url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | induction e.epsilon_arrow_list <;> simp [*, epsilon_arrow_list_to_graph, or_and_right, exists_or] | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔
∃ stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } ∈ e.epsilon_arrow_list ∧ c ∈ stop_state_list | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
⊢ ((∃ x, (b = head✝.start_state ∧ x = head✝.stop_state_list) ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list) ↔
(∃ x, { start_state := b, stop_state_list := x } = head✝ ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | apply or_congr_left <| exists_congr fun a => and_congr_left fun _ => ?_ | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
⊢ ((∃ x, (b = head✝.start_state ∧ x = head✝.stop_state_list) ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list) ↔
(∃ x, { start_state := b, stop_state_list := x } = head✝ ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a✝ b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
a : List σ
x✝ : c ∈ a
⊢ b = head✝.start_state ∧ a = head✝.stop_state_list ↔ { start_state := b, stop_state_list := a } = head✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | constructor <;> [rintro ⟨rfl, rfl⟩; rintro rfl] <;> [rfl; constructor <;> rfl] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a✝ b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
a : List σ
x✝ : c ∈ a
⊢ b = head✝.start_state ∧ a = head✝.stop_state_list ↔ { start_state := b, stop_state_list := a } = head✝ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | simp [epsilon_closure_iff] | case nil
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s ∈ M.epsilon_closure S, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s [] | case nil
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) ↔
∃ s ∈ S, M.toAbstract.eval s [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | constructor | case nil
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) ↔
∃ s ∈ S, M.toAbstract.eval s [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) →
∃ s ∈ S, M.toAbstract.eval s []
case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s []) →
∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s, ⟨s', h1, h2⟩, h3⟩ | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) →
∃ s ∈ S, M.toAbstract.eval s [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ ∃ s ∈ S, M.toAbstract.eval s [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | refine ⟨_, h1, ?_⟩ | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ ∃ s ∈ S, M.toAbstract.eval s [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 using Relation.ReflTransGen.head_induction_on with
| refl => exact .accept _ h3
| head h _ ih => exact .eps _ _ _ ih h | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact .accept _ h3 | case nil.mp.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact .eps _ _ _ ih h | case nil.mp.head
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h3 : s ∈ M.accepting_state_list
a✝ c✝ : σ
h : M.toAbstract.epsilon a✝ c✝
h✝ : Relation.ReflTransGen M.toAbstract.epsilon c✝ s
ih : M.toAbstract.eval c✝ []
⊢ M.toAbstract.eval a✝ [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s, h1, h2⟩ | case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s []) →
∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list | case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | obtain ⟨s', h3, h4⟩ : ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list := by
clear h1; generalize e : [] = l at h2
induction h2 with cases e
| accept _ h' => exact ⟨_, .refl, h'⟩
| eps _ _ _ _ h2 ih =>
have ⟨s', h3, h4⟩ := ih rfl
exact ⟨_, .head h2 h3, h4⟩ | case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list | case nil.mpr.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
s' : σ
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, ⟨_, h1, h3⟩, h4⟩ | case nil.mpr.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
s' : σ
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h2 : M.toAbstract.eval s []
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | generalize e : [] = l at h2 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h2 : M.toAbstract.eval s []
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
e : [] = l
h2 : M.toAbstract.eval s l
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 with cases e
| accept _ h' => exact ⟨_, .refl, h'⟩
| eps _ _ _ _ h2 ih =>
have ⟨s', h3, h4⟩ := ih rfl
exact ⟨_, .head h2 h3, h4⟩ | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
e : [] = l
h2 : M.toAbstract.eval s l
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, .refl, h'⟩ | case accept.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ : σ
h' : M.toAbstract.accepting start_state✝
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | have ⟨s', h3, h4⟩ := ih rfl | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
a✝ : M.toAbstract.eval stop_state✝ []
ih : [] = [] → ∃ s', M.toAbstract.EpsilonClosure stop_state✝ s' ∧ s' ∈ M.accepting_state_list
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
