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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h2 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u)
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h2 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2 : ¬isBoundIn r P_u ∧ ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | case intro
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
left✝ : ¬isBoundIn r P_u
right✝ : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h3 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬isBoundIn s (P_u.imp_ Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h3 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h3 : ¬(isBoundIn s P_u ∨ isBoundIn s Q_u)
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h3 | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3 : ¬isBoundIn s P_u ∧ ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | case intro
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
left✝ : ¬isBoundIn s P_u
right✝ : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | specialize h1_ih_1 h2_left h3_left | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | specialize h1_ih_2 h2_right h3_right | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ ((eq_ r s).imp_ (Q_u.iff_ Q_v)) | P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))) | case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))
case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ ((eq_ r s).imp_ (P_u.iff_ P_v)) | case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ (((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))) | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))))
case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_iff_] | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ (P_u.iff_ P_v)).imp_
(((eq_ r s).imp_ (Q_u.iff_ Q_v)).imp_ ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v))))) | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)))))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [def_and_] | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).and_ (Q_v.imp_ Q_u))).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).and_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)))))) | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | SC | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅
(((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_
(((eq_ r s).imp_ ((Q_u.imp_ Q_v).imp_ (Q_v.imp_ Q_u).not_).not_).imp_
((eq_ r s).imp_ (((P_u.imp_ Q_u).imp_ (P_v.imp_ Q_v)).imp_ ((P_v.imp_ Q_v).imp_ (P_u.imp_ Q_u)).not_).not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih_1 | case a.a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih_2 | case a
P_r P_s : Formula
r s : VarName
P_u Q_u P_v Q_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_2 : IsReplOfVarInFormula r s Q_u Q_v
h2_left : ¬isBoundIn r P_u
h2_right : ¬isBoundIn r Q_u
h3_left : ¬isBoundIn s P_u
h3_right : ¬isBoundIn s Q_u
h1_ih_1 : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h1_ih_2 : IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v))
⊢ IsDeduct ∅ ((eq_ r s).imp_ (Q_u.iff_ Q_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h2 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬isBoundIn r (forall_ x P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h2 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2 : ¬(r = x ∨ isBoundIn r P_u)
h3 : ¬isBoundIn s (forall_ x P_u)
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h2 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2 : r ≠ x ∧ ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case intro
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
left✝ : r ≠ x
right✝ : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isBoundIn] at h3 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬isBoundIn s (forall_ x P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg at h3 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h3 : ¬(s = x ∨ isBoundIn s P_u)
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | cases h3 | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3 : s ≠ x ∧ ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case intro
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
left✝ : s ≠ x
right✝ : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply deduction_theorem | P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct (∅ ∪ {eq_ r s}) ((forall_ x P_u).iff_ (forall_ x P_v)) | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (forall_ x (P_u.iff_ P_v)) | case h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x P_u).iff_ (forall_ x P_v)) | case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v)))
case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply proof_imp_deduct | case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply T_18_1 | case h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((forall_ x (P_u.iff_ P_v)).imp_ ((forall_ x P_u).iff_ (forall_ x P_v))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (eq_ r s) | case h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (forall_ x (P_u.iff_ P_v)) | case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))
