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/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
induction xs
|
D : Type
V : VarAssignment D
xs : List String
⊢ V ∘ Var.openList 0 (List.map free_ xs) = shiftList D V (List.map (V ∘ free_) xs)
|
case nil
D : Type
V : VarAssignment D
⊢ V ∘ Var.openList 0 (List.map free_ []) = shiftList D V (List.map (V ∘ free_) [])
case cons
D : Type
V : VarAssignment D
head✝ : String
tail✝ : List String
tail_ih✝ : V ∘ Var.openList 0 (List.map free_ tail✝) = shiftList D V (List.map (V ∘ free_) tail✝)
⊢ V ∘ Var.openList 0 (List.map free_ (head✝ :: tail✝)) = shiftList D V (List.map (V ∘ free_) (head✝ :: tail✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
xs : List String
⊢ V ∘ Var.openList 0 (List.map free_ xs) = shiftList D V (List.map (V ∘ free_) xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case nil =>
funext v
simp
simp only [shiftList]
cases v
case _ x =>
simp only [Var.openList]
case _ i =>
simp only [Var.openList]
split
case _ c1 =>
rfl
case _ c1 =>
simp
|
D : Type
V : VarAssignment D
⊢ V ∘ Var.openList 0 (List.map free_ []) = shiftList D V (List.map (V ∘ free_) [])
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
⊢ V ∘ Var.openList 0 (List.map free_ []) = shiftList D V (List.map (V ∘ free_) [])
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ hd tl ih =>
funext v
simp
simp only [shiftList]
cases v
case _ x =>
simp only [shift]
simp only [← ih]
simp only [Var.openList]
simp
case _ i =>
cases i
case zero =>
simp only [shift]
simp only [Var.openList]
simp
case succ i =>
simp only [shift]
simp only [← ih]
simp
simp only [Var.openList]
simp
split
case _ c1 =>
have s1 : i < tl.length
linarith
simp only [s1]
simp
case _ c1 =>
have s1 : ¬ i < tl.length
linarith
simp only [s1]
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V ∘ Var.openList 0 (List.map free_ (hd :: tl)) = shiftList D V (List.map (V ∘ free_) (hd :: tl))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V ∘ Var.openList 0 (List.map free_ (hd :: tl)) = shiftList D V (List.map (V ∘ free_) (hd :: tl))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
funext v
|
D : Type
V : VarAssignment D
⊢ V ∘ Var.openList 0 (List.map free_ []) = shiftList D V (List.map (V ∘ free_) [])
|
case h
D : Type
V : VarAssignment D
v : Var
⊢ (V ∘ Var.openList 0 (List.map free_ [])) v = shiftList D V (List.map (V ∘ free_) []) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
⊢ V ∘ Var.openList 0 (List.map free_ []) = shiftList D V (List.map (V ∘ free_) [])
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
case h
D : Type
V : VarAssignment D
v : Var
⊢ (V ∘ Var.openList 0 (List.map free_ [])) v = shiftList D V (List.map (V ∘ free_) []) v
|
case h
D : Type
V : VarAssignment D
v : Var
⊢ V (Var.openList 0 [] v) = shiftList D V [] v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
V : VarAssignment D
v : Var
⊢ (V ∘ Var.openList 0 (List.map free_ [])) v = shiftList D V (List.map (V ∘ free_) []) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [shiftList]
|
case h
D : Type
V : VarAssignment D
v : Var
⊢ V (Var.openList 0 [] v) = shiftList D V [] v
|
case h
D : Type
V : VarAssignment D
v : Var
⊢ V (Var.openList 0 [] v) = V v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
V : VarAssignment D
v : Var
⊢ V (Var.openList 0 [] v) = shiftList D V [] v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
cases v
|
case h
D : Type
V : VarAssignment D
v : Var
⊢ V (Var.openList 0 [] v) = V v
|
case h.free_
D : Type
V : VarAssignment D
a✝ : String
⊢ V (Var.openList 0 [] (free_ a✝)) = V (free_ a✝)
case h.bound_
D : Type
V : VarAssignment D
a✝ : ℕ
⊢ V (Var.openList 0 [] (bound_ a✝)) = V (bound_ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
V : VarAssignment D
v : Var
⊢ V (Var.openList 0 [] v) = V v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ x =>
simp only [Var.openList]
|
D : Type
V : VarAssignment D
x : String
⊢ V (Var.openList 0 [] (free_ x)) = V (free_ x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
x : String
⊢ V (Var.openList 0 [] (free_ x)) = V (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ i =>
simp only [Var.openList]
split
case _ c1 =>
rfl
case _ c1 =>
simp
|
D : Type
V : VarAssignment D
i : ℕ
⊢ V (Var.openList 0 [] (bound_ i)) = V (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
⊢ V (Var.openList 0 [] (bound_ i)) = V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [Var.openList]
|
D : Type
V : VarAssignment D
x : String
⊢ V (Var.openList 0 [] (free_ x)) = V (free_ x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
x : String
⊢ V (Var.openList 0 [] (free_ x)) = V (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [Var.openList]
|
D : Type
V : VarAssignment D
i : ℕ
⊢ V (Var.openList 0 [] (bound_ i)) = V (bound_ i)
|
D : Type
V : VarAssignment D
i : ℕ
⊢ V (if i < 0 then bound_ i else if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) =
