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/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | case cons ys_hd ys_tl =>
simp only [Function.updateListITE]
simp only [Function.updateITE]
split_ifs
case pos c1 =>
subst c1
simp at h1
case neg c1 =>
simp at h1
push_neg at h1
cases h1
case intro h1_left h1_right =>
apply xs_ih ys_tl h1_right | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
⊢ updateListITE f (xs_hd :: xs_tl) (ys_hd :: ys_tl) v = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | simp only [Function.updateListITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
⊢ updateListITE f (xs_hd :: xs_tl) [] v = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | simp only [Function.updateListITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
⊢ updateListITE f (xs_hd :: xs_tl) (ys_hd :: ys_tl) v = f v | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
⊢ updateITE (updateListITE f xs_tl ys_tl) xs_hd ys_hd v = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | simp only [Function.updateITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
⊢ updateITE (updateListITE f xs_tl ys_tl) xs_hd ys_hd v = f v | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
⊢ (if v = xs_hd then ys_hd else updateListITE f xs_tl ys_tl v) = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | split_ifs | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
⊢ (if v = xs_hd then ys_hd else updateListITE f xs_tl ys_tl v) = f v | case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
h✝ : v = xs_hd
⊢ ys_hd = f v
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
h✝ : ¬v = xs_hd
⊢ updateListITE f xs_tl ys_tl v = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | case pos c1 =>
subst c1
simp at h1 | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
c1 : v = xs_hd
⊢ ys_hd = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | case neg c1 =>
simp at h1
push_neg at h1
cases h1
case intro h1_left h1_right =>
apply xs_ih ys_tl h1_right | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
⊢ updateListITE f xs_tl ys_tl v = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | subst c1 | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
c1 : v = xs_hd
⊢ ys_hd = f v | α β : Type
inst✝ : DecidableEq α
f : α → β
v : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
h1 : v ∉ v :: xs_tl
⊢ ys_hd = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | simp at h1 | α β : Type
inst✝ : DecidableEq α
f : α → β
v : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
h1 : v ∉ v :: xs_tl
⊢ ys_hd = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | simp at h1 | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
h1 : v ∉ xs_hd :: xs_tl
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
⊢ updateListITE f xs_tl ys_tl v = f v | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
h1 : ¬v = xs_hd ∧ v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | push_neg at h1 | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
h1 : ¬v = xs_hd ∧ v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
h1 : v ≠ xs_hd ∧ v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | cases h1 | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
h1 : v ≠ xs_hd ∧ v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v | case intro
α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
left✝ : v ≠ xs_hd
right✝ : v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | case intro h1_left h1_right =>
apply xs_ih ys_tl h1_right | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
h1_left : v ≠ xs_hd
h1_right : v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_not_mem | [519, 1] | [548, 37] | apply xs_ih ys_tl h1_right | α β : Type
inst✝ : DecidableEq α
f : α → β
v xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), v ∉ xs_tl → updateListITE f xs_tl ys v = f v
ys_hd : β
ys_tl : List β
c1 : ¬v = xs_hd
h1_left : v ≠ xs_hd
h1_right : v ∉ xs_tl
⊢ updateListITE f xs_tl ys_tl v = f v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | induction xs generalizing ys | α β : Type
inst✝ : DecidableEq α
f : α → β
xs : List α
ys : List β
x y : α
z : β
h1 : ¬x = y
⊢ updateListITE (updateITE f y z) xs ys x = updateListITE f xs ys x | case nil
α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ updateListITE (updateITE f y z) [] ys x = updateListITE f [] ys x
case cons
α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (ys : List β), updateListITE (updateITE f y z) tail✝ ys x = updateListITE f tail✝ ys x
ys : List β
⊢ updateListITE (updateITE f y z) (head✝ :: tail✝) ys x = updateListITE f (head✝ :: tail✝) ys x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | case nil =>
simp only [Function.updateListITE]
simp only [Function.updateITE]
simp only [if_neg h1] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ updateListITE (updateITE f y z) [] ys x = updateListITE f [] ys x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | case cons xs_hd xs_tl xs_ih =>
cases ys
case nil =>
simp only [Function.updateListITE]
simp only [Function.updateITE]
simp only [if_neg h1]
case cons ys_hd ys_tl =>
simp only [Function.updateListITE]
simp only [Function.updateITE]
split_ifs
case pos c1 =>
rfl
case neg c1 =>
apply xs_ih | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys : List β
