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

State before:
F : Formula
v u : VarName
S T : Finset VarName
h1 : admitsAux v u S F
h2 : u ∉ T
⊢ admitsAux v u (S ∪ T) F
Tactic:
induction F generalizing S
State after:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_const_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)

case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_var_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)

case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : admitsAux v u S (eq_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (eq_ a✝¹ a✝)

case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S true_
⊢ admitsAux v u (S ∪ T) true_

case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S false_
⊢ admitsAux v u (S ∪ T) false_

case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝.not_
⊢ admitsAux v u (S ∪ T) a✝.not_

case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.imp_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.imp_ a✝)

case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.and_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.and_ a✝)

case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.or_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.or_ a✝)

case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.iff_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.iff_ a✝)

case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (forall_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (forall_ a✝¹ a✝)

case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (exists_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (exists_ a✝¹ a✝)

case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (def_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)

State before:
case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_const_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)

case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_var_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)

case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : admitsAux v u S (eq_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (eq_ a✝¹ a✝)

case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S true_
⊢ admitsAux v u (S ∪ T) true_

case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S false_
⊢ admitsAux v u (S ∪ T) false_

case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝.not_
⊢ admitsAux v u (S ∪ T) a✝.not_

case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.imp_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.imp_ a✝)

case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.and_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.and_ a✝)

case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.or_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.or_ a✝)

case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.iff_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.iff_ a✝)

case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (forall_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (forall_ a✝¹ a✝)

case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (exists_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (exists_ a✝¹ a✝)

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

case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T

case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ S → u ∉ S
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ S ∪ T → u ∉ S ∪ T

case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝

case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ {a✝¹}) a✝
⊢ admitsAux v u (S ∪ T ∪ {a✝¹}) a✝

case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ {a✝¹}) a✝
⊢ admitsAux v u (S ∪ T ∪ {a✝¹}) a✝

case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T

State before:
v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T
Tactic:
case pred_const_ X xs | pred_var_ X xs | eq_ x y |def_ X xs =>
  simp
  tauto
State after:
no goals

State before:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ T ∪ {x}) phi
Tactic:
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
  simp only [Finset.union_right_comm S T {x}]
  tauto
State after:
no goals

State before:
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝

case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝

case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
Tactic:
all_goals
  tauto
State after:
no goals

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

State before:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)
Tactic:
simp only [admitsAux]
State after:
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T

State before:
v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T
Tactic:
simp
State after:
v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T

State before:
v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T
Tactic:
tauto
State after:
no goals

State before:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ T ∪ {x}) phi
Tactic:
simp only [Finset.union_right_comm S T {x}]
State after:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ {x} ∪ T) phi

State before:
v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ {x} ∪ T) phi
Tactic:
tauto
State after:
no goals

State before:
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
Tactic:
tauto
State after:
no goals