a✝ : M.toAbstract.eval stop_state✝ []
ih : [] = [] → ∃ s', M.toAbstract.EpsilonClosure stop_state✝ s' ∧ s' ∈ M.accepting_state_list
s' : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, .head h2 h3, h4⟩ | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
a✝ : M.toAbstract.eval stop_state✝ []
ih : [] = [] → ∃ s', M.toAbstract.EpsilonClosure stop_state✝ s' ∧ s' ∈ M.accepting_state_list
s' : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | simp [IH, epsilon_closure_iff, eval_one_no_eps_iff] | case cons
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s ∈ M.eval_from (M.eval_one_no_eps (M.epsilon_closure S) a) as, s ∈ M.accepting_state_list) ↔
∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) ↔
∃ s ∈ S, M.toAbstract.eval s (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | constructor | case cons
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) ↔
∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) →
∃ s ∈ S, M.toAbstract.eval s (a :: as)
case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s (a :: as)) →
∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s₁, ⟨s₂, ⟨s₃, h1, h2⟩, h3⟩, h4⟩ | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) →
∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ ∃ s ∈ S, M.toAbstract.eval s (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | refine ⟨_, h1, ?_⟩ | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ ∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 using Relation.ReflTransGen.head_induction_on with
| refl => {apply AbstractEpsilonNFA.eval.sym _ _ _ _ h4 h3; }
| head h _ ih => exact .eps _ _ _ ih h | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | apply AbstractEpsilonNFA.eval.sym _ _ _ _ h4 h3 | case cons.mp.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₂ (a :: as) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact .eps _ _ _ ih h | case cons.mp.head
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
a✝ c✝ : σ
h : M.toAbstract.epsilon a✝ c✝
h✝ : Relation.ReflTransGen M.toAbstract.epsilon c✝ s₂
ih : M.toAbstract.eval c✝ (a :: as)
⊢ M.toAbstract.eval a✝ (a :: as) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s, h1, h2⟩ | case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s (a :: as)) →
∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as | case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | obtain ⟨s₁, s₂, h3, h4, h5⟩ :
∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧
M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as := by
clear h1; generalize e : a::as = l at h2
induction h2 with cases e
| sym _ _ _ _ h1 h2 => exact ⟨_, _, .refl, h2, h1⟩
| eps _ _ _ h1 h2 ih =>
have ⟨s₁, s₂, h3, h4, h5⟩ := ih rfl
exact ⟨_, _, .head h2 h3, h4, h5⟩ | case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as | case cons.mpr.intro.intro.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure s s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, ⟨_, ⟨_, h1, h3⟩, h4⟩, h5⟩ | case cons.mpr.intro.intro.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure s s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | generalize e : a::as = l at h2 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
e : a :: as = l
h2 : M.toAbstract.eval s l
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 with cases e
| sym _ _ _ _ h1 h2 => exact ⟨_, _, .refl, h2, h1⟩
| eps _ _ _ h1 h2 ih =>
have ⟨s₁, s₂, h3, h4, h5⟩ := ih rfl
exact ⟨_, _, .head h2 h3, h4, h5⟩ | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
e : a :: as = l
h2 : M.toAbstract.eval s l
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, _, .refl, h2, h1⟩ | case sym.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.symbol start_state✝ a stop_state✝
h1 : M.toAbstract.eval stop_state✝ as
a_ih✝ :
a :: as = as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | have ⟨s₁, s₂, h3, h4, h5⟩ := ih rfl | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
h1 : M.toAbstract.eval stop_state✝ (a :: as)
ih :
a :: as = a :: as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
h1 : M.toAbstract.eval stop_state✝ (a :: as)
ih :
a :: as = a :: as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, _, .head h2 h3, h4, h5⟩ | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
h1 : M.toAbstract.eval stop_state✝ (a :: as)
ih :
a :: as = a :: as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.accepts_iff | [457, 1] | [468, 8] | simp [accepts, eval] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ e.accepts input ↔ e.toAbstract.accepts input | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.eval_from e.starting_state_list input, s ∈ e.accepting_state_list) ↔ e.toAbstract.accepts input |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.accepts_iff | [457, 1] | [468, 8] | rw [EpsilonNFA.eval_from_iff] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.eval_from e.starting_state_list input, s ∈ e.accepting_state_list) ↔ e.toAbstract.accepts input | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.starting_state_list, e.toAbstract.eval s input) ↔ e.toAbstract.accepts input |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.accepts_iff | [457, 1] | [468, 8] | rfl | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.starting_state_list, e.toAbstract.eval s input) ↔ e.toAbstract.accepts input | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | induction xs | α : Type
