Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right
[8, 1]
[23, 44]
trans C
case left α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C ∪ D
α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ C ⊆ C ∪ D
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right
[8, 1]
[23, 44]
exact h1
α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ A ⊆ C
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right
[8, 1]
[23, 44]
exact Finset.subset_union_left C D
α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ C ⊆ C ∪ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right
[8, 1]
[23, 44]
trans D
case right α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ B ⊆ C ∪ D
α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ B ⊆ D α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ D ⊆ C ∪ D
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right
[8, 1]
[23, 44]
exact h2
α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ B ⊆ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right
[8, 1]
[23, 44]
exact Finset.subset_union_right C D
α : Type inst✝ : DecidableEq α A B C D : Finset α h1 : A ⊆ C h2 : B ⊆ D ⊢ D ⊆ C ∪ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
apply Finset.union_subset_iff.mpr
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ∪ B ⊆ C ∪ D ∪ E
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
constructor
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E
case left α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ⊆ C ∪ D ∪ E case right α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ B ⊆ C ∪ D ∪ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
trans C ∪ E
case left α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ⊆ C ∪ D ∪ E
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ⊆ C ∪ E α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ C ∪ E ⊆ C ∪ D ∪ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
exact h1
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ A ⊆ C ∪ E
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
apply Finset.union_subset_union_left
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ C ∪ E ⊆ C ∪ D ∪ E
case h α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ C ⊆ C ∪ D
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
exact Finset.subset_union_left C D
case h α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ C ⊆ C ∪ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
trans D ∪ E
case right α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ B ⊆ C ∪ D ∪ E
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ B ⊆ D ∪ E α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ D ∪ E ⊆ C ∪ D ∪ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
exact h2
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ B ⊆ D ∪ E
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
apply Finset.union_subset_union_left
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ D ∪ E ⊆ C ∪ D ∪ E
case h α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ D ⊆ C ∪ D
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_union_left_right
[26, 1]
[43, 42]
exact Finset.subset_union_right C D
case h α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C ∪ E h2 : B ⊆ D ∪ E ⊢ D ⊆ C ∪ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
apply Finset.union_subset_iff.mpr
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ∪ B ⊆ (C ∪ D) \ E
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
constructor
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E
case left α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ⊆ (C ∪ D) \ E case right α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ B ⊆ (C ∪ D) \ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
trans C \ E
case left α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ⊆ (C ∪ D) \ E
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ⊆ C \ E α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ C \ E ⊆ (C ∪ D) \ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
exact h1
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ A ⊆ C \ E
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
apply Finset.sdiff_subset_sdiff
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ C \ E ⊆ (C ∪ D) \ E
case hst α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ C ⊆ C ∪ D case hvu α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ E ⊆ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
exact Finset.subset_union_left C D
case hst α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ C ⊆ C ∪ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
rfl
case hvu α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ E ⊆ E
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
trans D \ E
case right α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ B ⊆ (C ∪ D) \ E
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ B ⊆ D \ E α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ D \ E ⊆ (C ∪ D) \ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
exact h2
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ B ⊆ D \ E
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
apply Finset.sdiff_subset_sdiff
α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ D \ E ⊆ (C ∪ D) \ E
case hst α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ D ⊆ C ∪ D case hvu α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ E ⊆ E
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_diff
[46, 1]
[65, 12]
exact Finset.subset_union_right C D
case hst α : Type inst✝ : DecidableEq α A B C D E : Finset α h1 : A ⊆ C \ E h2 : B ⊆ D \ E ⊢ D ⊆ C ∪ D
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right_diff
[68, 1]
[89, 12]
apply Finset.union_subset_iff.mpr
α : Type inst✝ : DecidableEq α A B C D E F : Finset α h1 : A ⊆ E ∪ C \ F h2 : B ⊆ E ∪ D \ F ⊢ A ∪ 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 ⊢ A ⊆ E ∪ (C ∪ D) \ F ∧ B ⊆ E ∪ (C ∪ D) \ F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Finset.lean
Finset.union_subset_left_right_diff
[68, 1]
[89, 12]
constructor
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 ∧ B ⊆ E ∪ (C ∪ D) \ 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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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}
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
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}
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}
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}
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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})
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
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
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
