Theorem: FOL.NV.isBoundIn_iff_mem_boundVarSet (FOL/NV/Binders.lean)

State before:
v : VarName
F : Formula
⊢ isBoundIn v F ↔ v ∈ F.boundVarSet
Tactic:
induction F
State after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boundVarSet

case eq_
v a✝¹ a✝ : VarName
⊢ isBoundIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).boundVarSet

case true_
v : VarName
⊢ isBoundIn v true_ ↔ v ∈ true_.boundVarSet

case false_
v : VarName
⊢ isBoundIn v false_ ↔ v ∈ false_.boundVarSet

case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝.not_ ↔ v ∈ a✝.not_.boundVarSet

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).boundVarSet

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).boundVarSet

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).boundVarSet

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).boundVarSet

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet

State before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boundVarSet

case eq_
v a✝¹ a✝ : VarName
⊢ isBoundIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).boundVarSet

case true_
v : VarName
⊢ isBoundIn v true_ ↔ v ∈ true_.boundVarSet

case false_
v : VarName
⊢ isBoundIn v false_ ↔ v ∈ false_.boundVarSet

case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝.not_ ↔ v ∈ a✝.not_.boundVarSet

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).boundVarSet

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).boundVarSet

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).boundVarSet

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).boundVarSet

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
Tactic:
all_goals
  simp only [isBoundIn]
  simp only [Formula.boundVarSet]
State after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ False ↔ v ∈ ∅

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ False ↔ v ∈ ∅

case eq_
v a✝¹ a✝ : VarName
⊢ False ↔ v ∈ ∅

case true_
v : VarName
⊢ False ↔ v ∈ ∅

case false_
v : VarName
⊢ False ↔ v ∈ ∅

case not_
v : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ v = a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ∪ {a✝¹}

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet
⊢ v = a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ∪ {a✝¹}

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ v ∈ ∅

State before:
v : VarName
X : DefName
xs : List VarName
⊢ False ↔ v ∈ ∅
Tactic:
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
  simp
State after:
no goals

State before:
v x y : VarName
⊢ False ↔ v ∈ ∅
Tactic:
case eq_ x y =>
  simp
State after:
no goals

State before:
v : VarName
⊢ False ↔ v ∈ ∅
Tactic:
case true_ | false_ =>
  tauto
State after:
no goals

State before:
v : VarName
phi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
⊢ isBoundIn v phi ↔ v ∈ phi.boundVarSet
Tactic:
case not_ phi phi_ih =>
  tauto
State after:
no goals

State before:
v : VarName
phi psi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet
⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet
Tactic:
case
    imp_ phi psi phi_ih psi_ih
  | and_ phi psi phi_ih psi_ih
  | or_ phi psi phi_ih psi_ih
  | iff_ phi psi phi_ih psi_ih =>
  simp
  tauto
State after:
no goals

State before:
v x : VarName
phi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∪ {x}
Tactic:
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
  simp
  tauto
State after:
no goals

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
Tactic:
simp only [isBoundIn]
State after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
Tactic:
simp only [Formula.boundVarSet]
State after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ False ↔ v ∈ ∅

State before:
v : VarName
X : DefName
xs : List VarName
⊢ False ↔ v ∈ ∅
Tactic:
simp
State after:
no goals

State before:
v x y : VarName
⊢ False ↔ v ∈ ∅
Tactic:
simp
State after:
no goals

State before:
v : VarName
⊢ False ↔ v ∈ ∅
Tactic:
tauto
State after:
no goals

State before:
v : VarName
phi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
⊢ isBoundIn v phi ↔ v ∈ phi.boundVarSet
Tactic:
tauto
State after:
no goals

State before:
v : VarName
phi psi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet
⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet
Tactic:
simp
State after:
v : VarName
phi psi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet
⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet

State before:
v : VarName
phi psi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet
⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet
Tactic:
tauto
State after:
no goals

State before:
v x : VarName
phi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∪ {x}
Tactic:
simp
State after:
v x : VarName
phi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x

State before:
v x : VarName
phi : Formula
phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet
⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x
Tactic:
tauto
State after:
no goals