V (bound_ i)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
⊢ V (Var.openList 0 [] (bound_ i)) = V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
split
|
D : Type
V : VarAssignment D
i : ℕ
⊢ V (if i < 0 then bound_ i else if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) =
V (bound_ i)
|
case inl
D : Type
V : VarAssignment D
i : ℕ
h✝ : i < 0
⊢ V (bound_ i) = V (bound_ i)
case inr
D : Type
V : VarAssignment D
i : ℕ
h✝ : ¬i < 0
⊢ V (if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) = V (bound_ i)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
⊢ V (if i < 0 then bound_ i else if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) =
V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ c1 =>
rfl
|
D : Type
V : VarAssignment D
i : ℕ
c1 : i < 0
⊢ V (bound_ i) = V (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
c1 : i < 0
⊢ V (bound_ i) = V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ c1 =>
simp
|
D : Type
V : VarAssignment D
i : ℕ
c1 : ¬i < 0
⊢ V (if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) = V (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
c1 : ¬i < 0
⊢ V (if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) = V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
rfl
|
D : Type
V : VarAssignment D
i : ℕ
c1 : i < 0
⊢ V (bound_ i) = V (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
c1 : i < 0
⊢ V (bound_ i) = V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
D : Type
V : VarAssignment D
i : ℕ
c1 : ¬i < 0
⊢ V (if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) = V (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
i : ℕ
c1 : ¬i < 0
⊢ V (if x : i - 0 < [].length then [][i - 0] else bound_ (i - 0 - [].length + 0)) = V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
funext v
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V ∘ Var.openList 0 (List.map free_ (hd :: tl)) = shiftList D V (List.map (V ∘ free_) (hd :: tl))
|
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ (V ∘ Var.openList 0 (List.map free_ (hd :: tl))) v = shiftList D V (List.map (V ∘ free_) (hd :: tl)) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V ∘ Var.openList 0 (List.map free_ (hd :: tl)) = shiftList D V (List.map (V ∘ free_) (hd :: tl))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ (V ∘ Var.openList 0 (List.map free_ (hd :: tl))) v = shiftList D V (List.map (V ∘ free_) (hd :: tl)) v
|
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) v) = shiftList D V (V (free_ hd) :: List.map (V ∘ free_) tl) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ (V ∘ Var.openList 0 (List.map free_ (hd :: tl))) v = shiftList D V (List.map (V ∘ free_) (hd :: tl)) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [shiftList]
|
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) v) = shiftList D V (V (free_ hd) :: List.map (V ∘ free_) tl) v
|
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) v) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) v) = shiftList D V (V (free_ hd) :: List.map (V ∘ free_) tl) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
cases v
|
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) v) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) v
|
case h.free_
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
a✝ : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ a✝)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (free_ a✝)
case h.bound_
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
a✝ : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ a✝)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
v : Var
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) v) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ x =>
simp only [shift]
simp only [← ih]
simp only [Var.openList]
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (free_ x)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (free_ x) = V (Var.openList 0 (List.map free_ tl) (free_ x))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case _ i =>
cases i
case zero =>
simp only [shift]
simp only [Var.openList]
simp
case succ i =>
simp only [shift]
simp only [← ih]
simp
simp only [Var.openList]
simp
split
case _ c1 =>
have s1 : i < tl.length
linarith
simp only [s1]
simp
case _ c1 =>
have s1 : ¬ i < tl.length
linarith
simp only [s1]
