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.ShiftVarOpenList | [815, 1] | [854, 17] | cases i | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ i) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ i)) | case zero
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ 0) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ 0))
case succ
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
n✝ : ℕ
⊢ shift D (V ∘ Var.openList j (List.map ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ i) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ i))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | case zero =>
simp only [shift]
simp only [Var.openList]
simp | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ 0) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ 0)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ 0) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ 0))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | case succ i =>
simp only [shift]
simp only [Var.openList]
simp
split
case _ c1 =>
have s1 : i + 1 < j + 1
linarith
simp only [if_pos s1]
case _ c1 =>
have s1 : ¬ i + 1 < j + 1
linarith
simp only [if_neg s1]
split
case _ c2 =>
simp
cas... | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ (i + 1)) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1))) | case h_3
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
i✝ : ℕ
heq✝ :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ i✝.succ
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = V (bound_... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ (i + 1)) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1)))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ 0) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ 0)) | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ d =
match Var.openList (j + 1) (List.map free_ zs) (bound_ 0) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ 0) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ 0))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | simp only [Var.openList] | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ d =
match Var.openList (j + 1) (List.map free_ zs) (bound_ 0) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ d =
match
if 0 < j + 1 then bound_ 0
else
if x : 0 - (j + 1) < (List.map free_ zs).length then (List.map free_ zs)[0 - (j + 1)]
else bound_ (0 - (j + 1) - (List.map free_ zs).length + (j + 1)) with
| free_ x => V (free_ x)
... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ d =
match Var.openList (j + 1) (List.map free_ zs) (bound_ 0) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | simp | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ d =
match
if 0 < j + 1 then bound_ 0
else
if x : 0 - (j + 1) < (List.map free_ zs).length then (List.map free_ zs)[0 - (j + 1)]
else bound_ (0 - (j + 1) - (List.map free_ zs).length + (j + 1)) with
| free_ x => V (free_ x)
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
⊢ d =
match
if 0 < j + 1 then bound_ 0
else
if x : 0 - (j + 1) < (List.map free_ zs).length then (List.map free_ zs)[0 - (j + 1)]
else bound_ (0 - (j + 1) - (List.map fr... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | simp only [shift] | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ (i + 1)) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1))) | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ (V ∘ Var.openList j (List.map free_ zs)) (bound_ i) =
match Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1)) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ shift D (V ∘ Var.openList j (List.map free_ zs)) d (bound_ (i + 1)) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1)))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | simp only [Var.openList] | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ (V ∘ Var.openList j (List.map free_ zs)) (bound_ i) =
match Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1)) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ (V ∘ Var.openList j (List.map free_ zs)) (bound_ i) =
match
if i + 1 < j + 1 then bound_ (i + 1)
else
if x : i + 1 - (j + 1) < (List.map free_ zs).length then (List.map free_ zs)[i + 1 - (j + 1)]
else bound_ (i + 1 - (j + ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ (V ∘ Var.openList j (List.map free_ zs)) (bound_ i) =
match Var.openList (j + 1) (List.map free_ zs) (bound_ (i + 1)) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.su... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | simp | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ (V ∘ Var.openList j (List.map free_ zs)) (bound_ i) =
match
if i + 1 < j + 1 then bound_ (i + 1)
else
if x : i + 1 - (j + 1) < (List.map free_ zs).length then (List.map free_ zs)[i + 1 - (j + 1)]
else bound_ (i + 1 - (j + ... | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) =
match
if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1)) with
| free_ x => V (free_ x)
| bound_ 0... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ (V ∘ Var.openList j (List.map free_ zs)) (bound_ i) =
match
if i + 1 < j + 1 then bound_ (i + 1)
else
if x : i + 1 - (j + 1) < (List.map free_ zs).length then (List.map ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | split | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) =
match
if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1)) with
| free_ x => V (free_ x)
| bound_ 0... | case h_1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
heq✝ :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = V (free_ ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) =
match
if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.lengt... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | case _ c1 =>
have s1 : i + 1 < j + 1
linarith
simp only [if_pos s1] | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = V (free_ x✝) | case s1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ i + 1 < j + 1
D : Type
V : VarAssignment D
j : ℕ
zs : List Stri... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ V (Var.open... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | case _ c1 =>
have s1 : ¬ i + 1 < j + 1
linarith
simp only [if_neg s1]
split
case _ c2 =>
simp
case _ c2 =>
simp | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ 0
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = d | case s1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ 0
⊢ ¬i + 1 < j + 1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i :... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ 0
⊢ V (Var.openList j (List.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | have s1 : i + 1 < j + 1 | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = V (free_ x✝) | case s1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ i + 1 < j + 1
D : Type
V : VarAssignment D
j : ℕ
zs : List Stri... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝¹ : Var
x✝ : String
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
free_ x✝
⊢ V (Var.open... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.ShiftVarOpenList | [815, 1] | [854, 17] | have s1 : ¬ i + 1 < j + 1 | D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ 0
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = d | case s1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ 0
⊢ ¬i + 1 < j + 1
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i :... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
c1 :
(if i < j then bound_ (i + 1)
else if h : i - j < zs.length then free_ (zs.get ⟨i - j, ⋯⟩) else bound_ (i - j - zs.length + (j + 1))) =
bound_ 0
⊢ V (Var.openList j (List.... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | induction F generalizing V j | D : Type
I : Interpretation D
V : VarAssignment D
j : ℕ
zs : List String
F : Formula
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) F ↔ Holds D I V (Formula.openList j (List.map free_ zs) F) | case pred_
D : Type
I : Interpretation D
zs : List String
a✝¹ : String
a✝ : List Var
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (pred_ a✝¹ a✝) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝))
case not_
D : Type
I : Interpretation D
zs : List String
a✝ : Form... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
j : ℕ
zs : List String
F : Formula
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) F ↔ Holds D I V (Formula.openList j (List.map free_ zs) F)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | case pred_ X vs =>
simp only [Holds]
congr! 1
simp | D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (pred_ X vs) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (pred_ X vs)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (pred_ X vs) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (pred_ X vs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | case not_ phi phi_ih =>
simp only [Holds]
congr! 1
apply phi_ih | D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi.not_ ↔
Hol... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | case forall_ _ phi phi_ih =>
simp only [Holds]
apply forall_congr'
intro d
simp only [← phi_ih]
congr!
