url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
obtain s1 := Function.right_id_right_inverse f finv h2
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
⊢ f (finv a) = a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : RightInverse finv f
⊢ f (finv a) = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
⊢ f (finv a) = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [Function.RightInverse] at s1
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : RightInverse finv f
⊢ f (finv a) = a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : LeftInverse f finv
⊢ f (finv a) = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : RightInverse finv f
⊢ f (finv a) = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
simp only [Function.LeftInverse] at s1
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : LeftInverse f finv
⊢ f (finv a) = a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : ∀ (x : β), f (finv x) = x
⊢ f (finv a) = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : LeftInverse f finv
⊢ f (finv a) = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right
|
[142, 1]
|
[171, 15]
|
exact s1 a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : ∀ (x : β), f (finv x) = x
⊢ f (finv a) = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
finv : β → α
g : β → γ
a : β
b : γ
h1 : finv ∘ f = id
h2 : f ∘ finv = id
x : α
a1 : x = finv a
s1 : ∀ (x : β), f (finv x) = x
⊢ f (finv a) = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
simp only [Function.Injective] at h1
|
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : Injective f
⊢ updateITE g (f a) b ∘ f = updateITE (g ∘ f) a b
|
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
⊢ updateITE g (f a) b ∘ f = updateITE (g ∘ f) a b
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : Injective f
⊢ updateITE g (f a) b ∘ f = updateITE (g ∘ f) a b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
funext x
|
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
⊢ updateITE g (f a) b ∘ f = updateITE (g ∘ f) a b
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ (updateITE g (f a) b ∘ f) x = updateITE (g ∘ f) a b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
⊢ updateITE g (f a) b ∘ f = updateITE (g ∘ f) a b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
simp
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ (updateITE g (f a) b ∘ f) x = updateITE (g ∘ f) a b x
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ updateITE g (f a) b (f x) = updateITE (g ∘ f) a b x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ (updateITE g (f a) b ∘ f) x = updateITE (g ∘ f) a b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
simp only [Function.updateITE]
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ updateITE g (f a) b (f x) = updateITE (g ∘ f) a b x
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ (if f x = f a then b else g (f x)) = if x = a then b else (g ∘ f) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ updateITE g (f a) b (f x) = updateITE (g ∘ f) a b x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
congr!
|
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ (if f x = f a then b else g (f x)) = if x = a then b else (g ∘ f) x
|
case h.h₁.a
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f a ↔ x = a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ (if f x = f a then b else g (f x)) = if x = a then b else (g ∘ f) x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
constructor
|
case h.h₁.a
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f a ↔ x = a
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f a → x = a
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ x = a → f x = f a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f a ↔ x = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
apply h1
|
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f a → x = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mp
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f a → x = a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
intro a1
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ x = a → f x = f a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
a1 : x = a
⊢ f x = f a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ x = a → f x = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
subst a1
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
a1 : x = a
⊢ f x = f a
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
a : α
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
a1 : x = a
⊢ f x = f a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp_right_injective
|
[174, 1]
|
[196, 8]
|
rfl
|
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h₁.a.mpr
α β γ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
g : β → γ
b : γ
h1 : ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂
x : α
⊢ f x = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
funext x
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
⊢ updateITE (updateITE f a d) b c = updateITE (updateITE f b c) a d
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ updateITE (updateITE f a d) b c x = updateITE (updateITE f b c) a d x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
⊢ updateITE (updateITE f a d) b c = updateITE (updateITE f b c) a d
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ updateITE (updateITE f a d) b c x = updateITE (updateITE f b c) a d x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ (if x = b then c else if x = a then d else f x) = if x = a then d else if x = b then c else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ updateITE (updateITE f a d) b c x = updateITE (updateITE f b c) a d x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
split_ifs
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ (if x = b then c else if x = a then d else f x) = if x = a then d else if x = b then c else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : x = b