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ i)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ i)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [shift]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (free_ x)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) = shiftList D V (List.map (V ∘ free_) tl) (free_ x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [← ih]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) = shiftList D V (List.map (V ∘ free_) tl) (free_ x)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) = (V ∘ Var.openList 0 (List.map free_ tl)) (free_ x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) = shiftList D V (List.map (V ∘ free_) tl) (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [Var.openList]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) = (V ∘ Var.openList 0 (List.map free_ tl)) (free_ x)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (free_ x) = (V ∘ Var.openList 0 (List.map free_ tl)) (free_ x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (free_ x)) = (V ∘ Var.openList 0 (List.map free_ tl)) (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (free_ x) = (V ∘ Var.openList 0 (List.map free_ tl)) (free_ x)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (free_ x) = V (Var.openList 0 (List.map free_ tl) (free_ x))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
x : String
⊢ V (free_ x) = (V ∘ Var.openList 0 (List.map free_ tl)) (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
cases i
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ i)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ i)
|
case zero
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ 0)
case succ
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
n✝ : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (n✝ + 1))) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ (n✝ + 1))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ i)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case zero =>
simp only [shift]
simp only [Var.openList]
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ 0)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
case succ i =>
simp only [shift]
simp only [← ih]
simp
simp only [Var.openList]
simp
split
case _ c1 =>
have s1 : i < tl.length
linarith
simp only [s1]
simp
case _ c1 =>
have s1 : ¬ i < tl.length
linarith
simp only [s1]
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ (i + 1))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ (i + 1))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [shift]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ 0)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) = V (free_ hd)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ 0)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [Var.openList]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) = V (free_ hd)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V
(if 0 < 0 then bound_ 0
else
if x : 0 - 0 < (free_ hd :: List.map free_ tl).length then (free_ hd :: List.map free_ tl)[0 - 0]
else bound_ (0 - 0 - (free_ hd :: List.map free_ tl).length + 0)) =
V (free_ hd)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ 0)) = V (free_ hd)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V
(if 0 < 0 then bound_ 0
else
if x : 0 - 0 < (free_ hd :: List.map free_ tl).length then (free_ hd :: List.map free_ tl)[0 - 0]
else bound_ (0 - 0 - (free_ hd :: List.map free_ tl).length + 0)) =
V (free_ hd)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
⊢ V
(if 0 < 0 then bound_ 0
else
if x : 0 - 0 < (free_ hd :: List.map free_ tl).length then (free_ hd :: List.map free_ tl)[0 - 0]
else bound_ (0 - 0 - (free_ hd :: List.map free_ tl).length + 0)) =
V (free_ hd)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [shift]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ (i + 1))
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shiftList D V (List.map (V ∘ free_) tl) (bound_ i)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shift D (shiftList D V (List.map (V ∘ free_) tl)) (V (free_ hd)) (bound_ (i + 1))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [← ih]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shiftList D V (List.map (V ∘ free_) tl) (bound_ i)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
(V ∘ Var.openList 0 (List.map free_ tl)) (bound_ i)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
shiftList D V (List.map (V ∘ free_) tl) (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
(V ∘ Var.openList 0 (List.map free_ tl)) (bound_ i)
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
V (Var.openList 0 (List.map free_ tl) (bound_ i))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
(V ∘ Var.openList 0 (List.map free_ tl)) (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp only [Var.openList]
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
V (Var.openList 0 (List.map free_ tl) (bound_ i))
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < (free_ hd :: List.map free_ tl).length then (free_ hd :: List.map free_ tl)[i + 1 - 0]
else bound_ (i + 1 - 0 - (free_ hd :: List.map free_ tl).length + 0)) =
V
(if i < 0 then bound_ i
else
if x : i - 0 < (List.map free_ tl).length then (List.map free_ tl)[i - 0]
else bound_ (i - 0 - (List.map free_ tl).length + 0))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V (Var.openList 0 (free_ hd :: List.map free_ tl) (bound_ (i + 1))) =
