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
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
trans E ∪ C \ F
|
case left
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ C \ F
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ E ∪ C \ F ⊆ E ∪ (C ∪ D) \ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
exact h1
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ C \ F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ C \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
apply Finset.union_subset_union_right
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ E ∪ C \ F ⊆ E ∪ (C ∪ D) \ F
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ C \ F ⊆ (C ∪ D) \ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ E ∪ C \ F ⊆ E ∪ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
apply Finset.sdiff_subset_sdiff
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ C \ F ⊆ (C ∪ D) \ F
|
case h.hst
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ C ⊆ C ∪ D
case h.hvu
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ F ⊆ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ C \ F ⊆ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
exact Finset.subset_union_left C D
|
case h.hst
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ C ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.hst
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ C ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
rfl
|
case h.hvu
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ F ⊆ F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.hvu
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ F ⊆ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
trans E ∪ D \ F
|
case right
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ (C ∪ D) \ F
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ D \ F
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ E ∪ D \ F ⊆ E ∪ (C ∪ D) \ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
exact h2
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ D \ F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ D \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
apply Finset.union_subset_union_right
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ E ∪ D \ F ⊆ E ∪ (C ∪ D) \ F
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ D \ F ⊆ (C ∪ D) \ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ E ∪ D \ F ⊆ E ∪ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
apply Finset.sdiff_subset_sdiff
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ D \ F ⊆ (C ∪ D) \ F
|
case h.hst
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ D ⊆ C ∪ D
case h.hvu
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ F ⊆ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ D \ F ⊆ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
exact Finset.subset_union_right C D
|
case h.hst
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ D ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.hst
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ D ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.diff_union_subset
|
[92, 1]
|
[105, 67]
|
have s1 : (A ∪ B) \ E = (A \ E) ∪ (B \ E)
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
⊢ (A ∪ B) \ E ⊆ C ∪ D
|
case s1
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
⊢ (A ∪ B) \ E = A \ E ∪ B \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
⊢ (A ∪ B) \ E ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.diff_union_subset
|
[92, 1]
|
[105, 67]
|
exact Finset.union_sdiff_distrib A B E
|
case s1
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
⊢ (A ∪ B) \ E = A \ E ∪ B \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ C ∪ D
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
⊢ (A ∪ B) \ E = A \ E ∪ B \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.diff_union_subset
|
[92, 1]
|
[105, 67]
|
trans (A \ E) ∪ (B \ E)
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ C ∪ D
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ A \ E ∪ B \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ A \ E ∪ B \ E ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.diff_union_subset
|
[92, 1]
|
[105, 67]
|
simp only [s1]
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ A \ E ∪ B \ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ A \ E ∪ B \ E ⊆ A \ E ∪ B \ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ (A ∪ B) \ E ⊆ A \ E ∪ B \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.diff_union_subset
|
[92, 1]
|
[105, 67]
|
rfl
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ A \ E ∪ B \ E ⊆ A \ E ∪ B \ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ A \ E ∪ B \ E ⊆ A \ E ∪ B \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.diff_union_subset
|
[92, 1]
|
[105, 67]
|
exact Finset.union_subset_left_right (A \ E) (B \ E) C D h1 h2
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ A \ E ∪ B \ E ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A \ E ⊆ C
h2 : B \ E ⊆ D
s1 : (A ∪ B) \ E = A \ E ∪ B \ E
⊢ A \ E ∪ B \ E ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_right_comm_assoc
|
[108, 1]
|
[116, 41]
|
simp only [Finset.union_right_comm S {x} T]
|
α : Type
inst✝ : DecidableEq α
x : α
S T : Finset α
⊢ S ∪ (T ∪ {x}) = S ∪ {x} ∪ T
|
α : Type
inst✝ : DecidableEq α
x : α
S T : Finset α
⊢ S ∪ (T ∪ {x}) = S ∪ T ∪ {x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
