0 + - sat_gen_form + - eval_mulT + - rstab_submod + - card_gen + - cycle_repr_structure + - mx_rsim_iso + - map_reprE + - hom_envelop_mxC + - cyclic_mx_sub + - kermx_hom_module + - Clifford_hom + - map_gring_proj + - DecSocleType + - mx_iso_component + - mxsimple_map + - eval_gen_term + - mxmodule_rowval_gen + - gring_valK + - cyclic_mx_id + - eqmx_rstab + - gring_row_mul + - Socle_iso + - in_genN + - cyclic_mx_eq0 + - in_genJ + - hom_mxsemisimple + - envelop_mxP + - normal_rfix_mx_module + - subSocle_module + - mxval0 + - der1_sub_rker + - mxmodule_form_qf + - mxsemisimple_reducible + - subg_mx_faithful + - Wedderburn_is_ring + - rker_norm + - mx_Schur_inj_iso + - scalar_mx_hom + - mxrank_iso + - rconj_mx_repr + - mxrank_rsim + - mx_series_repr_irr + - rfix_mx_conjsg + - in_factmodsK + - dec_mx_reducible_semisimple + - map_regular_repr + - rfix_subg + - mx_Schur_iso + - submx_in_gen_eq + - gen_mx_repr + - rfix_mxS + - rstabs_sub + - group_closure_closed_field + - Socle_semisimple + - rstab_factmod + - envelop_mx_ring + - mxval_centg + - mx_series_lt + - irr_comp'_op0 + - max_submodP + - mx_rsim_factmod + - gen_ntriv + - mxval_genM + - memmx_cent_envelop + - last_mod + - val_genJ + - morphim_mx_repr + - irr_degreeE + - centgmxP + - irr_mode1 + - nz_socle + - eqmx_iso + - mxvalD + - base_full + - component_mx_def + - rstab_morphpre + - morphim_mx_abs_irr + - in_submodE + - irr_modeX + - PackSocleK + - rank_irr_comp + - rfix_conj + - map_gring_mx + - eval_mxmodule + - mxmodule_map + - linear_irr_comp + - mx_Schur + - mx_JordanHolder_exists + - eval_mxT + - rstab_normal + - val_submod_module + - mxtrace_rsim + - envelop_mxM + - Clifford_astab + - component_mx_disjoint + - mxval_genV + - rconj_mxJ + - hom_component_mx_iso + - rsim_submod1 + - Clifford_iso + - mxtrace_component + - rstabs_eqg + - gen_invr0 + - linear_mx_abs_irr + - mx_reducible_semisimple + - rker_submod + - mx_rsim_trans + - rfix_regular + - gen_dim_ub_proof + - proj_factmodS + - mx_rsim_map + - gen_dim_factor + - repr_mx_unitr + - row_full_dom_hom + - dec_mxsimple_exists + - conj_mx_faithful + - intro_mxsemisimple + - mxsimple_iso_simple + - rstabS + - mxtrace_dsum_mod + - repr_mx_unit + - mxmodule_morphim + - gring_op_mx + - mx_iso_sym + - genK + - mxmodule_subg + - group_closure_field_exists + - rfix_mx_module + - val_submodK + - section_eqmx_add + - repr_mx_free + - map_mx_abs_irr + - in_factmodK + - in_submodJ + - val_genZ + - rstabs_map + - subg_mx_irr + - rstab_subg + - morphim_mxE + - repr_mxV + - rker_eqg + - mx_subseries_module + - rcent_conj + - mxmodule0 + - rker_conj + - quo_mx_repr + - component_mx_iso + - non_linear_gen_reducible + - Clifford_iso2 + - regular_mx_faithful + - gring_opE + - submx_rowval_gen + - rank_Wedderburn_subring + - morphpre_mx_repr + - repr_mxVr + - rsim_regular_submod + - val_submodJ + - mx_abs_irrW + - gring_mxP + - Wedderburn_center + - envelop_mx_id + - valWact + - val_genN + - gen_add0r + - row_hom_mxP + - groupCl + - rsim_irr_comp + - subSocle_iso + - dom_hom_mx_module + - Clifford_Socle1 + - mx_subseries_module' + - gen_mulVr + - semisimple_Socle + - irr_mode_neq0 + - centgmx_map + - val_factmodK + - mxtrace_regular + - rstab_quo + - irr_comp_id + - sum_irr_degree + - centgmx_hom + - addsmx_semisimple + - irr1_repr + - repr_mxKV + - rcent_group_set + - mxrank_in_factmod + - set_nth_map_rVval + - rfix_quo + - sumsmx_semisimple + - mxmodule_trans + - mx_factmod_sub + - mx_rsim_irr + - path: mathcomp/algebra/polydiv.v + theorems: + - dvdp_eq_mul + - ltn_divpr + - divp_eq + - divpK + - size2_dvdp_gdco + - divpD + - dvdp_Pexp2l + - eqp_mull + - coprimep_dvdr + - rmodpZ + - divp_small + - Gauss_dvdp + - modp_eq0 + - rdivp_mull + - divpZr + - size_gcdp1 + - edivp_def + - coprimep_XsubC2 + - divp_eq + - gcdp_mull + - rgdcop0 + - dvdpP + - coprimep_modr + - gcdp_scaler + - eqp_rgdco_gdco + - eqp_trans + - divpp + - egcdp_recP + - coprimepMl + - irredp_neq0 + - rmodp_addl_mul_small + - redivp_key + - mulKp + - dvdp1 + - rdivpDl + - dvd1p + - divpP + - polyXsubCP + - modp_mull + - modpZl + - leq_modp + - Bezout_eq1_coprimepP + - Nrdvdp_small + - dvdp_eq_mul + - eqp_div + - leq_gcdpr + - modNp + - uniq_roots_rdvdp + - modpZr + - take_poly_rmodp + - rdvdp_XsubCl + - eqp_coprimepr + - eqp0 + - eqp_mul2l + - eqp_exp + - modp1 + - dvdp_eq_div + - rdvdpP + - dvdpZl + - map_divp + - divp_modpP + - leq_divp + - egcdpE + - divpKC + - gcdp_eq0 + - rdivp_eq + - dvdp_mul + - gcdpE + - map_modp + - dvdp_exp2l + - mulpK + - divpKC + - divpE + - modp0 + - eqp_modpl + - coprimep_def + - dvdp_subr + - coprimep_comp_poly + - modpP + - mulKp + - eqp_map + - irredp_XaddC + - eq_dvdp + - dvdp_mod + - mupMr + - dvdpE + - mup_XsubCX + - gdcopP + - root_dvdp + - modp_mul + - dvdp_eq + - rdvdp_eqP + - root_bigmul + - divp_mulCA + - eqp_rgcd_gcd + - divpp + - dvdp_mul2r + - divpAC + - dvdp_exp2r + - eqpxx + - dvdp_comp_poly + - mu_prod_XsubC + - eqp_divl + - divpK + - divpE + - divp_dvd + - egcdp0 + - XsubC_dvd + - coprimep_expl + - mup_ltn + - divp_addl_mul + - gcdp_scalel + - dvdp_addr + - edivp_def + - divp_addl_mul_small + - modp_coprime + - dvdp_leq + - dvdp_div_eq0 + - dvdp_eq_div + - rmod0p + - dvdp_prod_XsubC + - dvdp_add_eq + - rdvdp_leq + - polyXsubC_eqp1 + - dvdp_eq + - rgcd0p + - gcdpp + - mup_leq + - eqp_divr + - rdvdpN0 + - rdvdp_mull + - rmodpp + - leq_rmodp + - gdcop0 + - rdvdpp + - eqp_size + - dvdp_exp_XsubCP + - modpP + - polyC_eqp1 + - rdivpDr + - modpZl + - scalp0 + - mulKp + - divp_mulA + - rdivpK + - edivp_redivp + - rmodpN + - coprimep_map + - dvdp_gcd_idl + - coprimep_pexpl + - rdvdp0 + - gdcop_map + - ltn_rmodpN0 + - divpN + - dvdp_gdcor + - eqp_eq + - egcdpP + - mulKp + - modpD + - rmodp1 + - dvdp_eq + - divp_pmul2l + - dvdp_addl + - eqp_gcd + - Gauss_gcdpl + - eqp_mod + - eqp_monic + - rmodp_mull + - ltn_divpl + - ulc_eqpP + - divpD + - dvdpP + - rdiv0p + - gcdp_mul2r + - gcdp_exp + - rmodp_mulml + - root_biggcd + - mulpK + - redivp_eq + - redivp_map + - rdvdp1 + - gcdp0 + - edivp_map + - divpE + - rdvdp_eq + - rgcdp0 + - divp_mulAC + - eqp_gcdr + - divp_mulA + - dvdp_pexp2r + - divpK + - edivp_eq + - rdvd0pP + - coprime1p + - prod_XsubC_eq + - edivpP + - root_gdco + - eq_rdvdp + - dvdp_mulIl + - gcdp_comp_poly + - drop_poly_divp + - drop_poly_rdivp + - Bezout_coprimepP + - coprimepZl + - gcdp_eqp1 + - divp_addl_mul + - coprimep_sym + - dvdp_add + - coprimep_addl_mul + - irredp_XsubC + - modpC + - coprimep_pexpr + - divp0 + - divp_divl + - dvdpZr + - comm_redivpP + - dvdp_map + - divp_eq + - irredp_XsubCP + - mupM + - divpP + - rdivp_eq + - coprimepPn + - mupMl + - div0p + - modp_addl_mul_small + - expp_sub + - dvdp_trans + - dvdp_gcdr + - gcdp_modl + - rmodp_mulmr + - coprimep_XsubC + - redivp_def + - eqp_modpl + - rmodp_small + - coprimep_size_gcd + - modpN + - coprimep0 + - modp_id + - coprimep_root + - eqp_coprimepl + - edivp_key + - dvdp_gcdl + - rmodpp + - divp1 + - rdivp_small + - root_factor_theorem + - scalp_map + - eqp_div_XsubC + - dvdp_gcdlr + - eqpf_eq + - eqpP + - rmodpp + - divp_pmul2l + - leq_divpr + - eqp_gdcol + - coprimepMr + - coprimep_gdco + - take_poly_modp + - gcdp_def + - gcdp_addl + - rdivp_addl_mul_small + - rgcdpE + - coprimepP + - divpKC + - scalpE + - modp_small + - rdvd1p + - gcdp_mul2l + - divpZr + - dvdp_gcd + - mupNroot + - divp_mulCA + - Gauss_dvdpr + - dvdUp + - divp_pmul2r + - coprimepX + - rdivpp + - eqp_divl + - eqp_rmod_mod + - rscalp_small + - eqp01 + - eqp_dvdr + - leq_rdivp + - modpE + - leq_trunc_divp + - gtNdvdp + - rmodp_mull + - root_factor_theorem + - rmodpB + - eqp_mul2r + - rmodp_eq0 + - divpN + - coprimep_dvdl + - divpN0 + - dvdp_mulIr + - divp_divl + - leq_divpl + - gcdp_addr + - dvdp_exp_sub + - dvdp_eqp1 + - Bezout_coprimepPn + - Gauss_gcdpr + - dvdp_eq + - scalpE + - expp_sub + - dvdp_gcd_idr + - eqp_modpr + - root_coprimep + - gcdpC + - mulp_gcdl + - eqpfP + - modp_XsubC + - dvdpP + - eqp_mulr + - rdivp_addl_mul + - modpE + - dvd0p + - redivp_eq + - modpZr + - ltn_modpN0 + - root_gcd + - mod0p + - rdivpp + - eqp_gdcor + - size_divp + - dvdp_subl + - horner_mod + - rdivp_eq + - eqp_dvdl + - divp_mulAC + - dvdpE + - modpp + - rmodpC + - Bezoutp + - rdivp0 + - modpE + - eqp_rdiv_div + - mulpK + - mulp_gcdr + - rcoprimep_coprimep + - rmodp_compr + - divp_pmul2r + - edivpP + - coprimep1 + - modpD + - modp_addl_mul_small + - egcdp_map + - redivpP + - dvdp_size_eqp + - Gauss_dvdpl + - rmodp0 + - uniq_roots_dvdp + - coprime0p + - divpp + - divp_eq + - divpZl + - leq_gcdpl + - dvdp_mul_XsubC + - rdivp1 + - mulpK + - size_poly_eq1 + - rdvdp_mull + - modp_mulr + - coprimep_modl + - dvdpP + - gcdp1 + - dvdp_mul2l + - dvd_eqp_divl + - dvdp_mulr + - eqp_ltrans + - eqp_gcdl + - rdivpK + - divpZl + - dvdpNl + - rmodpD + - scalpE + - ltn_modp + - modpN + - dvdp_sub + - modp_mul + - ucl_eqp_eq + - coprimep_div_gcd + - dvdp0 + - divpAC + - path: mathcomp/solvable/frobenius.v + theorems: + - FrobeniusJ + - partition_class_support + - Frobenius_subl + - Frobenius_reg_compl + - Frobenius_partition + - semiprimeJ + - normedTI_memJ_P + - Frobenius_Ldiv + - regular_norm_coprime + - semiregularS + - Frobenius_ker_dvd_ker1 + - Frobenius_ker_coprime + - Frobenius_index_coprime + - semiregular_prime + - normedTI_S + - partition_normedTI + - FrobeniusJcompl + - Frobenius_context + - semiregular1r + - semiregularJ + - normedTI_J + - semiprimeS + - Frobenius_action_kernel_def + - normedTI_P + - injm_Frobenius_compl + - Frobenius_coprime + - Frobenius_ker_Hall + - Frobenius_subr + - Frobenius_trivg_cent + - FrobeniusJker + - FrobeniusWker + - injm_Frobenius_ker + - Frobenius_kerP + - cent_semiprime + - Frobenius_actionP + - semiregular_sym + - semiregular1l + - regular_norm_dvd_pred + - set_Frobenius_compl + - Frobenius_index_dvd_ker1 + - ltn_odd_Frobenius_ker + - semiprime_regular + - Frobenius_dvd_ker1 + - cent1_normedTI + - FrobeniusWcompl + - injm_Frobenius_group + - Frobenius_reg_ker + - FrobeniusW + - cent_semiregular + - Frobenius_compl_Hall + - path: mathcomp/algebra/archimedean.v + theorems: + - natr_mul_eq1 + - floor1 + - floorX + - conj_natr + - intrKfloor + - sum_truncK + - floorK + - natr_sum_eq1 + - intrEfloor + - trunc0Pn + - floor_itv + - trunc0 + - floorP + - ceilX + - raddfZ_nat + - gt_pred_ceil + - rpredZ_nat + - floor_le + - ceil_itv + - floorD + - truncX + - sqr_intr_ge1 + - floorpK + - norm_intr_ge1 + - truncM + - Rreal_int + - ceil_le + - floor_def + - int_num_subring + - rpred_nat_num + - floor_subproof + - raddfZ_int + - ceil_le_int + - intr_aut + - truncK + - intr_ler_sqr + - intr_nat + - le_ceil + - ceilM + - natr_normK + - intrP + - conj_intr + - ceil0 + - rpredZ_int + - natr_ge0 + - intrEsign + - intrKceil + - natr_exp_even + - truncD + - aut_intr + - trunc_gt0 + - floor0 + - ceilD + - intr_int + - floorpP + - intrEge0 + - rpred_int_num + - ceilN + - Rreal_nat + - ceilK + - ceil1 + - intrEceil + - natr_prod_eq1 + - floorM + - trunc_floor + - natr_aut + - ge_floor + - natr_gt0 + - natrEint + - intr_normK + - path: mathcomp/fingroup/perm.v + theorems: + - perm_onM + - porbitPmin + - permKV + - porbitV + - lift_perm_id + - odd_permV + - odd_mul_tperm + - perm1 + - cast_perm_id + - permS0 + - cast_perm_comp + - perm_onto + - prod_tpermP + - perm_on_id + - tpermV + - odd_perm_prod + - card_porbit_neq0 + - cast_ord_permE + - cast_permE + - porbit_sym + - lift_permV + - perm_oneP + - lift_permM + - tpermC + - porbit_traject + - cast_permK + - perm_on1 + - tpermL + - perm_proof + - tpermK + - perm_onV + - Sym_group_set + - odd_permJ + - tpermR + - tuple_permP + - perm_invP + - im_perm_on + - permX + - card_Sn + - permX_fix + - tpermD + - isom_cast_perm + - eq_porbit_mem + - card_Sym + - tperm_on + - apermE + - iter_porbit + - imset_perm1 + - uniq_traject_porbit + - odd_perm1 + - porbits_mul_tperm + - cast_perm_sym + - tperm2 + - porbitsV + - lift_perm1 + - tpermP + - tperm1 + - tpermJ + - permK + - perm_closed + - cast_permKV + - odd_lift_perm + - im_permV + - permJ + - lift_perm_lift + - cast_perm_morphM + - pvalE + - lift_permK + - card_perm + - odd_permM + - perm_inj + - perm_invK + - permP + - permS01 + - tperm_proof + - permS1 + - mem_porbit + - permM + - permE + - perm_onC + - path: mathcomp/fingroup/morphism.v + theorems: + - morphpreP + - morphpreV + - im_sgval + - injm_morphim_inj + - morphim_ker + - injm_sgval + - morphim_setIpre + - invm_subker + - injm_subcent1 + - morphimGK + - morphimU + - card_im_injm + - mkerr + - morphim_subnorm + - injm_cent1 + - kerP + - morphpreI + - morphim_cent1 + - morphJ + - morphV + - domP + - injm_norms + - morphim_cents + - ker_norm + - mem_morphpre + - morphim_restrm + - ker_injm + - injm_normal + - card_isog + - isog_transr + - eq_in_morphim + - isog_trans + - injm_cent + - invmK + - injm_comp + - isom_isog + - morphpreIim + - sgval_sub + - injm_invm + - morphmE + - morph_injm_eq1 + - dom_ker + - morphimSGK + - morphim_invm + - morphim_trivm + - morphpre_set1 + - morphR + - morphim_eq0 + - isog_subg + - morphimIG + - morphim_abelian + - morphpreSK + - rcoset_kerP + - isogEhom + - injm_subnorm + - morphpre_cent + - idm_isom + - injm_restrm + - morphpreS + - nclasses_isog + - morphpre0 + - invmE + - morphpre_cent1 + - morphim_subcent + - trivm_morphM + - restrmP + - sub_isom + - im_restrm + - morph_dom_groupset + - ker_rcoset + - morphim_class + - injm_abelian + - morphpreD + - morphim_cent + - morphimE + - morphpre_subcent1 + - morphim_invmE + - isog_isom + - morphpreMr + - morphpre_idm + - ker_sgval + - morph1 + - morphim_factm + - classes_morphim + - morphimD1 + - injmP + - eq_morphim + - ker_sub_pre + - ker_ifactm + - morphim_inj + - misomP + - order_injm + - morphim1 + - isom_sub_im + - injm_subg + - morphimDG + - mkerl + - restr_isom_to + - homgP + - morphpre_cent1s + - morphim_norms + - leq_morphim + - morphim_injG + - morphim_set1 + - homg_trans + - morphim_cycle + - morphpre_ifactm + - isog_eq1 + - morphim_isom + - morphim_injm_eq1 + - comp_morphM + - subgmK + - im_invm + - morphpre_normal + - morphimSK + - morphimIim + - restr_isom + - sub_morphpre_injm + - injm_eq + - ker_subg + - injm_factm + - morphimI + - morphpre_inj + - morphimMl + - morphim_cent1s + - injm_cents + - morphpre_subnorm + - morphim_subcent1 + - morphpreMl + - injmD1 + - isogP + - ker_factm_loc + - injm_idm + - morph_prod + - morphimT + - morphpre_factm + - isom_subg + - morphM + - morphimD + - morphim_sub + - im_subg + - ker_restrm + - injmK + - morphpre_restrm + - morphim_idm + - ker_comp + - morphimIdom + - morphimJ + - morphpre_proper + - morphimV + - isomP + - isog_transl + - morphpreIdom + - isom_card + - morphimY + - idm_morphM + - im_idm + - morphim_norm + - morphX + - ker_idm + - morphimP + - isog_abelian + - isom_im + - morphim_subnormG + - injm_proper + - injm_norm + - morphpre_groupset + - homg_refl + - injmI + - morphpre_invm + - injm1 + - morphpre_cents + - sub_isog + - morphim_normG + - restrmEsub + - mem_morphim + - morphpre_norm + - card_injm + - isog_hom + - isog_refl + - eq_homgr + - isom_inj + - injm_factmP + - mker + - ifactmE + - morphim_homg + - im_ifactm + - ker_normal_pre + - morphpreU + - isom_sym + - morphim_gen + - morphim_ifactm + - morphimR + - leq_homg + - morphimS + - morphpreJ + - morphimEsub + - ker_invm + - injm_subcent + - morphimMr + - ker_factm + - sub_morphim_pre + - isom_sgval + - morphim_groupset + - morphpre_norms + - morphpreE + - morphpre_comp + - nclasses_injm + - morphim_comp + - factm_morphM + - injmSK + - ltn_morphim + - ker_normal + - path: mathcomp/fingroup/quotient.v + theorems: + - coset_idr + - quotient_norm + - morphpre_qisom + - card_homg + - injm_qisom + - sub_cosetpre_quo + - homg_quotientS + - quotient_class + - index_injm + - cosetpre_cent1 + - weak_second_isog + - im_qisom_proof + - quotient_cents + - quotient0 + - quotientGI + - quotient_inj + - quotientE + - dvdn_morphim + - quotientMidr + - kercoset_rcoset + - norm_quotient_pre + - im_qisom + - coset_default + - morphim_qisom_inj + - quotient_abelian + - quotientS + - coprime_morph + - ltn_quotient + - qisom_inj + - quotientMr + - card_quotient_subnorm + - second_isog + - inv_quotientN + - index_quotient_eq + - cosetpre_cent + - cosetpreK + - coset_kerl + - divg_normal + - quotient_proper + - third_isog + - quotient1_isom + - quotientMl + - quotientU + - quotient_gen + - coset_norm + - sub_quotient_pre + - third_isom + - val_coset + - quotientJ + - qisom_isog + - coprime_morphl + - index_morphim + - qisom_isom + - cosetpre_proper + - cosetpre_cent1s + - coset_mem + - quotient1 + - qisom_ker_proof + - cosetpre_set1 + - logn_morphim + - quotient_subcent + - cosetP + - im_coset + - trivg_quotient + - quotient_isog + - quotientSK + - cosetpre_subcent + - quotient_norms + - coset_mulP + - card_cosetpre + - quotmE + - coset_one_proof + - mem_repr_coset + - quotient_sub1 + - quotientS1 + - qisomE + - coset1 + - first_isom + - coset_range_mul + - coset_reprK + - quotientD1 + - quotient_subnormG + - first_isog + - quotient_subnorm + - cosetpre_normal + - quotientGK + - first_isog_loc + - imset_coset + - quotient_cent1s + - cosetpre_set1_coset + - qisom_restr_proof + - morphim_qisom + - quotientR + - sub_im_coset + - quotientV + - cosetpreM + - quotientInorm + - coset1_injm + - coset_kerr + - val_coset_prim + - quotm_dom_proof + - quotientSGK + - quotientY + - normal_cosetpre + - ker_coset + - inv_quotientS + - card_morphpre + - coset_invP + - quotientIG + - cosetpre_subcent1 + - dvdn_quotient + - coset_oneP + - coset_morphM + - index_morphpre + - ker_quotm + - repr_coset_norm + - coset_id + - coprime_morphr + - quotient_isom + - quotientD + - coset_range_inv + - index_quotient_ker + - quotientK + - restrm_quotientE + - quotient_homg + - morphim_quotm + - im_quotient + - card_quotient + - quotientT + - val_quotient + - index_cosetpre + - repr_coset1 + - cosetpreSK + - quotientYidr + - index_quotient + - mem_quotient + - card_morphim + - quotient_injG + - quotient_cent1 + - quotient_set1 + - leq_quotient + - first_isom_loc + - injm_quotm + - sub_cosetpre + - cosetpre_gen + - quotientI + - quotm_ker_proof + - quotient_setIpre + - quotient_subcent1 + - quotientDG + - quotient1_isog + - quotientYidl + - quotient_cent + - classes_quotient + - cosetpre_cents + - ker_coset_prim + - quotient_normG + - val_qisom + - char_from_quotient + - quotientYK + - quotientMidl + - quotient_neq1 + - second_isom + - path: mathcomp/ssreflect/fintype.v + theorems: + - existsb + - negb_exists_in + - proper_card + - enumP + - ordS_subproof + - predX_prod_enum + - exists_inPn + - mem_sub_enum + - exists_eq_inP + - eq_rlshift + - eq_card_trans + - lift_max + - bij_on_image + - disjointU + - card2 + - proper_subn + - subxx_hint + - eqfun_inP + - flatten_imageP + - cardC + - f_iinv + - ord_pred_bij + - card_sig + - bumpS + - tag_enumP + - in_iinv_f + - eq_lrshift + - ltn_ord + - subxx + - enum_val_nth + - subset_leq_card + - rev_ord_proof + - subset_cons + - card0 + - ord_pred_subproof + - unit_enumP + - size_enum_ord + - fin_all_exists2 + - image_injP + - enum_default + - card_option + - sub_enum_uniq + - mem_ord_enum + - eq_disjoint + - negb_forall + - exists_inP + - card_gt0P + - forallPP + - disjoint_cat + - card_prod + - inordK + - dinjectiveP + - seq_sub_axiom + - unlift_subproof + - enum_rankK_in + - enum_rank_ord + - leq_ord + - eq_existsb + - cast_ord_inj + - enum_uniq + - f_invF + - void_enumP + - codomP + - eq_disjoint_r + - card_image + - splitK + - invF_f + - subset_catl + - card_uniqP + - codom_val + - eq_disjoint1 + - disjoint0 + - disjoint_has + - eq_card + - sum_enum_uniq + - bumpK + - size_codom + - eq_card + - lift_subproof + - max_card + - card_ord + - card_seq_sub + - enum_ordSr + - leq_bump + - codom_f + - rshift_subproof + - subset_pred1 + - card_gt2P + - pred0P + - subset_all + - eq_lshift + - card_void + - card1 + - cardX + - eq_forallb_in + - ord_predK + - fin_all_exists + - size_image + - seq_subE + - subset_cons2 + - image_f + - ord_enum_uniq + - enum_val_bij + - iinv_f + - canF_invF + - bij_on_codom + - image_iinv + - canF_RL + - subset_eqP + - cast_ord_proof + - card_size + - injectivePn + - cardD1 + - cardT + - pred0Pn + - proper_trans + - preim_iinv + - eq_card0 + - enum_val_bij_in + - extremumP + - uniq_enumP + - count_enumP + - rshift_inj + - lift_eqF + - eq_invF + - subsetP + - lift0 + - mem_image + - mem_iinv + - ordSK + - eq_pick + - imageP + - forallPn + - lshift_inj + - pcan_enumP + - disjointU1 + - filter_subset + - leq_card_in + - forallP + - enum1 + - pre_image + - prod_enumP + - eq_disjoint0 + - disjointWr + - leq_image_card + - seq_sub_pickleK + - nth_codom + - enum_ord0 + - eq_subset_r + - val_ord_enum + - arg_minnP + - fin_pickleK + - eq_enum_rank_in + - eq_rshift + - nth_enum_rank_in + - bumpDl + - existsP + - exists_inb + - lift_inj + - unlift_some + - eq_card1 + - cardU1 + - nth_ord_enum + - sub_ordK + - mem_seq_sub_enum + - option_enumP + - splitP + - disjointW + - subset_leqif_card + - disjoint_sym + - cast_ordK + - properE + - unbumpS + - card_le1P + - fintype1P + - nth_enum_rank + - fintype0 + - sub_ord_proof + - enum_val_inj + - eq_codom + - mask_enum_ord + - card_sum + - card_le1_eqP + - unlift_none + - enum_ordSl + - val_sub_enum + - unbumpDl + - card_sub + - ord_pred_inj + - enum0 + - widen_ord_proof + - map_preim + - rev_ordK + - enum_rank_subproof + - unbumpKcond + - liftK + - enum_valK_in + - inord_val + - seq_sub_default + - mem_sum_enum + - ord_inj + - ordS_bij + - sub_proper_trans + - canF_LR + - rev_ord_inj + - card_gt1P + - card_codom + - subset_trans + - eq_existsb_in + - inj_leq + - card_preim + - enum_rank_bij + - subsetE + - bij_eq_card + - nth_image + - index_enum_ord + - subset_cardP + - inj_card_bij + - codomE + - properP + - enum_rankK + - inj_card_onto + - card1P + - properxx + - image_codom + - eq_card_sub + - card_tagged + - eq_enum + - existsPP + - enumT + - enum_rank_inj + - card0_eq + - image_pred0 + - disjointFl + - eq_card_prod + - canF_sym + - dinjectivePn + - leq_card + - eq_subset + - card_in_image + - eq_cardT + - forall_inPP + - disjoint_cons + - subset_predT + - eq_image + - eqfunP + - unliftP + - card_unit + - mem_enum + - proper_sub_trans + - disjoint_subset + - subset_disjoint + - eq_subxx + - injF_bij + - cardE + - negb_forall_in + - subsetPn + - cast_ord_id + - split_ordP + - cast_ordKV + - neq_bump + - bool_enumP + - mem_card1 + - existsPn + - val_enum_ord + - cardC1 + - unbumpK + - injectiveP + - subset_filter + - iinv_proof + - disjointFr + - map_subset + - leq_bump2 + - ord1 + - subset_cat2 + - enum_valK + - enum_val_ord + - enum_valP + - disjoint1 + - path: mathcomp/ssreflect/bigop.v + theorems: + - sub_le_big_seq_cond + - pair_big_idem + - big_enum_val_cond + - leq_bigmax_seq + - big_ord_narrow_cond + - big_distrl + - big_enum_cond + - big_split_ord + - big_has + - big_ord1_cond + - some_big_AC_mk_monoid + - sum1_count + - big_geq_mkord + - big_ord_narrow + - big_mkcondl_idem + - addmC + - exchange_big_dep + - big_allpairs_idem + - big_nat_rev + - exchange_big_dep_nat + - sum_nat_seq_eq1 + - big_mkcondr + - big_allpairs_dep + - mulmAC + - big_nat_widenl + - oopC_subdef + - le_big_nat_cond + - deprecated_filter_index_enum + - big_ord_recr + - big_cat_idem + - big_nat1_cond_eq + - big_nat_mul + - mulmCA + - partition_big + - bigmax_eq_arg + - sum1_card + - exchange_big_nat_idem + - big_ltn + - mulm1 + - idem_sub_le_big_cond + - mem_index_iota + - big_image_cond + - sum_nat_eq0 + - big_ord_narrow_leq + - prodn_gt0 + - bigA_distr_big_dep + - big_mask_tuple + - big_add1 + - big_mkord + - eq_big_op + - oop1x_subdef + - big_rmcond_in_idem + - big_enum_rank + - eq_big_idx_seq + - mulmACA + - mem_index_enum + - prod_nat_const_nat + - big_rem_AC + - big_rmcond_idem + - eq_bigmax_cond + - bigID_idem + - le_big_nat + - prod_nat_seq_eq1 + - big_map + - foldl_idx + - oopA_subdef + - perm_big_supp_cond + - big_rcons_op + - eq_big_nat + - big_geq + - bigD1 + - bigmax_sup + - sub_le_big + - big_bool + - uniq_sub_le_big + - oACE + - big_condT + - big_rec + - oopx1_subdef + - idem_sub_le_big + - big_nat_cond + - big_allpairs + - big_andE + - uniq_sub_le_big_cond + - leq_sum + - eq_big_idx + - card_bseq + - sig_big_dep + - big_id_idem_AC + - congr_big_nat + - big_seq_cond + - big_nth + - big_split + - mul1m + - big_change_idx + - iteropE + - telescope_sumn_in + - bigmax_leqP + - mulC_dist + - big_undup + - big_mkcond + - big_mask + - big1_idem + - big_rec3 + - pair_bigA_idem + - perm_big + - big_andbC + - big_distr_big_dep + - big1 + - big_AC_mk_monoid + - subset_le_big + - prodn_cond_gt0 + - index_enum_uniq + - leq_prod + - big_cat_nested + - big_image + - opCA + - big_load + - sub_in_le_big + - prod_nat_const + - big_allpairs_dep_idem + - reindex + - mul0m + - big_ind3 + - big_pred0_eq + - big_rmcond + - big_enumP + - expn_sum + - pair_big_dep + - big_pred1_eq_id + - big_cat_nat_idem + - big_enum_val + - exchange_big_idem + - telescope_big + - sum_nat_eq1 + - exchange_big_nat + - pair_big + - big_undup_iterop_count + - sum_nat_const + - exchange_big + - big_flatten + - big_const_nat + - big_cat_nat + - biglcmn_sup + - big_distr_big + - big_pred1_id + - big_ind2 + - big_hasC + - big_map_id + - big_catl + - biggcdn_inf + - reindex_onto + - foldrE + - bigU + - big_distrlr + - mulC_zero + - sum1_size + - mulmDr + - big_const_idem + - big_ind + - prod_nat_seq_neq1 + - sub_le_big_seq + - big_has_cond + - big_filter + - big_ord0 + - prod_nat_seq_eq0 + - eq_big_seq + - big_rev_mkord + - bigmax_leqP_seq + - mulmDl + - big_endo + - big_const_ord + - big_nat1 + - congr_big + - big1_eq + - dvdn_biglcmP + - bigD1_seq + - big_const + - exchange_big_dep_idem + - mulC_id + - mulmA + - sumnE + - big_catr + - sum_nat_seq_eq0 + - big_rmcond_in + - big_id_idem + - leqif_sum + - pair_big_dep_idem + - big_enum + - perm_big_supp + - leq_bigmax_cond + - big_cons + - bigA_distr_big + - big_ord_recl + - foldlE + - reindex_inj + - big_seq1 + - leq_bigmax + - big_mkcondl + - big1_seq + - big_mkcondr_idem + - big_filter_cond + - mulm0 + - prod_nat_seq_neq0 + - sum_nat_const_nat + - eq_bigl_supp + - sum_nat_seq_neq0 + - bigID + - big_rem + - sumnB + - big_ord_widen_cond + - big_ord1 + - big_seq + - addmAC + - big_nat_recr + - big_pred1 + - big_addn + - big_morph + - big_index_uniq + - big_pred0 + - big_ltn_cond + - bigU_idem + - telescope_sumn + - big_ord_widen_leq + - big_pmap + - big_pred1_eq + - big_if + - index_enum_key + - bigmax_sup_seq + - big_only1 + - mulmC + - big_nil + - addmCA + - big_nat_widen + - eq_bigr + - add0m + - pair_bigA + - big_nat1_id + - big_seq1_id + - big_split_idem + - big_nseq + - le_big_ord + - big_all + - big_const_seq + - big_mkcond_idem + - partition_big_idem + - bigD1_ord + - big_ord_widen + - cardD1x + - exchange_big_dep_nat_idem + - path: mathcomp/algebra/rat.v + theorems: + - normr_num_div + - rat_linear + - denq_mulr_sign + - Qint_def + - mulVq + - le_rat0 + - fracq_eq + - sgr_denq + - frac0q + - oppq_frac + - fracq_opt_subdefE + - lerq0 + - invq_frac + - mulqC + - mulq_def + - ler_rat + - ler0q + - coprimeq_den + - mulqA + - divqP + - sgr_scalq + - sgr_numq_div + - norm_ratN + - numqN + - ratzE + - coprime_num_den + - rat0 + - rat_eq + - sgr_numq + - minr_rat + - le_rat0M + - rat_vm_compute + - addq_subdefC + - mulq_addl + - fracqE + - denq_gt0 + - ratzM + - gt_rat0 + - absz_denq + - add0q + - rat_eqE + - is_natE + - lt_ratE + - addq_def + - invq0 + - numq_sign_mul + - Qnat_def + - addq_frac + - truncP + - numqK + - valq_frac + - numq_lt0 + - ratr_is_additive + - ratr_norm + - rpred_rat + - signr_scalq + - mulq_subdefE + - fracq_eq0 + - denqVz + - nonzero1q + - ratzN + - intq_eq0 + - divq_num_den + - rat1 + - ratz_frac + - floor_rat + - fmorph_eq_rat + - numq_div_lt0 + - RatK + - le_ratE + - QnatP + - ltr_rat + - fracq_subproof + - fracqMM + - ltr0q + - denqN + - invq_def + - den_fracq + - numq_int + - numq_ge0 + - valqK + - fracqP + - ge_rat0_norm + - coprimeq_num + - denq_norm + - ratr_int + - le_rat0_anti + - scalq_def + - normr_denq + - denq_lt0 + - is_intE + - addNq + - addqA + - le_rat_total + - denq_eq0 + - numq_eq0 + - ratr_sg + - maxr_rat + - ratr_nat + - subq_ge0 + - mulq_frac + - oppq_def + - ratzD + - addq_subdefA + - mulq_subdefC + - ratr_is_multiplicative + - addq_subdefE + - scalq_eq0 + - ge_rat0 + - lt_rat_def + - numq_gt0 + - fmorph_rat + - val_fracq + - rat_ring_theory + - denq_int + - num_fracq + - fracq_opt_subdef_id + - le_rat0D + - denq_neq0 + - lt_rat0 + - ratP + - ceil_rat + - numqE + - denqP + - path: mathcomp/solvable/alt.v + theorems: + - Alt_index + - rfd_odd + - trivial_Alt_2 + - rgdP + - not_simple_Alt_4 + - Alt_subset + - simple_Alt5_base + - Sym_trans + - Alt_trans + - Alt_normal + - Alt_norm + - Alt_even + - rfd_funP + - rfd_iso + - card_Sym + - aperm_faithful + - rfdP + - path: mathcomp/solvable/cyclic.v + theorems: + - morph_generator + - orderXpnat + - has_prim_root_subproof + - quotient_cyclic + - cyclicY + - morphim_cyclic + - order_inj_cyclic + - Zp_unit_isom + - cyclicP + - expg_cardG + - order_inf + - cyclic_dprod + - field_mul_group_cyclic + - Aut_prime_cyclic + - im_cyclem + - cyclic1 + - Zp_unitmM + - eltmM + - card_Aut_cycle + - totient_gen + - cycle_generator + - order_dvdG + - isog_cyclic + - orderXexp + - eltmE + - cycleMsub + - div_ring_mul_group_cyclic + - sub_cyclic_char + - eq_subG_cyclic + - morph_order + - Aut_cyclic_abelian + - im_Zp_unitm + - cardSg_cyclic + - cyclemM + - Zp_unit_isog + - injm_generator + - im_Zpm + - cycle_cyclic + - cyclicJ + - ZpmM + - metacyclicP + - Euler_exp_totient + - generator_cycle + - Aut_prime_cycle_cyclic + - generator_coprime + - cyclic_abelian + - expgK + - Aut_cycle_abelian + - isog_cyclic_card + - metacyclic1 + - Zp_isom + - injm_cyclem + - cycleM + - orderXgcd + - cyclicS + - nt_prime_order + - sum_totient_dvd + - cyclic_metacyclic + - Zp_isog + - field_unit_group_cyclic + - quotient_cycle + - expg_znat + - orderXdvd + - cyclic_small + - expg_zneg + - im_eltm + - cycle_sub_group + - orderXpfactor + - nt_gen_prime + - eltm_id + - has_prim_root + - orderM + - injm_Zpm + - eq_expg_mod_order + - cyclicM + - units_Zp_cyclic + - injm_eltm + - order_dvdn + - dvdn_prime_cyclic + - metacyclicS + - card_Aut_cyclic + - generator_order + - orderXprime + - injm_cyclic + - cycle_subgroup_char + - sum_ncycle_totient + - path: mathcomp/field/algC.v + theorems: + - Cint_Cnat + - conjL_nt + - Cnat_sum_eq1 + - minCpoly_subproof + - Crat_divring_closed + - eqCmod_refl + - mulA + - sqrtK + - floorCK + - conj_is_semi_additive + - natCK + - floorC0 + - norm_Cint_ge1 + - norm_eq0 + - minCpoly_eq0 + - truncCX + - CratP + - dvdC_zmod + - Creal1 + - algC_invautK + - getCratK + - eqCmod_nat + - Cint_int + - minCpoly_aut + - addA + - LtoC_K + - dvdC0 + - sposD + - algCreal_Im + - eqCmodDr + - dvdC_trans + - truncCK + - eqCmod0_nat + - rpred_Crat + - Creal_Crat + - conjK + - Crat1 + - nz2 + - truncC_def + - Cnat_gt0 + - posJ + - Cnat_norm_Cint + - floorCM + - conj_Cnat + - floorCD + - truncC1 + - dvdCP_nat + - eq_root_is_equiv + - aut_Cnat + - truncCD + - dvdC_mul2l + - mulC + - Creal0 + - rpredZ_Cint + - raddfZ_Cint + - eqCmod_transl + - add0 + - normM + - sposDl + - eqCmod_transr + - truncC0Pn + - zCdivE + - dvdC_mulr + - CnatEint + - Crat_aut + - eqCmodN + - Creal_Cnat + - posE + - Cnat_nat + - minCpoly_monic + - addN + - algC_autK + - Cint_rat + - eqCmodMl0 + - leB + - intCK + - eqCmodDl + - truncC_gt0 + - CintP + - normD + - mul2I + - algCi_subproof + - conj_is_additive + - Cnat_aut + - dvdC_mul2r + - eqCmod0 + - algCrect + - floorCX + - pos_linear + - Cnat0 + - eqCmodm0 + - iJ + - algCreal_Re + - CtoL_inj + - CnatP + - CtoL_K + - eqCmodMr0 + - Cnat_mul_eq1 + - Cint_normK + - conj_is_multiplicative + - size_minCpoly + - conj_Crat + - dvdCP + - Cint0 + - Crat_rat + - CintE + - ratCK + - conj_Cint + - dvdC_refl + - normN + - addC + - Cint_aut + - Cint_ler_sqr + - eqCmodM + - algC_invaut_subproof + - archimedean + - nz2 + - getCrat_subproof + - rpredZ_Cnat + - dvd0C + - mulD + - raddfZ_Cnat + - dvdC_mull + - Cnat_exp_even + - norm_Cnat + - dvdC_int + - rpred_Cnat + - sqrMi + - algC_invaut_is_additive + - minCpolyP + - algebraic + - normE + - conj_subproof + - root_minCpoly + - eqCmod_sym + - nCdivE + - CintEsign + - normK + - CtoL_P + - truncC_itv + - one_nz + - floorC_def + - floorCN + - LtoC_subproof + - sqr_Cint_ge1 + - inv0 + - CtoL_is_additive + - closedFieldAxiom + - Cnat1 + - Cnat_ge0 + - dvdC_nat + - floorC1 + - Creal_Cint + - floorCpP + - conj_nt + - sqrtE + - Crat0 + - path: mathcomp/ssreflect/generic_quotient.v + theorems: + - left_trans + - encModRelP + - pi_DC + - pi_morph1 + - pi_mono2 + - equal_toE + - mpiE + - reprP + - quotW + - encoded_equivP + - eqquotE + - equiv_refl + - sortPx + - reprK + - pi_morph11 + - eq_op_trans + - eqmodP + - equiv_sym + - ereprK + - eqmodP + - eq_lock + - equivQTP + - equiv_rtrans + - encModEquivP + - equiv_ltrans + - canon_id + - piP + - repr_ofK + - pi_CD + - encoded_equivE + - quotP + - encModRelE + - sort_Sub + - eqmodE + - equiv_trans + - encoded_equiv_is_equiv + - eqmodE + - eqquotP + - qreprK + - path: mathcomp/solvable/extraspecial.v + theorems: + - gtype_key + - exponent_pX1p2n + - card_pX1p2n + - DnQ_extraspecial + - isog_pX1p2 + - isog_2extraspecial + - rank_DnQ + - exponent_pX1p2 + - card_pX1p2 + - pX1p2S + - card_DnQ + - DnQ_pgroup + - isog_pX1p2n + - rank_Dn + - DnQ_P + - Q8_extraspecial + - gactP + - Grp_pX1p2 + - pX1p2n_extraspecial + - actP + - isog_2X1p2 + - not_isog_Dn_DnQ + - pX1p2_extraspecial + - Ohm1_extraspecial_odd + - pX1p2id + - pX1p2n_pgroup + - path: mathcomp/algebra/ssrnum.v + theorems: + - gtr_pMr + - ler_nat + - ger_pMl + - lteifNl + - lteifNr0 + - lerDl + - deg_le2_poly_ge0 + - minr_nMr + - rootC_gt0 + - normC_sum_eq1 + - lerN10 + - ltr_wpMn2r + - ler_ndivrMr + - nmulr_llt0 + - trunc_itv + - degpN + - nmulr_lge0 + - deg2_poly_minE + - lteifN2 + - real_lerNnormlW + - sqa2 + - oppr_min + - ler_distlBl + - real_normrEsign + - normr0 + - ler_prod + - natf_div + - normrV + - ler0_norm + - lteif_nM2l + - invCi + - ImMr + - le00 + - mulr_sign_norm + - rootC_ge0 + - ltr0_ge_norm + - invf_nlt + - gtr0_sg + - nat_num1 + - mulrn_wge0 + - rootCK + - oppr_lt0 + - pnatr_eq1 + - ltr_nat + - ImM + - ler_distlDr + - ltrDl + - le0_add + - ltr_pMn2r + - natrK + - conj_Creal + - eqrXn2 + - exprn_odd_le0 + - agt0 + - deg2_poly_factor + - deltam + - lteif_pM2l + - normC2_rect + - sqrtC0 + - real_arg_minP + - r1N + - ler_normr + - ltr_prod + - deg2_poly_noroot + - lerD2r + - maxr_pMl + - normrMsign + - deg2_poly_ge0 + - real_ler_distlCDr + - ler_iXn2l + - normr_real + - ltrD2l + - normr_unit + - real_exprn_even_lt0 + - ltr_pDr + - subr_ge0 + - conjC_ge0 + - nneg_divr_closed + - ler_wpMn2l + - ler_pM2r + - ler0_ge_norm + - deg2_poly_root1 + - deg_le2_poly_delta_le0 + - posrE + - leif_mean_square_scaled + - trunc_subproof + - Im_div + - sqrtrV + - deg2_poly_gt0l + - ltr_distlCDr + - big_real + - subr_gt0 + - real_mono + - ltr1n + - normrN1 + - real_nmono + - ger0_real + - ler10 + - poly_ivt + - eqr_pMn2r + - lef_nV2 + - ger1_real + - mul_conjC_ge0 + - Nreal_ltF + - subr_lteifr0 + - eqr_norm2 + - lerD + - real_ler_distlCBl + - subC_rect + - ler_wnMn2l + - ImE + - exprn_even_gt0 + - real_ler_normlW + - leif_Re_Creal + - ltr0Sn + - invr_gt1 + - ltr0_sqrtr + - lteif_ndivlMr + - real_ltgtP + - imaginaryCE + - ltr_distl + - eqr_nat + - mulr_ege1 + - pmulr_lgt0 + - real_maxNr + - real_nmono_in + - ler0_sqrtr + - lerBlDl + - pmulrn_rgt0 + - leif_AGM_scaled + - real_leif_norm + - conjC_rect + - lteif_norml + - ltr01 + - deg2_poly_le0 + - le0N + - subr_lt0 + - geC0_conj + - ler1n + - sgr_gt0 + - sqrtC_inj + - lerMn2r + - rootC_subproof + - aa4gt0 + - normrEsg + - a4gt0 + - normr_le0 + - pmulrn_lle0 + - real_oppr_max + - deg2_poly_lt0m + - rootC_gt1 + - intrE + - ImV + - sqrtC1 + - conjC1 + - ler_rootC + - leifBRL + - ltr_nnorml + - aneq0 + - le_total + - ler_normlP + - eqC_semipolar + - ler_leVge + - ler_wnM2l + - exprn_even_lt0 + - ltf_nV2 + - ltr0n + - ler_pMn2r + - prod_real + - ltr_nDl + - mulr_gt0 + - real0 + - ltr_eXnr + - nat_num0 + - deg2_poly_lt0r + - gtr_nMr + - aneq0 + - deg2_poly_le0r + - poly_itv_bound + - invf_ngt + - lteif_ndivrMl + - lerD2l + - real_mono_in + - minr_to_max + - real_lteif_norml + - normM + - normr1 + - rootCMr + - real_ltr_distlDr + - sqrtr_gt0 + - conjCN1 + - Creal_Im + - gtrBl + - gtr0_le_norm + - pmulr_rlt0 + - oppr_ge0 + - normM + - nz2 + - lt0_cp + - sqrtC_eq0 + - realNEsign + - nposrE + - deg2_poly_max + - a4gt0 + - lerNnormlW + - nmulr_rge0 + - ler_sqr + - b2a + - pexpIrn + - ler_pMl + - deg2_poly_root1 + - ler_ltB + - ler_nMr + - minrN + - real_ltgt0P + - ltr_prod_nat + - lteif_pdivrMr + - le_total + - ltrN10 + - ler_pMr + - ler_peMr + - int_num1 + - exprn_gt0 + - ltrBrDl + - leif_sum + - sqr_sqrtr + - sgr_norm + - lerBrDl + - normr_nneg + - deltaN + - mulrIn + - sgr0 + - normr0P + - mulr_Nsign_norm + - realN + - ler_wnDl + - deg2_poly_ge0r + - deg2_poly_gt0r + - lteif01 + - mulr_sg_eqN1 + - conjC0 + - pmulr_rle0 + - exprn_even_ge0 + - invf_gt1 + - real_leP + - eq0_norm + - maxNr + - ImMl + - rootC0 + - gt_ge + - normr_prod + - minr_nMl + - real_leif_mean_square + - natrG_neq0 + - ltrB + - Crect + - prodr_gt0 + - invf_le1 + - lteifBrDr + - deltam + - sgrP + - addr_max_min + - degpN + - eqr_normN + - rootC_lt0 + - deg2_poly_root2 + - ler_neMr + - lt_le + - ler_nMn2l + - sgrV + - ler_nV2 + - root1C + - nz2 + - neq0Ci + - mulrn_wlt0 + - leif_0_sum + - signr_le0 + - mulr_le0 + - eqr_norm_id + - nmulrn_rgt0 + - invr_lt0 + - eqr_sqrtC + - deg2_poly_factor + - ler_wpDr + - ltr0_sg + - expr_ge1 + - ltr0_real + - Re_lock + - leN_total + - ltr_iXnr + - leif_pM + - real_addr_minl + - ler_ltD + - sqrtC_ge0 + - ltr_normr + - sqrtCM + - invf_plt + - nnegrE + - real_leif_AGM2 + - invr_gt0 + - lteifD2l + - r2N + - exprn_egt1 + - argCleP + - real_addr_closed + - lteifD2r + - mulrn_wgt0 + - normr_nat + - lerB_dist + - sqrtC_gt0 + - deg2_poly_le0l + - lef_pV2 + - splitr + - deg2_poly_minE + - deg2_poly_lt0 + - nmulr_rgt0 + - ler_wpM2r + - gtr_pMl + - posrE + - ltr_distlDr + - gerBl + - addC_rect + - ler_distD + - sgr_id + - gtr_nMl + - ltr_pM2l + - lern0 + - expr_lt1 + - realB + - leif_normC_Re_Creal + - le00 + - ImMil + - normCBeq + - rootC1 + - int_num_subring + - CrealE + - num_real + - deg2_poly_gt0l + - real_exprn_odd_lt0 + - ltr_nMl + - natf_indexg + - leif_rootC_AGM + - pmulrn_rge0 + - sqrp_eq1 + - leif_nM + - real1 + - addr_ge0 + - neqr0_sign + - invr_ge1 + - real_ler_norm + - lerDr + - ltr_pMr + - ltr_wnDl + - oppr_gt0 + - char_num + - mulr_ge0_le0 + - deg2_poly_gt0r + - invr_ge0 + - leif_pprod + - ltr_normlP + - ler_pdivlMl + - invf_ple + - signr_lt0 + - real_addr_maxr + - real_ler_distl + - normCDeq + - lerNr + - poly_disk_bound + - truncP + - lt0r_neq0 + - pos_divr_closed + - real_ltr_distlCBl + - Creal_ImP + - numNEsign + - ger0_norm + - ltrgt0P + - lteif_normr + - addr_ss_eq0 + - le01 + - ltr_nwDl + - realE + - bigmin_real + - mulr_egt1 + - ler_iXnr + - normCKC + - nz2 + - trunc_def + - natrP + - lerN2 + - ger_pMr + - ltr01 + - naddr_eq0 + - ieexprn_weq1 + - ltr_wnDr + - ler_norm + - sqrtr_ge0 + - deg2_poly_lt0m + - sqrtrM + - ltr_leD + - ler_nnorml + - ltrXn2r + - ltr_ndivrMl + - Nreal_gtF + - deg2_poly_ge0 + - lt0_add + - ReMir + - real_minr_nMr + - gt0_cp + - ler_eXnr + - sgr_def + - ltrr + - rootC_eq1 + - ltr0N1 + - normC_sum_upper + - le0r + - addr_ge0 + - paddr_eq0 + - lerr + - conj_normC + - mulr_ge0 + - pmulrnI + - natrE + - ltrBrDr + - eqNr + - real_addr_maxl + - pmulr_rge0 + - eqC + - ltrn0 + - lteif_pdivrMl + - realn + - lerB_real + - min_real + - pneq0 + - minNr + - numEsign + - le0r + - ltr_wpXn2r + - leif_nat_r + - invr_sg + - boundP + - leif_AGM2_scaled + - real_leNgt + - sqrtC_lt0 + - sgrN + - deg2_poly_ge0l + - gtr0_real + - ReMl + - truncP + - ler_wMn2r + - real_comparable + - rootC_le0 + - ler_niMl + - realEsqr + - lteifBlDl + - ltr_pM2r + - maxr_nMl + - max_real + - leif_AGM2 + - ler_normlW + - normrEsign + - Re_is_additive + - rootC_Re_max + - leifBLR + - rootC_le1 + - rootC_lt1 + - real_ltr_normr + - invC_Crect + - lteifNr + - real_ltP + - ltr_nDr + - deg2_poly_ge0r + - exprn_odd_gt0 + - mulr_ge0_gt0 + - nnegrE + - deg2_poly_lt0l + - realn_mono_in + - ler_pM + - real_minrN + - ler_norml + - ler_distlCDr + - deg2_poly_le0m + - signr_gt0 + - nmulr_lgt0 + - sqr_ge0 + - deg2_poly_factor + - pmulr_rgt0 + - ger_nMr + - ltrD + - norm_conjC + - realn_nmono + - real_ltr_distlCDr + - rectC_mulr + - ltr_distlBl + - ltr_pdivlMr + - ltrMn2r + - realn_nmono_in + - ger0P + - invf_nge + - realV + - ger0_def + - ler_pdivrMl + - midf_lt + - sgr1 + - deg2_poly_root2 + - real_ler_normr + - sqrtr0 + - normr_id + - ler_pV2 + - nmulrn_rge0 + - neg_unity_root + - ltrNr + - ler_weXn2l + - pexpr_eq1 + - real_leif_AGM2_scaled + - real_exprn_odd_ge0 + - deg2_poly_root2 + - ltr_nM2l + - mulrn_eq0 + - sqrCK + - sgr_nat + - sgr_le0 + - le0_cp + - le0_mul + - ltrN2 + - pmulrn_lgt0 + - ltr_normlW + - exprn_ile1 + - ltrNnormlW + - eqr_norml + - ler_eXn2l + - real_maxrN + - real_minr_nMl + - midf_le + - conjCi + - Re_conj + - subr_comparable0 + - eqr_rootC + - ler_pM2l + - deg2_poly_gt0m + - deg2_poly_maxE + - lteif_nM2r + - nmulr_lle0 + - real_ltr_normlP + - invf_pgt + - lteif_pdivlMl + - nonRealCi + - mul_conjC_gt0 + - le_trans + - ler_ndivrMl + - ltr_pV2 + - nmulrn_rle0 + - lerBrDr + - normC_rect + - real_ler_distlDr + - le_def + - invf_ge1 + - Creal_Re + - sum_real + - ltrBlDr + - ltrgtP + - deg2_poly_gt0 + - CrealP + - normr_gt0 + - normC2_Re_Im + - ler_sqrtC + - deg2_poly_factor + - realD + - prodr_ge0 + - real_wlog_ltr + - ltr_leB + - realrM + - ltr_nV2 + - real_ge0P + - ler1_real + - Creal_ReP + - ler_real + - deg2_poly_min + - ltr_pDl + - ler0_def + - deg2_poly_root1 + - ler_distlC + - sgrM + - realEsg + - Im_rect + - real_mulr_sign_norm + - eqrMn2r + - rootCMl + - real_ltr_norml + - Im_is_additive + - deg2_poly_ge0l + - addr_min_max + - mulr_lt0 + - pmulr_lge0 + - ler_peMl + - ltr0_neq0 + - maxrN + - ltr_ndivlMl + - le_normD + - real_wlog_ler + - invC_norm + - pmulr_rgt0 + - comparable0r + - divC_rect + - real_minNr + - exprCK + - pexprn_eq1 + - ler_wpDl + - eqr_sqrt + - CrealJ + - comparabler_trans + - exprn_ege1 + - real_exprn_odd_gt0 + - ler_distlCBl + - real_maxr_nMr + - natr_nat + - sqrtrP + - ltf_pV2 + - conjC_nat + - ler_distl + - ler_addgt0Pl + - realEsign + - ler_piMr + - ger_nMl + - Cauchy_root_bound + - pmulr_llt0 + - normCi + - ltr_pwDr + - ltr_wpDl + - sgr_eq0 + - a2 + - lteif_nnormr + - leifD + - ltr_pM + - exprn_ge0 + - realX + - subr_le0 + - ler_addgt0Pr + - expr_le1 + - ler_wpM2l + - ler0N1 + - real_maxr_nMl + - lteif_ndivlMl + - normC_sum_eq + - xb4 + - real_exprn_even_ge0 + - ler01 + - sqrtCK + - ltr_ndivrMr + - lerP + - lerB + - deg_le2_poly_delta_ge0 + - lteif_pdivlMr + - ltr_pMl + - exprn_ilt1 + - ltr_rootC + - ImMir + - lteif_ndivrMr + - ltrBlDl + - addr_maxr + - real_leVge + - ler_nMl + - rootC_inj + - lteifBrDl + - ltW + - ltr_norml + - ltr_pMn2l + - mulr_sg_eq1 + - normr_lt0 + - addr_maxl + - lerNl + - ler_ndivlMl + - ler_wsqrtr + - conjC_eq0 + - numEsg + - natrG_gt0 + - Im_conj + - ler_normB + - natr_indexg_neq0 + - sgr_lt0 + - nat_num_semiring + - divr_ge0 + - normr_sg + - rootC_ge1 + - distrC + - ltr_wMn2r + - ltr_iXn2l + - sqrtr_subproof + - real_divr_closed + - pmulrn_lge0 + - signr_inj + - real_ler_distlBl + - ler_nM2l + - normrN + - real_mulr_Nsign_norm + - ReMr + - upper_nthrootP + - deg2_poly_noroot + - archi_boundP + - num_real + - divr_gt0 + - ler0_real + - real_ler_norml + - ler_psqrt + - normr_idP + - ltr_sqr + - a4 + - oppr_le0 + - minr_pMr + - expr_gt1 + - mulrn_wle0 + - maxr_to_min + - real_exprn_even_le0 + - normr_sign + - invC_rect + - real_oppr_min + - sgrX + - Nreal_geF + - lteifBlDr + - psumr_eq0P + - real_addr_minr + - ler0n + - exprn_even_le0 + - invf_lt1 + - geC0_unit_exp + - lteif_distl + - ler_sqrt + - realMr + - ltr_nM2r + - realrMn + - unitf_lt0 + - pmulrn_llt0 + - ger0_le_norm + - Re_rect + - sgr_smul + - ReV + - sqrCK_P + - ltr_wpDr + - mulr_le0_ge0 + - real_eqr_norml + - gtrDl + - signr_ge0 + - lt_def + - normC_def + - ler_neMl + - oppC_rect + - nmulr_rlt0 + - rootCX + - norm_rootC + - deg2_poly_le0m + - ler_wnDr + - Im_lock + - invr_le1 + - ger0_def + - Re_i + - ltr_rootCl + - sqrn_eq1 + - sgr_cp0 + - ltr_sqrtC + - ler_dist_normD + - real_lteif_normr + - ltr_eXn2l + - sqrtC_le0 + - ler_dist_dist + - ler_pXn2r + - ltr_pdivlMl + - mulC_rect + - gtrN + - invr_le0 + - rootCV + - subr_lteif0r + - maxr_pMr + - realM + - deg2_poly_ge0m + - lerBlDr + - real_le0P + - rootCpX + - real_leif_mean_square_scaled + - normf_div + - rectC_mull + - ler_pdivlMr + - sumr_ge0 + - deltaN + - mul_conjC_eq0 + - pmulr_lle0 + - deg2_poly_root1 + - gerDr + - le_normD + - lt01 + - a2gt0 + - mulCii + - invr_lt1 + - sqr_sg + - eqCP + - neq0_mulr_lt0 + - lteif_pM2r + - real_ler_normlP + - subr_gt0 + - a1 + - real_ltNge + - addr_minl + - ltr10 + - invf_pge + - ReE + - gerDl + - ler_niMr + - ltr_pXn2r + - ler_pdivrMr + - maxr_nMr + - ltrD2r + - pmulrn_rlt0 + - ler_piMl + - ltr_nMr + - sqrtr1 + - ltr_ndivlMr + - rootC_eq0 + - lteif0Nr + - real_exprn_even_gt0 + - eq0_norm + - Re_div + - ler_sum + - pneq0 + - sgrMn + - path: mathcomp/solvable/sylow.v + theorems: + - nilpotent_maxp_normal + - card_Syl_dvd + - Baer_Suzuki + - Sylow_exists + - nil_class3 + - pgroup_nil + - nil_Zgroup_cyclic + - Hall_pJsub + - card_p2group_abelian + - trivg_center_pgroup + - morphim_Zgroup + - Hall_psubJ + - sub_nilpotent_cent2 + - pgroup_fix_mod + - nil_class2 + - Syl_trans + - pcore_sub_astab_irr + - nilpotent_Hall_pcore + - Sylow_setI_normal + - Sylow_trans + - Hall_setI_normal + - nontrivial_gacent_pgroup + - pgroup_sol + - nil_class_pgroup + - pcore_faithful_irr_act + - p2group_abelian + - Sylow's_theorem + - small_nil_class + - Sylow_subJ + - normal_pgroup + - Frattini_arg + - max_pgroup_Sylow + - nilpotent_pcoreC + - card_Syl + - coprime_mulG_setI_norm + - pi_center_nilpotent + - Sylow_subnorm + - normal_sylowP + - Sylow_gen + - Sylow_transversal_gen + - path: mathcomp/field/fieldext.v + theorems: + - size_Fadjoin_poly + - Fadjoin0 + - mulfxC + - prodvAC + - base_aspaceOver + - minPolyxx + - field_subvMr + - field_module_eq + - monic_minPoly + - base_moduleOver + - root_minPoly + - mem1v + - p0z0 + - Fadjoin_nil + - field_module_semisimple + - sub1v + - irredp_FAdjoin + - baseField_scaleDr + - aspaceOver_suproof + - nz_p0 + - Fadjoin_polyX + - nonzero1fx + - field_mem_algid + - adjoin0_deg + - subfx_irreducibleP + - subfield_closed + - subfx_inj_is_additive + - map_minPoly + - vspaceOver_refBase + - subfx_scalerDr + - Fadjoin_poly_is_linear + - field_dimS + - Fadjoin_polyC + - pi_subfx_inj + - minPolyOver + - AEnd_lker0 + - fieldExt_hornerX + - modp_polyOver + - subfx_poly_invE + - dim_sup_field + - poly_rV_modp_K + - vsval_invf + - Fadjoin_eq_sum + - aimg_is_aspace + - pi_subfext_add + - subfx_scaleAr + - subfx_inj_eval + - dim_Fadjoin + - AHom_lker0 + - subfx_fieldAxiom + - fieldOver_scaleAl + - subfx_scalerA + - polyOver_subvs + - subfx_inj_base + - polyOverSv + - subfx_scalerDl + - fieldOver_scaleE + - subfx_inj_root + - mem_aspaceOver + - sup_field_module + - baseField_scale1 + - addfxC + - Fadjoin_idP + - baseField_vectMixin + - pi_subfext_inv + - addfxA + - dim_vspaceOver + - nz_p + - baseField_scaleDl + - Fadjoin_polyP + - iotaPz_repr + - adjoin_deg_eq1 + - z0Ciota + - iotaPz_modp + - sub_baseField + - add0fx + - aspaceOverP + - sub_adjoin1v + - subfx_eval_is_additive + - p0_mon + - dim_aspaceOver + - subfxEroot + - subvs_fieldMixin + - mempx_Fadjoin + - fieldOver_scaleDl + - field_module_dimS + - prodvCA + - gcdp_polyOver + - fieldOver_scaleDr + - root_small_adjoin_poly + - aspace_divr_closed + - baseVspace_module + - mul1fx + - prodvC + - fieldOver_scaleA + - FadjoinP + - minPoly_irr + - addfxN + - mulfxA + - fieldExt_hornerC + - baseField_scaleA + - F0ZEZ + - pi_subfext_opp + - equiv_subfext_is_equiv + - adjoin_degree_aimg + - subfx_scaleAl + - prodv_is_aspace + - subfx_eval_is_multiplicative + - fieldExt_hornerZ + - pi_subfext_mul + - min_subfx_vect + - subfx_evalZ + - field_subvMl + - algid1 + - mem_baseVspace + - baseField_scaleE + - dim_cosetv + - alg_polyOver + - dim_baseVspace + - trivial_fieldOver + - nz_x_i + - mulfx_addl + - dim_field_module + - subfx_inv0 + - fieldOver_vectMixin + - subfxE + - Fadjoin_poly_mod + - baseField_scaleAr + - fieldOver_scaleAr + - size_minPoly + - Fadjoin1_polyP + - vspaceOverP + - minPolyS + - Fadjoin_poly_eq + - baseField_scaleAl + - baseAspace_suproof + - Fadjoin_polyOver + - module_baseAspace + - adjoin_degreeE + - minPoly_XsubC + - Fadjoin_seqP + - n_gt0 + - Fadjoin_sum_direct + - base_vspaceOver + - fieldOver_scale1 + - subfx_injZ + - Fadjoin_poly_unique + - path: mathcomp/character/character.v + theorems: + - cfBigdprodi_lin_char + - cfker_constt + - cfcenter_sub + - lin_charV_conj + - cfDprodr_lin_char + - irr_inv + - cfMorph_charE + - subGcfker + - cfAut_lin_char + - cfDetRes + - xcfunZr + - cfDetMorph + - cfQuo_irr + - cfRepr_dsum + - dsumx_mul + - cap_cfker_normal + - neq0_has_constt + - cfConjC_irr1 + - cfdot_sum_irr + - irr_prime_injP + - conjC_IirrK + - trow_is_linear + - cfAut_irr1 + - Iirr_cast + - add_mx_repr + - cfker_reg_quo + - cfdot_Res_ge_constt + - irr_free + - tprodE + - cfRepr_inj + - dprodr_IirrE + - cfConjC_lin_char + - sdprod_IirrE + - TI_cfker_irr + - irr_classP + - card_afix_irr_classes + - mxtrace_prod + - irr_basis + - cfcenter_repr + - lin_char_unity_root + - cfRegE + - socle_of_Iirr_bij + - irr_eq1 + - cfReg_sum + - cfRepr_standard + - isom_IirrE + - cfkerEirr + - cap_cfcenter_irr + - conjC_Iirr0 + - cfaithful_reg + - dprod_IirrEl + - cfSdprod_irr + - cfun1_irr + - aut_IirrE + - irr1_gt0 + - Res_irr_neq0 + - cfnorm_Res_leif + - irrWnorm + - Iirr1_neq0 + - cfDet_order_dvdG + - lin_char_prod + - Res_sdprod_irr + - cap_cfker_lin_irr + - lin_char_group + - char1_ge_constt + - reindex_irr_class + - Ind_irr_neq0 + - cfnorm_irr + - cfRepr_subproof + - dprod_Iirr0r + - eq_irr_mem_classP + - conjC_IirrE + - cfInd_eq0 + - cfRepr_sub + - lin_char_neq0 + - eq_subZnat_irr + - cfMorph_char + - isom_Iirr0 + - cfcenter_cyclic + - cfRes_lin_char + - char_sum_irr + - lin_charX + - dprodr_Iirr0 + - prod_mx_repr + - trowbE + - cfMod_charE + - cfBigdprodi_char + - irr_of_socle_bij + - mod_Iirr0 + - irr_cfcenterE + - cfRepr1 + - cfExp_prime_transitive + - socle_Iirr0 + - cfDprodr_irr + - cfker_irr0 + - cfMod_irr + - Wedderburn_id_expansion + - cfMod_char + - dprod_Iirr0l + - lin_charM + - cfcenter_normal + - cfcenter_group_set + - irr_faithful_center + - max_cfRepr_mx1 + - linear_char_divr + - dprod_Iirr_onto + - dprod_Iirr0 + - cfRepr_char + - irr1_bound + - constt_Res_trans + - quo_Iirr_eq0 + - dprod_IirrEr + - cfMorph_lin_char + - cfRepr_rsimP + - mod_IirrK + - eq_addZ_irr + - morph_Iirr_eq0 + - cfBigdprod_irr + - cfConjC_irr + - cfBigdprodi_lin_charE + - cfQuo_lin_charE + - irrP + - cforder_lin_char + - constt_ortho_char + - cfdot_aut_char + - groupC + - constt_cfInd_irr + - cfkerE + - cfRes_char + - aut_Iirr_inj + - cfBigdprod_Res_lin + - cfBigdprod_lin_char + - irr_orthonormal + - sdprod_Iirr0 + - cfBigdprod_char + - cfIsom_char + - cfRepr_morphim + - xcfun_id + - cfker_Res + - sAG + - dprod_IirrK + - dprodl_Iirr0 + - irr_char + - cfIsom_irr + - dprodr_IirrK + - mul_conjC_lin_char + - eq_scaled_irr + - cfDetD + - card_Iirr_abelian + - dprod_IirrE + - irr_cyclic_lin + - cfDprodKl_abelian + - cfun1_char + - sdprod_Iirr_eq0 + - morph_Iirr_inj + - quo_IirrK + - quo_IirrE + - cfBigdprodi_charE + - normC_lin_char + - cfDet_order_lin + - trow0 + - cfIirrE + - aut_Iirr0 + - cfcenter_eq_center + - isom_IirrKV + - cfDet_lin_char + - irr1_neq0 + - cfun1_lin_char + - second_orthogonality_relation + - nKG + - cfBigdprodKabelian + - trowb_is_linear + - cfun0_char + - cfun_sum_cfdot + - class_IirrK + - character_table_unit + - cfDprod_irr + - congr_irr + - conjC_irrAut + - first_orthogonality_relation + - dprod_Iirr_inj + - isom_IirrK + - trow_mul + - cfcenter_fful_irr + - cfdot_dprod_irr + - cfMorph_irr + - cfSdprod_char + - detRepr_lin_char + - cfIirr_key + - mod_Iirr_eq0 + - cfDprod_char + - cfkerEchar + - char_sum_irrP + - cfIsom_lin_char + - prod_repr_lin + - cfRes_lin_lin + - cfDetMn + - cfConjC_char1 + - NirrE + - cfIirrPE + - dprod_Iirr_eq0 + - mul_lin_irr + - Cnat_cfdot_char_irr + - sdprod_Res_IirrE + - cforder_irr_eq1 + - lin_char_der1 + - sdprod_IirrK + - irr_sum_square + - cfBigdprod_eq1 + - xcfun_mul_id + - cfRepr_dadd + - eq_signed_irr + - irr1_degree + - isom_Iirr_eq0 + - repr_rsim_diag + - cfDet_id + - cfDprodKr_abelian + - char1_ge_norm + - irr_reprP + - irr1_abelian_bound + - has_nonprincipal_irr + - xcfun_is_additive + - mx_rsim_socle + - irr_prime_lin + - mod_IirrE + - repr_irr_classK + - inv_dprod_Iirr0 + - irr0 + - morph_Iirr0 + - cfRepr_sim + - cforder_lin_char_gt0 + - cfker_center_normal + - isom_Iirr_inj + - cfQuo_charE + - lin_charW + - cfdot_irr + - cfAut_char1 + - irr_neq0 + - cfBigdprodi_irr + - cfDprod_eq1 + - cfdot_char_r + - solvable_has_lin_char + - cfInd_char + - cfAut_irr + - coord_cfdot + - cfQuo_lin_char + - mx_rsim_dsum + - cfcenter_subset_center + - Cnat_irr1 + - irrK + - cfDet0 + - cfBigdprodKlin + - cfSdprod_lin_char + - card_subcent1_coset + - cfker_Ind + - xcfunG + - mx_rsim_dadd + - cfDprodl_char + - conjC_Iirr_eq0 + - lin_char_irr + - lin_irr_der1 + - dprodl_IirrE + - card_Iirr_cyclic + - invr_lin_char + - Cnat_char1 + - generalized_orthogonality_relation + - cfRepr0 + - mem_irr + - dprodl_IirrK + - char_reprP + - morph_IirrE + - mx_rsim_standard + - cfDetIsom + - Nxi + - cfDprodl_lin_char + - irrWchar + - usumx_mul + - char1_eq0 + - cfMod_lin_charE + - constt_irr + - cfConjC_char + - Res_Iirr0 + - aut_Iirr_eq0 + - quo_Iirr0 + - fful_lin_char_inj + - cfDet_mul_lin + - XX'_1 + - conjC_charAut + - irrRepr + - xiMV + - socle_of_IirrK + - irrEchar + - xcfun_repr + - cfDprodr_char + - cfDetRepr + - tprod_tr + - cfReg_char + - cfMorph_lin_charE + - mul_char + - cfQuo_char + - char_abelianP + - card_lin_irr + - mx_repr0 + - quo_IirrKeq + - lin_char_unitr + - eq_scale_irr + - char_inv + - cfun_sum_constt + - cforder_lin_char_dvdG + - char_cfcenterE + - cfDprod_lin_char + - char1_ge0 + - dprodr_Iirr_eq0 + - cfcenter_Res + - det_is_repr + - irr_aut_closed + - irr_of_socleK + - constt_cfRes_irr + - lin_char1 + - sdprod_Res_IirrK + - cfker_nzcharE + - cfReprReg + - mod_Iirr_bij + - add_char + - path: mathcomp/field/algnum.v + theorems: + - Aint_aut + - Crat_spanP + - eqAmodMr0 + - eqAmod_refl + - eqAmod_addl_mul + - restrict_aut_to_normal_num_field + - eqAmod0_rat + - Crat_span_zmod_closed + - eqAmodMl0 + - dec_Cint_span + - eqAmodN + - restrict_aut_to_num_field + - fin_Csubring_Aint + - Aint0 + - eqAmodMl + - Cint_span_zmod_closed + - eqAmod0_nat + - eqAmodD + - dvdA_zmod_closed + - num_field_exists + - mem_Cint_span + - Aint_prim_root + - Aint1 + - eqAmod0 + - Aint_subring + - eqAmod_rat + - rmorphZ_num + - eqAmodm0 + - eqAmodMr + - dvdn_orderC + - Aint_Cint + - eqAmod_transl + - exp_orderC + - eqAmod_sym + - root_monic_Aint + - Crat_spanM + - Aint_unity_root + - eqAmod_transr + - alg_num_field + - mem_Crat_span + - Crat_spanZ + - Aint_Cnat + - map_Qnum_poly + - eqAmod_trans + - num_field_proj + - eqAmod_nat + - eqAmodM + - Cint_spanP + - Crat_span_subproof + - fmorph_numZ + - Aint_int + - extend_algC_subfield_aut + - eqAmodDl + - eqAmodDr + - algC_PET + - Cint_rat_Aint + - path: mathcomp/algebra/poly.v + theorems: + - drop_poly_is_linear + - comm_poly_exp + - multiplicity_XsubC + - mul_0poly + - coefXM + - root_ZXsubC + - polyOverZ + - comp_poly0 + - mul_poly_key + - size_map_polyC + - size_Poly + - fmorph_unity_root + - nderivnC + - monicXnaddC + - prim_root_dvd_eq0 + - map_polyXaddC + - nderivnMn + - size_polyC_leq1 + - odd_polyE + - commr_polyXn + - dvdn_prim_root + - lead_coefM + - aa4 + - polySpred + - polyOverNr + - comm_polyX + - rpred_horner + - size_polyXn + - size_exp_leq + - prim_root_natf_neq0 + - derivnC + - derivnB + - scale_poly_eq0 + - rootE + - comp_poly_eq0 + - nderivnXn + - lead_coefDr + - size_poly_eq + - mul_polyDr + - derivMXaddC + - poly2_root + - comp_polyXaddC_K + - horner_eval_is_linear + - prim_order_dvd + - scale_polyC + - mul_poly0 + - derivn1 + - coefMXn + - horner_coef_wide + - lead_coefX + - nderivn_def + - polyOver0 + - size_exp + - polyseqXn + - rreg_polyMC_eq0 + - hornerN + - prim_expr_order + - lead_coef_monicM + - root_exp_XsubC + - scale_1poly + - polyC0 + - root_polyC + - deg2_poly_root1 + - size1_polyC + - even_polyD + - monic_neq0 + - coefXn + - coef_opp_poly + - derivnXn + - lead_coef_map_inj + - closed_nonrootP + - size_odd_poly + - rmorph_root + - lead_coefMX + - polyOver_addr_closed + - monic_lreg + - polyX_key + - map_polyZ + - commr_horner + - monic_exp + - aneq0 + - polyXsubC_eq0 + - deg2_poly_canonical + - lead_coef_exp + - polyOverC + - closed_rootP + - rootN + - coef0 + - polyOverXaddC + - map_poly_inj + - mul_polyC + - rmorph_unity_root + - hornerXn + - poly_inj + - polyCM + - mapf_root + - coef0_prod_XsubC + - polyOver_poly + - factor_Xn_sub_1 + - size_prod_seq + - comp_poly_multiplicative + - scale_poly_key + - lead_coefXnsubC + - factor_theorem + - prim_root_charF + - size_exp_XsubC + - take_polyDMXn + - odd_polyD + - comm_polyM + - unity_rootE + - root0 + - horner_prod + - deg2_poly_root1 + - polyOver_deriv + - root_prod_XsubC + - drop_polyDMXn + - size_prod_seq_eq1 + - size_prod_leq + - coefB + - derivnMNn + - derivMNn + - even_polyZ + - lead_coef_map + - rootPt + - polyOverXnsubC + - polyCV + - comm_poly1 + - polyCMn + - derivMn + - take_poly0l + - coef_map_id0 + - dec_factor_theorem + - odd_polyZ + - size_drop_poly + - map_poly_is_multiplicative + - prim_rootP + - coef_drop_poly + - sum_odd_poly + - sum_even_poly + - coef_comp_poly_Xn + - poly_mulVp + - polyseq_cons + - hornerMX + - rootPf + - map_poly_com + - prim_expr_mod + - size_prod_eq1 + - polyCK + - derivXsubC + - size_add + - size_comp_poly2 + - coefMn + - polyOverXn + - map_polyC + - comp_poly_is_linear + - nderivn0 + - hornerD + - size_opp + - coefCM + - nderivnMNn + - comp_polyM + - prim_root_eq0 + - commr_polyX + - map_prod_XsubC + - prim_root_exp_coprime + - roots_geq_poly_eq0 + - lead_coefDl + - poly_take_drop + - unity_rootP + - derivnMXaddC + - hornerXsubC + - mul_lead_coef + - deg2_poly_root2 + - map_polyE + - map_comm_coef + - deriv_exp + - map_poly_comp_id0 + - map_poly_is_additive + - root_exp + - horner_map + - coefXnM + - coefPn_prod_XsubC + - poly_intro_unit + - monic_map + - size_polyC + - poly_inv_out + - even_polyE + - eqp_take_drop + - polyOverXsubC + - size_poly + - size_poly0 + - size_Mmonic + - size_polyX + - lead_coefE + - comp_polyX + - rootZ + - derivM + - max_poly_roots + - odd_polyMX + - size_even_poly_eq + - even_polyC + - polyC_inj + - polyseqXaddC + - comp_polyZ + - monic_mulr_closed + - monic_prod_XsubC + - polyseqMX + - polyOver_mulr_2closed + - polyC_eq0 + - take_polyMXn_0 + - horner_algX + - nderiv_taylor_wide + - polyX_eq0 + - poly_even_odd + - rreg_size + - comp_poly_MXaddC + - prim_root_pi_eq0 + - polyseqC + - coef_mul_poly + - comp_polyB + - lead_coef_Mmonic + - comp_poly0r + - derivXn + - poly_idomainAxiom + - horner0 + - size_map_inj_poly + - polyC1 + - nderivn_map + - mem_root + - coef_odd_poly + - map_comm_poly + - polyP + - deg2_poly_canonical + - drop_polyZ + - mul_polyA + - size_XmulC + - derivnZ + - size_sum + - root_XaddC + - coef_cons + - polyseqK + - aut_unity_rootC + - horner_coef0 + - scale_polyAl + - comp_polyXr + - add_poly0 + - sqa2neq0 + - poly_mul_comm + - max_unity_roots + - hornerCM + - coef0_prod + - multiplicity_XsubC + - monicXaddC + - fmorph_root + - lead_coef_eq0 + - derivSn + - nderiv_taylor + - horner_comp + - monic1 + - size_poly_gt0 + - coefMC + - nderivnB + - horner_is_linear + - monicXn + - poly_initial + - size_map_poly + - in_alg_comm + - polyOver_derivn + - hornerX + - size_mulXn + - deriv0 + - rootM + - comm_coef_poly + - lead_coef_lreg + - size_prod_XsubC + - multiplicity_XsubC + - polyseqMXn + - polyseq0 + - polyC_natr + - lead_coef1 + - derivn_is_linear + - polyseqXsubC + - horner_exp + - polyCD + - coef0M + - prim_order_gt0 + - coef_derivn + - lead_coefN + - aut_unity_rootP + - nderivn_is_linear + - coef_deriv + - coefMr + - eq_map_poly + - rreg_lead + - map_diff_roots + - comm_polyD + - opp_poly_key + - drop_poly0r + - size_MXaddC + - coefX + - map_Poly + - comp_polyE + - coefC + - monicMl + - size_mul + - coef_nderivn + - horner_morphX + - coef_poly + - cons_poly_def + - deriv_mulC + - lt_size_deriv + - horner_is_multiplicative + - polyseq_poly + - derivnS + - polyOverXnaddC + - even_polyMX + - deriv_comp + - polyC_multiplicative + - closed_rootP + - map_polyXn + - polyOver_comp + - take_poly0r + - derivn0 + - aut_prim_rootP + - comp_poly2_eq0 + - prod_map_poly + - a1 + - polyOver_mul1_closed + - poly0Vpos + - size_Cmul + - horner_algC + - hornerC + - size_prod + - nderivnZ + - deg2_poly_factor + - map_inj_poly + - monicP + - size_scale_leq + - hornerM_comm + - map_poly_id + - eq_poly + - poly_key + - size_proper_mul + - polyseqX + - map_polyK + - size_cons_poly + - drop_poly_sum + - comp_polyD + - derivn_map + - max_ring_poly_roots + - root_size_gt1 + - eq_prim_root_expr + - deriv_map + - take_polyD + - polyOverX + - eq_in_map_poly_id0 + - size_poly1 + - eq_in_map_poly + - lead_coefXsubC + - size_even_poly + - poly_morphX_comm + - coefZ + - monicE + - coef_map + - lead_coef0 + - pneq0 + - splitr + - poly_invE + - poly_unitE + - coef_take_poly + - rreg_div0 + - derivC + - monic_prod + - map_uniq_roots + - lead_coefZ + - coefp0_multiplicative + - monic_rreg + - all_roots_prod_XsubC + - polyOver_nderivn + - a2neq0 + - lreg_lead0 + - coef_Poly + - lreg_lead + - pE + - monicXnsubC + - rootX + - lead_coef_map_eq + - horner_cons + - derivnD + - size_mul_eq1 + - char_poly + - coefD + - drop_poly0l + - poly_def + - lead_coefXnaddC + - coef_sum + - rootP + - horner_exp_comm + - PolyK + - coefMNn + - map_poly0 + - add_polyA + - derivn_poly0 + - gt_size_poly_neq0 + - lead_coefC + - size_take_poly + - lead_coef_comp + - derivB + - nderivn_poly0 + - size_map_poly_id0 + - coef_even_poly + - scale_polyA + - lreg_polyZ_eq0 + - size_poly1P + - lead_coefXn + - root_XsubC + - drop_polyMXn + - odd_poly_is_linear + - lead_coef_prod_XsubC + - derivnMn + - lead_coef_poly + - horner_morphC + - nderivnMXaddC + - exp_prim_root + - size_monicM + - map_poly_comp + - alg_polyC + - comp_Xn_poly + - sum_drop_poly + - map_polyX + - take_poly_id + - fmorph_primitive_root + - size_comp_poly + - comp_poly_Xn + - horner_eval_is_multiplicative + - mem_unity_roots + - monicX + - size_XnsubC + - lead_coef_proper_mul + - poly1_neq0 + - hornerZ + - map_polyC_eq0 + - lead_coef_prod + - coefK + - derivD + - nderivnN + - coef_mul_poly_rev + - add_polyN + - monicXsubC + - mul_poly1 + - hornerMXaddC + - size_XsubC + - take_poly_sum + - uniq_roots_prod_XsubC + - horner_sum + - uniq_rootsE + - size_mul_leq + - nderivnD + - mul_1poly + - coefM + - nderivn1 + - mul_polyDl + - drop_polyMXn_id + - nil_poly + - horner_Poly + - odd_polyC + - polyCN + - take_polyZ + - comp_polyC + - monic_comreg + - horner_poly + - even_poly_is_linear + - hornerMn + - polyOverS + - scale_polyDr + - root_comp + - path: mathcomp/field/separable.v + theorems: + - extendDerivation_id + - separable_generatorP + - adjoin_separable_eq + - separable_polyP + - extendDerivation_horner + - separable_deriv_eq0 + - separable_refl + - poly_square_freeP + - strong_Primitive_Element_Theorem + - make_separable + - eqp_separable + - separableS + - Derivation_scalar + - Derivation_separable + - sub_inseparable + - charf_n_separable + - separableP + - adjoin_separableP + - Primitive_Element_Theorem + - separable_Fadjoin_seq + - cyclic_or_large + - extendDerivation_scalable_subproof + - extendDerivationP + - separable_generator_mem + - separable_poly_neq0 + - charf0_separable + - separable_map + - separable_root + - separableSl + - purely_inseparableP + - purely_inseparable_trans + - separable_root_der + - finite_PET + - eq_adjoin_separable_generator + - adjoin_separable + - Derivation_exp + - separablePn + - separable_elementP + - inseparable_sum + - sub_adjoin_separable_generator + - separable_sum + - separable_inseparable_decomposition + - inseparable_add + - extendDerivation_additive_subproof + - Derivation1 + - separable_coprime + - separable_mul + - separable_elementS + - separable_nz_der + - charf_p_separable + - Derivation_separableP + - purely_inseparable_elementP + - DerivationS + - separableSr + - Derivation_mul + - Derivation_horner + - separable_exponent + - purely_inseparable_refl + - large_field_PET + - separable_nosquare + - separable_trans + - path: mathcomp/solvable/extremal.v + theorems: + - r_gt0 + - cyclic_SCN + - odd_pgroup_rank1_cyclic + - Grp_2dihedral + - defQ + - dihedral2_structure + - modular_group_classP + - card_quaternion + - def2qr + - card_ext_dihedral + - def_q + - prime_Ohm1P + - involutions_gen_dihedral + - maximal_cycle_extremal + - ltqm + - card_2dihedral + - modular_group_structure + - Grp_ext_dihedral + - generators_modular_group + - cyclic_pgroup_Aut_structure + - Grp_quaternion + - r_gt0 + - def_r + - dihedral_classP + - def_p + - bound_extremal_groups + - card_modular_group + - card_semidihedral + - card + - generators_quaternion + - generators_semidihedral + - quaternion_structure + - aut_dvdn + - Grp'_dihedral + - semidihedral_structure + - card_dihedral + - act_dom + - extremal2_structure + - eq_Mod8_D8 + - ltrq + - Grp_dihedral + - Grp + - Grp_modular_group + - q_gt1 + - q_gt0 + - symplectic_type_group_structure + - Grp_semidihedral + - semidihedral_classP + - quaternion_classP + - cancel_index_extremal_groups + - path: mathcomp/solvable/maximal.v + theorems: + - injm_Fitting + - Fitting_group_set + - SCN_P + - der1_stab_Ohm1_SCN_series + - Ohm1_stab_Ohm1_SCN_series + - card_extraspecial + - p_index_maximal + - Phi_Mho + - p3group_extraspecial + - charsimple_dprod + - isog_extraspecial + - index_maxnormal_sol_prime + - p_core_Fitting + - Phi_quotient_abelem + - trivg_Phi + - Fitting_sub + - pcore_Fitting + - Phi_nongen + - injm_special + - sol_prime_factor_exists + - Phi_joing + - extraspecial_prime + - solvable_norm_abelem + - exponent_special + - Phi_normal + - Phi_sub + - SCN_max + - simple_sol_prime + - charsimpleP + - Phi_quotient_cyclic + - maxnormal_charsimple + - cprod_extraspecial + - injm_extraspecial + - center_special_abelem + - Fitting_pcore + - Fitting_normal + - p_maximal_index + - minnormal_solvable + - Fitting_nil + - Fitting_eq_pcore + - FittingEgen + - Fitting_char + - split1_extraspecial + - Phi_char + - PhiJ + - card_subcent_extraspecial + - p_abelem_split1 + - Frattini_continuous + - abelem_split_dprod + - Phi_sub_max + - critical_extraspecial + - cent1_extraspecial_maximal + - Phi_min + - quotient_Phi + - Thompson_critical + - Phi_cprod + - Phi_mulg + - critical_class2 + - p_maximal_normal + - injm_Phi + - extraspecial_nonabelian + - trivg_Fitting + - morphim_Fitting + - abelem_charsimple + - charsimple_solvable + - center_aut_extraspecial + - pmaxElem_extraspecial + - isog_Phi + - PhiS + - card_center_extraspecial + - Ohm1_cent_max_normal_abelem + - Phi_proper + - max_SCN + - isog_Fitting + - exponent_Ohm1_class2 + - morphim_Phi + - abelian_charsimple_special + - FittingS + - minnormal_charsimple + - exponent_2extraspecial + - Phi_quotient_id + - extraspecial_structure + - SCN_abelian + - critical_p_stab_Aut + - path: mathcomp/field/falgebra.v + theorems: + - prodv_sub + - agenvX + - vsval_invr + - memv_adjoin + - id_is_ahom + - prodvSr + - memv_algid + - expv_line + - adim1P + - adjoinSl + - ker_sub_ahom_is_aspace + - centraliser_is_aspace + - prod1v + - adjoin_seqSl + - FalgType_proper + - agenvS + - agenvE + - subvs_mulDr + - skew_field_algid1 + - agenvM + - adjoin_seq1 + - aimg_adjoin_seq + - prodv1 + - aimgM + - lfun_mulE + - adjoin_seqSr + - prodvS + - subvs_mul1 + - adjoin_nil + - dim_cosetv_unit + - sub_agenv + - unitrP + - expv_id + - mulVr + - amull1 + - aspacef_subproof + - algid_eq1 + - prodv0 + - expv2 + - Falgebra_FieldMixin + - subvs_scaleAr + - amulr_inj + - prodvDl + - skew_field_dimS + - centv1 + - aimgX + - aspace1_subproof + - cent1v1 + - agenvEr + - not_asubv0 + - amulr_is_linear + - regular_fullv + - lfun_invr_out + - memvM + - limg_amulr + - unitr_algid1 + - prodv_line + - prodvP + - dim_algid + - centraliser1_is_aspace + - adim_gt0 + - subvs_mu1l + - algid_neq0 + - memvV + - amE + - aimg_agen + - asubv + - aspace_cap_subproof + - lker0_amulr + - skew_field_module_semisimple + - divrr + - subv_adjoin + - centv_algid + - algidl + - expv0n + - expvD + - amullM + - has_algid1 + - cent1v_id + - agenv_modl + - prodvSl + - centvP + - prod0v + - expvS + - subvs_mulA + - prodv_id + - lfun_compE + - vbasis1 + - subv_cent1 + - linfun_is_ahom + - amull_inj + - cent1vC + - algid_subproof + - centvsP + - polyOver1P + - expvSr + - subvs_scaleAl + - adjoinC + - aimg_adjoin + - cent1vX + - agenv_modr + - skew_field_module_dimS + - ahomWin + - agenv_is_aspace + - memv_mul + - lfun_unitrP + - prodvA + - lker0_amull + - agenv_id + - amulr_is_multiplicative + - algid_center + - subvP_adjoin + - lfun1_poly + - vsval_unitr + - ahomP + - subvs_mulDl + - comp_is_ahom + - expvSl + - invr_out + - lfun_mulrV + - agenv_sub_modr + - adjoin_rcons + - seqv_sub_adjoin + - subv_adjoin_seq + - has_algidP + - centvC + - dim_prodv + - ahom_is_multiplicative + - ahom_inP + - vspace1_neq0 + - dimv1 + - centvX + - agenvEl + - agenv_add_id + - expv1 + - expvM + - memv_cosetP + - aimg1 + - prodv_key + - algid_decidable + - lfun_mulVr + - cent1vP + - path: mathcomp/algebra/mxpoly.v + theorems: + - geigenspaceE + - eigenpoly_map + - codiagonalizable1 + - sub_kermxpoly_conjmx + - eigenvalue_conjmx + - mxminpoly_linear_is_scalar + - submx_form_qf + - integral_root + - kermxpolyX + - integral_nat + - degree_mxminpoly_proof + - nth_row_env + - mx_root_minpoly + - resultant_eq0 + - char_poly_monic + - diagonalizablePeigen + - horner_rVpolyK + - conjmx_scalar + - conjmxK + - map_resultant + - integral_add + - size_mod_mxminpoly + - rVpolyK + - mxminpoly_conj + - diagonalizableP + - algebraic_sub + - diagonalizable_for_sum + - integral_sub + - char_block_diag_mx + - diagonalizable0 + - conj1mx + - stablemx_restrict + - sub_eigenspace_conjmx + - mxdirect_sum_geigenspace + - eval_col_mx + - kermxpolyM + - companion_map_poly + - coef_rVpoly_ord + - horner_mx_C + - conjuMumx + - integral_rmorph + - map_geigenspace + - size_seq_of_rV + - Exists_rowP + - integral0 + - mxminpoly_dvd_char + - simmxP + - minpoly_mx_free + - eigenpolyP + - mulmx_delta_companion + - intR_XsubC + - integral_horner_root + - algebraic0 + - diagonalizable_diag + - eval_mulmx + - diagonalizable_for_mxminpoly + - conjmx_eigenvalue + - row'_col'_char_poly_mx + - diagonalizable_scalar + - conjMmx + - size_char_poly + - map_kermxpoly + - horner_mx_conj + - algebraic_div + - mxminpoly_min + - poly_rV_K + - eigenvalue_root_min + - kermxpolyC + - comm_horner_mx + - companionmxK + - integral_opp + - horner_mx_mem + - comm_mx_stable_kermxpoly + - codiagonalizable_on + - root_mxminpoly + - integral_div + - size_diagA + - minpoly_mx_ring + - mxminpoly_uconj + - algebraic_opp + - algebraic_id + - codiagonalizablePfull + - mx_poly_ring_isom + - map_rVpoly + - integral1 + - size_mxminpoly + - mxminpoly_map + - horner_mx_stable + - comm_mx_stable_geigenspace + - integral_inv + - map_powers_mx + - integral_mul + - integral_root_monic + - eigenspace_sub_geigen + - conjMumx + - comm_mx_horner + - rVpoly_delta + - stablemx_comp + - eval_row_var + - minpoly_mx1 + - char_poly_trig + - eval_mxrank + - eigenvalue_poly + - char_poly_det + - poly_rV_is_linear + - algebraic_mul + - diagonalizable_for_row_base + - mxminpoly_monic + - horner_mx_X + - mxminpoly_minP + - simmx_minpoly + - conjuMmx + - integral_poly + - map_mx_inv_horner + - Cayley_Hamilton + - integral_id + - diagonalizable_forPp + - eigenpoly_conjmx + - degree_mxminpoly_map + - mxminpoly_nonconstant + - eval_vec_mx + - conjVmx + - Sylvester_mxE + - kermxpoly_prod + - mxdirect_kermxpoly + - mxminpoly_diag + - simmxLR + - eval_submx + - conjmx0 + - kermxpoly_min + - minpoly_mxM + - integral_algebraic + - nth_seq_of_rV + - eigenvalue_root_char + - mx_inv_hornerK + - mxdirect_sum_kermx + - map_poly_rV + - horner_mx_uconj + - eigenspace_poly + - simmxPp + - eval_mx_term + - codiagonalizableP + - horner_rVpoly + - simmxRL + - diagonalizable_forP + - dvd_mxminpoly + - diagonalizable_conj_diag + - kermxpoly1 + - XsubC0 + - resultant_in_ideal + - algebraic1 + - diagonalizable_forLR + - mxrank_form_qf + - horner_mxZ + - horner_mxK + - conjmxVK + - path: mathcomp/field/closed_field.v + theorems: + - qf_cps_if + - eval_amulXnT + - rgdcop_recT_qf + - holds_ex_elim + - redivp_rec_loopP + - rgcdp_loopT_qf + - redivpTP + - rgcdpTP + - rgdcopTP + - abstrX1 + - rgcdpTsP + - rgcdpT_qf + - rseq_poly_map + - isnull_qf + - rpoly_map_mul + - rgdcop_recTP + - redivp_rec_loopT_qf + - sizeTP + - abstrXP + - holds_conjn + - rsumpT + - eval_poly_mulM + - lead_coefTP + - countable_algebraic_closure + - rgdcopT_qf + - redivp_rec_loopTP + - eval_lift + - ex_elim_seqP + - ex_elim_seq_qf + - eval_poly1 + - wf_ex_elim + - redivpT_qf + - eval_opppT + - holds_conj + - qf_simpl + - eval_mulpT + - isnullP + - countable_field_extension + - ramulXnT + - rabstrX + - lead_coefT_qf + - rgcdpTs_qf + - qf_cps_ret + - abstrX_mulM + - qf_cps_bind + - eval_sumpT + - rgcdp_loopP + - sizeT_qf + - eval_natmulpT + - path: mathcomp/fingroup/automorphism.v + theorems: + - Aut_conj_aut + - Aut_morphic + - Aut_isomM + - char_norm_trans + - conj_isom + - char_normal + - Aut_aut + - eq_Aut + - im_Aut_isom + - injm_char + - im_autm + - conj_autE + - perm_in_inj + - perm_inE + - Aut_Aut_isom + - Aut_isomP + - char_injm + - conj_aut_morphM + - charI + - morphim_conj + - Aut_closed + - autmE + - imset_autE + - perm_in_on + - lone_subgroup_char + - char_refl + - char_norm + - conjgmE + - charP + - Aut_isom_subproof + - char_sub + - injm_Aut_isom + - morphim_fixP + - char_norms + - Aut1 + - preim_autE + - ker_conj_aut + - char_trans + - injm_autm + - charM + - Aut_isomE + - norm_conjg_im + - out_Aut + - norm_conj_isom + - norm_conj_autE + - char_normal_trans + - aut_closed + - conj_isog + - path: mathcomp/ssreflect/fingraph.v + theorems: + - same_connect + - order_gt0 + - finv_inv + - fconnect_invariant + - iter_findex + - eq_n_comp_r + - connect_closed + - eq_fcard + - predC_closed + - eq_order_cycle + - injectivePcycle + - fconnect_cycle + - fconnect1 + - fpath_finv_cycle + - fpath_finv_in + - size_orbit + - finv_in + - connect_cycle + - connect_sub + - fcycle_consEflatten + - fcycle_consE + - orbit_uniq + - subset_dfs + - same_fconnect_finv + - fcard_id + - connect_trans + - rgraphK + - eq_roots + - fclosed1 + - order_finv + - image_orbit + - fconnect_finv + - same_fconnect1 + - eq_n_comp + - iter_finv_in + - eq_root + - orbit_id + - closure_closed + - fconnect_id + - f_finv_in + - eq_connect0 + - fcard_order_set + - orbitPcycle + - root_root + - subset_closure + - order_cycle + - finv_inj_cycle + - froots_id + - prevE + - connect1 + - looping_order + - mem_orbit + - fpath_f_finv_cycle + - froot_id + - finv_bij + - finv_cycle + - adjunction_closed + - fpath_finv + - undup_cycle_cons + - intro_closed + - cycle_orbit + - connectP + - fpath_finv_f_cycle + - path_connect + - fconnect_f + - strict_adjunction + - cycle_orbit_cycle + - eq_fconnect + - in_orbit_cycle + - fcycleEflatten + - fconnect_iter + - root_connect + - n_comp_connect + - iter_order + - finv_f_in + - iter_order_cycle + - n_comp_closure2 + - same_connect_r + - same_connect1r + - fconnect_findex + - order_id_cycle + - closed_connect + - fcard_gt0P + - dfsP + - intro_adjunction + - orbitE + - fconnect_orbit + - same_connect1 + - findex_eq0 + - n_compC + - findex_max + - connect_root + - f_finv + - finv_inj + - orderPcycle + - mem_closure + - iter_finv_cycle + - same_connect_rev + - f_finv_cycle + - fconnect_sym + - eq_finv + - findex_iter + - iter_order_in + - fconnect_eqVf + - dfs_pathP + - rootP + - fcard_gt1P + - in_orbit + - order_le_cycle + - eq_connect + - fconnect_sym_in + - same_fconnect1_r + - finv_f + - fpath_f_finv_in + - fpath_finv_f_in + - iter_finv + - connect_rev + - fcard_finv + - fcycle_rconsE + - orderSpred + - path: mathcomp/algebra/ssrint.v + theorems: + - nmulrz_rlt0 + - exprz_pintl + - mulr0z + - distn_eq1 + - rpredMz + - sgz_odd + - ltr_piXz2l + - ltr1z + - ler_int + - pmulrz_llt0 + - exprzD_ss + - NegzE + - distn_eq0 + - scalerMzr + - lez_total + - nonzero1z + - mulzn_eq1 + - mulrz_le0 + - intr_norm + - abszMsign + - ltr_int + - sgzX + - ltr_nXz2r + - expfz_eq0 + - abszN1 + - ler_pMz2l + - ltr_pXz2r + - mulrzAC + - distnn + - ltr0_sgz + - oppzK + - rmorphMz + - mulr_absz + - exprSzr + - commr_int + - intrV + - fmorphXz + - exprzMzl + - PoszD + - lerz0 + - mul2z + - eqz_nat + - subSz1 + - natr_absz + - exprnP + - ler_wpXz2r + - pmulrz_lgt0 + - mulrbz + - mulrz_nat + - ltz1D + - Frobenius_aut_int + - mulrz_suml + - rpredZint + - realz + - commrXz + - nmulrz_rgt0 + - ffunMzE + - rmorphXz + - unitr_n0expz + - derivMz + - mulz_Nsign_abs + - pexprz_eq1 + - is_intE + - leqifD_distz + - linearMn + - abszM + - normr_sgz + - natz + - sgz_le0 + - sgzP + - ltNz_nat + - ler_wpMz2l + - sgz_smul + - exprzD_Nnat + - invz_out + - absz_eq0 + - lez_abs + - lez0_abs + - expfz_neq0 + - mulr1z + - nmulrz_lge0 + - horner_int + - oppzD + - distnEl + - sgrMz + - ler_niXz2l + - commrMz + - intr_sign + - mulrz_int + - mul0rz + - invr_expz + - raddfMz + - mulNrNz + - abszX + - lez_anti + - sgz_sgr + - mulrzDr_tmp + - gtz0_ge1 + - mulr2z + - ltr_eXz2l + - ler_wnXz2r + - ler_wpiXz2l + - is_natE + - exprMz_comm + - ltzD1 + - truncP + - Znat_def + - expfzMl + - ler_nMz2r + - intr_sg + - ler_weXz2l + - mulzA + - mulrz_neq0 + - rmorphzP + - distn0 + - mulz0 + - absz1 + - pexpIrz + - ieexprIz + - Frobenius_autMz + - absz0 + - exprnN + - mul0z + - mulrzr + - mulrz_le0_ge0 + - addNz + - exprzMl + - lez1D + - exprz_pMzl + - abszEsg + - ltr_pMz2r + - ler_eXz2l + - distSn + - lez_nat + - intS + - lez_mul + - abszN + - ltr_nMz2l + - sgz_def + - mulrzA_C + - intmul1_is_multiplicative + - pmulrz_rle0 + - nmulrz_lgt0 + - eqr_int + - leqD_dist + - ltzN_nat + - rpredXz + - expNrz + - lerz1 + - lez_add + - ler1z + - ltz_nat + - ltrz1 + - le0z_nat + - sgz_eq0 + - unitrXz + - eqrXz2 + - ler0z + - scalezrE + - distnS + - nmulrz_rle0 + - addzC + - mulrzz + - ltr0z + - polyCMz + - mulNrz + - unitzPl + - mulrzAl + - mulpz + - intr_eq0 + - raddf_int_scalable + - abszE + - natsum_of_intK + - PoszM + - mulzC + - normzN + - sgz_gt0 + - mulz_sign_abs + - absz_gt0 + - mulrz_ge0 + - exp1rz + - sgrEz + - mulrzBr + - sgz_eq + - sgz1 + - prodMz + - nmulrn + - nmulrz_llt0 + - sgz_cp0 + - mulz_addl + - gez0_norm + - distnC + - exprN1 + - sumMz + - intrM + - mulNz + - mulrz_ge0_le0 + - distnDl + - pmulrz_lle0 + - normrMz + - ler_wpMz2r + - mulz_sg + - expr0z + - rpred_int + - pmulrz_rgt0 + - sgz_ge0 + - intz + - absz_sign + - subz_ge0 + - commrXz_wmulls + - int_rect + - leNz_nat + - intP + - exprzDr + - exprz_inv + - scalerMzl + - mulrNz + - intEsg + - mulrzBl + - intrB + - mulz_sg_eq1 + - ler_wniXz2l + - ltr_nMz2r + - sgzN + - abszEsign + - subzSS + - lez0_nat + - ZnatP + - ltz0_abs + - sgz_int + - ler_nMz2l + - ltr_niXz2l + - ler_pXz2r + - mulrzDl_tmp + - mulzN + - lezD1 + - mul1z + - expfz_n0addr + - mulrz_sumr + - mulrzA + - sgzM + - expfzDr + - nmulrz_rge0 + - idomain_axiomz + - gtr0_sgz + - sgrz + - add1Pz + - add0z + - subzn + - ler_wnMz2r + - predn_int + - mulVz + - ler_wneXz2l + - ltrz0 + - pmulrz_rge0 + - mulrIz + - pmulrz_lge0 + - int_rect + - ler_wpeXz2l + - dist0n + - mulrzl + - intrD + - expr1z + - sgz_id + - exprzAC + - exprz_gt0 + - predn_int + - intrN + - ler_nXz2r + - NegzE + - expfV + - nmulrz_lle0 + - ltz_def + - gez0_abs + - mulrzAr + - addzA + - exprzD_nat + - mulz_sg_eqN1 + - absz_sg + - sgz_lt0 + - normr_sg + - ler_piXz2l + - exprz_ge0 + - pmulrn + - exprz_exp + - sgz0 + - addPz + - absz_nat + - hornerMz + - intEsign + - rmorph_int + - absz_id + - lezN_nat + - gtz0_abs + - path: mathcomp/algebra/vector.v + theorems: + - vsof_sub + - limg_line + - capv_idPl + - addvA + - lfun_vect_iso + - vsproj_key + - cat_basis + - limg_dim_eq + - b2mxK + - memv_span1 + - vsprojK + - vs2mxI + - span_lfunP + - dimv_add_leqif + - gen_vs2mx + - lker0_compfK + - lfun_img_key + - eq_limg_ker0 + - add_lfunE + - span_subvP + - memv_cap + - funmx_linear + - memv0 + - vsvalK + - comp_lfun0r + - lpreimK + - subv_anti + - memvB + - freeP + - span_def + - free_cons + - lfun_key + - lker0_compVKf + - subvsP + - mem0v + - congr_subvs + - mxof_comp + - SubvsE + - rVof_sub + - diffvSl + - capv_compl + - coord_free + - dimv_sum_leqif + - memvN + - lpreimS + - mxof1 + - memv_line + - comp_lfunDr + - memvD + - directv_addP + - v2r_inj + - fixedSpace_id + - subv_sumP + - comp_lfunNr + - nil_free + - limg_cap + - dim_vline + - sumv_pi_sum + - fixedSpace_limg + - limgE + - coord_sum_free + - basis_free + - ffun_vect_iso + - span_seq1 + - capfv + - subvP + - vecof_delta + - addv_pi2_proj + - msofK + - basis_not0 + - span_key + - subv0 + - lfun_scale1 + - hommxE + - addv0 + - directv_sumE + - basisEdim + - subvv + - coord_vbasis + - vlineP + - lker_proj + - vecof_eq0 + - lfun_addA + - addvS + - capv_idPr + - memvf + - rVof_linear + - vecof_linear + - subvs_vect_iso + - lker0_lfunK + - addv_pi1_pi2 + - v2rK + - capvv + - dimv_cap_compl + - coord_is_scalar + - span_nil + - directv_addE + - mul_b2mx + - memv_ker + - lim1g + - leigenspaceE + - limgD + - memv_suml + - lfun_is_linear + - vsof_eq0 + - mxof_eq0 + - projv_id + - nil_basis + - capv_diff + - subvPn + - sumv_pi_uniq_sum + - basisEfree + - dimv_leqif_eq + - limg_span + - comp_lfun1r + - comp_lfunZl + - scale_lfunE + - hommx_linear + - rVof_mul + - vsproj_is_linear + - span_b2mx + - rVof_app + - bigcat_basis + - addvv + - catr_free + - lker0_compfVK + - perm_free + - dimvf + - subvf + - msof_sub + - fixedSpacesP + - vs2mxK + - mx2vs_subproof + - addv_diff_cap + - subv_bigcapP + - bigcat_free + - addv_complf + - vecof_mul + - addvC + - vs2mxF + - dimvS + - vs2mxD + - addv_pi2_id + - msof0 + - eqEdim + - vs2mx0 + - free_b2mx + - hommx_eq0 + - dimv_eq0 + - eq_in_limg + - vspaceP + - add0v + - sum_lfunE + - lkerE + - comp_lfunZr + - filter_free + - directvP + - hommxK + - directvEgeq + - subvs_inj + - mxofK + - span_cons + - subv_trans + - limg_ker_compl + - limg_ker0 + - vs2mx_sum_expr_subproof + - comp_lfunA + - vsolve_eqP + - vspace_modl + - bigcapv_inf + - memv_sumP + - lker0_limgf + - matrix_vect_iso + - directv_trivial + - sub_vsof + - lpreim0 + - directv_add_unique + - projv_proj + - limg_basis_of + - daddv_pi_id + - vbasis_mem + - memv_projC + - comp_lfun1l + - free_uniq + - lker_ker + - opp_lfunE + - inv_lfun_def + - binary_addv_subproof + - lpreim_cap_limg + - limg_sum + - subv_cap + - vecofK + - subv_add + - size_basis + - mxof_linear + - eqEsubv + - nary_addv_subproof + - memv_sumr + - lim0g + - lfun_scaleDr + - sub0v + - sumfv + - memv_pi2 + - rVof_eq0 + - lfun1_neq0 + - lker0_lfunVK + - addv_idPl + - directv_sum_unique + - addv_diff + - pair_vect_iso + - limgS + - capvf + - coord_vecof + - capvSr + - limg_lfunVK + - memv_sum_pi + - id_lfunE + - lfun_scaleA + - basis_mem + - coord_rVof + - memv_span + - r2v_inj + - vpick0 + - rVofK + - memv_proj + - dimv_leqif_sup + - memv_addP + - addvSr + - sumv_pi_nat_sum + - linear_of_free + - r2vK + - limg_ker_dim + - daddv_pi_proj + - free_not0 + - memv_submod_closed + - lfunE + - lker0_compKf + - vbasisP + - lfun_add0 + - capvC + - hom_vecof + - directv_sum_independent + - sub_msof + - dimv_disjoint_sum + - vsof0 + - mem_vecof + - memvZ + - cap0v + - r2v_subproof + - limg_proj + - free_directv + - directv_sumP + - capv0 + - addv_pi1_proj + - memv_preim + - subset_limgP + - dimv_compl + - comp_lfunE + - rVofE + - memv_img + - addvSl + - coord_basis + - lker0_compVf + - fullv_lfunP + - regular_vect_iso + - coord0 + - dimv_sum_cap + - directvE + - memv_pi + - addvf + - limg0 + - fixedSpaceP + - dim_matrix + - memvE + - lfun_scaleDl + - lfunPn + - lin_b2mx + - v2r_subproof + - addv_idPr + - capvS + - memv_pi1 + - limg_bigcap + - eq_span + - hommx_mul + - vsval_is_linear + - coord_span + - row_b2mx + - catl_free + - vsofK + - memv_imgP + - lker0P + - limg_comp + - eqlfun_inP + - msof_eq0 + - eqlfunP + - perm_basis + - dimv_leq_sum + - freeE + - mx2vsK + - diffv_eq0 + - mem_r2v + - lker0_img_cap + - freeNE + - memv_pick + - span_cat + - daddv_pi_add + - path: mathcomp/ssreflect/path.v + theorems: + - suffix_sorted + - e'_e + - eq_in_path + - undup_sorted + - cycle_path + - homo_sorted_in + - homo_path_in + - all_sort + - mem2_map + - sort_stable_in + - cycle_from_next + - order_path_min_in + - filter_sort + - nextE + - loopingP + - mono_cycle_in + - sub_in_cycle + - merge_uniq + - path_relI + - take_sorted + - map_merge + - perm_sort_inP + - pop_stable + - size_traject + - sorted_leq_nth + - cat_path + - merge_stable_sorted + - rcons_path + - e_e' + - take_traject + - sorted_mask_in + - inj_cycle + - cycle_from_prev + - right_arc + - sort_pairwise_stable + - mem2_last + - sub_in_path + - perm_sortP + - subseq_sorted + - count_sort + - sorted_filter + - homo_path + - sub_in_sorted + - mem_next + - mem2_seq1 + - subseq_sort_in + - homo_sort_map_in + - eq_in_cycle + - nth_traject + - rev_path + - mem_fcycle + - leElex + - path_filter_in + - next_prev + - mem_sort + - sorted_ltn_nth_in + - mem2_cat + - mono_sorted + - next_nth + - merge_path + - all_merge + - path_mask_in + - homo_sorted + - prefix_sorted + - ucycle_uniq + - path_sorted + - looping_uniq + - sort_map + - rev_sorted + - mem_prev + - mono_path + - path_le + - irr_sorted_eq + - homo_cycle_in + - homo_cycle + - eq_cycle + - leT_tr' + - infix_sorted + - prev_nth + - cycle_map + - pathP + - sorted_mask_sort_in + - prev_rot + - sort_uniq + - eq_in_sorted + - perm_sort + - perm_iota_sort + - sorted_uniq + - sorted_sort_in + - mono_sorted_in + - ucycle_cycle + - pairwise_sorted + - subseq_sort + - sortedP + - map_sort + - merge_sorted + - mem2r + - path_pairwise + - size_sort + - cycle_catC + - sub_path + - path_filter + - undup_path + - pairwise_sort + - rot_cycle + - splitP + - cat_sorted2 + - prev_rev + - prev_next + - merge_map + - fpath_traject + - traject_iteri + - eq_path + - sort_sorted + - count_merge + - mono_cycle + - sorted_ltn_index + - mem2_splice1 + - size_merge + - sort_sorted_in + - path_sortedE + - prev_map + - path_map + - subseq_path_in + - sorted_filter_in + - sorted_uniq_in + - path_sorted_inE + - size_merge_sort_push + - sorted_relI + - sorted_merge + - fpathE + - sorted_leq_nth_in + - eq_fcycle + - sorted_subseq_sort + - mask_sort_in + - merge_stable_path + - cycle_all2rel + - irr_sorted_eq_in + - next_cycle + - subseq_sorted_in + - mem2_sort + - mem2lf + - mono_path_in + - sorted_pairwise_in + - eq_count_merge + - next_rev + - sorted_leq_index_in + - next_rotr + - mem2_cons + - mem2l_cat + - last_traject + - prefix_path + - mask_sort + - trajectSr + - trajectP + - path_pairwise_in + - fpathP + - sub_cycle + - left_arc + - sorted_leq_index + - mergeA + - path_mask + - splitPl + - sort_iota_stable + - sorted_ltn_nth + - cycle_all2rel_in + - next_rot + - mem2l + - sorted_eq + - eq_sorted + - sortE + - iota_ltn_sorted + - mem2_splice + - sub_sorted + - rot_ucycle + - sorted_pairwise + - cycle_next + - order_path_min + - next_map + - sorted_mask_sort + - ltn_sorted_uniq_leq + - filter_sort_in + - sorted_mask + - push_stable + - rev_cycle + - trajectD + - mem2_sort_in + - prev_cycle + - path: mathcomp/field/finfield.v + theorems: + - finDomain_mulrC + - card_finField_unit + - order_primeChar + - card_finCharP + - finField_galois_generator + - primeChar_scaleDl + - natrFp + - Fermat's_little_theorem + - finField_galois + - primeChar_dimf + - lregR + - ffT_splitting_subproof + - galLgen + - expf_card + - finRing_gt1 + - card_primeChar + - card_vspacef + - card_vspace + - primeChar_scaleDr + - finField_is_abelem + - primeChar_pgroup + - pr_p + - FinSplittingFieldFor + - card_vspace1 + - finField_genPoly + - primeChar_vectAxiom + - PrimePowerField + - galL + - primeChar_scaleAr + - primeChar_scaleA + - finDomain_field + - finCharP + - path: mathcomp/solvable/gseries.v + theorems: + - quotient_subnormal + - subnormalP + - quotient_simple + - normal_subnormal + - subnormalEsupport + - setI_subnormal + - cosetpre_maximal + - isog_simple + - invariant_subnormal + - maximal_exists + - maxnormal_minnormal + - maximalJ + - mulg_normal_maximal + - subnormal_refl + - subnormal_trans + - central_central_factor + - cosetpre_maximal_eq + - injm_maxnormal + - maxnormal_normal + - simple_maxnormal + - chief_series_exists + - maxnormal_sub + - quotient_maximal_eq + - injm_minnormal + - subnormal_sub + - path_setIgr + - morphim_subnormal + - injm_maximal_eq + - maximal_eqP + - subnormalEl + - chief_factor_minnormal + - maxnormalM + - central_factor_central + - injm_maximal + - ex_maxnormal_ntrivg + - maxnormal_proper + - acts_irrQ + - path: mathcomp/algebra/zmodp.v + theorems: + - Fp_nat_mod + - add_1_Zp + - char_Fp_0 + - Zp_addC + - unitZpE + - Fp_fieldMixin + - add_N1_Zp + - Zp_nat + - Zp1_expgz + - rshift1 + - card_Fp + - card_Zp + - Zp_nontrivial + - Zp_mul1z + - val_Fp_nat + - valZpK + - split1 + - order_Zp1 + - Zp_inv_out + - Zp_addA + - Zp_intro_unit + - natr_Zp + - Zp_cycle + - char_Zp + - Zp_mulA + - Zp_mul_addl + - Zp_mul_addr + - modZp + - unitFpE + - Zp_mulz1 + - card_units_Zp + - unit_Zp_expg + - Zp_expg + - units_Zp_abelian + - Zp_add0z + - Zp_mulgC + - add_Zp_1 + - Zp_mulrn + - Zp_mulC + - lshift0 + - Zp_nat_mod + - val_Zp_nat + - char_Fp + - Zp_cast + - Zp_mulzV + - mem_Zp + - ord1 + - Zp_addNz + - natr_negZp + - path: mathcomp/character/integral_char.v + theorems: + - mxZn_inj + - Burnside_p_a_q_b + - group_num_field_exists + - faithful_degree_p_part + - gring_class_sum_central + - gring_classM_coef_sum_eq + - nonlinear_irr_vanish + - gring_mode_class_sum_eq + - Aint_char + - mx_irr_gring_op_center_scalar + - index_support_dvd_degree + - dvd_irr1_index_center + - Aint_irr + - gring_classM_expansion + - Aint_gring_mode_class_sum + - coprime_degree_support_cfcenter + - sum_norm2_char_generators + - set_gring_classM_coef + - cfRepr_gring_center + - gring_irr_modeM + - Aint_class_div_irr1 + - dvd_irr1_cardG + - primes_class_simple_gt1 + - path: mathcomp/ssreflect/prime.v + theorems: + - mem_primes + - logn_gt0 + - primeNsig + - pfactorKpdiv + - sub_in_partn + - up_log_gt0 + - partnC + - prime_nt_dvdP + - primePns + - pdiv_gt0 + - partn_lcm + - dvdn_partP + - ltn_log0 + - primeP + - up_logMp + - ltn_logl + - Euclid_dvdM + - pfactor_gt0 + - max_pdiv_dvd + - partn1 + - elogn2P + - pdiv_leq + - trunc_log_eq + - Euclid_dvdX + - p_natP + - up_log_min + - totient_pfactor + - partn0 + - p'natEpi + - Euclid_dvd1 + - trunc_log1 + - pi_pnat + - partn_biggcd + - pi_pdiv + - logn_count_dvd + - primes_part + - p_part_eq1 + - logn_lcm + - pi_p'nat + - pnat_div + - eq_partn_from_log + - up_log_eq0 + - up_log_bounds + - odd_prime_gt2 + - part_p'nat + - pnat_pi + - sub_pnat_coprime + - pi_of_dvd + - pnatX + - sorted_divisors_ltn + - ifnzP + - sorted_primes + - p'natE + - prime_oddPn + - p'nat_coprime + - prime_decompE + - logn_prime + - primes_prime + - pnatI + - filter_pi_of + - up_log_trunc_log + - prime_gt0 + - coprime_has_primes + - pfactorK + - eq_in_pnat + - divisors_uniq + - widen_partn + - mem_prime_decomp + - logn_coprime + - part_pnat_id + - part_gt0 + - eq_partn + - trunc_expnK + - trunc_log1n + - all_prime_primes + - primesM + - trunc_logMp + - eq_negn + - pi'_p'nat + - pi_max_pdiv + - coprime_pi' + - pdiv_dvd + - divisors_correct + - trunc_lognn + - dvdn_pfactor + - totientE + - pi_of_exp + - leq_trunc_log + - up_expnK + - up_log0 + - prime_coprime + - up_log2S + - logn0 + - trunc_log_gt0 + - logn_part + - dvdn_sum + - p_part_gt1 + - partnM + - coprime_partC + - pnat_1 + - pfactor_dvdn + - logn_gcd + - primePn + - modn_partP + - max_pdiv_prime + - pnatPpi + - logn1 + - max_pdiv_gt0 + - pnat_dvd + - p_part + - up_lognn + - prime_above + - max_pdiv_max + - negnK + - lognE + - trunc_log_eq0 + - prime_gt1 + - pnatNK + - partn_eq1 + - pi_ofM + - trunc_log0n + - partnNK + - pnatE + - prime_decomp_correct + - up_log_gtn + - trunc_log2_double + - partn_gcd + - up_log_eq + - trunc_log0 + - trunc_log2S + - dvdn_part + - edivn2P + - divisor1 + - odd_2'nat + - pnat_coprime + - pdivP + - primes_eq0 + - primes_uniq + - trunc_log_bounds + - even_prime + - eqn_from_log + - leq_up_log + - eq_pnat + - pnatP + - partn_dvd + - pi_of_part + - lognX + - pnat_id + - eq_piP + - pdiv_prime + - divisors_id + - pfactor_coprime + - partnT + - ltn_pdiv2_prime + - up_log1 + - pi_of_prime + - sub_in_pnat + - totient_coprime + - up_logP + - eq_primes + - sorted_divisors + - partn_pi + - partnI + - partn_biglcm + - totient_gt1 + - dvdn_leq_log + - trunc_log_max + - prod_prime_decomp + - logn_Gauss + - totient_gt0 + - pdiv_min_dvd + - pfactor_dvdnn + - part_pnat + - dvdn_divisors + - path: mathcomp/character/mxabelem.v + theorems: + - rVabelemN + - GLmx_faithful + - mx_group_homocyclic + - rowg_mxS + - faithful_repr_extraspecial + - abelem_mx_faithful + - GL_mx_repr + - abelem_rV_S + - abelem_rV_X + - rstabs_abelem + - pcore_faithful_mx_irr + - comp_reprGLm + - rowg_mx1 + - im_abelem_rV + - rfix_pgroup_char + - scale_actE + - abelian_type_mx_group + - card_rowg + - rVabelem_minj + - astab_rowg_repr + - rVabelemD + - mxsimple_abelemP + - mxrank_rowg + - sub_rVabelem + - abelem_rV_V + - exponent_mx_group + - mx_repr_is_groupAction + - abelem_rV_injm + - rowg_mx_eq0 + - pcore_sub_rstab_mxsimple + - eq_rowg + - card_rVabelem + - abelem_mx_irrP + - val_reprGLm + - rowgS + - rker_abelem + - rVabelem_injm + - im_rVabelem + - rfix_abelem + - rVabelem0 + - mxmodule_abelem_subg + - rowg0 + - rowg_stable + - afix_repr + - astab_setT_repr + - scale_is_groupAction + - abelem_rV_isom + - mxsimple_abelem_subg + - p_pr + - gacent_repr + - mx_repr_is_action + - abelem_rV_K + - dim_abelemE + - extraspecial_repr_structure + - rowgI + - eq_abelem_subg_repr + - rVabelemK + - mx_Fp_stable + - abelem_rV_inj + - dprod_rowg + - astab1_scale_act + - mx_Fp_abelem + - stable_rowg_mxK + - card_abelem_rV + - rowgK + - abelem_rV_M + - mx_repr_actE + - rowg_mxSK + - rVabelem_inj + - abelem_rV_1 + - abelem_rowgJ + - reprGLmM + - sub_abelem_rV_im + - bigdprod_rowg + - mxmodule_abelemG + - abelem_mx_linear_proof + - rVabelem_mK + - rVabelemS + - sub_rVabelem_im + - cprod_rowg + - rowgD + - mem_rowg + - rsim_abelem_subg + - mxmodule_abelem + - mxsimple_abelemGP + - trivg_rowg + - abelem_rV_J + - sub_im_abelem_rV + - isog_abelem_rV + - modIp' + - abelem_mx_repr + - ker_reprGLm + - abelem_rV_mK + - bigcprod_rowg + - pcore_sub_rker_mx_irr + - rstab_abelem + - acts_rowg + - mem_rVabelem + - rstabs_abelemG + - rank_mx_group + - rVabelemJ + - mem_im_abelem_rV + - rowg_mxK + - scale_is_action + - rV_abelem_sJ + - rVabelemZ + - path: mathcomp/character/vcharacter.v + theorems: + - zchar_split + - zchar_onS + - dirr_constt_oppr + - dirr_dchi + - dirr_norm1 + - cfnorm_map_orthonormal + - irr_constt_to_dirr + - Aint_vchar + - cfdot_sum_orthonormal + - Z_S + - ndirr_inj + - Zisometry_of_cfnorm + - dirr_opp + - cfdot_dirr_eq1 + - mul_vchar + - char_vchar + - isometry_in_zchar + - vchar_mulr_closed + - zchar_trans + - zchar_small_norm + - notS0 + - cfdot_add_dirr_eq1 + - cfdot_dirr + - ndirrK + - dchi_ndirrE + - cfnorm_orthonormal + - sub_aut_zchar + - dirr_dIirrE + - cnorm_dconstt + - dirr_constt_oppl + - zcharW + - dIrrP + - cfInd_vchar + - Cnat_cfnorm_vchar + - dirrP + - cfdot_sum_orthogonal + - zchar_on + - cfproj_sum_orthonormal + - zcharD1 + - Zchar_zmod + - cfnorm_sum_orthogonal + - dchi_vchar + - cfnorm_sum_orthonormal + - zchar_tuple_expansion + - zchar_nth_expansion + - map_pairwise_orthogonal + - cfdot_dchi + - dirr_constt_oppI + - zchar_filter + - zchar_span + - orthonormal_span + - Cnat_dirr + - dirr_small_norm + - dirrE + - cfAut_vchar + - cfRes_vchar + - scale_zchar + - vchar_norm1P + - Zisometry_inj + - vchar_aut + - cfdot_sum_dchi + - conjC_vcharAut + - zchar_subseq + - cfproj_sum_orthogonal + - dirr_aut + - support_zchar + - dirr_consttE + - zchar_onG + - dirr_sign + - Zisometry_of_iso + - dirr_dIirrPE + - cfdot_vchar_r + - mem_zchar_on + - cfun0_zchar + - irr_dirr + - sub_conjC_vchar + - Frobenius_kernel_exists + - dchi1 + - map_orthonormal + - vchar_orthonormalP + - dirr_oppr_closed + - ndirr_diff + - irr_vchar + - cfun_sum_dconstt + - cfRes_vchar_on + - cfAut_zchar + - zchar_trans_on + - cfdot_todirrE + - Cint_cfdot_vchar_irr + - of_irrK + - to_dirrK + - cfdot_aut_vchar + - Cint_cfdot_vchar + - nS1 + - Cint_vchar1 + - zchar_sub_irr + - zchar_expansion + - cfnorm_orthogonal + - irr_vchar_on + - path: