url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
simp only [Formula.freeVarSet]
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆
(forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList (j + 1) (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenListFreeVarSet
|
[306, 1]
|
[338, 17]
|
apply phi_ih
|
zs : List String
x : String
phi : Formula
phi_ih :
∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList (j + 1) (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
cases v
|
j : ℕ
z : String
v : Var
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
|
case free_
j : ℕ
z a✝ : String
⊢ (free_ a✝).freeVarSet ⊆ (Var.open j (free_ z) (free_ a✝)).freeVarSet
case bound_
j : ℕ
z : String
a✝ : ℕ
⊢ (bound_ a✝).freeVarSet ⊆ (Var.open j (free_ z) (bound_ a✝)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case free_ x =>
simp only [Var.open]
simp only [Var.freeVarSet]
simp
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (Var.open j (free_ z) (free_ x)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case bound_ i =>
simp only [Var.open]
split_ifs
case _ c1 =>
simp only [Var.freeVarSet]
simp
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (Var.open j (free_ z) (bound_ i)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.open]
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (Var.open j (free_ z) (free_ x)).freeVarSet
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet
|
j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.open]
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (Var.open j (free_ z) (bound_ i)).freeVarSet
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
split_ifs
|
j : ℕ
z : String
i : ℕ
⊢ (bound_ i).freeVarSet ⊆ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet
|
case pos
j : ℕ
z : String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
case pos
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
case neg
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case _ c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
case _ c1 c2 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (bound_ i).freeVarSet ⊆ (free_ z).freeVarSet
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ ∅ ⊆ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ ∅ ⊆ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ i).freeVarSet ⊆ (bound_ (i - 1)).freeVarSet
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarOpenFreeVarSet'
|
[344, 1]
|
[366, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
induction F generalizing j
|
j : ℕ
z : String
F : Formula
⊢ F.freeVarSet ⊆ (Formula.open j (free_ z) F).freeVarSet
|
case pred_
z a✝¹ : String
a✝ : List Var
j : ℕ
⊢ (pred_ a✝¹ a✝).freeVarSet ⊆ (Formula.open j (free_ z) (pred_ a✝¹ a✝)).freeVarSet
case not_
z : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), a✝.freeVarSet ⊆ (Formula.open j (free_ z) a✝).freeVarSet
j : ℕ
⊢ a✝.not_.freeVarSet ⊆ (Formula.open j (free_ z) a✝.not_).freeVarSet
case imp_
z : String
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (j : ℕ), a✝¹.freeVarSet ⊆ (Formula.open j (free_ z) a✝¹).freeVarSet
a_ih✝ : ∀ (j : ℕ), a✝.freeVarSet ⊆ (Formula.open j (free_ z) a✝).freeVarSet
j : ℕ
⊢ (a✝¹.imp_ a✝).freeVarSet ⊆ (Formula.open j (free_ z) (a✝¹.imp_ a✝)).freeVarSet
case forall_
z a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), a✝.freeVarSet ⊆ (Formula.open j (free_ z) a✝).freeVarSet
j : ℕ
⊢ (forall_ a✝¹ a✝).freeVarSet ⊆ (Formula.open j (free_ z) (forall_ a✝¹ a✝)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
case not_ phi phi_ih =>
simp only [Formula.open]
simp only [Formula.freeVarSet]
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi.not_).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.open]
simp only [Formula.freeVarSet]
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ (forall_ x phi).freeVarSet ⊆ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.open]
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.freeVarSet]
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X vs).freeVarSet ⊆ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet
|
z X : String
vs : List Var
j : ℕ
⊢ vs.toFinset.biUnion Var.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp
|
z X : String
vs : List Var
j : ℕ
⊢ vs.toFinset.biUnion Var.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ x ∈ vs, x.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
intro v a1
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ x ∈ vs, x.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
trans Var.freeVarSet (Var.open j (free_ z) v)
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply VarOpenFreeVarSet'
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply Finset.subset_biUnion_of_mem Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.open j (free_ z) v).freeVarSet ⊆ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ (List.map (Var.open j (free_ z)) vs).toFinset
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply List.mem_toFinset.mpr
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ (List.map (Var.open j (free_ z)) vs).toFinset