x : α
S T : Finset α
⊢ S ∪ (T ∪ {x}) = S ∪ {x} ∪ T
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_right_comm_assoc
|
[108, 1]
|
[116, 41]
|
simp only [Finset.union_assoc S T {x}]
|
α : Type
inst✝ : DecidableEq α
x : α
S T : Finset α
⊢ S ∪ (T ∪ {x}) = S ∪ T ∪ {x}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
x : α
S T : Finset α
⊢ S ∪ (T ∪ {x}) = S ∪ T ∪ {x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
subst h1
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
h1 : f x = x'
⊢ image f S \ {x'} = image f (S \ {x}) \ {x'}
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
⊢ image f S \ {f x} = image f (S \ {x}) \ {f x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
h1 : f x = x'
⊢ image f S \ {x'} = image f (S \ {x}) \ {x'}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
apply Finset.ext
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
⊢ image f S \ {f x} = image f (S \ {x}) \ {f x}
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
⊢ ∀ (a : β), a ∈ image f S \ {f x} ↔ a ∈ image f (S \ {x}) \ {f x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
⊢ image f S \ {f x} = image f (S \ {x}) \ {f x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
intro a
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
⊢ ∀ (a : β), a ∈ image f S \ {f x} ↔ a ∈ image f (S \ {x}) \ {f x}
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
⊢ a ∈ image f S \ {f x} ↔ a ∈ image f (S \ {x}) \ {f x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
⊢ ∀ (a : β), a ∈ image f S \ {f x} ↔ a ∈ image f (S \ {x}) \ {f x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
simp
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
⊢ a ∈ image f S \ {f x} ↔ a ∈ image f (S \ {x}) \ {f x}
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
⊢ ¬a = f x → ((∃ a_2 ∈ S, f a_2 = a) ↔ ∃ a_2, (a_2 ∈ S ∧ ¬a_2 = x) ∧ f a_2 = a)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
⊢ a ∈ image f S \ {f x} ↔ a ∈ image f (S \ {x}) \ {f x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
intro a1
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
⊢ ¬a = f x → ((∃ a_2 ∈ S, f a_2 = a) ↔ ∃ a_2, (a_2 ∈ S ∧ ¬a_2 = x) ∧ f a_2 = a)
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1 ∈ S, f a_1 = a) ↔ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
⊢ ¬a = f x → ((∃ a_2 ∈ S, f a_2 = a) ↔ ∃ a_2, (a_2 ∈ S ∧ ¬a_2 = x) ∧ f a_2 = a)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
constructor
|
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1 ∈ S, f a_1 = a) ↔ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1 ∈ S, f a_1 = a) → ∃ a_2, (a_2 ∈ S ∧ ¬a_2 = x) ∧ f a_2 = a
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a) → ∃ a_2 ∈ S, f a_2 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1 ∈ S, f a_1 = a) ↔ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
intro a2
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1 ∈ S, f a_1 = a) → ∃ a_2, (a_2 ∈ S ∧ ¬a_2 = x) ∧ f a_2 = a
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
⊢ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1 ∈ S, f a_1 = a) → ∃ a_2, (a_2 ∈ S ∧ ¬a_2 = x) ∧ f a_2 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
apply Exists.elim a2
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
⊢ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
⊢ ∀ (a_1 : α), a_1 ∈ S ∧ f a_1 = a → ∃ a_3, (a_3 ∈ S ∧ ¬a_3 = x) ∧ f a_3 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
⊢ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
intro b a3
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
⊢ ∀ (a_1 : α), a_1 ∈ S ∧ f a_1 = a → ∃ a_3, (a_3 ∈ S ∧ ¬a_3 = x) ∧ f a_3 = a
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3 : b ∈ S ∧ f b = a
⊢ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
⊢ ∀ (a_1 : α), a_1 ∈ S ∧ f a_1 = a → ∃ a_3, (a_3 ∈ S ∧ ¬a_3 = x) ∧ f a_3 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
apply Exists.intro b
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3 : b ∈ S ∧ f b = a
⊢ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3 : b ∈ S ∧ f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3 : b ∈ S ∧ f b = a
⊢ ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
cases a3
|
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3 : b ∈ S ∧ f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
|
case a.mp.intro
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
left✝ : b ∈ S
right✝ : f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3 : b ∈ S ∧ f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
case _ a3_left a3_right =>
subst a3_right
tauto
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3_left : b ∈ S
a3_right : f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3_left : b ∈ S
a3_right : f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
subst a3_right
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3_left : b ∈ S
a3_right : f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
b : α
a3_left : b ∈ S
a1 : ¬f b = f x
a2 : ∃ a ∈ S, f a = f b
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = f b
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1 ∈ S, f a_1 = a
b : α
a3_left : b ∈ S
a3_right : f b = a
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
tauto
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
b : α
a3_left : b ∈ S
a1 : ¬f b = f x
a2 : ∃ a ∈ S, f a = f b
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = f b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
