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

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

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ admitsAux v u binders (eq_ a✝¹ a✝)

case true_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v true_
⊢ admitsAux v u binders true_

case false_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v false_
⊢ admitsAux v u binders false_

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

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

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

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

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

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

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

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝)

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

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
⊢ admitsAux v u binders (eq_ a✝¹ a✝)

case true_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v true_
⊢ admitsAux v u binders true_

case false_
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v false_
⊢ admitsAux v u binders false_

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

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

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

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

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

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

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

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

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

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders

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

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

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

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

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

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

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

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

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

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

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(v = a✝¹ ∨ v = a✝)
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders

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

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

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

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

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

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

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

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
by_cases c1 : v = x
State after:
case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi

case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ admitsAux v u (binders ∪ {x}) phi

State before:
case pos
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
apply mem_binders_imp_admitsAux
State after:
case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∪ {x}

State before:
case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∪ {x}
Tactic:
simp
State after:
case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∨ v = x

State before:
case pos.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : v = x
⊢ v ∈ binders ∨ v = x
Tactic:
tauto
State after:
no goals

State before:
case neg
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
apply phi_ih
State after:
case neg.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ ¬isFreeIn v phi

State before:
case neg.h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(¬v = x ∧ isFreeIn v phi)
c1 : ¬v = x
⊢ ¬isFreeIn v phi
Tactic:
tauto
State after:
no goals

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