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ List.map (Var.open j (free_ z)) vs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
exact List.mem_map_of_mem (Var.open j (free_ z)) a1
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ Var.open j (free_ z) v ∈ List.map (Var.open j (free_ z)) vs
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.open]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi.not_).freeVarSet
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi).not_.freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.not_.freeVarSet ⊆ (Formula.open j (free_ z) phi).not_.freeVarSet
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.open]
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ (phi.imp_ psi).freeVarSet ⊆ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ (phi.imp_ psi).freeVarSet ⊆ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ (phi.imp_ psi).freeVarSet ⊆ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ∪ psi.freeVarSet ⊆
(Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply Finset.union_subset_left_right
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ∪ psi.freeVarSet ⊆
(Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet
|
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
exact phi_ih j
|
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
exact psi_ih j
|
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
psi_ih : ∀ (j : ℕ), psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
j : ℕ
⊢ psi.freeVarSet ⊆ (Formula.open j (free_ z) psi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.open]
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ (forall_ x phi).freeVarSet ⊆ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ (forall_ x phi).freeVarSet ⊆ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
simp only [Formula.freeVarSet]
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ (forall_ x phi).freeVarSet ⊆ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open (j + 1) (free_ z) phi).freeVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaOpenFreeVarSet'
|
[369, 1]
|
[400, 17]
|
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), phi.freeVarSet ⊆ (Formula.open j (free_ z) phi).freeVarSet
j : ℕ
⊢ phi.freeVarSet ⊆ (Formula.open (j + 1) (free_ z) phi).freeVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
cases v
|
j : ℕ
z : String
v : Var
⊢ (Var.close j (free_ z) v).freeVarSet ⊆ v.freeVarSet \ {free_ z}
|
case free_
j : ℕ
z a✝ : String
⊢ (Var.close j (free_ z) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet \ {free_ z}
case bound_
j : ℕ
z : String
a✝ : ℕ
⊢ (Var.close j (free_ z) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
case free_ x =>
simp only [Var.close]
split_ifs
case _ c1 =>
simp only [Var.freeVarSet]
simp
case _ c1 =>
simp only [Var.freeVarSet]
simp
simp at c1
exact c1
|
j : ℕ
z x : String
⊢ (Var.close j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
case bound_ i =>
simp only [Var.close]
split_ifs
case _ c1 =>
simp only [Var.freeVarSet]
simp
case _ c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
⊢ (Var.close j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp only [Var.close]
|
j : ℕ
z x : String
⊢ (Var.close j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
j : ℕ
z x : String
⊢ (if free_ x = free_ z then bound_ j else free_ x).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
split_ifs
|
j : ℕ
z x : String
⊢ (if free_ x = free_ z then bound_ j else free_ x).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
case pos
j : ℕ
z x : String
h✝ : free_ x = free_ z
⊢ (bound_ j).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
case neg
j : ℕ
z x : String
h✝ : ¬free_ x = free_ z
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
case _ c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z x : String
c1 : free_ x = free_ z
⊢ (bound_ j).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
case _ c1 =>
simp only [Var.freeVarSet]
simp
simp at c1
exact c1
|
j : ℕ
z x : String
c1 : ¬free_ x = free_ z
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z x : String
c1 : free_ x = free_ z
⊢ (bound_ j).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
j : ℕ
z x : String
c1 : free_ x = free_ z
⊢ ∅ ⊆ {free_ x} \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp
|
j : ℕ
z x : String
c1 : free_ x = free_ z
⊢ ∅ ⊆ {free_ x} \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z x : String
c1 : ¬free_ x = free_ z
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet \ {free_ z}
|
j : ℕ
z x : String
c1 : ¬free_ x = free_ z
⊢ {free_ x} ⊆ {free_ x} \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp
|
j : ℕ
z x : String
c1 : ¬free_ x = free_ z
⊢ {free_ x} ⊆ {free_ x} \ {free_ z}
|
j : ℕ
z x : String
c1 : ¬free_ x = free_ z
⊢ ¬x = z
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp at c1
|
j : ℕ
z x : String
c1 : ¬free_ x = free_ z
⊢ ¬x = z
|
j : ℕ
z x : String
c1 : ¬x = z
⊢ ¬x = z
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
exact c1
|
j : ℕ
z x : String