b : α
a3_left : b ∈ S
a1 : ¬f b = f x
a2 : ∃ a ∈ S, f a = f b
⊢ (b ∈ S ∧ ¬b = x) ∧ f b = f b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
intro a2
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a) → ∃ a_2 ∈ S, f a_2 = a
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
⊢ ∃ a_1 ∈ S, f a_1 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
⊢ (∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a) → ∃ a_2 ∈ S, f a_2 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
apply Exists.elim a2
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
⊢ ∃ a_1 ∈ S, f a_1 = a
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
⊢ ∀ (a_1 : α), (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a → ∃ a_3 ∈ S, f a_3 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
⊢ ∃ a_1 ∈ S, f a_1 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
intro b a3
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
⊢ ∀ (a_1 : α), (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a → ∃ a_3 ∈ S, f a_3 = a
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3 : (b ∈ S ∧ ¬b = x) ∧ f b = a
⊢ ∃ a_1 ∈ S, f a_1 = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
⊢ ∀ (a_1 : α), (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a → ∃ a_3 ∈ S, f a_3 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
apply Exists.intro b
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3 : (b ∈ S ∧ ¬b = x) ∧ f b = a
⊢ ∃ a_1 ∈ S, f a_1 = a
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3 : (b ∈ S ∧ ¬b = x) ∧ f b = a
⊢ b ∈ S ∧ f b = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3 : (b ∈ S ∧ ¬b = x) ∧ f b = a
⊢ ∃ a_1 ∈ S, f a_1 = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
cases a3
|
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3 : (b ∈ S ∧ ¬b = x) ∧ f b = a
⊢ b ∈ S ∧ f b = a
|
case a.mpr.intro
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
left✝ : b ∈ S ∧ ¬b = x
right✝ : f b = a
⊢ b ∈ S ∧ f b = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3 : (b ∈ S ∧ ¬b = x) ∧ f b = a
⊢ b ∈ S ∧ f b = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
case _ a3_left a3_right =>
subst a3_right
tauto
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3_left : b ∈ S ∧ ¬b = x
a3_right : f b = a
⊢ b ∈ S ∧ f b = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3_left : b ∈ S ∧ ¬b = x
a3_right : f b = a
⊢ b ∈ S ∧ f b = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
subst a3_right
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3_left : b ∈ S ∧ ¬b = x
a3_right : f b = a
⊢ b ∈ S ∧ f b = a
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
b : α
a3_left : b ∈ S ∧ ¬b = x
a1 : ¬f b = f x
a2 : ∃ a, (a ∈ S ∧ ¬a = x) ∧ f a = f b
⊢ b ∈ S ∧ f b = f b
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
a : β
a1 : ¬a = f x
a2 : ∃ a_1, (a_1 ∈ S ∧ ¬a_1 = x) ∧ f a_1 = a
b : α
a3_left : b ∈ S ∧ ¬b = x
a3_right : f b = a
⊢ b ∈ S ∧ f b = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton
|
[119, 1]
|
[152, 12]
|
tauto
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
b : α
a3_left : b ∈ S ∧ ¬b = x
a1 : ¬f b = f x
a2 : ∃ a, (a ∈ S ∧ ¬a = x) ∧ f a = f b
⊢ b ∈ S ∧ f b = f b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
f : α → β
b : α
a3_left : b ∈ S ∧ ¬b = x
a1 : ¬f b = f x
a2 : ∃ a, (a ∈ S ∧ ¬a = x) ∧ f a = f b
⊢ b ∈ S ∧ f b = f b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
apply Finset.image_congr
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ image (Function.updateITE f x x') (S \ {x}) = image f (S \ {x})
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ Set.EqOn (Function.updateITE f x x') f ↑(S \ {x})
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ image (Function.updateITE f x x') (S \ {x}) = image f (S \ {x})
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
simp only [Set.EqOn]
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ Set.EqOn (Function.updateITE f x x') f ↑(S \ {x})
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ ∀ ⦃x_1 : α⦄, x_1 ∈ ↑(S \ {x}) → Function.updateITE f x x' x_1 = f x_1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ Set.EqOn (Function.updateITE f x x') f ↑(S \ {x})
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
intro a a1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ ∀ ⦃x_1 : α⦄, x_1 ∈ ↑(S \ {x}) → Function.updateITE f x x' x_1 = f x_1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ ↑(S \ {x})
⊢ Function.updateITE f x x' a = f a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
⊢ ∀ ⦃x_1 : α⦄, x_1 ∈ ↑(S \ {x}) → Function.updateITE f x x' x_1 = f x_1
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
simp at a1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ ↑(S \ {x})
⊢ Function.updateITE f x x' a = f a
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ S ∧ ¬a = x
⊢ Function.updateITE f x x' a = f a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ ↑(S \ {x})
⊢ Function.updateITE f x x' a = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ S ∧ ¬a = x
⊢ Function.updateITE f x x' a = f a
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ S ∧ ¬a = x
⊢ (if a = x then x' else f a) = f a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ S ∧ ¬a = x
⊢ Function.updateITE f x x' a = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
cases a1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ S ∧ ¬a = x
⊢ (if a = x then x' else f a) = f a
|