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
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
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
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
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})
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
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
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
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
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
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
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
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
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
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
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
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
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
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
induction h1 generalizing V V'
D : Type I : Interpretation D V V' : VarAssignment D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula h1 : IsSubAux σ binders F F' h2 : ∀ v ∈ binders, V v = V' (σ v) h3 : ∀ (v : VarName), σ v ∉ binders → V v = V' (σ v) h4 : ∀ v ∈ binders, v = σ v ⊢ Holds D I V E F ↔ Holds D I V' E F'
case pred_const_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName X✝ : PredName xs✝ : List VarName a✝ : ∀ v ∈ xs✝, v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (pred_const_ X✝ xs✝) ↔ Holds D I V' E (pred_const_ X✝ (List.map σ✝ xs✝)) case pred_var_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName X✝ : PredName xs✝ : List VarName a✝ : ∀ v ∈ xs✝, v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (pred_var_ X✝ xs✝) ↔ Holds D I V' E (pred_var_ X✝ (List.map σ✝ xs✝)) case eq_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName x✝ y✝ : VarName a✝ : ∀ (v : VarName), v = x✝ ∨ v = y✝ → v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (eq_ x✝ y✝) ↔ Holds D I V' E (eq_ (σ✝ x✝) (σ✝ y✝)) case true_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E true_ ↔ Holds D I V' E true_ case false_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E false_ ↔ Holds D I V' E false_ case not_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ phi'✝ : Formula a✝ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E phi✝.not_ ↔ Holds D I V' E phi'✝.not_ case imp_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.imp_ psi✝) ↔ Holds D I V' E (phi'✝.imp_ psi'✝) case and_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.and_ psi✝) ↔ Holds D I V' E (phi'✝.and_ psi'✝) case or_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.or_ psi✝) ↔ Holds D I V' E (phi'✝.or_ psi'✝) case iff_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName phi✝ psi✝ phi'✝ psi'✝ : Formula a✝¹ : IsSubAux σ✝ binders✝ phi✝ phi'✝ a✝ : IsSubAux σ✝ binders✝ psi✝ psi'✝ a_ih✝¹ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝, V v = V' (σ✝ v)) → (∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v)) → (∀ v ∈ binders✝, v = σ✝ v) → (Holds D I V E psi✝ ↔ Holds D I V' E psi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (phi✝.iff_ psi✝) ↔ Holds D I V' E (phi'✝.iff_ psi'✝) case forall_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName x✝ : VarName phi✝ phi'✝ : Formula a✝ : IsSubAux (Function.updateITE σ✝ x✝ x✝) (binders✝ ∪ {x✝}) phi✝ phi'✝ a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝ ∪ {x✝}, V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ (v : VarName), Function.updateITE σ✝ x✝ x✝ v ∉ binders✝ ∪ {x✝} → V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ v ∈ binders✝ ∪ {x✝}, v = Function.updateITE σ✝ x✝ x✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (forall_ x✝ phi✝) ↔ Holds D I V' E (forall_ x✝ phi'✝) case exists_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName x✝ : VarName phi✝ phi'✝ : Formula a✝ : IsSubAux (Function.updateITE σ✝ x✝ x✝) (binders✝ ∪ {x✝}) phi✝ phi'✝ a_ih✝ : ∀ (V V' : VarAssignment D), (∀ v ∈ binders✝ ∪ {x✝}, V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ (v : VarName), Function.updateITE σ✝ x✝ x✝ v ∉ binders✝ ∪ {x✝} → V v = V' (Function.updateITE σ✝ x✝ x✝ v)) → (∀ v ∈ binders✝ ∪ {x✝}, v = Function.updateITE σ✝ x✝ x✝ v) → (Holds D I V E phi✝ ↔ Holds D I V' E phi'✝) V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (exists_ x✝ phi✝) ↔ Holds D I V' E (exists_ x✝ phi'✝) case def_ D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName X✝ : DefName xs✝ : List VarName a✝ : ∀ v ∈ xs✝, v ∉ binders✝ → σ✝ v ∉ binders✝ V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E (def_ X✝ xs✝) ↔ Holds D I V' E (def_ X✝ (List.map σ✝ xs✝))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case true_ | false_ => simp only [Holds]
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ✝ : VarName → VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : ∀ v ∈ binders✝, V v = V' (σ✝ v) h3 : ∀ (v : VarName), σ✝ v ∉ binders✝ → V v = V' (σ✝ v) h4 : ∀ v ∈ binders✝, v = σ✝ v ⊢ Holds D I V E false_ ↔ Holds D I V' E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case not_ σ' binders' phi phi' _ ih_2 => simp only [Holds] congr! 1 exact ih_2 V V' h2 h3 h4
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux σ' binders' phi phi' ih_2 : ∀ (V V' : VarAssignment D), (∀ v ∈ binders', V v = V' (σ' v)) → (∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)) → (∀ v ∈ binders', v = σ' v) → (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E phi.not_ ↔ Holds D I V' E phi'.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E (pred_var_ X' xs') ↔ Holds D I V' E (pred_var_ X' (List.map σ' xs'))
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map V' (List.map σ' xs'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map V' (List.map σ' xs'))
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map (V' ∘ σ') xs')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
congr! 1
D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ I.pred_var_ X' (List.map V xs') ↔ I.pred_var_ X' (List.map (V' ∘ σ') xs')
case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ List.map V xs' = List.map (V' ∘ σ') xs'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [List.map_eq_map_iff]
case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ List.map V xs' = List.map (V' ∘ σ') xs'
case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ ∀ x ∈ xs', V x = (V' ∘ σ') x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro x a1
case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ ∀ x ∈ xs', V x = (V' ∘ σ') x
case a.h.e'_4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v x : VarName a1 : x ∈ xs' ⊢ V x = (V' ∘ σ') x