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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} |