internlm/Lean-Github huggingface dataset:
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
Proof state before:
case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ fastAdmitsAux v u binders phi
Tactic:
exact fastAdmitsAux_del_binders phi v u binders {x} h1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
Proof state before:
case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → isFreeIn v phi → u ∉ binders
binders : Finset VarName
h2_left : ¬v = x
h2_right : isFreeIn v phi
h1 : fastAdmitsAux v u (binders ∪ {x}) phi
⊢ isFreeIn v phi
Tactic:
exact h2_right
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_isFreeIn
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ → u ∉ binders
h2 : v ∈ a✝
⊢ u ∉ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_mem_binders
Proof state before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : u ∈ binders
⊢ ¬isFreeIn v F
Tactic:
contrapose! h2
Proof state after:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_mem_binders
Proof state before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : fastAdmitsAux v u binders F
h2 : isFreeIn v F
⊢ u ∉ binders
Tactic:
exact fastAdmitsAux_isFreeIn F v u binders h1 h2
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
    BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
Tactic:
case eq_ x y =>
  simp
  constructor
  case left | right =>
    split_ifs
    case pos c1 =>
      subst c1
      tauto
    case neg c1 =>
      rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi
⊢ (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
Tactic:
case not_ phi phi_ih =>
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ (toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
    (toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
Tactic:
case
    imp_ phi psi phi_ih psi_ih
  | and_ phi psi phi_ih psi_ih
  | or_ phi psi phi_ih psi_ih
  | iff_ phi psi phi_ih psi_ih =>
  simp
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders (def_ a✝¹ a✝)
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
Tactic:
simp only [fastAdmitsAux] at h2
Proof state after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (fastReplaceFree v u (def_ a✝¹ a✝))
Tactic:
simp only [fastReplaceFree]
Proof state after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ toIsBoundAux binders (def_ a✝¹ a✝) = toIsBoundAux binders (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
Tactic:
simp only [toIsBoundAux]
Proof state after:
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ a✝ → u ∉ binders
⊢ BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) a✝) =
    BoolFormula.def_ a✝¹ (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) a✝))
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) xs) =
    BoolFormula.def_ X (List.map (fun v => decide (v ∈ binders)) (List.map (fun x => if v = x then u else x) xs))
Tactic:
simp
Proof state after:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ List.map (fun v => decide (v ∈ binders)) xs =
    List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ List.map (fun v => decide (v ∈ binders)) xs =
    List.map ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) xs
Tactic:
simp only [List.map_eq_map_iff]
Proof state after:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
⊢ ∀ x ∈ xs, decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
Tactic:
intro x a1
Proof state after:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ decide (x ∈ binders) = ((fun v => decide (v ∈ binders)) ∘ fun x => if v = x then u else x) x
Tactic:
simp
Proof state after:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
Tactic:
by_cases c1 : v = x
Proof state after:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders

case neg
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
Tactic:
subst c1
Proof state after:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ (if v = v then u else v) ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ (if v = v then u else v) ∈ binders
Tactic:
simp
Proof state after:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ u ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case pos
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
a1 : v ∈ xs
⊢ v ∈ binders ↔ u ∈ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case neg
v u : VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v ∈ xs → u ∉ binders
x : VarName
a1 : x ∈ xs
c1 : ¬v = x
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders
Tactic:
simp only [if_neg c1]
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ BoolFormula.eq_ (decide (x ∈ binders)) (decide (y ∈ binders)) =
    BoolFormula.eq_ (decide ((if v = x then u else x) ∈ binders)) (decide ((if v = y then u else y) ∈ binders))
Tactic:
simp
Proof state after:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ (x ∈ binders ↔ (if v = x then u else x) ∈ binders) ∧ (y ∈ binders ↔ (if v = y then u else y) ∈ binders)
Tactic:
constructor
Proof state after:
case left
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ x ∈ binders ↔ (if v = x then u else x) ∈ binders

case right
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
Tactic:
case left | right =>
  split_ifs
  case pos c1 =>
    subst c1
    tauto
  case neg c1 =>
    rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
⊢ y ∈ binders ↔ (if v = y then u else y) ∈ binders
Tactic:
split_ifs
Proof state after:
case pos
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
h✝ : v = y
⊢ y ∈ binders ↔ u ∈ binders

case neg
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
h✝ : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : v = y
⊢ y ∈ binders ↔ u ∈ binders
Tactic:
case pos c1 =>
  subst c1
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
Tactic:
case neg c1 =>
  rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : v = y
⊢ y ∈ binders ↔ u ∈ binders
Tactic:
subst c1
Proof state after:
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = v → u ∉ binders
⊢ v ∈ binders ↔ u ∈ binders
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = v → u ∉ binders
⊢ v ∈ binders ↔ u ∈ binders
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x y : VarName
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ v = y → u ∉ binders
c1 : ¬v = y
⊢ y ∈ binders ↔ y ∈ binders
Tactic:
rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi
⊢ (toIsBoundAux binders phi).not_ = (toIsBoundAux binders (fastReplaceFree v u phi)).not_
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ (toIsBoundAux binders phi).iff_ (toIsBoundAux binders psi) =
    (toIsBoundAux binders (fastReplaceFree v u phi)).iff_ (toIsBoundAux binders (fastReplaceFree v u psi))
Tactic:
simp
Proof state after:
v u : VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
    toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u : VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
psi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders psi → toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
binders : Finset VarName
h1 : v ∉ binders
h2 : fastAdmitsAux v u binders phi ∧ fastAdmitsAux v u binders psi
⊢ toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) ∧
    toIsBoundAux binders psi = toIsBoundAux binders (fastReplaceFree v u psi)
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v u phi))
Tactic:
split_ifs
Proof state after:
case pos
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h✝ : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)

case neg
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
h✝ : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
Tactic:
case pos c1 =>
  rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi)
Tactic:
rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    toIsBoundAux binders (exists_ x (fastReplaceFree v u phi))
Tactic:
simp only [toIsBoundAux]
Proof state after:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) =
    BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi))
Tactic:
simp
Proof state after:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
Tactic:
apply phi_ih
Proof state after:
case h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}

case h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∪ {x}
Tactic:
simp
Proof state after:
case h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case h1
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ v ∉ binders ∧ ¬v = x
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_free_and_bound_unchanged
Proof state before:
case h2
v u x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    v ∉ binders →
      fastAdmitsAux v u binders phi → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi)
binders : Finset VarName
h1 : v ∉ binders
h2 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
c1 : ¬v = x
⊢ fastAdmitsAux v u (binders ∪ {x}) phi
Tactic:
tauto
Proof state after:
no goals