case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply proof_imp_deduct | case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))) | case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.mp_ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))) | case h1.a.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsProof ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))) | case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v))))
case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply T_19_TS_21_left | case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))).imp_ ((eq_ r s).imp_ (forall_ x (P_u.iff_ P_v)))) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [isFreeIn] | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬isFreeIn x (eq_ r s) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | push_neg | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ¬(x = r ∨ x = s) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | constructor | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r ∧ x ≠ s | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r
case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [ne_comm] | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ r | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h2_left | case h1.a.a.h1.a.h1.left
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ r ≠ x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp only [ne_comm] | case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ x ≠ s | case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h3_left | case h1.a.a.h1.a.h1.right
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ s ≠ x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply generalization | case h1.a.a.h1.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ (forall_ x ((eq_ r s).imp_ (P_u.iff_ P_v))) | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | exact h1_ih h2_right h3_right | case h1.a.a.h1.a.h1
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | intro H a1 | case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ ∀ H ∈ ∅, ¬isFreeIn x H | case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp at a1 | case h1.a.a.h1.a.h2
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | apply IsDeduct.assume_ | case h1.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ IsDeduct {eq_ r s} (eq_ r s) | case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | simp | case h1.a.a.a
P_r P_s : Formula
r s x : VarName
P_u P_v : Formula
h1_1 : IsReplOfVarInFormula r s P_u P_v
h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v))
h2_left : r ≠ x
h2_right : ¬isBoundIn r P_u
h3_left : s ≠ x
h3_right : ¬isBoundIn s P_u
⊢ eq_ r s ∈ {eq_ r s} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_21_8 | [1404, 1] | [1543, 10] | sorry | case exists_
P_r P_s : Formula
r s x✝ : VarName
P_u✝ P_v✝ : Formula
a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝
a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝))
h2 : ¬isBoundIn r (exists_ x✝ P_u✝)
h3 : ¬isBoundIn s (exists_ x✝ P_u✝)
⊢ IsProof ((eq_ r s).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | induction F | F : Formula
τ : PredName → PredName
h1 : F.predVarSet = ∅
⊢ sub τ F = F | case pred_const_
τ : PredName → PredName
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
τ : PredName → PredName
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
τ : PredName → PredName
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
τ : PredName → PredName
h1 : true_.predVarSet = ∅
⊢ sub τ true_ = true_
case false_
τ : PredName → PredName
h1 : false_.predVarSet = ∅
⊢ sub τ false_ = false_
case not_
τ : PredName → PredName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : a✝.not_.predVarSet = ∅
⊢ sub τ a✝.not_ = a✝.not_
case imp_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.imp_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.and_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.or_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
τ : PredName → PredName
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → sub τ a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (a✝¹.iff_ a✝).predVarSet = ∅
⊢ sub τ (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
τ : PredName → PredName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (forall_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
τ : PredName → PredName
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → sub τ a✝ = a✝
h1 : (exists_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
τ : PredName → PredName
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).predVarSet = ∅
⊢ sub τ (def_ a✝¹ a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case pred_const_ X xs =>
simp only [sub] | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ sub τ (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case pred_var_ X xs =>
simp only [predVarSet] at h1
simp at h1 | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case eq_ x y =>
simp only [sub] | τ : PredName → PredName
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ sub τ (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case true_ | false_ =>
simp only [sub] | τ : PredName → PredName
h1 : false_.predVarSet = ∅
⊢ sub τ false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case not_ phi phi_ih =>
simp only [predVarSet] at h1
simp only [sub]
congr!
exact phi_ih h1 | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [predVarSet] at h1
simp only [sub]
congr!
