Theorem: LN.FormulaCloseFreeVarSet (FOL/LN/Paper.lean)

State before:
j : ℕ
z : String
F : Formula
⊢ (Formula.close j (free_ z) F).freeVarSet ⊆ F.freeVarSet \ {free_ z}
Tactic:
induction F generalizing j
State after:
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}

State before:
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}
Tactic:
case not_ phi phi_ih =>
  simp only [Formula.close]
  simp only [Formula.freeVarSet]
  apply phi_ih
State after:
no goals

State before:
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}
Tactic:
case forall_ x phi phi_ih =>
  simp only [Formula.close]
  simp only [Formula.freeVarSet]
  apply phi_ih
State after:
no goals

State before:
z X : String
vs : List Var
j : ℕ
⊢ (Formula.close j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet \ {free_ z}
Tactic:
simp only [Formula.close]
State after:
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}

State before:
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}
Tactic:
simp only [Formula.freeVarSet]
State after:
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}

State before:
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}
Tactic:
simp
State after:
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.close j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}

State before:
z X : String
vs : List Var
j : ℕ
⊢ ∀ a ∈ vs, (Var.close j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
Tactic:
intro v a1
State after:
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}

State before:
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}
Tactic:
trans Var.freeVarSet v \ {free_ z}
State after:
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}

State before:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ (Var.close j (free_ z) v).freeVarSet ⊆ v.freeVarSet \ {free_ z}
Tactic:
exact VarCloseFreeVarSet j z v
State after:
no goals

State before:
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet \ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet \ {free_ z}
Tactic:
apply Finset.sdiff_subset_sdiff
State after:
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}

State before:
case hst
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
Tactic:
apply Finset.subset_biUnion_of_mem
State after:
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset

State before:
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
Tactic:
simp
State after:
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs

State before:
case hst.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
Tactic:
exact a1
State after:
no goals

State before:
case hvu
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ {free_ z} ⊆ {free_ z}
Tactic:
simp
State after:
no goals

State before:
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}
Tactic:
simp only [Formula.close]
State after:
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}

State before:
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}
Tactic:
simp only [Formula.freeVarSet]
State after:
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}

State before:
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}
Tactic:
apply phi_ih
State after:
no goals

State before:
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}
Tactic:
simp only [Formula.close]
State after:
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}

State before:
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}
Tactic:
simp only [Formula.freeVarSet]
State after:
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}

State before:
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}
Tactic:
apply Finset.union_subset_diff
State after:
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}

State before:
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}
Tactic:
exact phi_ih j
State after:
no goals

State before:
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}
Tactic:
exact psi_ih j
State after:
no goals

State before:
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}
Tactic:
simp only [Formula.close]
State after:
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}

State before:
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}
Tactic:
simp only [Formula.freeVarSet]
State after:
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}

State before:
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}
Tactic:
apply phi_ih
State after:
no goals