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) ys x = updateListITE f (xs_hd :: xs_tl) ys x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [Function.updateListITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ updateListITE (updateITE f y z) [] ys x = updateListITE f [] ys x | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ updateITE f y z x = f x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [Function.updateITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ updateITE f y z x = f x | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ (if x = y then z else f x) = f x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [if_neg h1] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
ys : List β
⊢ (if x = y then z else f x) = f x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | cases ys | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys : List β
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) ys x = updateListITE f (xs_hd :: xs_tl) ys x | case nil
α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) [] x = updateListITE f (xs_hd :: xs_tl) [] x
case cons
α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
head✝ : β
tail✝ : List β
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) (head✝ :: tail✝) x =
updateListITE f (xs_hd :: xs_tl) (head✝ :: tail✝) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | case nil =>
simp only [Function.updateListITE]
simp only [Function.updateITE]
simp only [if_neg h1] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) [] x = updateListITE f (xs_hd :: xs_tl) [] x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | case cons ys_hd ys_tl =>
simp only [Function.updateListITE]
simp only [Function.updateITE]
split_ifs
case pos c1 =>
rfl
case neg c1 =>
apply xs_ih | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) (ys_hd :: ys_tl) x =
updateListITE f (xs_hd :: xs_tl) (ys_hd :: ys_tl) x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [Function.updateListITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) [] x = updateListITE f (xs_hd :: xs_tl) [] x | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ updateITE f y z x = f x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [Function.updateITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ updateITE f y z x = f x | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ (if x = y then z else f x) = f x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [if_neg h1] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
⊢ (if x = y then z else f x) = f x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [Function.updateListITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
⊢ updateListITE (updateITE f y z) (xs_hd :: xs_tl) (ys_hd :: ys_tl) x =
updateListITE f (xs_hd :: xs_tl) (ys_hd :: ys_tl) x | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
⊢ updateITE (updateListITE (updateITE f y z) xs_tl ys_tl) xs_hd ys_hd x =
updateITE (updateListITE f xs_tl ys_tl) xs_hd ys_hd x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | simp only [Function.updateITE] | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
⊢ updateITE (updateListITE (updateITE f y z) xs_tl ys_tl) xs_hd ys_hd x =
updateITE (updateListITE f xs_tl ys_tl) xs_hd ys_hd x | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
⊢ (if x = xs_hd then ys_hd else updateListITE (updateITE f y z) xs_tl ys_tl x) =
if x = xs_hd then ys_hd else updateListITE f xs_tl ys_tl x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | split_ifs | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
⊢ (if x = xs_hd then ys_hd else updateListITE (updateITE f y z) xs_tl ys_tl x) =
if x = xs_hd then ys_hd else updateListITE f xs_tl ys_tl x | case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
h✝ : x = xs_hd
⊢ ys_hd = ys_hd
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
h✝ : ¬x = xs_hd
⊢ updateListITE (updateITE f y z) xs_tl ys_tl x = updateListITE f xs_tl ys_tl x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | case pos c1 =>
rfl | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
c1 : x = xs_hd
⊢ ys_hd = ys_hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | case neg c1 =>
apply xs_ih | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
c1 : ¬x = xs_hd
⊢ updateListITE (updateITE f y z) xs_tl ys_tl x = updateListITE f xs_tl ys_tl x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | rfl | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
c1 : x = xs_hd
⊢ ys_hd = ys_hd | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_updateIte | [551, 1] | [581, 20] | apply xs_ih | α β : Type
inst✝ : DecidableEq α
f : α → β
x y : α
z : β
h1 : ¬x = y
xs_hd : α
xs_tl : List α
xs_ih : ∀ (ys : List β), updateListITE (updateITE f y z) xs_tl ys x = updateListITE f xs_tl ys x
ys_hd : β
ys_tl : List β
c1 : ¬x = xs_hd
⊢ updateListITE (updateITE f y z) xs_tl ys_tl x = updateListITE f xs_tl ys_tl x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | have s1 : List.map f ys = List.map g ys | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
⊢ List.map f ys = List.map g ys
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | simp only [List.map_eq_map_iff] | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
⊢ List.map f ys = List.map g ys
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
⊢ ∀ x ∈ ys, f x = g x
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | exact h1 | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