V (Var.openList 0 (List.map free_ tl) (bound_ i))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.ShiftListVarOpenList
|
[891, 1]
|
[945, 15]
|
simp
|
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < (free_ hd :: List.map free_ tl).length then (free_ hd :: List.map free_ tl)[i + 1 - 0]
else bound_ (i + 1 - 0 - (free_ hd :: List.map free_ tl).length + 0)) =
V
(if i < 0 then bound_ i
else
if x : i - 0 < (List.map free_ tl).length then (List.map free_ tl)[i - 0]
else bound_ (i - 0 - (List.map free_ tl).length + 0))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
hd : String
tl : List String
ih : V ∘ Var.openList 0 (List.map free_ tl) = shiftList D V (List.map (V ∘ free_) tl)
i : ℕ
⊢ V
(if i + 1 < 0 then bound_ (i + 1)
else
if x : i + 1 - 0 < (free_ hd :: List.map free_ tl).length then (free_ hd :: List.map free_ tl)[i + 1 - 0]
else bound_ (i + 1 - 0 - (free_ hd :: List.map free_ tl).length + 0)) =
V
(if i < 0 then bound_ i
else
if x : i - 0 < (List.map free_ tl).length then (List.map free_ tl)[i - 0]
else bound_ (i - 0 - (List.map free_ tl).length + 0))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_iff_lc
|
[948, 1]
|
[956, 10]
|
constructor
|
F : Formula
⊢ Formula.lc_at 0 F ↔ F.lc
|
case mp
F : Formula
⊢ Formula.lc_at 0 F → F.lc
case mpr
F : Formula
⊢ F.lc → Formula.lc_at 0 F
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
⊢ Formula.lc_at 0 F ↔ F.lc
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_iff_lc
|
[948, 1]
|
[956, 10]
|
intro a1
|
case mp
F : Formula
⊢ Formula.lc_at 0 F → F.lc
|
case mp
F : Formula
a1 : Formula.lc_at 0 F
⊢ F.lc
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
F : Formula
⊢ Formula.lc_at 0 F → F.lc
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_iff_lc
|
[948, 1]
|
[956, 10]
|
sorry
|
case mp
F : Formula
a1 : Formula.lc_at 0 F
⊢ F.lc
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
F : Formula
a1 : Formula.lc_at 0 F
⊢ F.lc
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_iff_lc
|
[948, 1]
|
[956, 10]
|
intro a1
|
case mpr
F : Formula
⊢ F.lc → Formula.lc_at 0 F
|
case mpr
F : Formula
a1 : F.lc
⊢ Formula.lc_at 0 F
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
F : Formula
⊢ F.lc → Formula.lc_at 0 F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_iff_lc
|
[948, 1]
|
[956, 10]
|
sorry
|
case mpr
F : Formula
a1 : F.lc
⊢ Formula.lc_at 0 F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
F : Formula
a1 : F.lc
⊢ Formula.lc_at 0 F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
induction F generalizing V V'
|
D : Type
I : Interpretation D
V V' : VarAssignment D
F : Formula
h1 : ∀ (v : Var), occursFreeIn v F → V v = V' v
⊢ Holds D I V F ↔ Holds D I V' F
|
case pred_
D : Type
I : Interpretation D
a✝¹ : String
a✝ : List Var
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (pred_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (pred_ a✝¹ a✝) ↔ Holds D I V' (pred_ a✝¹ a✝)
case not_
D : Type
I : Interpretation D
a✝ : Formula
a_ih✝ : ∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v a✝ → V v = V' v) → (Holds D I V a✝ ↔ Holds D I V' a✝)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v a✝.not_ → V v = V' v
⊢ Holds D I V a✝.not_ ↔ Holds D I V' a✝.not_
case imp_
D : Type
I : Interpretation D
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v a✝¹ → V v = V' v) → (Holds D I V a✝¹ ↔ Holds D I V' a✝¹)
a_ih✝ : ∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v a✝ → V v = V' v) → (Holds D I V a✝ ↔ Holds D I V' a✝)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (a✝¹.imp_ a✝) → V v = V' v
⊢ Holds D I V (a✝¹.imp_ a✝) ↔ Holds D I V' (a✝¹.imp_ a✝)
case forall_
D : Type
I : Interpretation D
a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v a✝ → V v = V' v) → (Holds D I V a✝ ↔ Holds D I V' a✝)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (forall_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (forall_ a✝¹ a✝) ↔ Holds D I V' (forall_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V V' : VarAssignment D
F : Formula
h1 : ∀ (v : Var), occursFreeIn v F → V v = V' v
⊢ Holds D I V F ↔ Holds D I V' F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case pred_ X vs =>
simp only [occursFreeIn] at h1
simp only [Holds]
congr! 1
simp only [List.map_eq_map_iff]
exact h1
|
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (pred_ X vs) → V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (pred_ X vs) → V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case not_ phi phi_ih =>
simp only [occursFreeIn] at h1
simp only [Holds]
congr! 1
exact phi_ih V V' h1
|
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi.not_ → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi.not_ → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case forall_ x phi phi_ih =>
simp only [occursFreeIn] at h1
simp only [Holds]
apply forall_congr'
intro d
apply phi_ih
intro v a1
cases v
case free_ x =>
simp only [shift]
apply h1
exact a1
case bound_ i =>
cases i