h✝ : x = a
⊢ c = d
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : x = b
h✝ : ¬x = a
⊢ c = c
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : ¬x = b
h✝ : x = a
⊢ d = d
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : ¬x = b
h✝ : ¬x = a
⊢ f x = f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ (if x = b then c else if x = a then d else f x) = if x = a then d else if x = b then c else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
case _ c1 c2 =>
subst c1 c2
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = b
c2 : x = a
⊢ c = d
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = b
c2 : x = a
⊢ c = d
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
case _ | _ | _ =>
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : ¬x = b
h✝ : ¬x = a
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : ¬x = b
h✝ : ¬x = a
⊢ f x = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
subst c1 c2
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = b
c2 : x = a
⊢ c = d
|
α β : Type
inst✝ : DecidableEq α
f : α → β
c d : β
x : α
h1 : ¬x = x
⊢ c = d
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = b
c2 : x = a
⊢ c = d
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
contradiction
|
α β : Type
inst✝ : DecidableEq α
f : α → β
c d : β
x : α
h1 : ¬x = x
⊢ c = d
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
c d : β
x : α
h1 : ¬x = x
⊢ c = d
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm
|
[199, 1]
|
[216, 8]
|
rfl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : ¬x = b
h✝ : ¬x = a
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
h✝¹ : ¬x = b
h✝ : ¬x = a
⊢ f x = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_idem
|
[219, 1]
|
[232, 8]
|
funext x
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
⊢ updateITE (updateITE f a b) a c = updateITE f a c
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
⊢ updateITE (updateITE f a b) a c x = updateITE f a c x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
⊢ updateITE (updateITE f a b) a c = updateITE f a c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_idem
|
[219, 1]
|
[232, 8]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
⊢ updateITE (updateITE f a b) a c x = updateITE f a c x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
⊢ (if x = a then c else if x = a then b else f x) = if x = a then c else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
⊢ updateITE (updateITE f a b) a c x = updateITE f a c x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_idem
|
[219, 1]
|
[232, 8]
|
split_ifs
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
⊢ (if x = a then c else if x = a then b else f x) = if x = a then c else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
h✝ : x = a
⊢ c = c
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
h✝ : ¬x = a
⊢ f x = f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
⊢ (if x = a then c else if x = a then b else f x) = if x = a then c else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_idem
|
[219, 1]
|
[232, 8]
|
rfl
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
h✝ : x = a
⊢ c = c
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
h✝ : x = a
⊢ c = c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_idem
|
[219, 1]
|
[232, 8]
|
rfl
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
h✝ : ¬x = a
⊢ f x = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b c : β
x : α
h✝ : ¬x = a
⊢ f x = f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
funext x
|
α : Type
inst✝ : DecidableEq α
a : α
⊢ updateITE id a a = id
|
case h
α : Type
inst✝ : DecidableEq α
a x : α
⊢ updateITE id a a x = id x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
⊢ updateITE id a a = id
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
simp only [Function.updateITE]
|
case h
α : Type
inst✝ : DecidableEq α
a x : α
⊢ updateITE id a a x = id x
|
case h
α : Type
inst✝ : DecidableEq α
a x : α
⊢ (if x = a then a else id x) = id x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
a x : α
⊢ updateITE id a a x = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
split_ifs
|
case h
α : Type
inst✝ : DecidableEq α
a x : α
⊢ (if x = a then a else id x) = id x
|
case pos
α : Type
inst✝ : DecidableEq α
a x : α
h✝ : x = a
⊢ a = id x
case neg
α : Type
inst✝ : DecidableEq α
a x : α
h✝ : ¬x = a
⊢ id x = id x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
a x : α
⊢ (if x = a then a else id x) = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
case _ c1 =>
subst c1
simp
|
α : Type
inst✝ : DecidableEq α
a x : α
c1 : x = a
⊢ a = id x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a x : α
c1 : x = a
⊢ a = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
case _ =>
rfl
|
α : Type
inst✝ : DecidableEq α
a x : α
h✝ : ¬x = a
⊢ id x = id x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a x : α
h✝ : ¬x = a
⊢ id x = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
subst c1
|
α : Type
inst✝ : DecidableEq α
a x : α
c1 : x = a
⊢ a = id x
|
α : Type
inst✝ : DecidableEq α
x : α
⊢ x = id x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a x : α
c1 : x = a
⊢ a = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
simp
|
α : Type
inst✝ : DecidableEq α
x : α
⊢ x = id x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
x : α
⊢ x = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_id
|
[235, 1]
|
[248, 8]
|
rfl
|
α : Type
inst✝ : DecidableEq α
a x : α
h✝ : ¬x = a
⊢ id x = id x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a x : α
h✝ : ¬x = a
⊢ id x = id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm_id
|
[251, 1]
|
[264, 24]
|
funext x
|
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
⊢ updateITE (updateITE id b c) a a = updateITE id b c