f : α → α
xs : List α
⊢ map f xs = xs ↔ ∀ x ∈ xs, f x = x | case nil
α : Type
f : α → α
⊢ map f [] = [] ↔ ∀ x ∈ [], f x = x
case cons
α : Type
f : α → α
head✝ : α
tail✝ : List α
tail_ih✝ : map f tail✝ = tail✝ ↔ ∀ x ∈ tail✝, f x = x
⊢ map f (head✝ :: tail✝) = head✝ :: tail✝ ↔ ∀ x ∈ head✝ :: tail✝, f x = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | case nil =>
simp | α : Type
f : α → α
⊢ map f [] = [] ↔ ∀ x ∈ [], f x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | case cons hd tl ih =>
simp
intro _
exact ih | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ map f (hd :: tl) = hd :: tl ↔ ∀ x ∈ hd :: tl, f x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | simp | α : Type
f : α → α
⊢ map f [] = [] ↔ ∀ x ∈ [], f x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | simp | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ map f (hd :: tl) = hd :: tl ↔ ∀ x ∈ hd :: tl, f x = x | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ f hd = hd → (map f tl = tl ↔ ∀ a ∈ tl, f a = a) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | intro _ | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
⊢ f hd = hd → (map f tl = tl ↔ ∀ a ∈ tl, f a = a) | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
a✝ : f hd = hd
⊢ map f tl = tl ↔ ∀ a ∈ tl, f a = a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_eq_self_iff | [7, 1] | [20, 13] | exact ih | α : Type
f : α → α
hd : α
tl : List α
ih : map f tl = tl ↔ ∀ x ∈ tl, f x = x
a✝ : f hd = hd
⊢ map f tl = tl ↔ ∀ a ∈ tl, f a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | induction xs | α : Type
f : α → α
xs : List α
h1 : ∀ x ∈ xs, f x = x
⊢ map f xs = xs | case nil
α : Type
f : α → α
h1 : ∀ x ∈ [], f x = x
⊢ map f [] = []
case cons
α : Type
f : α → α
head✝ : α
tail✝ : List α
tail_ih✝ : (∀ x ∈ tail✝, f x = x) → map f tail✝ = tail✝
h1 : ∀ x ∈ head✝ :: tail✝, f x = x
⊢ map f (head✝ :: tail✝) = head✝ :: tail✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | case nil =>
simp | α : Type
f : α → α
h1 : ∀ x ∈ [], f x = x
⊢ map f [] = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | simp | α : Type
f : α → α
h1 : ∀ x ∈ [], f x = x
⊢ map f [] = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | simp at h1 | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1 : ∀ x ∈ hd :: tl, f x = x
⊢ map f (hd :: tl) = hd :: tl | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1 : f hd = hd ∧ ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | cases h1 | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1 : f hd = hd ∧ ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl | case intro
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
left✝ : f hd = hd
right✝ : ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | simp | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ map f (hd :: tl) = hd :: tl | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd ∧ map f tl = tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | constructor | α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd ∧ map f tl = tl | case left
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd
case right
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ map f tl = tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | exact h1_left | case left
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ f hd = hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.map_mem_id | [23, 1] | [40, 26] | exact ih h1_right | case right
α : Type
f : α → α
hd : α
tl : List α
ih : (∀ x ∈ tl, f x = x) → map f tl = tl
h1_left : f hd = hd
h1_right : ∀ a ∈ tl, f a = a
⊢ map f tl = tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | induction xs generalizing ys | α : Type
inst✝ : DecidableEq α
xs ys : List α
h1 : xs.length = ys.length
h2 : xs.Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f xs ∧ map f xs = ys | case nil
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f [] ∧ map f [] = ys
case cons
α : Type
inst✝ : DecidableEq α
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (ys : List α), tail✝.length = ys.length → tail✝.Nodup → ys.Nodup → ∃ f, InjOn f tail✝ ∧ map f tail✝ = ys
ys : List α
h1 : (head✝ :: tail✝).length = ys.length
h2 : (head✝ :: tail✝).Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f (head✝ :: tail✝) ∧ map f (head✝ :: tail✝) = ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | have s1 : ys = [] | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f [] ∧ map f [] = ys | case s1
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys = []
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | {
apply List.eq_nil_of_length_eq_zero
simp only [← h1]
simp
} | case s1
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys = []
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [s1] | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = ys | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply Exists.intro id | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∃ f, InjOn f [] ∧ map f [] = [] | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id [] ∧ map id [] = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | constructor | α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id [] ∧ map id [] = [] | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id []
case right
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ map id [] = [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply List.eq_nil_of_length_eq_zero | case s1
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys = [] | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys.length = 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [← h1] | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ ys.length = 0 | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ [].length = 0 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | case s1.x
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
⊢ [].length = 0 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [List.InjOn] | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ InjOn id [] | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∀ ⦃x₁ : α⦄, x₁ ∈ [] → ∀ ⦃x₂ : α⦄, x₂ ∈ [] → id x₁ = id x₂ → x₁ = x₂ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | case left
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ ∀ ⦃x₁ : α⦄, x₁ ∈ [] → ∀ ⦃x₂ : α⦄, x₂ ∈ [] → id x₁ = id x₂ → x₁ = x₂ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | case right
α : Type
inst✝ : DecidableEq α
ys : List α
h1 : [].length = ys.length
h2 : [].Nodup
h3 : ys.Nodup
s1 : ys = []
⊢ map id [] = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases ys | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys : List α
h1 : (xs_hd :: xs_tl).length = ys.length
h2 : (xs_hd :: xs_tl).Nodup
h3 : ys.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys | case nil
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
h1 : (xs_hd :: xs_tl).length = [].length
h3 : [].Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = []
case cons
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
head✝ : α
tail✝ : List α
h1 : (xs_hd :: xs_tl).length = (head✝ :: tail✝).length
h3 : (head✝ :: tail✝).Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = head✝ :: tail✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case nil =>
simp at h1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
h1 : (xs_hd :: xs_tl).length = [].length
h3 : [].Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
h1 : (xs_hd :: xs_tl).length = [].length
h3 : [].Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
ys_hd : α
ys_tl : List α
h1 : (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length
h3 : (ys_hd :: ys_tl).Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
h2 : (xs_hd :: xs_tl).Nodup
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at h3 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h3 : (ys_hd :: ys_tl).Nodup
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases h2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2 : xs_hd ∉ xs_tl ∧ xs_tl.Nodup
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | case intro
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
left✝ : xs_hd ∉ xs_tl
right✝ : xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases h3 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h3 : ys_hd ∉ ys_tl ∧ ys_tl.Nodup
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | case intro
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
left✝ : ys_hd ∉ ys_tl
right✝ : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ map f (xs_hd :: xs_tl) = ys_hd :: ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | specialize xs_ih ys_tl h1 h2_right h3_right | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List α), xs_tl.length = ys.length → xs_tl.Nodup → ys.Nodup → ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply Exists.elim xs_ih | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∀ (a : α → α),
InjOn a xs_tl ∧ map a xs_tl = ys_tl → ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro f a1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∀ (a : α → α),
InjOn a xs_tl ∧ map a xs_tl = ys_tl → ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | clear xs_ih | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
xs_ih : ∃ f, InjOn f xs_tl ∧ map f xs_tl = ys_tl
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [List.InjOn] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f, InjOn f (xs_hd :: xs_tl) ∧ f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases a1 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1 : InjOn f xs_tl ∧ map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | case intro
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
left✝ : InjOn f xs_tl
right✝ : map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | apply Exists.intro (Function.updateITE f xs_hd ys_hd) | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ ∃ f,