case h.intro
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
left✝ : a ∈ S
right✝ : ¬a = x
⊢ (if a = x then x' else f a) = f a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1 : a ∈ S ∧ ¬a = x
⊢ (if a = x then x' else f a) = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
case _ a1_left a1_right =>
simp only [if_neg a1_right]
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1_left : a ∈ S
a1_right : ¬a = x
⊢ (if a = x then x' else f a) = f a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1_left : a ∈ S
a1_right : ¬a = x
⊢ (if a = x then x' else f a) = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_sdiff_singleton_updateITE
|
[155, 1]
|
[173, 32]
|
simp only [if_neg a1_right]
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1_left : a ∈ S
a1_right : ¬a = x
⊢ (if a = x then x' else f a) = f a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
x' : β
f : α → β
a : α
a1_left : a ∈ S
a1_right : ¬a = x
⊢ (if a = x then x' else f a) = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
apply Finset.image_congr
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ image (Function.updateITE f x a) (S \ {x}) = image (Function.updateITE f x b) (S \ {x})
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ Set.EqOn (Function.updateITE f x a) (Function.updateITE f x b) ↑(S \ {x})
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ image (Function.updateITE f x a) (S \ {x}) = image (Function.updateITE f x b) (S \ {x})
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
simp only [Set.EqOn]
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ Set.EqOn (Function.updateITE f x a) (Function.updateITE f x b) ↑(S \ {x})
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ ∀ ⦃x_1 : α⦄, x_1 ∈ ↑(S \ {x}) → Function.updateITE f x a x_1 = Function.updateITE f x b x_1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ Set.EqOn (Function.updateITE f x a) (Function.updateITE f x b) ↑(S \ {x})
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
intro v a1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ ∀ ⦃x_1 : α⦄, x_1 ∈ ↑(S \ {x}) → Function.updateITE f x a x_1 = Function.updateITE f x b x_1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ ↑(S \ {x})
⊢ Function.updateITE f x a v = Function.updateITE f x b v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
⊢ ∀ ⦃x_1 : α⦄, x_1 ∈ ↑(S \ {x}) → Function.updateITE f x a x_1 = Function.updateITE f x b x_1
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
simp at a1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ ↑(S \ {x})
⊢ Function.updateITE f x a v = Function.updateITE f x b v
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ S ∧ ¬v = x
⊢ Function.updateITE f x a v = Function.updateITE f x b v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ ↑(S \ {x})
⊢ Function.updateITE f x a v = Function.updateITE f x b v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ S ∧ ¬v = x
⊢ Function.updateITE f x a v = Function.updateITE f x b v
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ S ∧ ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ S ∧ ¬v = x
⊢ Function.updateITE f x a v = Function.updateITE f x b v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
cases a1
|
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ S ∧ ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
|
case h.intro
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
left✝ : v ∈ S
right✝ : ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1 : v ∈ S ∧ ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
case intro a1_left a1_right =>
simp only [if_neg a1_right]
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1_left : v ∈ S
a1_right : ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1_left : v ∈ S
a1_right : ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.image_congr_update_ite
|
[176, 1]
|
[194, 32]
|
simp only [if_neg a1_right]
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1_left : v ∈ S
a1_right : ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
x : α
a b : β
f : α → β
v : α
a1_left : v ∈ S
a1_right : ¬v = x
⊢ (if v = x then a else f v) = if v = x then b else f v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.mem_image_update
|
[197, 1]
|
[212, 26]
|
simp only [Finset.mem_image]
|
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ f y ∈ image (Function.updateITE f x x) S
|
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ ∃ a ∈ S, Function.updateITE f x x a = f y
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ f y ∈ image (Function.updateITE f x x) S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.mem_image_update
|
[197, 1]
|
[212, 26]
|
apply Exists.intro y
|
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ ∃ a ∈ S, Function.updateITE f x x a = f y
|
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ y ∈ S ∧ Function.updateITE f x x y = f y
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ ∃ a ∈ S, Function.updateITE f x x a = f y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.mem_image_update
|
[197, 1]
|
[212, 26]
|
constructor
|
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ y ∈ S ∧ Function.updateITE f x x y = f y
|
case left
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ y ∈ S
case right
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ Function.updateITE f x x y = f y