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ updateITE (updateITE id b c) a a x = updateITE id b c x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
⊢ updateITE (updateITE id b c) a a = updateITE id b c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm_id
|
[251, 1]
|
[264, 24]
|
simp only [Function.updateITE]
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ updateITE (updateITE id b c) a a x = updateITE id b c x
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ (if x = a then a else if x = b then c else id x) = if x = b then c else id x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ updateITE (updateITE id b c) a a x = updateITE id b c x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm_id
|
[251, 1]
|
[264, 24]
|
simp
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ (if x = a then a else if x = b then c else id x) = if x = b then c else id x
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ x = a → a = if x = b then c else x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ (if x = a then a else if x = b then c else id x) = if x = b then c else id x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm_id
|
[251, 1]
|
[264, 24]
|
intro a1
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ x = a → a = if x = b then c else x
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
a1 : x = a
⊢ a = if x = b then c else x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
⊢ x = a → a = if x = b then c else x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm_id
|
[251, 1]
|
[264, 24]
|
subst a1
|
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
a1 : x = a
⊢ a = if x = b then c else x
|
case h
α : Type
inst✝ : DecidableEq α
b c x : α
h1 : ¬x = b
⊢ x = if x = b then c else x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
a b c : α
h1 : ¬a = b
x : α
a1 : x = a
⊢ a = if x = b then c else x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comm_id
|
[251, 1]
|
[264, 24]
|
simp only [if_neg h1]
|
case h
α : Type
inst✝ : DecidableEq α
b c x : α
h1 : ¬x = b
⊢ x = if x = b then c else x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
b c x : α
h1 : ¬x = b
⊢ x = if x = b then c else x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
funext x
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
⊢ updateITE f a (g a) = g
|
case h
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
⊢ updateITE f a (g a) x = g x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
⊢ updateITE f a (g a) = g
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
⊢ updateITE f a (g a) x = g x
|
case h
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
⊢ (if x = a then g a else f x) = g x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
⊢ updateITE f a (g a) x = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
split_ifs
|
case h
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
⊢ (if x = a then g a else f x) = g x
|
case pos
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
h✝ : x = a
⊢ g a = g x
case neg
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
h✝ : ¬x = a
⊢ f x = g x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
⊢ (if x = a then g a else f x) = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
case _ c1 =>
subst c1
rfl
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : x = a
⊢ g a = g x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : x = a
⊢ g a = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
case _ c1 =>
exact h1 x c1
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : ¬x = a
⊢ f x = g x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : ¬x = a
⊢ f x = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
subst c1
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : x = a
⊢ g a = g x
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
h1 : ∀ (x_1 : α), ¬x_1 = x → f x_1 = g x_1
⊢ g x = g x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : x = a
⊢ g a = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
rfl
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
h1 : ∀ (x_1 : α), ¬x_1 = x → f x_1 = g x_1
⊢ g x = g x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f g : α → β
x : α
h1 : ∀ (x_1 : α), ¬x_1 = x → f x_1 = g x_1
⊢ g x = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_coincide
|
[267, 1]
|
[282, 18]
|
exact h1 x c1
|
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : ¬x = a
⊢ f x = g x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f g : α → β
a : α
h1 : ∀ (x : α), ¬x = a → f x = g x
x : α
c1 : ¬x = a
⊢ f x = g x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
induction xs
|
α β : Type
inst✝ : DecidableEq α
xs : List α
f : α → β
a : α
b : β
h1 : a ∉ xs
⊢ List.map (updateITE f a b) xs = List.map f xs
|
case nil
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
h1 : a ∉ []
⊢ List.map (updateITE f a b) [] = List.map f []
case cons
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
head✝ : α
tail✝ : List α
tail_ih✝ : a ∉ tail✝ → List.map (updateITE f a b) tail✝ = List.map f tail✝
h1 : a ∉ head✝ :: tail✝
⊢ List.map (updateITE f a b) (head✝ :: tail✝) = List.map f (head✝ :: tail✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
xs : List α
f : α → β
a : α
b : β
h1 : a ∉ xs
⊢ List.map (updateITE f a b) xs = List.map f xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
case nil =>
simp
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
h1 : a ∉ []
⊢ List.map (updateITE f a b) [] = List.map f []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
h1 : a ∉ []
⊢ List.map (updateITE f a b) [] = List.map f []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
case cons hd tl ih =>
simp at h1
push_neg at h1
cases h1
case intro h1_left h1_right =>
simp
simp only [Function.updateITE]
split_ifs <;> tauto
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ∉ hd :: tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ∉ hd :: tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
simp