apply ShiftVarOpenList | D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (foral... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | simp only [Holds] | D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (pred_ X vs) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (pred_ X vs)) | D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ I.pred_ X (List.map (V ∘ Var.openList j (List.map free_ zs)) vs) ↔
I.pred_ X (List.map V (List.map (Var.openList j (List.map free_ zs)) vs)) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (pred_ X vs) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (pred_ X vs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | congr! 1 | D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ I.pred_ X (List.map (V ∘ Var.openList j (List.map free_ zs)) vs) ↔
I.pred_ X (List.map V (List.map (Var.openList j (List.map free_ zs)) vs)) | case a.h.e'_4
D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ List.map (V ∘ Var.openList j (List.map free_ zs)) vs = List.map V (List.map (Var.openList j (List.map free_ zs)) vs) | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ I.pred_ X (List.map (V ∘ Var.openList j (List.map free_ zs)) vs) ↔
I.pred_ X (List.map V (List.map (Var.openList j (List.map free_ zs)) vs))
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | simp | case a.h.e'_4
D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ List.map (V ∘ Var.openList j (List.map free_ zs)) vs = List.map V (List.map (Var.openList j (List.map free_ zs)) vs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_4
D : Type
I : Interpretation D
zs : List String
X : String
vs : List Var
V : VarAssignment D
j : ℕ
⊢ List.map (V ∘ Var.openList j (List.map free_ zs)) vs = List.map V (List.map (Var.openList j (List.map free_ zs)) vs)
TACTIC:
|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | simp only [Holds] | D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi.not_ ↔
Hol... | D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ ¬Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ ¬Holds D I ... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | congr! 1 | D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ ¬Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ ¬Holds D I ... | case a.h.e'_1.a
D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) ph... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ ¬Holds D... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) ph... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
zs : List String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | simp only [Holds] | D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_... | D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | congr! 1 | D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_... | case a.h.a
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (Lis... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | apply phi_ih | case a.h.a
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (Lis... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : Var... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | apply psi_ih | case a.h'.a
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (Li... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h'.a
D : Type
I : Interpretation D
zs : List String
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
psi_ih :
∀ (V : Va... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | simp only [Holds] | D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (foral... | D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ (∀ (d : D), Holds D I (shift D (V ∘ Var.openList j (List.... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | apply forall_congr' | D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ (∀ (d : D), Holds D I (shift D (V ∘ Var.openList j (List.... | case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ ∀ (a : D),
Holds D I (shift D (V ∘ Var.openLis... | Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | intro d | case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
⊢ ∀ (a : D),
Holds D I (shift D (V ∘ Var.openLis... | case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
d : D
⊢ Holds D I (shift D (V ∘ Var.openList j (List... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | simp only [← phi_ih] | case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
d : D
⊢ Holds D I (shift D (V ∘ Var.openList j (List... | case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
d : D
⊢ Holds D I (shift D (V ∘ Var.openList j (List... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | congr! | case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
d : D
⊢ Holds D I (shift D (V ∘ Var.openList j (List... | case h.a.h.e'_3
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
d : D
⊢ shift D (V ∘ Var.openList j (List.m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment... |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.HoldsOpenList | [857, 1] | [887, 27] | apply ShiftVarOpenList | case h.a.h.e'_3
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarAssignment D
j : ℕ
d : D
⊢ shift D (V ∘ Var.openList j (List.m... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a.h.e'_3
D : Type
I : Interpretation D
zs : List String
a✝ : String
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi)
V : VarA... |
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... | 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]
... | 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... |
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 ... |
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... | 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 ∘ f... |
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... |
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 _ c... | 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 ∘ f... |
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... |
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)... |
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)... |
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
... | 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 ∘ f... |
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_) ... |
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 ... |
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_) ... |
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 - ... | 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 - ... | 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 (fr... |
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 ... |
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... |
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 fre... |
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]
... | 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_... |
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]
... | 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_... |
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), occur... | 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 ... |
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 ... | 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... | 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 =... |
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 ... |
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... |
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 ph... |
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
⊢... |
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... | 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... | 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) ... |
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... | 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' : VarA... | 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) ... |
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' : VarA... | 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' : 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... |
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' : 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' : 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 ... |
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' : 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' :... | 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 ... |
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' :... | 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' :... | 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 ps... |
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' :... | 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 ps... |
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' : Var... | 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' : ... | 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 → ... |
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' : ... | 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' : ... | 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... |
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' : ... | 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' ... | 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... |
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' ... | 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' ... | 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 p... |
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