c1 : ¬x = z
⊢ ¬x = z
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp only [Var.close]
|
j : ℕ
z : String
i : ℕ
⊢ (Var.close j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
j : ℕ
z : String
i : ℕ
⊢ (if i < j then bound_ i else bound_ (i + 1)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
split_ifs
|
j : ℕ
z : String
i : ℕ
⊢ (if i < j then bound_ i else bound_ (i + 1)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
case pos
j : ℕ
z : String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
case neg
j : ℕ
z : String
i : ℕ
h✝ : ¬i < j
⊢ (bound_ (i + 1)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
case _ c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
case _ c1 =>
simp only [Var.freeVarSet]
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
⊢ (bound_ (i + 1)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp only [Var.freeVarSet]
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
⊢ (bound_ (i + 1)).freeVarSet ⊆ (bound_ i).freeVarSet \ {free_ z}
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
⊢ ∅ ⊆ ∅ \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarCloseFreeVarSet
|
[406, 1]
|
[432, 11]
|
simp
|
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
⊢ ∅ ⊆ ∅ \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
induction F generalizing j
|
j : ℕ
z : String
F : Formula
⊢ (Formula.close j (free_ z) F).freeVarSet ⊆ F.freeVarSet \ {free_ z}
|
case pred_
z a✝¹ : String
a✝ : List Var
j : ℕ
⊢ (Formula.close j (free_ z) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet \ {free_ z}
case not_
z : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.close j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet \ {free_ z}
case imp_
z : String
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (j : ℕ), (Formula.close j (free_ z) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet \ {free_ z}
a_ih✝ : ∀ (j : ℕ), (Formula.close j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet \ {free_ z}
case forall_
z a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.close j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
case not_ phi phi_ih =>
simp only [Formula.close]
simp only [Formula.freeVarSet]
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
case forall_ x phi phi_ih =>
simp only [Formula.close]
simp only [Formula.freeVarSet]
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.close]
|
z X : String
vs : List Var
j : ℕ
⊢ (Formula.close j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet \ {free_ z}
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.close j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.freeVarSet]
|
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.close j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet \ {free_ z}
|
z X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.close j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp
|
z X : String
vs : List Var
j : ℕ
⊢ (List.map (Var.close j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.close j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
intro v a1
|
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.close j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.close j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
trans Var.freeVarSet v \ {free_ z}
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.close j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.close j (free_ z) v).freeVarSet ⊆ v.freeVarSet \ {free_ z}
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet \ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
exact VarCloseFreeVarSet j z v
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.close j (free_ z) v).freeVarSet ⊆ v.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
apply Finset.sdiff_subset_sdiff
|
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet \ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
|
case hst
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
case hvu
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ {free_ z} ⊆ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
apply Finset.subset_biUnion_of_mem
|
case hst
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
|
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp
|
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
|
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
exact a1
|
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp
|
case hvu
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ {free_ z} ⊆ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.close]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet \ {free_ z}
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet \ {free_ z}
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
apply phi_ih
|
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.close]
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet \ {free_ z}
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ ((Formula.close j (free_ z) phi).imp_ (Formula.close j (free_ z) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.freeVarSet]