⊢ ∀ x ∈ ys, f x = g x
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | simp only [s1] | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map g ys) x | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map g ys) x = updateListITE g xs (List.map g ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | apply Function.updateListITE_mem_eq_len | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ updateListITE f xs (List.map g ys) x = updateListITE g xs (List.map g ys) x | case h1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ x ∈ xs
case h2
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ xs.length = (List.map g ys).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | exact h2 | case h1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ x ∈ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | simp | case h2
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ xs.length = (List.map g ys).length | case h2
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ xs.length = ys.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_fun_coincide_mem_eq_len | [584, 1] | [604, 13] | exact h3 | case h2
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
x : α
h1 : ∀ v ∈ ys, f v = g v
h2 : x ∈ xs
h3 : xs.length = ys.length
s1 : List.map f ys = List.map g ys
⊢ xs.length = ys.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | have s1 : List.map h ys = List.map h' ys | α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x | case s1
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ List.map h ys = List.map h' ys
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | simp only [List.map_eq_map_iff] | case s1
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ List.map h ys = List.map h' ys
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x | case s1
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ ∀ x ∈ ys, h x = h' x
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | exact h1 | case s1
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ ∀ x ∈ ys, h x = h' x
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x | α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | simp only [s1] | α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h ys) x = updateListITE g xs (List.map h' ys) x | α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h' ys) x = updateListITE g xs (List.map h' ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | apply Function.updateListITE_mem_eq_len | α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ updateListITE f xs (List.map h' ys) x = updateListITE g xs (List.map h' ys) x | case h1
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ x ∈ xs
case h2
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ xs.length = (List.map h' ys).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | exact h3 | case h1
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ x ∈ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | simp | case h2
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ xs.length = (List.map h' ys).length | case h2
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ xs.length = ys.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem_ext | [607, 1] | [627, 13] | exact h2 | case h2
α β : Type
inst✝ : DecidableEq α
xs ys : List α
f g h h' : α → β
x : α
h1 : ∀ y ∈ ys, h y = h' y
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : List.map h ys = List.map h' ys
⊢ xs.length = ys.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | induction xs | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs : List α
x : α
h1 : x ∈ xs
⊢ updateListITE f xs (List.map g xs) x = g x | case nil
α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
h1 : x ∈ []
⊢ updateListITE f [] (List.map g []) x = g x
case cons
α β : Type
inst✝ : DecidableEq α
f g : α → β
x head✝ : α
tail✝ : List α
tail_ih✝ : x ∈ tail✝ → updateListITE f tail✝ (List.map g tail✝) x = g x
h1 : x ∈ head✝ :: tail✝
⊢ updateListITE f (head✝ :: tail✝) (List.map g (head✝ :: tail✝)) x = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | case nil =>
simp at h1 | α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
h1 : x ∈ []
⊢ updateListITE f [] (List.map g []) x = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | case cons hd tl ih =>
simp at h1
simp
simp only [Function.updateListITE]
simp only [Function.updateITE]
split_ifs
case _ c1 =>
subst c1
rfl
case _ c1 =>
tauto | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x ∈ hd :: tl
⊢ updateListITE f (hd :: tl) (List.map g (hd :: tl)) x = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | simp at h1 | α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
h1 : x ∈ []
⊢ updateListITE f [] (List.map g []) x = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | simp at h1 | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x ∈ hd :: tl
