internlm/Lean-Github huggingface dataset:
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z x : String
⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}
Tactic:
simp
Proof state after:
no goals
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
simp only [Var.open]
Proof state after:
j : ℕ
z : String
i : ℕ
⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
split_ifs
Proof state after:
case pos
j : ℕ
z : String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

case pos
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : i = j
⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

case neg
j : ℕ
z : String
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i = j
⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
case _ c1 =>
  simp only [Var.freeVarSet]
  simp
Proof state after:
no goals
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
case _ c1 c2 =>
  simp only [Var.freeVarSet]
  simp
Proof state after:
no goals
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
case _ c1 c2 =>
  simp only [Var.freeVarSet]
  simp
Proof state after:
no goals
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ {free_ z}
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ {free_ z}
Tactic:
simp
Proof state after:
no goals
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ {free_ z} ⊆ ∅ ∪ {free_ z}
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : i = j
⊢ {free_ z} ⊆ ∅ ∪ {free_ z}
Tactic:
simp
Proof state after:
no goals
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅ ∪ {free_ z}
Theorem: LN.VarOpenFreeVarSet
Proof state before:
j : ℕ
z : String
i : ℕ
c1 : ¬i < j
c2 : ¬i = j
⊢ ∅ ⊆ ∅ ∪ {free_ z}
Tactic:
simp
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof state before:
j : ℕ
z : String
F : Formula
⊢ (Formula.open j (free_ z) F).freeVarSet ⊆ F.freeVarSet ∪ {free_ z}
Tactic:
induction F generalizing j
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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}
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
case h
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
Theorem: LN.FormulaOpenFreeVarSet
Proof state before:
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs.toFinset
Tactic:
simp
Proof state after:
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
Theorem: LN.FormulaOpenFreeVarSet
Proof state before:
case h.xs
z X : String
vs : List Var
j : ℕ
v : Var
a1 : v ∈ vs
⊢ v ∈ vs
Tactic:
exact a1
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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]
Proof 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}
Theorem: LN.FormulaOpenFreeVarSet
Proof 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
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
v : Var
⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
cases v
Proof state after:
case free_
j : ℕ
zs : List String
a✝ : String
⊢ (Var.openList j (List.map free_ zs) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset

case bound_
j : ℕ
zs : List String
a✝ : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
x : String
⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case free_ x =>
  simp only [Var.openList]
  simp only [Var.freeVarSet]
  simp
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case bound_ i =>
  simp only [Var.openList]
  split_ifs
  case pos c1 =>
    simp only [Var.freeVarSet]
    simp
  case pos c1 c2 =>
    simp
    simp only [Var.freeVarSet]
    simp
    apply List.get_mem
  case neg c1 c2 =>
    simp only [Var.freeVarSet]
    simp
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
x : String
⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp only [Var.openList]
Proof state after:
j : ℕ
zs : List String
x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
x : String
⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
zs : List String
x : String
⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
x : String
⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
Tactic:
simp
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp only [Var.openList]
Proof state after:
j : ℕ
zs : List String
i : ℕ
⊢ (if i < j then bound_ i
      else
        if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
        else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆
    (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
⊢ (if i < j then bound_ i
      else
        if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
        else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆
    (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
split_ifs
Proof state after:
case pos
j : ℕ
zs : List String
i : ℕ
h✝ : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

case pos
j : ℕ
zs : List String
i : ℕ
h✝¹ : ¬i < j
h✝ : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

case neg
j : ℕ
zs : List String
i : ℕ
h✝¹ : ¬i < j
h✝ : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case pos c1 =>
  simp only [Var.freeVarSet]
  simp
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case pos c1 c2 =>
  simp
  simp only [Var.freeVarSet]
  simp
  apply List.get_mem
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case neg c1 c2 =>
  simp only [Var.freeVarSet]
  simp
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : i < j
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
Tactic:
simp
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp
Proof state after:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
Tactic:
simp
Proof state after:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : i - j < (List.map free_ zs).length
⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
Tactic:
apply List.get_mem
Proof state after:
no goals
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
simp only [Var.freeVarSet]
Proof state after:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
Theorem: LN.VarOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
i : ℕ
c1 : ¬i < j
c2 : ¬i - j < (List.map free_ zs).length
⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
Tactic:
simp
Proof state after:
no goals
Theorem: LN.FormulaOpenListFreeVarSet
Proof state before:
j : ℕ
zs : List String
F : Formula
⊢ (Formula.openList j (List.map free_ zs) F).freeVarSet ⊆ F.freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
induction F generalizing j
Proof state after:
case pred_
zs : List String
a✝¹ : String
a✝ : List Var
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝)).freeVarSet ⊆
    (pred_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset

case not_
zs : List String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ (List.map free_ zs).toFinset

case imp_
zs : List String
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ (List.map free_ zs).toFinset
a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝)).freeVarSet ⊆
    (a✝¹.imp_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset

case forall_
zs : List String
a✝¹ : String
a✝ : Formula
a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ
⊢ (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)).freeVarSet ⊆
    (forall_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
Theorem: LN.FormulaOpenListFreeVarSet
Proof state before:
zs : List 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 (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case not_ phi phi_ih =>
  simp only [Formula.openList]
  simp only [Formula.freeVarSet]
  apply phi_ih
Proof state after:
no goals
Theorem: LN.FormulaOpenListFreeVarSet
Proof state before:
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 (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆
    (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
Tactic:
case forall_ x phi phi_ih =>
  simp only [Formula.openList]
  simp only [Formula.freeVarSet]
  apply phi_ih
Proof state after:
no goals