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

State before:
j : ℕ
z : String
F : Formula
⊢ (Formula.open 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.open 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.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ {free_ z}
a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
Tactic:
case not_ phi phi_ih =>
  simp only [Formula.open]
  simp only [Formula.freeVarSet]
  apply phi_ih
State after:
no goals

State before:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
Tactic:
case forall_ x phi phi_ih =>
  simp only [Formula.open]
  simp only [Formula.freeVarSet]
  apply phi_ih
State after:
no goals

State before:
z X : String
vs : List Var
j : ℕ
⊢ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
Tactic:
simp only [Formula.open]
State after:
z X : String
vs : List Var
j : ℕ
⊢ (pred_ X (List.map (Var.open 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.open 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.open 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.open 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.open 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.open 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.open 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.open 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.open 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.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}
Tactic:
exact VarOpenFreeVarSet 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.union_subset_union_left
State after:
case h
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet

State before:
case h
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 h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset

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

State before:
case h.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:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
Tactic:
simp only [Formula.open]
State after:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}

State before:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

State before:
z : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
Tactic:
simp only [Formula.open]
State after:
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆
    (phi.imp_ psi).freeVarSet ∪ {free_ z}

State before:
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open 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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆
    phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}

State before:
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆
    phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
Tactic:
apply Finset.union_subset_union_left_right
State after:
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

case h2
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}

State before:
case h1
z : String
phi psi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open 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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
Tactic:
simp only [Formula.open]
State after:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}

State before:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (forall_ x (Formula.open (j + 1) (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.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

State before:
z x : String
phi : Formula
phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
j : ℕ
⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
Tactic:
apply phi_ih
State after:
no goals