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
h1 : a ∉ []
⊢ List.map (updateITE f a b) [] = List.map f []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
h1 : a ∉ []
⊢ List.map (updateITE f a b) [] = List.map f []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
simp at h1
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ∉ hd :: tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : ¬a = hd ∧ a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ∉ hd :: tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
push_neg at h1
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : ¬a = hd ∧ a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ≠ hd ∧ a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : ¬a = hd ∧ a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
cases h1
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ≠ hd ∧ a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
case intro
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
left✝ : a ≠ hd
right✝ : a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1 : a ≠ hd ∧ a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
case intro h1_left h1_right =>
simp
simp only [Function.updateITE]
split_ifs <;> tauto
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
simp
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ updateITE f a b hd = f hd ∧ List.map (updateITE f a b) tl = List.map f tl
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ List.map (updateITE f a b) (hd :: tl) = List.map f (hd :: tl)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
simp only [Function.updateITE]
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ updateITE f a b hd = f hd ∧ List.map (updateITE f a b) tl = List.map f tl
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ (if hd = a then b else f hd) = f hd ∧ List.map (updateITE f a b) tl = List.map f tl
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ updateITE f a b hd = f hd ∧ List.map (updateITE f a b) tl = List.map f tl
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_list
|
[285, 1]
|
[306, 26]
|
split_ifs <;> tauto
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ (if hd = a then b else f hd) = f hd ∧ List.map (updateITE f a b) tl = List.map f tl
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
b : β
hd : α
tl : List α
ih : a ∉ tl → List.map (updateITE f a b) tl = List.map f tl
h1_left : a ≠ hd
h1_right : a ∉ tl
⊢ (if hd = a then b else f hd) = f hd ∧ List.map (updateITE f a b) tl = List.map f tl
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
induction S using Finset.induction_on
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
f : α → β
a : α
b : β
h1 : a ∉ S
⊢ Finset.image (updateITE f a b) S = Finset.image f S
|
case empty
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
h1 : a ∉ ∅
⊢ Finset.image (updateITE f a b) ∅ = Finset.image f ∅
case insert
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
a✝² : α
s✝ : Finset α
a✝¹ : a✝² ∉ s✝
a✝ : a ∉ s✝ → Finset.image (updateITE f a b) s✝ = Finset.image f s✝
h1 : a ∉ insert a✝² s✝
⊢ Finset.image (updateITE f a b) (insert a✝² s✝) = Finset.image f (insert a✝² s✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
S : Finset α
f : α → β
a : α
b : β
h1 : a ∉ S
⊢ Finset.image (updateITE f a b) S = Finset.image f S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
case empty =>
simp
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
h1 : a ∉ ∅
⊢ Finset.image (updateITE f a b) ∅ = Finset.image f ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
h1 : a ∉ ∅
⊢ Finset.image (updateITE f a b) ∅ = Finset.image f ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
simp
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
h1 : a ∉ ∅
⊢ Finset.image (updateITE f a b) ∅ = Finset.image f ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
h1 : a ∉ ∅
⊢ Finset.image (updateITE f a b) ∅ = Finset.image f ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
simp at h1
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : a ∉ insert S_a S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : ¬a = S_a ∧ a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : a ∉ insert S_a S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
push_neg at h1
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : ¬a = S_a ∧ a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : a ≠ S_a ∧ a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : ¬a = S_a ∧ a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
cases h1
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : a ≠ S_a ∧ a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
case intro
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
left✝ : a ≠ S_a
right✝ : a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1 : a ≠ S_a ∧ a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
simp
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ insert (updateITE f a b S_a) (Finset.image (updateITE f a b) S_S) = insert (f S_a) (Finset.image f S_S)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ Finset.image (updateITE f a b) (insert S_a S_S) = Finset.image f (insert S_a S_S)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
congr! 1
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ insert (updateITE f a b S_a) (Finset.image (updateITE f a b) S_S) = insert (f S_a) (Finset.image f S_S)
|
case h.e'_4
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ updateITE f a b S_a = f S_a
case h.e'_5
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ Finset.image (updateITE f a b) S_S = Finset.image f S_S
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ insert (updateITE f a b S_a) (Finset.image (updateITE f a b) S_S) = insert (f S_a) (Finset.image f S_S)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
simp only [Function.updateITE]
|
case h.e'_4
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ updateITE f a b S_a = f S_a