case zero =>
simp only [shift]
case succ i =>
simp only [shift]
apply h1
simp only [lift]
exact a1
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (forall_ x phi) → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (forall_ x phi) → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [occursFreeIn] at h1
|
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (pred_ X vs) → V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
|
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (pred_ X vs) → V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
|
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ I.pred_ X (List.map V vs) ↔ I.pred_ X (List.map V' vs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V (pred_ X vs) ↔ Holds D I V' (pred_ X vs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
congr! 1
|
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ I.pred_ X (List.map V vs) ↔ I.pred_ X (List.map V' vs)
|
case a.h.e'_4
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ List.map V vs = List.map V' vs
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ I.pred_ X (List.map V vs) ↔ I.pred_ X (List.map V' vs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [List.map_eq_map_iff]
|
case a.h.e'_4
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ List.map V vs = List.map V' vs
|
case a.h.e'_4
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ ∀ x ∈ vs, V x = V' x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ List.map V vs = List.map V' vs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
exact h1
|
case a.h.e'_4
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ ∀ x ∈ vs, V x = V' x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
X : String
vs : List Var
V V' : VarAssignment D
h1 : ∀ v ∈ vs, V v = V' v
⊢ ∀ x ∈ vs, V x = V' x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [occursFreeIn] at h1
|
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi.not_ → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
|
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi.not_ → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
|
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ ¬Holds D I V phi ↔ ¬Holds D I V' phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ Holds D I V phi.not_ ↔ Holds D I V' phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
congr! 1
|
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ ¬Holds D I V phi ↔ ¬Holds D I V' phi
|
case a.h.e'_1.a
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ Holds D I V phi ↔ Holds D I V' phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ ¬Holds D I V phi ↔ ¬Holds D I V' phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
exact phi_ih V V' h1
|
case a.h.e'_1.a
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ Holds D I V phi ↔ Holds D I V' phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v phi → V v = V' v
⊢ Holds D I V phi ↔ Holds D I V' phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I V' (phi.imp_ psi)
|
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V phi → Holds D I V psi ↔ Holds D I V' phi → Holds D I V' psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V (phi.imp_ psi) ↔ Holds D I V' (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
congr! 1
|
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V phi → Holds D I V psi ↔ Holds D I V' phi → Holds D I V' psi
|
case a.h.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V phi ↔ Holds D I V' phi
case a.h'.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
⊢ Holds D I V psi ↔ Holds D I V' psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V phi → Holds D I V psi ↔ Holds D I V' phi → Holds D I V' psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply phi_ih
|
case a.h.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V phi ↔ Holds D I V' phi
|
case a.h.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ ∀ (v : Var), occursFreeIn v phi → V v = V' v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ Holds D I V phi ↔ Holds D I V' phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
intro v a1
|
case a.h.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ ∀ (v : Var), occursFreeIn v phi → V v = V' v
|
case a.h.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ V v = V' v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
⊢ ∀ (v : Var), occursFreeIn v phi → V v = V' v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply h1
|
case a.h.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ V v = V' v
|
case a.h.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ occursFreeIn v (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ V v = V' v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [occursFreeIn]
|
case a.h.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ occursFreeIn v (phi.imp_ psi)
|
case a.h.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ occursFreeIn v (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
