internlm/Lean-Github huggingface dataset:
Theorem: FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
Proof state before:
case h.h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
  BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∪ {x}
Tactic:
simp
Proof state after:
case h.h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
  BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∧ ¬v = x
Theorem: FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
Proof state before:
case h.h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
  BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ v ∉ binders ∧ ¬v = x
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
Proof state before:
case h.h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
  BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Tactic:
simp only [toIsBoundAux] at h2
Proof state after:
case h.h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
  BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Theorem: FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
Proof state before:
case h.h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 :
  BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Tactic:
simp at h2
Proof state after:
case h.h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 : toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Theorem: FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux
Proof state before:
case h.h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
h2 : toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Tactic:
exact h2
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.admitsAux_self
Proof state before:
F : Formula
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders F
Tactic:
induction F generalizing binders
Proof state after:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (pred_const_ a✝¹ a✝)

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

case eq_
v a✝¹ a✝ : VarName
binders : Finset VarName
⊢ admitsAux v v binders (eq_ a✝¹ a✝)

case true_
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders true_

case false_
v : VarName
binders : Finset VarName
⊢ admitsAux v v binders false_

case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.imp_ a✝)

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.and_ a✝)

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.or_ a✝)

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (a✝¹.iff_ a✝)

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (forall_ a✝¹ a✝)

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders (exists_ a✝¹ a✝)

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (def_ a✝¹ a✝)
Theorem: FOL.NV.Sub.Var.One.Rec.admitsAux_self
Proof state before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders

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

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

case not_
v : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v (binders ∪ {a✝¹}) a✝

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝
binders : Finset VarName
⊢ admitsAux v v (binders ∪ {a✝¹}) a✝

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
Tactic:
all_goals
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.admitsAux_self
Proof state before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ admitsAux v v binders (def_ a✝¹ a✝)
Tactic:
simp only [admitsAux]
Proof state after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
Theorem: FOL.NV.Sub.Var.One.Rec.admitsAux_self
Proof state before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.admits_self
Proof state before:
F : Formula
v : VarName
⊢ admits v v F
Tactic:
simp only [admits]
Proof state after:
F : Formula
v : VarName
⊢ admitsAux v v ∅ F
Theorem: FOL.NV.Sub.Var.One.Rec.admits_self
Proof state before:
F : Formula
v : VarName
⊢ admitsAux v v ∅ F
Tactic:
apply admitsAux_self
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof state before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v F
⊢ admitsAux v u binders F
Tactic:
induction F generalizing binders
Proof 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✝)
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof 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✝)
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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]
Proof 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
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof 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
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof 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}
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof 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
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof 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
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux
Proof 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
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admits
Proof state before:
F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ admits v u F
Tactic:
simp only [admits]
Proof state after:
F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ admitsAux v u ∅ F
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admits
Proof state before:
F : Formula
v u : VarName
h1 : ¬isFreeIn v F
⊢ admitsAux v u ∅ F
Tactic:
exact not_isFreeIn_imp_admitsAux F v u ∅ h1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof 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
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof 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
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof 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
Proof 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✝)
Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof 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]
Proof 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
Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof 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
Proof 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
Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof 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
Proof 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