|
case h.e'_4
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ (if S_a = a then b else f S_a) = f S_a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ updateITE f a b S_a = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
split_ifs
|
case h.e'_4
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ (if S_a = a then b else f S_a) = f S_a
|
case pos
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
h✝ : S_a = a
⊢ b = f S_a
case neg
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
h✝ : ¬S_a = a
⊢ f S_a = f S_a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ (if S_a = a then b else f S_a) = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
case _ c1 =>
subst c1
contradiction
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : S_a = a
⊢ b = f S_a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : S_a = a
⊢ b = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
case _ c1 =>
rfl
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : ¬S_a = a
⊢ f S_a = f S_a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : ¬S_a = a
⊢ f S_a = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
subst c1
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : S_a = a
⊢ b = f S_a
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : S_a ∉ S_S → Finset.image (updateITE f S_a b) S_S = Finset.image f S_S
h1_left : S_a ≠ S_a
h1_right : S_a ∉ S_S
⊢ b = f S_a
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : S_a = a
⊢ b = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
contradiction
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : S_a ∉ S_S → Finset.image (updateITE f S_a b) S_S = Finset.image f S_S
h1_left : S_a ≠ S_a
h1_right : S_a ∉ S_S
⊢ b = f S_a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : S_a ∉ S_S → Finset.image (updateITE f S_a b) S_S = Finset.image f S_S
h1_left : S_a ≠ S_a
h1_right : S_a ∉ S_S
⊢ b = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
rfl
|
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : ¬S_a = a
⊢ f S_a = f S_a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
c1 : ¬S_a = a
⊢ f S_a = f S_a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_not_mem_set
|
[309, 1]
|
[339, 28]
|
exact S_ih h1_right
|
case h.e'_5
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ Finset.image (updateITE f a b) S_S = Finset.image f S_S
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α β : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
a : α
b : β
S_a : α
S_S : Finset α
a✝ : S_a ∉ S_S
S_ih : a ∉ S_S → Finset.image (updateITE f a b) S_S = Finset.image f S_S
h1_left : a ≠ S_a
h1_right : a ∉ S_S
⊢ Finset.image (updateITE f a b) S_S = Finset.image f S_S
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
funext x
|
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
⊢ updateITE (updateITE f a c) b d = updateITE (updateITE f b d) a c
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ updateITE (updateITE f a c) b d x = updateITE (updateITE f b d) a c x
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
⊢ updateITE (updateITE f a c) b d = updateITE (updateITE f b d) a c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
simp only [Function.updateITE]
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ updateITE (updateITE f a c) b d x = updateITE (updateITE f b d) a c x
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ updateITE (updateITE f a c) b d x = updateITE (updateITE f b d) a c x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
by_cases c1 : x = a
|
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
by_cases c2 : x = b
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
c2 : x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
c2 : ¬x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
subst c1
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
c2 : x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 : x = b
h1 : ¬x = b
⊢ (if x = b then d else if x = x then c else f x) = if x = x then c else if x = b then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
c2 : x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
subst c2
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 : x = b
h1 : ¬x = b
⊢ (if x = b then d else if x = x then c else f x) = if x = x then c else if x = b then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
c d : β
x : α
h1 : ¬x = x
⊢ (if x = x then d else if x = x then c else f x) = if x = x then c else if x = x then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 : x = b
h1 : ¬x = b
⊢ (if x = b then d else if x = x then c else f x) = if x = x then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
contradiction
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
c d : β
x : α
h1 : ¬x = x
⊢ (if x = x then d else if x = x then c else f x) = if x = x then c else if x = x then d else f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
c d : β
x : α
h1 : ¬x = x
⊢ (if x = x then d else if x = x then c else f x) = if x = x then c else if x = x then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
subst c1
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
c2 : ¬x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
⊢ (if x = b then d else if x = x then c else f x) = if x = x then c else if x = b then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : x = a
c2 : ¬x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
simp
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
⊢ (if x = b then d else if x = x then c else f x) = if x = x then c else if x = b then d else f x
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
⊢ x = b → d = c
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
⊢ (if x = b then d else if x = x then c else f x) = if x = x then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
intro a1
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
⊢ x = b → d = c
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