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ y ∈ S ∧ Function.updateITE f x x y = f y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.mem_image_update
|
[197, 1]
|
[212, 26]
|
exact h2
|
case left
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ y ∈ S
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ y ∈ S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.mem_image_update
|
[197, 1]
|
[212, 26]
|
simp only [Function.updateITE]
|
case right
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ Function.updateITE f x x y = f y
|
case right
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ (if y = x then x else f y) = f y
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ Function.updateITE f x x y = f y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.mem_image_update
|
[197, 1]
|
[212, 26]
|
simp only [if_neg h1]
|
case right
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ (if y = x then x else f y) = f y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
x y : α
f : α → α
S : Finset α
h1 : ¬y = x
h2 : y ∈ S
⊢ (if y = x then x else f y) = f y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.left_id_left_inverse
|
[74, 1]
|
[83, 22]
|
simp only [Function.LeftInverse]
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ LeftInverse g f
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ LeftInverse g f
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.left_id_left_inverse
|
[74, 1]
|
[83, 22]
|
intro x
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
⊢ ∀ (x : α), g (f x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.left_id_left_inverse
|
[74, 1]
|
[83, 22]
|
exact congrFun h1 x
|
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : g ∘ f = id
x : α
⊢ g (f x) = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.right_id_right_inverse
|
[86, 1]
|
[94, 45]
|
simp only [Function.RightInverse]
|
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ RightInverse g f
|
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ RightInverse g f
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.right_id_right_inverse
|
[86, 1]
|
[94, 45]
|
exact Function.left_id_left_inverse g f h1
|
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
f : α → β
g : β → α
h1 : f ∘ g = id
⊢ LeftInverse f g
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
funext x
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
⊢ updateITE f a b = Function.updateITE' f a b
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
⊢ updateITE f a b = Function.updateITE' f a b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ updateITE f a b x = Function.updateITE' f a b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
simp only [Function.updateITE']
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = Function.updateITE' f a b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
split_ifs
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : x = a
h✝ : a = x
⊢ b = b
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : x = a
h✝ : ¬a = x
⊢ b = f x
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : ¬x = a
h✝ : a = x
⊢ f x = b
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
h✝¹ : ¬x = a
h✝ : ¬a = x
⊢ f x = f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
⊢ (if x = a then b else f x) = if a = x then b else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
subst c1
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
subst c2
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
case _ c1 c2 =>
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : a = x
⊢ b = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
subst c1
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x
|
α β : Type
inst✝ : DecidableEq α
f : α → β
b : β
x : α
c2 : ¬x = x
⊢ b = f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : x = a
c2 : ¬a = x
⊢ b = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
b : β
x : α
c2 : ¬x = x
⊢ b = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
b : β
x : α
c2 : ¬x = x
⊢ b = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
subst c2
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
c1 : ¬a = a
⊢ f a = b
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : a = x
⊢ f x = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
c1 : ¬a = a
⊢ f a = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
c1 : ¬a = a
⊢ f a = b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_eq_Function.updateITE'
|
[99, 1]
|
[121, 8]
|
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
x : α
c1 : ¬x = a
c2 : ¬a = x
⊢ f x = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_left
|
[124, 1]
|
[139, 8]
|
funext x
|
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
⊢ f ∘ updateITE g a b = updateITE (f ∘ g) a (f b)
|
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ (f ∘ updateITE g a b) x = updateITE (f ∘ g) a (f b) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
⊢ f ∘ updateITE g a b = updateITE (f ∘ g) a (f b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_left
|
[124, 1]
|
[139, 8]
|
simp
|
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ (f ∘ updateITE g a b) x = updateITE (f ∘ g) a (f b) x
|
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ f (updateITE g a b x) = updateITE (f ∘ g) a (f b) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ (f ∘ updateITE g a b) x = updateITE (f ∘ g) a (f b) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_left
|
[124, 1]
|
[139, 8]
|
simp only [Function.updateITE]
|