⊢ updateListITE f (hd :: tl) (List.map g (hd :: tl)) x = g x | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ updateListITE f (hd :: tl) (List.map g (hd :: tl)) x = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | simp | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ updateListITE f (hd :: tl) (List.map g (hd :: tl)) x = g x | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ updateListITE f (hd :: tl) (g hd :: List.map g tl) x = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | simp only [Function.updateListITE] | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ updateListITE f (hd :: tl) (g hd :: List.map g tl) x = g x | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ updateITE (updateListITE f tl (List.map g tl)) hd (g hd) x = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | simp only [Function.updateITE] | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ updateITE (updateListITE f tl (List.map g tl)) hd (g hd) x = g x | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ (if x = hd then g hd else updateListITE f tl (List.map g tl) x) = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | split_ifs | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
⊢ (if x = hd then g hd else updateListITE f tl (List.map g tl) x) = g x | case pos
α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
h✝ : x = hd
⊢ g hd = g x
case neg
α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
h✝ : ¬x = hd
⊢ updateListITE f tl (List.map g tl) x = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | case _ c1 =>
subst c1
rfl | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
c1 : x = hd
⊢ g hd = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | case _ c1 =>
tauto | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
c1 : ¬x = hd
⊢ updateListITE f tl (List.map g tl) x = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | subst c1 | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
c1 : x = hd
⊢ g hd = g x | α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = x ∨ x ∈ tl
⊢ g x = g x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | rfl | α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = x ∨ x ∈ tl
⊢ g x = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_mem | [630, 1] | [653, 12] | tauto | α β : Type
inst✝ : DecidableEq α
f g : α → β
x hd : α
tl : List α
ih : x ∈ tl → updateListITE f tl (List.map g tl) x = g x
h1 : x = hd ∨ x ∈ tl
c1 : ¬x = hd
⊢ updateListITE f tl (List.map g tl) x = g x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | have s1 : ∀ (y : α), y ∈ ys → f y =Function.updateITE f v a y | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ ∀ y ∈ ys, f y = updateITE f v a y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | intro y a1 | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
⊢ ∀ y ∈ ys, f y = updateITE f v a y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
⊢ f y = updateITE f v a y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | simp only [Function.updateITE] | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
⊢ f y = updateITE f v a y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
⊢ f y = if y = v then a else f y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | split_ifs | case s1
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
⊢ f y = if y = v then a else f y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x | case pos
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
h✝ : y = v
⊢ f y = a
case neg
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
h✝ : ¬y = v
⊢ f y = f y
α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | case _ c1 =>
specialize h1 y a1
contradiction | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
c1 : y = v
⊢ f y = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | case _ c2 =>
rfl | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
c2 : ¬y = v
⊢ f y = f y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | exact Function.updateListITE_map_mem_ext xs ys f g f (Function.updateITE f v a) x s1 h2 h3 | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
s1 : ∀ y ∈ ys, f y = updateITE f v a y
⊢ updateListITE f xs (List.map f ys) x = updateListITE g xs (List.map (updateITE f v a) ys) x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | specialize h1 y a1 | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
c1 : y = v
⊢ f y = a | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
c1 : y = v
h1 : ¬y = v
⊢ f y = a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | contradiction | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
c1 : y = v
h1 : ¬y = v
⊢ f y = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/FunctionUpdateITE.lean | Function.updateListITE_map_updateIte | [656, 1] | [680, 93] | rfl | α β : Type
inst✝ : DecidableEq α
f g : α → β
xs ys : List α
v : α
a : β
x : α
h1 : ∀ y ∈ ys, ¬y = v
h2 : xs.length = ys.length
h3 : x ∈ xs
y : α
a1 : y ∈ ys
c2 : ¬y = v
⊢ f y = f y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | induction F | P : PredName
zs : List VarName
H F : Formula
h1 : F.predVarSet = ∅
⊢ replace P zs H F = F | case pred_const_
P : PredName
zs : List VarName
H : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
P : PredName
zs : List VarName
H : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
P : PredName
zs : List VarName
H : Formula
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
P : PredName
zs : List VarName
H : Formula
h1 : true_.predVarSet = ∅
⊢ replace P zs H true_ = true_
case false_
P : PredName
zs : List VarName
H : Formula
h1 : false_.predVarSet = ∅
⊢ replace P zs H false_ = false_
case not_
P : PredName
zs : List VarName
H a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : a✝.not_.predVarSet = ∅
⊢ replace P zs H a✝.not_ = a✝.not_
case imp_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.imp_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.and_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.or_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