tauto
|
case a.h.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
v : Var
a1 : occursFreeIn v phi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply psi_ih
|
case a.h'.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
⊢ Holds D I V psi ↔ Holds D I V' psi
|
case a.h'.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
⊢ ∀ (v : Var), occursFreeIn v psi → V v = V' v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h'.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
⊢ Holds D I V psi ↔ Holds D I V' psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
intro v a1
|
case a.h'.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
⊢ ∀ (v : Var), occursFreeIn v psi → V v = V' v
|
case a.h'.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ V v = V' v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h'.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
⊢ ∀ (v : Var), occursFreeIn v psi → V v = V' v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply h1
|
case a.h'.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ V v = V' v
|
case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v (phi.imp_ psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h'.a.h1
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ V v = V' v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [occursFreeIn]
|
case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v (phi.imp_ psi)
|
case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v (phi.imp_ psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
tauto
|
case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h'.a.h1.a
D : Type
I : Interpretation D
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
psi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v psi → V v = V' v) → (Holds D I V psi ↔ Holds D I V' psi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (phi.imp_ psi) → V v = V' v
a✝ : Holds D I V' phi
v : Var
a1 : occursFreeIn v psi
⊢ occursFreeIn v phi ∨ occursFreeIn v psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [occursFreeIn] at h1
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (forall_ x phi) → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn v (forall_ x phi) → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔ ∀ (d : D), Holds D I (shift D V' d) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ Holds D I V (forall_ x phi) ↔ Holds D I V' (forall_ x phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply forall_congr'
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔ ∀ (d : D), Holds D I (shift D V' d) phi
|
case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ ∀ (a : D), Holds D I (shift D V a) phi ↔ Holds D I (shift D V' a) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ (∀ (d : D), Holds D I (shift D V d) phi) ↔ ∀ (d : D), Holds D I (shift D V' d) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
intro d
|
case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ ∀ (a : D), Holds D I (shift D V a) phi ↔ Holds D I (shift D V' a) phi
|
case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ Holds D I (shift D V d) phi ↔ Holds D I (shift D V' d) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
⊢ ∀ (a : D), Holds D I (shift D V a) phi ↔ Holds D I (shift D V' a) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply phi_ih
|
case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ Holds D I (shift D V d) phi ↔ Holds D I (shift D V' d) phi
|
case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ ∀ (v : Var), occursFreeIn v phi → shift D V d v = shift D V' d v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ Holds D I (shift D V d) phi ↔ Holds D I (shift D V' d) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
intro v a1
|
case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ ∀ (v : Var), occursFreeIn v phi → shift D V d v = shift D V' d v
|
case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
v : Var
a1 : occursFreeIn v phi
⊢ shift D V d v = shift D V' d v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
⊢ ∀ (v : Var), occursFreeIn v phi → shift D V d v = shift D V' d v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
cases v
|
case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
v : Var
a1 : occursFreeIn v phi
⊢ shift D V d v = shift D V' d v
|
case h.h1.free_
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a✝ : String
a1 : occursFreeIn (free_ a✝) phi
⊢ shift D V d (free_ a✝) = shift D V' d (free_ a✝)
case h.h1.bound_
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a✝ : ℕ
a1 : occursFreeIn (bound_ a✝) phi
⊢ shift D V d (bound_ a✝) = shift D V' d (bound_ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
v : Var
a1 : occursFreeIn v phi
⊢ shift D V d v = shift D V' d v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case free_ x =>
simp only [shift]
apply h1
exact a1
