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stringclasses 147
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stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
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stringlengths 1
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| state_after
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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 free_ zs)) d (bound_ (n✝ + 1)) =
shift D V d (Var.openList (j + 1) (List.map free_ zs) (bound_ (n✝ + 1))) |
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 |
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
case _ c2 =>
simp | 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_ i✝) |
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) |
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)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) |
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)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) | no goals |
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) |
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 + 1) - (List.map free_ zs).length + (j + 1)) with
| free_ x => V (free_ x)
| bound_ 0 => d
| bound_ i.succ => V (bound_ i) |
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 + 1) - (List.map free_ zs).length + (j + 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 < 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 => d
| bound_ i.succ => V (bound_ i) |
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 => d
| bound_ i.succ => V (bound_ i) | 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_ x✝)
case h_2
D : Type
V : VarAssignment D
j : ℕ
zs : List String
d : D
i : ℕ
x✝ : Var
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_ 0
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = d
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_ i✝) |
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 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✝
s1 : i + 1 < j + 1
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = V (free_ x✝) |
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 : 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
s1 : ¬i + 1 < j + 1
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = d |
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 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✝
s1 : i + 1 < j + 1
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = V (free_ x✝) |
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 : 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
s1 : ¬i + 1 < j + 1
⊢ V (Var.openList j (List.map free_ zs) (bound_ i)) = d |
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✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) a✝ ↔ Holds D I V (Formula.openList j (List.map free_ zs) a✝)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) a✝.not_ ↔
Holds D I V (Formula.openList j (List.map free_ zs) a✝.not_)
case imp_
D : Type
I : Interpretation D
zs : List String
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) a✝¹ ↔ Holds D I V (Formula.openList j (List.map free_ zs) a✝¹)
a_ih✝ :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) a✝ ↔ Holds D I V (Formula.openList j (List.map free_ zs) a✝)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (a✝¹.imp_ a✝) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝))
case forall_
D : Type
I : Interpretation D
zs : List String
a✝¹ : String
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (j : ℕ),
Holds D I (V ∘ Var.openList j (List.map free_ zs)) a✝ ↔ Holds D I V (Formula.openList j (List.map free_ zs) a✝)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (forall_ a✝¹ a✝) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)) |
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 |
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_ ↔
Holds D I V (Formula.openList j (List.map free_ zs) phi.not_) | no goals |
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)) (forall_ a✝ phi) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (forall_ a✝ phi)) | no goals |
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)) |
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) |
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 |
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_ ↔
Holds D I V (Formula.openList j (List.map free_ zs) phi.not_) | 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 V (Formula.openList j (List.map free_ zs) phi) |
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 V (Formula.openList j (List.map free_ zs) phi) | 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)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi) |
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)) phi ↔ Holds D I V (Formula.openList j (List.map free_ zs) phi) | no goals |
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_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) (phi.imp_ psi) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (phi.imp_ psi)) | 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_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi → Holds D I (V ∘ Var.openList j (List.map free_ zs)) psi ↔
Holds D I V (Formula.openList j (List.map free_ zs) phi) → Holds D I V (Formula.openList j (List.map free_ zs) psi) |
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_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
V : VarAssignment D
j : ℕ
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) phi → Holds D I (V ∘ Var.openList j (List.map free_ zs)) psi ↔
Holds D I V (Formula.openList j (List.map free_ zs) phi) → Holds D I V (Formula.openList j (List.map free_ zs) psi) | 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 (List.map free_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
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)
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 (List.map free_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
V : VarAssignment D
j : ℕ
a✝ : Holds D I V (Formula.openList j (List.map free_ zs) phi)
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi) |
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 (List.map free_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
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) | no goals |
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 (List.map free_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi)
V : VarAssignment D
j : ℕ
a✝ : Holds D I V (Formula.openList j (List.map free_ zs) phi)
⊢ Holds D I (V ∘ Var.openList j (List.map free_ zs)) psi ↔ Holds D I V (Formula.openList j (List.map free_ zs) psi) | no goals |
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)) (forall_ a✝ phi) ↔
Holds D I V (Formula.openList j (List.map free_ zs) (forall_ a✝ phi)) | 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.map free_ zs)) d) phi) ↔
∀ (d : D), Holds D I (shift D V d) (Formula.openList (j + 1) (List.map free_ zs) phi) |
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.map free_ zs)) d) phi) ↔
∀ (d : D), Holds D I (shift D V d) (Formula.openList (j + 1) (List.map free_ zs) phi) | 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.openList j (List.map free_ zs)) a) phi ↔
Holds D I (shift D V a) (Formula.openList (j + 1) (List.map free_ zs) phi) |
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.openList j (List.map free_ zs)) a) phi ↔
Holds D I (shift D V a) (Formula.openList (j + 1) (List.map free_ zs) phi) | 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.map free_ zs)) d) phi ↔
Holds D I (shift D V d) (Formula.openList (j + 1) (List.map free_ zs) phi) |
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.map free_ zs)) d) phi ↔
Holds D I (shift D V d) (Formula.openList (j + 1) (List.map free_ zs) phi) | 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.map free_ zs)) d) phi ↔
Holds D I (shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)) phi |
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.map free_ zs)) d) phi ↔
Holds D I (shift D V d ∘ Var.openList (j + 1) (List.map free_ zs)) phi | 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.map free_ zs)) d = shift D V d ∘ Var.openList (j + 1) (List.map free_ zs) |
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.map free_ zs)) d = shift D V d ∘ Var.openList (j + 1) (List.map free_ zs) | no goals |
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✝)) |
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 |
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 |
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 |
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 |
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 |
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✝) |
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 |
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 |
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 |
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) |
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) |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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✝) |
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)) |
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 |
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) |
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) |
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) |
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)) |
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)) |
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 |
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 |
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) |
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) |
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 |
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) |
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) |
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)) |
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)) |
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 |
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 |
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 |
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 |
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 |
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 |
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✝) |
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 |
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 |
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 |
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) |
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) |
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 |
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 |
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 |
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_ |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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) |
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 |
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 |
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 |
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 |
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) |
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 |