P : PredName
zs : List VarName
H a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.predVarSet = ∅ → replace P zs H a✝¹ = a✝¹
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (a✝¹.iff_ a✝).predVarSet = ∅
⊢ replace P zs H (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
P : PredName
zs : List VarName
H : Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (forall_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
P : PredName
zs : List VarName
H : Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.predVarSet = ∅ → replace P zs H a✝ = a✝
h1 : (exists_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
P : PredName
zs : List VarName
H : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).predVarSet = ∅
⊢ replace P zs H (def_ a✝¹ a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case pred_const_ X xs =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case pred_var_ X xs =>
simp only [predVarSet] at h1
simp at h1 | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case eq_ x y =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ replace P zs H (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case true_ | false_ =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
h1 : false_.predVarSet = ∅
⊢ replace P zs H false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case not_ phi phi_ih =>
simp only [predVarSet] at h1
simp only [replace]
congr!
exact phi_ih h1 | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [predVarSet] at h1
simp only [replace]
congr!
exact phi_ih h1 | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | case def_ X xs =>
simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : DefName
xs : List VarName
h1 : (def_ X xs).predVarSet = ∅
⊢ replace P zs H (def_ X xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_const_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_const_ X xs) = pred_const_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : (pred_var_ X xs).predVarSet = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp at h1 | P : PredName
zs : List VarName
H : Formula
X : PredName
xs : List VarName
h1 : {(X, xs.length)} = ∅
⊢ replace P zs H (pred_var_ X xs) = pred_var_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
x y : VarName
h1 : (eq_ x y).predVarSet = ∅
⊢ replace P zs H (eq_ x y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
h1 : false_.predVarSet = ∅
⊢ replace P zs H false_ = false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.not_.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi.not_ = phi.not_ | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ (replace P zs H phi).not_ = phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | congr! | P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ (replace P zs H phi).not_ = phi.not_ | case h.e'_1
P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact phi_ih h1 | case h.e'_1
P : PredName
zs : List VarName
H phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : (phi.iff_ psi).predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [Finset.union_eq_empty] at h1 | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet ∪ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | cases h1 | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1 : phi.predVarSet = ∅ ∧ psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | case intro
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
left✝ : phi.predVarSet = ∅
right✝ : psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H (phi.iff_ psi) = phi.iff_ psi | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (replace P zs H phi).iff_ (replace P zs H psi) = phi.iff_ psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | congr! | P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ (replace P zs H phi).iff_ (replace P zs H psi) = phi.iff_ psi | case h.e'_1
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H phi = phi
case h.e'_2
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H psi = psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact phi_ih h1_left | case h.e'_1
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | exact psi_ih h1_right | case h.e'_2
P : PredName
zs : List VarName
H phi psi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
psi_ih : psi.predVarSet = ∅ → replace P zs H psi = psi
h1_left : phi.predVarSet = ∅
h1_right : psi.predVarSet = ∅
⊢ replace P zs H psi = psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [predVarSet] at h1 | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : (exists_ x phi).predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | simp only [replace] | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H (exists_ x phi) = exists_ x phi | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (replace P zs H phi) = exists_ x phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/One/Rec/Sub.lean | FOL.NV.Sub.Pred.One.Rec.replace_no_predVar | [107, 1] | [152, 24] | congr! | P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ exists_ x (replace P zs H phi) = exists_ x phi | case h.e'_2
P : PredName
zs : List VarName
H : Formula
x : VarName
phi : Formula
phi_ih : phi.predVarSet = ∅ → replace P zs H phi = phi
h1 : phi.predVarSet = ∅
⊢ replace P zs H phi = phi |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.