|
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ shift D V d (free_ x) = shift D V' d (free_ x)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ shift D V d (free_ x) = shift D V' d (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case bound_ i =>
cases i
case zero =>
simp only [shift]
case succ i =>
simp only [shift]
apply h1
simp only [lift]
exact a1
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) phi
⊢ shift D V d (bound_ i) = shift D V' d (bound_ i)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) phi
⊢ shift D V d (bound_ i) = shift D V' d (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [shift]
|
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ shift D V d (free_ x) = shift D V' d (free_ x)
|
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ V (free_ x) = V' (free_ x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ shift D V d (free_ x) = shift D V' d (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply h1
|
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ V (free_ x) = V' (free_ x)
|
case a
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ occursFreeIn (lift (free_ x)) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ V (free_ x) = V' (free_ x)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
exact a1
|
case a
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ occursFreeIn (lift (free_ x)) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
x✝ : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
x : String
a1 : occursFreeIn (free_ x) phi
⊢ occursFreeIn (lift (free_ x)) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
cases i
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) phi
⊢ shift D V d (bound_ i) = shift D V' d (bound_ i)
|
case zero
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0)
case succ
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
n✝ : ℕ
a1 : occursFreeIn (bound_ (n✝ + 1)) phi
⊢ shift D V d (bound_ (n✝ + 1)) = shift D V' d (bound_ (n✝ + 1))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) phi
⊢ shift D V d (bound_ i) = shift D V' d (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case zero =>
simp only [shift]
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
case succ i =>
simp only [shift]
apply h1
simp only [lift]
exact a1
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ shift D V d (bound_ (i + 1)) = shift D V' d (bound_ (i + 1))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ shift D V d (bound_ (i + 1)) = shift D V' d (bound_ (i + 1))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [shift]
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
a1 : occursFreeIn (bound_ 0) phi
⊢ shift D V d (bound_ 0) = shift D V' d (bound_ 0)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [shift]
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ shift D V d (bound_ (i + 1)) = shift D V' d (bound_ (i + 1))
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ V (bound_ i) = V' (bound_ i)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ shift D V d (bound_ (i + 1)) = shift D V' d (bound_ (i + 1))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
apply h1
|
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ V (bound_ i) = V' (bound_ i)
|
case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (lift (bound_ i)) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ V (bound_ i) = V' (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
simp only [lift]
|
case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (lift (bound_ i)) phi
|
case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (bound_ (i + 1)) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (lift (bound_ i)) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsCoincideVar
|
[959, 1]
|
[1015, 17]
|
exact a1
|
case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (bound_ (i + 1)) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
x : String
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D), (∀ (v : Var), occursFreeIn v phi → V v = V' v) → (Holds D I V phi ↔ Holds D I V' phi)
V V' : VarAssignment D
h1 : ∀ (v : Var), occursFreeIn (lift v) phi → V v = V' v
d : D
i : ℕ
a1 : occursFreeIn (bound_ (i + 1)) phi
⊢ occursFreeIn (bound_ (i + 1)) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
apply forall_congr'
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (shift D V d) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F
|
case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D), Holds D I (shift D V a) F ↔ Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ (∀ (d : D), Holds D I (shift D V d) F) ↔
∀ (d : D), Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
intro d
|
case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D), Holds D I (shift D V a) F ↔ Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F
|
case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (shift D V d) F ↔ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