a1 : x = b
⊢ d = c
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
⊢ x = b → d = c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
contradiction
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
a1 : x = b
⊢ d = c
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
b : α
c d : β
x : α
c2 h1 : ¬x = b
a1 : x = b
⊢ d = c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
by_cases c2 : x = b
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
c2 : x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
c2 : ¬x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
subst c2
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
c2 : x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
c d : β
x : α
c1 : ¬x = a
h1 : ¬a = x
⊢ (if x = x then d else if x = a then c else f x) = if x = a then c else if x = x then d else f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
c2 : x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
simp
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
c d : β
x : α
c1 : ¬x = a
h1 : ¬a = x
⊢ (if x = x then d else if x = a then c else f x) = if x = a then c else if x = x then d else f x
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
c d : β
x : α
c1 : ¬x = a
h1 : ¬a = x
⊢ d = if x = a then c else d
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
c d : β
x : α
c1 : ¬x = a
h1 : ¬a = x
⊢ (if x = x then d else if x = a then c else f x) = if x = a then c else if x = x then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
simp only [if_neg c1]
|
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
c d : β
x : α
c1 : ¬x = a
h1 : ¬a = x
⊢ d = if x = a then c else d
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
f : α → β
a : α
c d : β
x : α
c1 : ¬x = a
h1 : ¬a = x
⊢ d = if x = a then c else d
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_symm
|
[342, 1]
|
[367, 28]
|
simp only [if_neg c1]
|
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
c2 : ¬x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
f : α → β
a b : α
c d : β
h1 : ¬a = b
x : α
c1 : ¬x = a
c2 : ¬x = b
⊢ (if x = b then d else if x = a then c else f x) = if x = a then c else if x = b then d else f x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
simp only [Function.updateITE]
|
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
⊢ updateITE g c d (updateITE f b c a) = updateITE (g ∘ f) b d a
|
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
⊢ updateITE g c d (updateITE f b c a) = updateITE (g ∘ f) b d a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
by_cases c1 : a = b
|
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
|
case pos
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : a = b
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
simp only [c1]
|
case pos
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : a = b
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
|
case pos
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : a = b
⊢ (if (if True then c else f b) = c then d else g (if True then c else f b)) = if True then d else (g ∘ f) b
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : a = b
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
simp
|
case pos
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : a = b
⊢ (if (if True then c else f b) = c then d else g (if True then c else f b)) = if True then d else (g ∘ f) b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : a = b
⊢ (if (if True then c else f b) = c then d else g (if True then c else f b)) = if True then d else (g ∘ f) b
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
simp only [c1]
|
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
|
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ (if (if False then c else f a) = c then d else g (if False then c else f a)) = if False then d else (g ∘ f) a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ (if (if a = b then c else f a) = c then d else g (if a = b then c else f a)) = if a = b then d else (g ∘ f) a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
simp
|
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ (if (if False then c else f a) = c then d else g (if False then c else f a)) = if False then d else (g ∘ f) a
|
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ f a = c → d = g (f a)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ (if (if False then c else f a) = c then d else g (if False then c else f a)) = if False then d else (g ∘ f) a
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateITE_comp
|
[370, 1]
|
[386, 13]
|
exact h1
|
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ f a = c → d = g (f a)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α β : Type
inst✝ : DecidableEq α
a b c : α
d : β
f : α → α
g : α → β
h1 : f a = c → d = g (f a)
c1 : ¬a = b
⊢ f a = c → d = g (f a)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/FunctionUpdateITE.lean
|
Function.updateListITE_comp
|
[391, 1]
|
[413, 41]
|
induction xs generalizing ys
|
α β γ : Type
inst✝ : DecidableEq α
f : α → β
g : β → γ
xs : List α
ys : List β
⊢ g ∘ updateListITE f xs ys = updateListITE (g ∘ f) xs (List.map g ys)
|
case nil
α β γ : Type
inst✝ : DecidableEq α
f : α → β
g : β → γ
ys : List β
⊢ g ∘ updateListITE f [] ys = updateListITE (g ∘ f) [] (List.map g ys)
case cons
α β γ : Type
inst✝ : DecidableEq α
f : α → β
g : β → γ
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ (ys : List β), g ∘ updateListITE f tail✝ ys = updateListITE (g ∘ f) tail✝ (List.map g ys)
ys : List β
⊢ g ∘ updateListITE f (head✝ :: tail✝) ys = updateListITE (g ∘ f) (head✝ :: tail✝) (List.map g ys)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α β γ : Type
inst✝ : DecidableEq α
f : α → β
g : β → γ
xs : List α
ys : List β
⊢ g ∘ updateListITE f xs ys = updateListITE (g ∘ f) xs (List.map g ys)
TACTIC:
|
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