internlm/Lean-Github huggingface dataset:
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi
Tactic:
apply psi_ih
Proof state after:
case h.e'_2.h1
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v
Tactic:
intro v a1
Proof state after:
case h.e'_2.h1
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ σ v = σ' v
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ σ v = σ' v
Tactic:
apply h1
Proof state after:
case h.e'_2.h1.a
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v phi ∨ isFreeIn v psi
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1.a
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v phi ∨ isFreeIn v psi
Tactic:
right
Proof state after:
case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v psi
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1.a.h
phi psi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v psi
⊢ isFreeIn v psi
Tactic:
exact a1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) =
    exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi)
Tactic:
congr! 1
Proof state after:
case h.e'_2
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi
Tactic:
apply phi_ih
Proof state after:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v
Tactic:
intro v a1
Proof state after:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v
Tactic:
simp only [Function.updateITE]
Proof state after:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ (if v = x then x else σ v) = if v = x then x else σ' v
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case h.e'_2.h1
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ (if v = x then x else σ v) = if v = x then x else σ' v
Tactic:
split_ifs
Proof state after:
case pos
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
h✝ : v = x
⊢ x = x

case neg
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
h✝ : ¬v = x
⊢ σ v = σ' v
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : v = x
⊢ x = x
Tactic:
case _ c1 =>
  rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ σ v = σ' v
Tactic:
case _ c1 =>
  apply h1
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : v = x
⊢ x = x
Tactic:
rfl
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ σ v = σ' v
Tactic:
apply h1
Proof state after:
case a
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ ¬v = x ∧ isFreeIn v phi
Theorem: FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
Proof state before:
case a
x : VarName
phi : Formula
phi_ih :
  ∀ (σ σ' : VarName → VarName),
    (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
c1 : ¬v = x
⊢ ¬v = x ∧ isFreeIn v phi
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) =
    def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs)
Tactic:
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
  congr! 1
  simp only [List.map_eq_map_iff]
  intro x _
  split_ifs <;> tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_
Tactic:
case not_ phi phi_ih =>
  simp
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) =
    (replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders 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.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)
Tactic:
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
  congr! 1
  apply phi_ih
  simp
  tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)
Tactic:
simp only [replaceFreeAux]
Proof state after:
case def_
σ σ' : VarName → VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) =
    def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) =
    def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs)
Tactic:
congr! 1
Proof state after:
case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs
Tactic:
simp only [List.map_eq_map_iff]
Proof state after:
case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
Tactic:
intro x _
Proof state after:
case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
x : VarName
a✝ : x ∈ xs
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2
σ σ' : VarName → VarName
X : DefName
xs : List VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
x : VarName
a✝ : x ∈ xs
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
Tactic:
split_ifs <;> tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ eq_ (if x ∉ binders then σ x else x) (if y ∉ binders then σ y else y) =
    eq_ (if x ∉ binders then σ' x else x) (if y ∉ binders then σ' y else y)
Tactic:
congr! 1
Proof state after:
case h.e'_1
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x

case h.e'_2
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_1
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
Tactic:
split_ifs <;> tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2
σ σ' : VarName → VarName
x y : VarName
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y
Tactic:
split_ifs <;> tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_
Tactic:
simp
Proof state after:
σ σ' : VarName → VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) =
    (replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi)
Tactic:
simp
Proof state after:
σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧
    replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
phi psi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
psi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧
    replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
Tactic:
tauto
Proof state after:
no goals
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)
Tactic:
congr! 1
Proof state after:
case h.e'_2
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi
Tactic:
apply phi_ih
Proof state after:
case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v
Tactic:
simp
Proof state after:
case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v
Theorem: FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free
Proof state before:
case h.e'_2.h1
σ σ' : VarName → VarName
x : VarName
phi : Formula
phi_ih :
  ∀ (binders : Finset VarName),
    (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
binders : Finset VarName
h1 : ∀ v ∉ binders, σ v = σ' v
⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v
Tactic:
tauto
Proof state after:
no goals