⊢ ∀ (a : D), Holds D I (shift D V a) F ↔ Holds D I (Function.updateITE V (free_ z) a ∘ Var.openList 0 [free_ z]) F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
apply HoldsCoincideVar
|
case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (shift D V d) F ↔ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ ∀ (v : Var), occursFreeIn v F → shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ Holds D I (shift D V d) F ↔ Holds D I (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
intro v a1
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ ∀ (v : Var), occursFreeIn v F → shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
⊢ ∀ (v : Var), occursFreeIn v F → shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
simp
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = Function.updateITE V (free_ z) d (Var.openList 0 [free_ z] v)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = (Function.updateITE V (free_ z) d ∘ Var.openList 0 [free_ z]) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
simp only [Function.updateITE]
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = Function.updateITE V (free_ z) d (Var.openList 0 [free_ z] v)
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = if Var.openList 0 [free_ z] v = free_ z then d else V (Var.openList 0 [free_ z] v)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = Function.updateITE V (free_ z) d (Var.openList 0 [free_ z] v)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
cases v
|
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = if Var.openList 0 [free_ z] v = free_ z then d else V (Var.openList 0 [free_ z] v)
|
case h.h1.free_
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a✝ : String
a1 : occursFreeIn (free_ a✝) F
⊢ shift D V d (free_ a✝) =
if Var.openList 0 [free_ z] (free_ a✝) = free_ z then d else V (Var.openList 0 [free_ z] (free_ a✝))
case h.h1.bound_
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
a✝ : ℕ
a1 : occursFreeIn (bound_ a✝) F
⊢ shift D V d (bound_ a✝) =
if Var.openList 0 [free_ z] (bound_ a✝) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ a✝))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
v : Var
a1 : occursFreeIn v F
⊢ shift D V d v = if Var.openList 0 [free_ z] v = free_ z then d else V (Var.openList 0 [free_ z] v)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
case _ x' =>
simp only [shift]
simp only [Var.openList]
split_ifs
case _ c1 =>
simp at c1
simp only [← c1] at h1
contradiction
case _ c1 =>
rfl
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ shift D V d (free_ x') =
if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ shift D V d (free_ x') =
if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
case _ i =>
cases i
case zero =>
simp only [shift]
simp only [Var.openList]
simp
case succ i =>
simp only [shift]
simp only [Var.openList]
simp
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) F
⊢ shift D V d (bound_ i) =
if Var.openList 0 [free_ z] (bound_ i) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ i))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
i : ℕ
a1 : occursFreeIn (bound_ i) F
⊢ shift D V d (bound_ i) =
if Var.openList 0 [free_ z] (bound_ i) = free_ z then d else V (Var.openList 0 [free_ z] (bound_ i))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
simp only [shift]
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ shift D V d (free_ x') =
if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ shift D V d (free_ x') =
if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
simp only [Var.openList]
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if free_ x' = free_ z then d else V (free_ x')
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if Var.openList 0 [free_ z] (free_ x') = free_ z then d else V (Var.openList 0 [free_ z] (free_ x'))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
split_ifs
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if free_ x' = free_ z then d else V (free_ x')
|
case pos
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
h✝ : free_ x' = free_ z
⊢ V (free_ x') = d
case neg
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
h✝ : ¬free_ x' = free_ z
⊢ V (free_ x') = V (free_ x')
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
⊢ V (free_ x') = if free_ x' = free_ z then d else V (free_ x')
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.HoldsShift
|
[1018, 1]
|
[1053, 11]
|
case _ c1 =>
simp at c1
simp only [← c1] at h1
contradiction
|
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : free_ x' = free_ z
⊢ V (free_ x') = d
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
F : Formula
z : String
h1 : ¬occursFreeIn (free_ z) F
d : D
x' : String
a1 : occursFreeIn (free_ x') F
c1 : free_ x' = free_ z
⊢ V (free_ x') = d
TACTIC:
|
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