Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn (FOL/NV/Sub/Var/One/Rec/Admits.lean)

State before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
Tactic:
induction F generalizing binders
State after:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ u ∉ binders

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ u ∉ binders

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
⊢ u ∉ binders

case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
h2 : isFreeIn v true_
⊢ u ∉ binders

case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
h2 : isFreeIn v false_
⊢ u ∉ binders

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
⊢ u ∉ binders

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
⊢ u ∉ binders

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
⊢ u ∉ binders

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
⊢ u ∉ binders

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
⊢ u ∉ binders

case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
⊢ u ∉ binders

case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
⊢ u ∉ binders

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders

State before:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
⊢ u ∉ binders

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
⊢ u ∉ binders

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
⊢ u ∉ binders

case true_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders true_
h2 : isFreeIn v true_
⊢ u ∉ binders

case false_
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders false_
h2 : isFreeIn v false_
⊢ u ∉ binders

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
⊢ u ∉ binders

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
⊢ u ∉ binders

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
⊢ u ∉ binders

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
⊢ u ∉ binders

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
⊢ u ∉ binders

case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
⊢ u ∉ binders

case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
⊢ u ∉ binders

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
Tactic:
all_goals
  simp only [fastAdmitsAux] at h1
  simp only [isFreeIn] at h2
State after:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
⊢ u ∉ binders

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝
⊢ u ∉ binders

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ u ∉ binders

case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
⊢ u ∉ binders

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders

State before:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
⊢ u ∉ binders

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝
⊢ u ∉ binders

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → isFreeIn v a✝¹ → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → isFreeIn v a✝ → u ∉ binders
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
⊢ u ∉ binders

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
Tactic:
all_goals
  tauto
State after:
no goals

State before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
Tactic:
simp only [fastAdmitsAux] at h1
State after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders

State before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
⊢ u ∉ binders
Tactic:
simp only [isFreeIn] at h2
State after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h2 : ¬v = x ∧ isFreeIn v phi
⊢ u ∉ binders
Tactic:
cases h2
State after:
case intro
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
left✝ : ¬v = x
right✝ : isFreeIn v phi
⊢ u ∉ binders

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ u ∉ binders
Tactic:
cases h1
State after:
case inl
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h✝ : v = x
⊢ u ∉ binders

case inr
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h✝ : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ u ∉ binders

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : v = x
⊢ u ∉ binders
Tactic:
case inl h1 =>
  contradiction
State after:
no goals

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : v = x
⊢ u ∉ binders
Tactic:
contradiction
State after:
no goals

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ u ∉ binders
Tactic:
apply phi_ih
State after:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u binders phi

case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ isFreeIn v phi

State before:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u binders phi
Tactic:
exact fastAdmitsAux_del_binders phi v u binders {x} h1
State after:
no goals

State before:
case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ isFreeIn v phi
Tactic:
exact h2_right
State after:
no goals

State before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
Tactic:
tauto
State after:
no goals