mathcomp/solvable/burnside_app.v + theorems: + - R50_inj + - F_Sv + - F_r034 + - Fid3 + - is_isoP + - r41_inv + - F_r32 + - dir_s0p + - burnside_app_iso_2_4col + - card_Fid + - F_s6 + - r14_inv + - act_f_morph + - F_r012 + - act_g_morph + - group_set_iso3 + - F_r013 + - card_n4 + - iso_eq_F0_F1_F2 + - R021_inj + - rot_eq_c0 + - s14 + - sd2_inv + - iso0_1 + - R32_inj + - Lcorrect + - card_n + - burnside_app_rot + - prod_t_correct + - ecubes_def + - r3_inv + - S2_inv + - F_r021 + - rot_is_rot + - F_Sh + - card_Fid3 + - isometries_iso + - r2_inv + - F_r05 + - burnside_app_iso + - sd1_inv + - act_g_1 + - s23_inv + - R043_inj + - R042_inj + - R1_inj + - group_set_diso3 + - R14_inj + - R013_inj + - Sh_inj + - F_r3 + - card_n3s + - S5_inv + - R2_inj + - R024_inj + - Sv_inj + - card_n2 + - group_set_iso + - r1_inv + - r50_inv + - F_s05 + - eqperm + - S0_inv + - S14_inj + - sop_inj + - R012_inj + - group_set_rot + - sop_spec + - Sd2_inj + - burnside_app2 + - F_r14 + - R031_inj + - card_n2_3 + - sv_inv + - F_r042 + - iso3_ndir + - seqs1 + - F_r41 + - rotations_is_rot + - group_set_iso2 + - F_r23 + - F_Sd2 + - F_Sd1 + - iso_eq_F0_F1 + - act_f_1 + - F_s1 + - F_s2 + - F_r2 + - F_s5 + - card_n3_3 + - L_iso + - stable + - group_set_rotations + - F_r024 + - dir_iso_iso3 + - R23_inj + - S4_inv + - uniq4_uniq6 + - R41_inj + - S6_inv + - R3_inj + - R05_inj + - F_r031 + - card_n3 + - burnside_app_iso_3_3col + - F_s4 + - eqperm_map + - F_r043 + - F_s3 + - F_r1 + - R034_inj + - Fid + - gen_diso3 + - ndir_s0p + - is_iso3P + - burnside_app_iso3 + - card_rot + - F_r50 + - card_iso2 + - diff_id_sh + - F_s14 + - ord_enum4 + - burnside_formula + - sop_morph + - path: mathcomp/algebra/finalg.v + theorems: + - unit_is_groupAction + - mulrV + - unit_actE + - zmod_mulgC + - val_unitV + - zmodXgE + - zmodVgE + - invr_out + - mulVr + - unit_mul_proof + - unit_mul1u + - unit_muluA + - decidable + - card_finRing_gt1 + - card_finField_unit + - zmod1gE + - val_unit1 + - intro_unit + - val_unitX + - val_unitM + - unit_mulVu + - zmodMgE + - path: mathcomp/ssreflect/binomial.v + theorems: + - bin2_sum + - binS + - fermat_little + - card_partial_ord_partitions + - ffact_small + - cards_draws + - bin_gt0 + - bin0 + - bin_sub + - binSn + - ffactnS + - bin2 + - bin_ffact + - bin1 + - prime_modn_expSn + - leq_bin2l + - bin_small + - mul_bin_left + - prime_dvd_bin + - ffact_factd + - binn + - modn_summ + - predn_exp + - bin2odd + - Wilson + - dvdn_pred_predX + - card_uniq_tuples + - mul_bin_down + - subn_exp + - ffactE + - ffact_fact + - bin_ffactd + - card_ltn_sorted_tuples + - expnDn + - logn_fact + - card_sorted_tuples + - card_ord_partitions + - ffactn1 + - ffact0n + - ffactnn + - Vandermonde + - binE + - bin0n + - card_inj_ffuns + - ffact_prod + - ffactn0 + - bin_fact + - card_inj_ffuns_on + - card_draws + - fact_prod + - mul_bin_diag + - path: mathcomp/ssreflect/div.v + theorems: + - dvdn_lcm + - divnMA + - lcmnAC + - gcdnMDl + - lcmn_gt0 + - divn1 + - divn0 + - edivnB + - modn1 + - divnK + - chinese_modr + - divnn + - modnMDl + - dvdn1 + - modnDmr + - coprime_dvdl + - gcdn_gt0 + - modn0 + - dvdnn + - gcdnMr + - dvdnP + - modnDml + - divn2 + - coprimen1 + - leq_div2r + - divnDl + - modn_small + - dvd1n + - gcdnC + - divnAC + - dvdn_add_eq + - dvdn_addr + - edivn_pred + - leq_mod + - gcdn_idPl + - modnD + - egcd0n + - edivn_eq + - gcdnDl + - divn_gt0 + - Bezoutl + - gcdnAC + - ltn_ceil + - Gauss_dvdl + - modn_pred + - dvdn_mull + - lcmnMl + - dvdn_exp2r + - dvdn_gcd + - leq_div + - coprimeXl + - gcdnDr + - gcdn_modr + - Gauss_dvd + - gcdnA + - dvdn_exp + - modnMl + - dvdn_gcdr + - gcdn0 + - divn_eq + - chinese_modl + - ltn_divRL + - divnDMl + - coprimeXr + - modnDl + - gcdnCA + - divnB + - dvdn_addl + - coprime_pexpr + - coprimen2 + - mulKn + - modn2 + - dvdn_pexp2r + - gcdnACA + - dvdn_gcdl + - gcd1n + - coprime_modr + - lcmn_idPr + - Gauss_gcdl + - ltn_Pdiv + - modnS + - lcmn_idPl + - leq_divDl + - dvdn_add + - gcdn_idPr + - leqDmod + - dvd0n + - expnB + - dvdn_pmul2l + - divnMBl + - lcmnCA + - dvdn_Pexp2l + - mod0n + - dvdn_subl + - geq_divBl + - eqn_modDr + - muln_lcm_gcd + - coprimeP + - coprime_dvdr + - modn_mod + - divn_modl + - coprimenS + - edivnS + - modn_def + - dvdn_mul + - dvdn_fact + - modnMml + - coprimeMl + - gcdn_def + - dvdn_odd + - divnMr + - coprimeMr + - expn_max + - muln_gcdr + - coprimeSn + - divn_pred + - dvdn_double_leq + - muln_divCA + - lcmnA + - modn_coprime + - muln_modr + - dvdn_exp2l + - lcmnACA + - coprime1n + - gcdnE + - modnMr + - edivn_def + - divn_small + - dvdn_pmul2r + - lcmn1 + - divnA + - leq_divLR + - dvdn_div + - dvdn_divRL + - muln_divA + - edivnD + - egcdnP + - eqn_mul + - coprimenP + - modnDr + - dvdn_gt0 + - modnMm + - gcd0n + - gcdn_modl + - leq_div2l + - coprime_modl + - coprime2n + - odd_mod + - divnDr + - modnn + - dvdn_double_ltn + - lcmnMr + - divnMl + - divnD + - lcm0n + - muln_divCA_gcd + - ltn_pmod + - muln_lcmr + - Gauss_gcdr + - divn_mulAC + - muln_gcdl + - muln_modl + - Bezoutr + - divnBMl + - lcmn0 + - gtnNdvd + - expn_min + - dvdn_leq + - gcdnMl + - eqn_dvd + - lcm1n + - chinese_mod + - dvdn2 + - chinese_remainder + - modnDm + - dvdn_trans + - modn_divl + - ltn_divLR + - div0n + - muln_lcml + - coprimePn + - coprime_egcdn + - ltn_mod + - dvdn_divLR + - Gauss_dvdr + - dvdn_mulr + - divnBl + - mulnK + - gcdnn + - divnS + - gcdn1 + - divnMDl + - path: mathcomp/algebra/interval.v + theorems: + - mem0_itvoo_xNx + - mid_in_itvcc + - BInfty_leE + - le_bound_refl + - itv_bound_can + - BRight_BLeft_leE + - itv_splitI + - oppr_itvcc + - subset_itv + - itv_meetA + - bound_lex1 + - subitvPl + - subitvP + - BLeft_ltE + - BInfty_le_eqE + - miditv_ge_right + - itv_splitU + - BLeft_BRight_ltE + - BInfty_BInfty_ltE + - itv_meetUl + - bound_meetA + - subset_itv_oo_cc + - itv_le0x + - in_segmentDgt0Pr + - in_segmentDgt0Pl + - miditv_le_left + - bound_lexx + - subitvPr + - bound_joinA + - mid_in_itv + - itv_ge + - itv_dec + - BInfty_geE + - mem_miditv + - subset_itv_co_cc + - ge_pinfty + - BInfty_gtF + - itv_meetKU + - itv_total_meet3E + - bound_leEmeet + - itvxx + - in_itv + - itvP + - oppr_itvoo + - bound_meetC + - bound_meetKU + - leBRight_ltBLeft + - lteif_in_itv + - subitv_trans + - BInfty_ltF + - boundr_in_itv + - lt_ninfty + - ltBSide + - subitvE + - predC_itv + - bound_le0x + - boundl_in_itv + - gt_pinfty + - bound_joinKI + - BLeft_BSide_leE + - BInfty_ltE + - predC_itvr + - itv_joinA + - ltBRight_leBLeft + - bound_ltxx + - itv_bound_total + - itv_splitU1 + - BSide_ltE + - lt_in_itv + - oppr_itvoc + - itv_bound_display + - in_itvI + - bound_joinC + - oppr_itv + - BInfty_gtE + - itv_splitUeq + - mid_in_itvoo + - BSide_leE + - subitv_anti + - itv_boundlr + - subset_itv_oo_oc + - itvxxP + - predC_itvl + - subset_itv_oc_cc + - lt_bound_def + - itv_lex1 + - BInfty_ge_eqE + - subset_itv_oo_co + - le_bound_anti + - itv_meetC + - interval_can + - itv_total_join3E + - itv_joinC + - BRight_leE + - BRight_BSide_ltE + - mem0_itvcc_xNx + - itv_xx + - subitv_refl + - interval_display + - path: mathcomp/solvable/center.v + theorems: + - center_class_formula + - xcprodmI + - xcprodmEl + - cprod_by_uniq + - subcentP + - ker_cprod_by_central + - subcent1_cycle_sub + - subcent1_id + - injm_cpairg1 + - xcprodm_cent + - im_cpair_cprod + - ncprod1 + - xcprodP + - center_bigdprod + - injm_cpair1g + - sub_center_normal + - injm_xcprodm + - xcprodmE + - cprod_center_id + - im_cpair_cent + - subcent1C + - cpairg1_dom + - center_bigcprod + - center_cprod + - ncprodS + - cpair1g_dom + - xcprod_subproof + - subcent_norm + - injgz + - subcent_sub + - cpair1g_center + - isog_xcprod + - center_normal + - Aut_cprod_by_full + - cpairg1_center + - ker_cprod_by_is_group + - gzZchar + - im_xcprodm + - center_char + - center_idP + - isog_cprod_by + - Aut_cprod_full + - subcent1_cycle_normal + - center_ncprod0 + - centerP + - cprod_by_key + - morphim_center + - subcent_normal + - center_abelian + - center1 + - subcent1_cycle_norm + - in_cprodM + - subcent1_sub + - xcprodmEr + - cyclic_center_factor_abelian + - cpair_center_id + - im_cpair + - isog_center + - eq_cpairZ + - gzZ_lone + - ncprod0 + - injm_center + - ker_in_cprod + - im_xcprodml + - centerC + - subcent1P + - subcent_char + - ncprod_key + - cyclic_factor_abelian + - Aut_ncprod_full + - setI_im_cpair + - gzZ + - im_xcprodmr + - path: mathcomp/solvable/jordanholder.v + theorems: + - maxainv_norm + - qacts_coset + - maxainvM + - asimpleP + - section_reprP + - asimpleI + - StrongJordanHolderUniqueness + - simple_compsP + - maxainvS + - maxainv_exists + - trivg_acomps + - asimple_acompsP + - qacts_cosetpre + - maxainv_ainvar + - section_repr_isog + - gactsM + - compsP + - exists_comps + - qact_dom_doms + - maxainv_sub + - asimple_quo_maxainv + - acomps_cons + - trivg_comps + - acts_qact_doms + - exists_acomps + - comps_cons + - path: mathcomp/solvable/commutator.v + theorems: + - derJ + - commXg + - quotient_cents2 + - sub_der1_abelian + - commg_subr + - dergS + - commgV + - normsRr + - commXXg + - commgMJ + - commg_normal + - normsRl + - commMgR + - der_normalS + - derg1 + - commg_normr + - der_abelian + - commgAC + - Hall_Witt_identity + - quotient_der + - sub_der1_norm + - commg_sub + - commG1 + - commg_norm + - commg_normSr + - commg_norml + - sub_der1_normal + - der1_min + - der_sub + - commg_normSl + - comm1G + - commMGr + - commVg + - charR + - der_normal + - expMg_Rmul + - commgMR + - der_cont + - morphim_der + - der_norm + - comm_norm_cent_cent + - der_group_set + - commMgJ + - dergSn + - conjg_mulR + - commg_subl + - commg_subI + - three_subgroup + - der1_joing_cycles + - derG1P + - conjg_Rmul + - path: mathcomp/ssreflect/finfun.v + theorems: + - card_pfamily + - tnth_fgraph + - ffunK + - card_ffun + - ffunE + - eq_dffun + - supportP + - FinfunK + - tuple_of_finfunK + - nth_fgraph_ord + - tfgraph_inj + - ffunP + - codom_ffun + - fgraphK + - tagged_tfgraph + - fgraph_codom + - familyP + - pffun_onP + - card_dep_ffun + - finfun_of_tupleK + - pfamilyP + - ffun_onP + - card_ffun_on + - card_family + - card_pffun_on + - eq_ffun + - codom_tffun + - fgraph_ffun0 + - tfgraphK + - ffun0 + - path: mathcomp/ssreflect/ssrfun.v + theorems: + - eq_omap + - inj_omap + - omapK + - omap_id + - path: mathcomp/ssreflect/choice.v + theorems: + - chooseP + - pair_of_tagK + - ltn_code + - seq_of_optK + - nat_pickleK + - gtn_decode + - codeK + - bool_of_unitK + - pickle_invK + - xchooseP + - pickleK_inv + - sigW + - eq_xchoose + - pickle_seqK + - sig_eqW + - PCanHasChoice + - nat_hasChoice + - opair_of_sumK + - codeK + - eq_choose + - tag_of_pairK + - sig2_eqW + - decodeK + - pcan_pickleK + - xchoose_subproof + - path: mathcomp/algebra/polyXY.v + theorems: + - map_div_annihilantP + - swapXYK + - swapXY_is_multiplicative + - size_poly_XaY + - max_size_lead_coefXY + - swapXY_comp_poly + - horner_polyC + - swapXY_poly_XaY + - swapXY_map + - horner_poly_XaY + - max_size_evalC + - div_annihilant_in_ideal + - lead_coef_poly_XaY + - sizeY_mulX + - poly_XaY_eq0 + - swapXY_key + - root_annihilant + - max_size_evalX + - poly_XmY0 + - coef_swapXY + - swapXY_map_polyC + - swapXY_polyC + - horner2_swapXY + - swapXY_eq0 + - sizeY_eq0 + - algebraic_root_polyXY + - horner_swapXY + - div_annihilant_neq0 + - size_poly_XmY + - swapXY_Y + - swapXY_poly_XmY + - map_sub_annihilantP + - poly_XmY_eq0 + - horner_poly_XmY + - sub_annihilant_neq0 + - div_annihilantP + - sub_annihilantP + - swapXY_is_additive + - sub_annihilant_in_ideal + - path: mathcomp/algebra/ring_quotient.v + theorems: + - rquot_IdomainAxiom + - nonzero1q + - idealMr + - mulqC + - idealr_closed_nontrivial + - idealr1 + - pi_is_multiplicative + - idealrDE + - addqC + - addNq + - idealr_closedB + - pi_is_additive + - pi_opp + - addqA + - mul1q + - idealrBE + - equivE + - pi_mul + - idealr0 + - mulq_addl + - add0q + - pi_add + - path: mathcomp/solvable/hall.v + theorems: + - strongest_coprime_quotient_cent + - coprime_Hall_trans + - ext_coprime_quotient_cent + - quotient_TI_subcent + - coprime_cent_mulG + - Hall_exists_subJ + - SchurZassenhaus_trans_actsol + - Hall_superset + - sol_coprime_Sylow_subset + - external_action_im_coprime + - ext_norm_conj_cent + - Hall_subJ + - SchurZassenhaus_split + - Hall_exists + - coprime_norm_cent + - ext_coprime_Hall_exists + - coprime_Hall_subset + - Hall_Frattini_arg + - sol_coprime_Sylow_trans + - norm_conj_cent + - coprime_norm_quotient_cent + - sol_coprime_Sylow_exists + - SchurZassenhaus_trans_sol + - ext_coprime_Hall_subset + - Hall_trans + - ext_coprime_Hall_trans + - coprime_Hall_exists + - path: mathcomp/ssreflect/ssrAC.v + theorems: + - serial_Op + - set_pos_trecE + - cforallP + - pos_set_pos + - proof + - path: mathcomp/field/cyclotomic.v + theorems: + - prod_cyclotomic + - size_cyclotomic + - Cintr_Cyclotomic + - root_cyclotomic + - Cyclotomic_monic + - C_prim_root_exists + - minCpoly_cyclotomic + - prod_Cyclotomic + - cyclotomic_monic + - separable_Xn_sub_1 + - size_Cyclotomic + - Cyclotomic0 + - path: mathcomp/solvable/primitive_action.v + theorems: + - n_act0 + - stab_ntransitive + - dtuple_on_add_D1 + - ntransitive_weak + - ntransitive1 + - n_act_add + - ntransitive_primitive + - trans_prim_astab + - ntransitive0 + - n_act_dtuple + - dtuple_on_add + - stab_ntransitiveI + - prim_trans_norm + - dtuple_on_subset + - dtuple_onP + - card_uniq_tuple + - path: mathcomp/ssreflect/ssrbool.v + theorems: + - if_add + - classic_ex + - classic_sigW + - if_or + - if_implybC + - if_and + - relpre_trans + - homo_mono1 + - path: mathcomp/field/algebraics_fundamentals.v + theorems: + - rat_algebraic_archimedean + - minPoly_decidable_closure + - rat_algebraic_decidable + - Fundamental_Theorem_of_Algebraics + - alg_integral diff --git a/multilingual/eval/local.meta.json b/multilingual/eval/local.meta.json new file mode 100644 index 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