exact phi_ih h1 | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | case def_ X xs =>
simp only [sub] | τ : PredName → PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ sub τ (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ sub τ (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp at h1 | τ : PredName → PredName
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ sub τ (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ sub τ (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
h1 : false_.predVarSet = ∅
⊢ sub τ false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi.not_ = phi.not_ | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ (sub τ phi).not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ (sub τ phi).not_ = phi.not_ | case h.e'_1
τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1 | case h.e'_1
τ : PredName → PredName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : (phi.iff_ psi).predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [Finset.union_eq_empty] at h1 | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | cases h1 | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | case intro
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
left✝ : phi.predVarSet = ∅
right✝ : psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ (phi.iff_ psi) = phi.iff_ psi | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (sub τ phi).iff_ (sub τ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (sub τ phi).iff_ (sub τ psi) = phi.iff_ psi | case h.e'_1
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ phi = phi
case h.e'_2
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1_left | case h.e'_1
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact psi_ih h1_right | case h.e'_2
τ : PredName → PredName
phi psi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
psi_ih : psi.predVarSet = ∅ → sub τ psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ sub τ psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [predVarSet] at h1 | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ (exists_ x phi) = exists_ x phi | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (sub τ phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | congr! | τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (sub τ phi) = exists_ x phi | case h.e'_2
τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | exact phi_ih h1 | case h.e'_2
τ : PredName → PredName
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → sub τ phi = phi
h1 : phi.predVarSet = ∅
⊢ sub τ phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.sub_no_predVar | [47, 1] | [91, 20] | simp only [sub] | τ : PredName → PredName
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ sub τ (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | induction E generalizing F V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
τ : PredName → PredName
F : Formula
⊢ Holds D I V E (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds }
V E F | case nil
D : Type
I : Interpretation D
τ : PredName → PredName
V : VarAssignment D
F : Formula
⊢ Holds D I V [] (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] F
case cons
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
F : Formula
⊢ Holds D I V (head✝ :: tail✝) (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case nil.def_ X xs =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case cons.def_ hd tl ih X xs =>
simp only [Holds] at ih
simp at ih
simp only [sub]
simp only [Holds]
split_ifs
case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q τ hd.h2] at ih
apply ih
case _ c1 =>
specialize ih V (def_ X xs)
simp only [sub] at ih
exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tl F
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | induction F generalizing V | case cons
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
F : Formula
⊢ Holds D I V (head✝ :: tail✝) (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) F | case cons.pred_const_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_const_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ a✝¹ a✝)
case cons.pred_var_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_var_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ a✝¹ a✝)
case cons.eq_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (eq_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ a✝¹ a✝)
case cons.true_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ true_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) true_
case cons.false_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ false_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_
case cons.not_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ a✝.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝.not_
case cons.imp_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.imp_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.imp_ a✝)
case cons.and_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.and_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.and_ a✝)
case cons.or_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.or_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.or_ a✝)
case cons.iff_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝¹) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝¹
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (a✝¹.iff_ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (a✝¹.iff_ a✝)
case cons.forall_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (forall_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (forall_ a✝¹ a✝)
case cons.exists_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ a✝) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) a✝
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ a✝¹ a✝)
case cons.def_
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (def_ a✝¹ a✝)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pred_const_ X xs =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_const_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pred_var_ X xs =>
simp only [sub]
split_ifs
case pos c1 =>
simp only [Holds]
simp
simp only [if_pos c1]
case neg c1 =>
simp only [Holds]
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_var_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case eq_ x y =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (eq_ x y)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case true_ | false_ =>
simp only [sub]
simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ false_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case not_ phi phi_ih =>
simp only [Holds] at phi_ih
simp only [sub]
simp only [Holds]
congr! 1
apply phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [Holds] at phi_ih
simp only [sub]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intros d
apply phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_const_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_const_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (pred_var_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (if xs = [] then pred_var_ (τ X) [] else pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (if xs = [] then pred_var_ (τ X) [] else pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | case pos
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
h✝ : xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ (τ X) []) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs)
case neg
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
h✝ : ¬xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case pos c1 =>
simp only [Holds]
simp
simp only [if_pos c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ (τ X) []) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case neg c1 =>
simp only [Holds]
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ (τ X) []) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) (List.map V []) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) (List.map V []) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) [] ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [if_pos c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : xs = []
⊢ I.pred_var_ (τ X) [] ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (pred_var_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔
if List.map V xs = [] then I.pred_var_ (τ X) (List.map V []) else I.pred_var_ X (List.map V xs) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [if_neg c1] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
X : PredName
xs : List VarName
V : VarAssignment D
c1 : ¬xs = []
⊢ I.pred_var_ X (List.map V xs) ↔ if xs = [] then I.pred_var_ (τ X) [] else I.pred_var_ X (List.map V xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (eq_ x y)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (eq_ x y) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x y : VarName
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (eq_ x y) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (eq_ x y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ false_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) false_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) false_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi.not_) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi).not_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ |
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