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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}