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ ((Formula.close j (free_ z) phi).imp_ (Formula.close j (free_ z) psi)).freeVarSet ⊆
(phi.imp_ psi).freeVarSet \ {free_ z}
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).freeVarSet ∪ (Formula.close j (free_ z) psi).freeVarSet ⊆
(phi.freeVarSet ∪ psi.freeVarSet) \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
apply Finset.union_subset_diff
|
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).freeVarSet ∪ (Formula.close j (free_ z) psi).freeVarSet ⊆
(phi.freeVarSet ∪ psi.freeVarSet) \ {free_ z}
|
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
exact phi_ih j
|
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
exact psi_ih j
|
case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.close]
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet \ {free_ z}
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.close (1 + j) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
simp only [Formula.freeVarSet]
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.close (1 + j) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet \ {free_ z}
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close (1 + j) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.FormulaCloseFreeVarSet
|
[435, 1]
|
[468, 17]
|
apply phi_ih
|
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.close j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
j : ℕ
⊢ (Formula.close (1 + j) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
cases v
|
z : String
t v : Var
⊢ ((free_ z).subst t v).freeVarSet ⊆ t.freeVarSet ∪ v.freeVarSet \ {free_ z}
|
case free_
z : String
t : Var
a✝ : String
⊢ ((free_ z).subst t (free_ a✝)).freeVarSet ⊆ t.freeVarSet ∪ (free_ a✝).freeVarSet \ {free_ z}
case bound_
z : String
t : Var
a✝ : ℕ
⊢ ((free_ z).subst t (bound_ a✝)).freeVarSet ⊆ t.freeVarSet ∪ (bound_ a✝).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
case free_ x =>
simp only [Var.subst]
split_ifs
case pos c1 =>
apply Finset.subset_union_left
case neg c1 =>
have s1 : Var.freeVarSet (free_ x) \ {free_ z} = {free_ x}
simp only [Var.freeVarSet]
simp
simp at c1
exact c1
simp only [s1]
apply Finset.subset_union_right
|
z : String
t : Var
x : String
⊢ ((free_ z).subst t (free_ x)).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
case bound_ i =>
simp only [Var.subst]
conv =>
lhs
simp only [Var.freeVarSet]
simp
|
z : String
t : Var
i : ℕ
⊢ ((free_ z).subst t (bound_ i)).freeVarSet ⊆ t.freeVarSet ∪ (bound_ i).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp only [Var.subst]
|
z : String
t : Var
x : String
⊢ ((free_ z).subst t (free_ x)).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
z : String
t : Var
x : String
⊢ (if free_ z = free_ x then t else free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
split_ifs
|
z : String
t : Var
x : String
⊢ (if free_ z = free_ x then t else free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
case pos
z : String
t : Var
x : String
h✝ : free_ z = free_ x
⊢ t.freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
case neg
z : String
t : Var
x : String
h✝ : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
case pos c1 =>
apply Finset.subset_union_left
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ t.freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
case neg c1 =>
have s1 : Var.freeVarSet (free_ x) \ {free_ z} = {free_ x}
simp only [Var.freeVarSet]
simp
simp at c1
exact c1
simp only [s1]
apply Finset.subset_union_right
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
apply Finset.subset_union_left
|
z : String
t : Var
x : String
c1 : free_ z = free_ x
⊢ t.freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
have s1 : Var.freeVarSet (free_ x) \ {free_ z} = {free_ x}
|
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
case s1
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} = {free_ x}
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp only [Var.freeVarSet]
|
case s1
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ (free_ x).freeVarSet \ {free_ z} = {free_ x}
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
case s1
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ {free_ x} \ {free_ z} = {free_ x}
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp
|
case s1
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ {free_ x} \ {free_ z} = {free_ x}
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
case s1
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ ¬z = x
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.VarSubstFreeVarSet
|
[474, 1]
|
[500, 9]
|
simp at c1
|
case s1
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
⊢ ¬z = x
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
case s1
z : String
t : Var
x : String
c1 : ¬z = x
⊢ ¬z = x
z : String
t : Var
x : String
c1 : ¬free_ z = free_ x
s1 : (free_ x).freeVarSet \ {free_ z} = {free_ x}
⊢ (free_ x).freeVarSet ⊆ t.freeVarSet ∪ (free_ x).freeVarSet \ {free_ z}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.