internlm/Lean-Github huggingface dataset:
Theorem: FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux
Proof state before:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders ∨ v = x
Tactic:
left
Proof state after:
case h1.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux
Proof state before:
case h1.h
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ binders
Tactic:
exact h1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
  cases h1
  case inl h1 =>
    apply mem_binders_imp_admitsAux
    subst h1
    simp
  case inr h1 =>
    apply phi_ih
    exact h1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders

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

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

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

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders 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), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders 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), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders 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), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝
binders : Finset VarName
h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders 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✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
Tactic:
all_goals
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof 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✝)
⊢ admitsAux v u binders (def_ a✝¹ a✝)
Tactic:
simp only [fastAdmitsAux] at h1
Proof state after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝)
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → 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 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
cases h1
Proof state after:
case inl
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h✝ : v = x
⊢ admitsAux v u (binders ∪ {x}) phi

case inr
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h✝ : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
case inl h1 =>
  apply mem_binders_imp_admitsAux
  subst h1
  simp
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
case inr h1 =>
  apply phi_ih
  exact h1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
apply mem_binders_imp_admitsAux
Proof state after:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ v ∈ binders ∪ {x}
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : v = x
⊢ v ∈ binders ∪ {x}
Tactic:
subst h1
Proof state after:
case h1
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
⊢ v ∈ binders ∪ {v}
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
case h1
v u : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
⊢ v ∈ binders ∪ {v}
Tactic:
simp
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ admitsAux v u (binders ∪ {x}) phi
Tactic:
apply phi_ih
Proof state after:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi
binders : Finset VarName
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
Tactic:
exact h1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits
Proof state before:
F : Formula
v u : VarName
⊢ admits v u F ↔ fastAdmits v u F
Tactic:
simp only [admits]
Proof state after:
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmits v u F
Theorem: FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits
Proof state before:
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmits v u F
Tactic:
simp only [fastAdmits]
Proof state after:
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmitsAux v u ∅ F
Theorem: FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits
Proof state before:
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F ↔ fastAdmitsAux v u ∅ F
Tactic:
constructor
Proof state after:
case mp
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F → fastAdmitsAux v u ∅ F

case mpr
F : Formula
v u : VarName
⊢ fastAdmitsAux v u ∅ F → admitsAux v u ∅ F
Theorem: FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits
Proof state before:
case mp
F : Formula
v u : VarName
⊢ admitsAux v u ∅ F → fastAdmitsAux v u ∅ F
Tactic:
apply admitsAux_imp_fastAdmitsAux
Proof state after:
case mp.h1
F : Formula
v u : VarName
⊢ v ∉ ∅
Theorem: FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits
Proof state before:
case mp.h1
F : Formula
v u : VarName
⊢ v ∉ ∅
Tactic:
simp
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits
Proof state before:
case mpr
F : Formula
v u : VarName
⊢ fastAdmitsAux v u ∅ F → admitsAux v u ∅ F
Tactic:
exact fastAdmitsAux_imp_admitsAux F v u ∅
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
F : Formula
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux 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
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝)

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

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

case true_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders true_

case false_
v : VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders false_

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

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

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

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

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

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

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

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders

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

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

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

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

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

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

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

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
Tactic:
all_goals
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)
Tactic:
simp only [fastAdmitsAux]
Proof state after:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
by_cases c1 : v = x
Proof state after:
case pos
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi

case neg
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case pos
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
left
Proof state after:
case pos.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case pos.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : v = x
⊢ v = x
Tactic:
exact c1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case neg
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
right
Proof state after:
case neg.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ fastAdmitsAux v v (binders ∪ {x}) phi
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case neg.h
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ fastAdmitsAux v v (binders ∪ {x}) phi
Tactic:
apply phi_ih
Proof state after:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
Tactic:
simp
Proof state after:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case neg.h.h1
v x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi
binders : Finset VarName
h1 : v ∉ binders
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self
Proof state before:
case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
⊢ v ∈ a✝ → v ∉ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmits_self
Proof state before:
F : Formula
v : VarName
⊢ fastAdmits v v F
Tactic:
simp only [fastAdmits]
Proof state after:
F : Formula
v : VarName
⊢ fastAdmitsAux v v ∅ F
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmits_self
Proof state before:
F : Formula
v : VarName
⊢ fastAdmitsAux v v ∅ F
Tactic:
apply fastAdmitsAux_self
Proof state after:
case h1
F : Formula
v : VarName
⊢ v ∉ ∅
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmits_self
Proof state before:
case h1
F : Formula
v : VarName
⊢ v ∉ ∅
Tactic:
simp
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux
Proof state before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isFreeIn v F
⊢ fastAdmitsAux 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✝)
⊢ fastAdmitsAux 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✝)
⊢ fastAdmitsAux 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✝)
⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝)

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

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

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

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

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

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