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

State before:
v : VarName
F : Formula
h1 : isFreeIn v F
⊢ occursIn v F
Tactic:
induction F
State after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)

case eq_
v a✝¹ a✝ : VarName
h1 : isFreeIn v (eq_ a✝¹ a✝)
⊢ occursIn v (eq_ a✝¹ a✝)

case true_
v : VarName
h1 : isFreeIn v true_
⊢ occursIn v true_

case false_
v : VarName
h1 : isFreeIn v false_
⊢ occursIn v false_

case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝.not_
⊢ occursIn v a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.imp_ a✝)
⊢ occursIn v (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.and_ a✝)
⊢ occursIn v (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.or_ a✝)
⊢ occursIn v (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.iff_ a✝)
⊢ occursIn v (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (forall_ a✝¹ a✝)
⊢ occursIn v (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (exists_ a✝¹ a✝)
⊢ occursIn v (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)

State before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ occursIn v (pred_const_ a✝¹ a✝)

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ occursIn v (pred_var_ a✝¹ a✝)

case eq_
v a✝¹ a✝ : VarName
h1 : isFreeIn v (eq_ a✝¹ a✝)
⊢ occursIn v (eq_ a✝¹ a✝)

case true_
v : VarName
h1 : isFreeIn v true_
⊢ occursIn v true_

case false_
v : VarName
h1 : isFreeIn v false_
⊢ occursIn v false_

case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝.not_
⊢ occursIn v a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.imp_ a✝)
⊢ occursIn v (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.and_ a✝)
⊢ occursIn v (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.or_ a✝)
⊢ occursIn v (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (a✝¹.iff_ a✝)
⊢ occursIn v (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (forall_ a✝¹ a✝)
⊢ occursIn v (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v (exists_ a✝¹ a✝)
⊢ occursIn v (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
Tactic:
all_goals
  simp only [isFreeIn] at h1
State after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_const_ a✝¹ a✝)

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_var_ a✝¹ a✝)

case eq_
v a✝¹ a✝ : VarName
h1 : v = a✝¹ ∨ v = a✝
⊢ occursIn v (eq_ a✝¹ a✝)

case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝
⊢ occursIn v a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)

State before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_const_ a✝¹ a✝)

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (pred_var_ a✝¹ a✝)

case eq_
v a✝¹ a✝ : VarName
h1 : v = a✝¹ ∨ v = a✝
⊢ occursIn v (eq_ a✝¹ a✝)

case not_
v : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝
⊢ occursIn v a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : isFreeIn v a✝¹ → occursIn v a✝¹
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ occursIn v (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : isFreeIn v a✝ → occursIn v a✝
h1 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ occursIn v (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)
Tactic:
all_goals
  simp only [occursIn]
  tauto
State after:
no goals

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : isFreeIn v (def_ a✝¹ a✝)
⊢ occursIn v (def_ a✝¹ a✝)
Tactic:
simp only [isFreeIn] at h1
State after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ occursIn v (def_ a✝¹ a✝)
Tactic:
simp only [occursIn]
State after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ v ∈ a✝

State before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : v ∈ a✝
⊢ v ∈ a✝
Tactic:
tauto
State after:
no goals