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ f (updateITE g a b x) = updateITE (f ∘ g) a (f b) x
|
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ f (if x = a then b else g x) = if x = a then f b else (f ∘ g) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ f (updateITE g a b x) = updateITE (f ∘ g) a (f b) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_left
|
[124, 1]
|
[139, 8]
|
split_ifs
|
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ f (if x = a then b else g x) = if x = a then f b else (f ∘ g) x
|
case pos
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
h✝ : x = a
⊢ f b = f b
case neg
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
h✝ : ¬x = a
⊢ f (g x) = (f ∘ g) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
⊢ f (if x = a then b else g x) = if x = a then f b else (f ∘ g) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_left
|
[124, 1]
|
[139, 8]
|
rfl
|
case pos
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
h✝ : x = a
⊢ f b = f b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
h✝ : x = a
⊢ f b = f b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_left
|
[124, 1]
|
[139, 8]
|
rfl
|
case neg
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
h✝ : ¬x = a
⊢ f (g x) = (f ∘ g) x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β γ : Type
inst✝ : DecidableEq α
f : β → γ
g : α → β
a : α
b : β
x : α
h✝ : ¬x = a
⊢ f (g x) = (f ∘ g) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
funext x
|
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
⊢ updateITE g a b ∘ f = updateITE (g ∘ f) (finv a) b
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ (updateITE g a b ∘ f) x = updateITE (g ∘ f) (finv a) b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
⊢ updateITE g a b ∘ f = updateITE (g ∘ f) (finv a) b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ (updateITE g a b ∘ f) x = updateITE (g ∘ f) (finv a) b x
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ updateITE g a b (f x) = updateITE (g ∘ f) (finv a) b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ (updateITE g a b ∘ f) x = updateITE (g ∘ f) (finv a) b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [Function.updateITE]
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ updateITE g a b (f x) = updateITE (g ∘ f) (finv a) b x
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ (if f x = a then b else g (f x)) = if x = finv a then b else (g ∘ f) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ updateITE g a b (f x) = updateITE (g ∘ f) (finv a) b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
congr!
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ (if f x = a then b else g (f x)) = if x = finv a then b else (g ∘ f) x
|
case h.h₁.a
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ f x = a ↔ x = finv a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ (if f x = a then b else g (f x)) = if x = finv a then b else (g ∘ f) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
constructor
|
case h.h₁.a
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ f x = a ↔ x = finv a
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ f x = a → x = finv a
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ x = finv a → f x = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ f x = a ↔ x = finv a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
intro a1
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ f x = a → x = finv a
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
⊢ x = finv a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ f x = a → x = finv a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [← a1]
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
⊢ x = finv a
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
⊢ x = finv (f x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
⊢ x = finv a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
obtain s1 := Function.left_id_left_inverse f finv h1
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
⊢ x = finv (f x)
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
s1 : LeftInverse finv f
⊢ x = finv (f x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
⊢ x = finv (f x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [Function.LeftInverse] at s1
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
s1 : LeftInverse finv f
⊢ x = finv (f x)
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
s1 : ∀ (x : α), finv (f x) = x
⊢ x = finv (f x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
s1 : LeftInverse finv f
⊢ x = finv (f x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [s1 x]
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
s1 : ∀ (x : α), finv (f x) = x
⊢ x = finv (f x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : f x = a
s1 : ∀ (x : α), finv (f x) = x
⊢ x = finv (f x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
intro a1
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ x = finv a → f x = a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
⊢ f x = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
⊢ x = finv a → f x = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [a1]
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
⊢ f x = a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
⊢ f (finv a) = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
⊢ f x = a
TACTIC:
|
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