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

State before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u F
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)

case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ admitsAux v u binders true_

case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ admitsAux v u binders false_

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

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

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

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

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

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

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

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)

case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ admitsAux v u binders true_

case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ admitsAux v u binders false_

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

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

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

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

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

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

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

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

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

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders

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

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {a✝¹}) a✝

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

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

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
  push_neg at h1
  cases h1
  case intro h1_left h1_right =>
    apply phi_ih (binders ∪ {x}) h1_right
    simp
    tauto
State after:
no goals

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

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

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders

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

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝)
h2 : u ∉ binders
⊢ 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 : ¬False
h2 : u ∉ binders
⊢ 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 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝)
Tactic:
simp only [isBoundIn] at h1
State after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬False
h2 : u ∉ binders
⊢ 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 : ¬False
h2 : u ∉ binders
⊢ 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 : ¬False
h2 : u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : ¬(u = x ∨ isBoundIn u phi)
h2 : u ∉ binders
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
push_neg at h1
State after:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1 : u ≠ x ∧ ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi

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

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
case intro h1_left h1_right =>
  apply phi_ih (binders ∪ {x}) h1_right
  simp
  tauto
State after:
no goals

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
apply phi_ih (binders ∪ {x}) h1_right
State after:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x}

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∪ {x}
Tactic:
simp
State after:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x

State before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi
binders : Finset VarName
h2 : u ∉ binders
h1_left : u ≠ x
h1_right : ¬isBoundIn u phi
⊢ u ∉ binders ∧ ¬u = x
Tactic:
tauto
State after:
no goals

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