(∀ ⦃x₁ : α⦄, x₁ ∈ xs_hd :: xs_tl → ∀ ⦃x₂ : α⦄, x₂ ∈ xs_hd :: xs_tl → f x₁ = f x₂ → x₁ = x₂) ∧
f xs_hd = ys_hd ∧ map f xs_tl = ys_tl | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ (∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂) ∧
Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | constructor | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ (∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂) ∧
Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ ∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂
case right
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ Function.updateITE f xs_hd ys_hd xs_hd = ys_hd ∧ map (Function.updateITE f xs_hd ys_hd) xs_tl = ys_tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro x1 x1_mem x2 x2_mem | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
⊢ ∀ ⦃x₁ : α⦄,
x₁ ∈ xs_hd :: xs_tl →
∀ ⦃x₂ : α⦄,
x₂ ∈ xs_hd :: xs_tl → Function.updateITE f xs_hd ys_hd x₁ = Function.updateITE f xs_hd ys_hd x₂ → x₁ = x₂ | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 : α
x1_mem : x1 ∈ xs_hd :: xs_tl
x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at x1_mem | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 : α
x1_mem : x1 ∈ xs_hd :: xs_tl
x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp at x2_mem | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x2_mem : x2 ∈ xs_hd :: xs_tl
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [Function.updateITE] | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ Function.updateITE f xs_hd ys_hd x1 = Function.updateITE f xs_hd ys_hd x2 → x1 = x2 | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ ((if x1 = xs_hd then ys_hd else f x1) = if x2 = xs_hd then ys_hd else f x2) → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | split_ifs | case left
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
⊢ ((if x1 = xs_hd then ys_hd else f x1) = if x2 = xs_hd then ys_hd else f x2) → x1 = x2 | case pos
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : x1 = xs_hd
h✝ : x2 = xs_hd
⊢ ys_hd = ys_hd → x1 = x2
case neg
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : x1 = xs_hd
h✝ : ¬x2 = xs_hd
⊢ ys_hd = f x2 → x1 = x2
case pos
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : ¬x1 = xs_hd
h✝ : x2 = xs_hd
⊢ f x1 = ys_hd → x1 = x2
case neg
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
h✝¹ : ¬x1 = xs_hd
h✝ : ¬x2 = xs_hd
⊢ f x1 = f x2 → x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro _
simp only [c1]
simp only [c2] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
⊢ ys_hd = ys_hd → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro a2
cases x2_mem
case inl c3 =>
contradiction
case inr c3 =>
obtain s1 := List.mem_map_of_mem f c3
simp only [a1_right] at s1
simp only [← a2] at s1
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ ys_hd = f x2 → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro a2
cases x1_mem
case inl c3 =>
contradiction
case inr c3 =>
obtain s1 := List.mem_map_of_mem f c3
simp only [a1_right] at s1
simp only [a2] at s1
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : x2 = xs_hd
⊢ f x1 = ys_hd → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case _ c1 c2 =>
intro a2
cases x1_mem
case inl x1_mem_left =>
contradiction
case inr x1_mem_right =>
cases x2_mem
case inl x2_mem_left =>
contradiction
case inr x2_mem_right =>
simp only [List.InjOn] at a1_left
apply a1_left x1_mem_right x2_mem_right a2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : ¬x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ f x1 = f x2 → x1 = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro _ | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
⊢ ys_hd = ys_hd → x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [c1] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ xs_hd = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | simp only [c2] | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : x2 = xs_hd
a✝ : ys_hd = ys_hd
⊢ xs_hd = x2 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | intro a2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
⊢ ys_hd = f x2 → x1 = x2 | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | cases x2_mem | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
x2_mem : x2 = xs_hd ∨ x2 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
⊢ x1 = x2 | case inl
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
h✝ : x2 = xs_hd
⊢ x1 = x2
case inr
α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
h✝ : x2 ∈ xs_tl
⊢ x1 = x2 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/List.lean | List.nodup_eq_len_imp_exists_bijon | [162, 1] | [256, 30] | case inl c3 =>
contradiction | α : Type
inst✝ : DecidableEq α
xs_hd : α
xs_tl : List α
ys_hd : α
ys_tl : List α
h1 : xs_tl.length = ys_tl.length
h2_left : xs_hd ∉ xs_tl
h2_right : xs_tl.Nodup
h3_left : ys_hd ∉ ys_tl
h3_right : ys_tl.Nodup
f : α → α
a1_left : InjOn f xs_tl
a1_right : map f xs_tl = ys_tl
x1 x2 : α
x1_mem : x1 = xs_hd ∨ x1 ∈ xs_tl
c1 : x1 = xs_hd
c2 : ¬x2 = xs_hd
a2 : ys_hd = f x2
c3 : x2 = xs_hd
⊢ x1 = x2 | no goals |
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