-- src/Iris/Algebra/OFE.lean

namespace Iris

class OFE (α : Type _) where
  Equiv : α → α → Prop
  Dist : Nat → α → α → Prop
  dist_eqv : Equivalence (Dist n)
  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y
  dist_lt : Dist n x y → m < n → Dist m x y

open OFE

scoped infix:40 " ≡ " => OFE.Equiv

scoped notation:40 x " ≡{" n "}≡ " y:41 => OFE.Dist n x y

namespace OFE

class NonExpansive [OFE α] [OFE β] (f : α → β) where
  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂

instance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩

class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where
  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂

def DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y

class Contractive [OFE α] [OFE β] (f : α → β) where
  distLater_dist : DistLater n x y → f x ≡{n}≡ f y

instance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where
  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)

instance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where
  distLater_dist := fun _ => Dist.rfl

@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where
  f : α → β
  ne : NonExpansive f

@[inherit_doc]
infixr:25 " -n> " => Hom

instance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne

@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where
  [contractive : Contractive f]
  ne := ne_of_contractive f

infixr:25 " -c> " => ContractiveHom

instance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive

instance : OFE Unit where
  Equiv _ _ := True
  Dist _ _ _ := True
  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩
  equiv_dist := by admit /- proof elided -/

instance [OFE α] : OFE (ULift α) where
  Equiv x y := x.down ≡ y.down
  Dist n x y := x.down ≡{n}≡ y.down
  dist_eqv := InvImage.equivalence dist_eqv
  equiv_dist := equiv_dist
  dist_lt := dist_lt

instance [OFE α] : OFE (Option α) where
  Equiv := Option.Forall₂ Equiv
  Dist n := Option.Forall₂ (Dist n)
  dist_eqv := Option.Forall₂.equivalence dist_eqv
  equiv_dist {x y} := by admit /- proof elided -/

instance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where
  discrete_0 {mx my} e :=
    match mx, my with
    | none,   none   => e
    | none,   some _ => e
    | some _, none   => e
    | some x, some y => show x ≡ y from discrete_0 e

instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩

instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :
    NonExpansive₂ (Option.merge op) where
  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/

instance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where
  Equiv f g := ∀ x, f x ≡ g x
  Dist n f g := ∀ x, f x ≡{n}≡ g x
  dist_eqv := {
    refl _ _ := dist_eqv.refl _
    symm h _ := dist_eqv.symm (h _)
    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)
  }
  equiv_dist {_ _} := by admit /- proof elided -/

instance [OFE α] [OFE β] : OFE (α -n> β) where
  Equiv f g := f.f ≡ g.f
  Dist n f g := f.f ≡{n}≡ g.f
  dist_eqv := {
    refl _ := dist_eqv.refl _
    symm h := dist_eqv.symm h
    trans h1 h2 := dist_eqv.trans h1 h2
  }
  equiv_dist := equiv_dist
  dist_lt := dist_lt

instance [OFE α] [OFE β] : OFE (α -c> β) where
  Equiv f g := Equiv f.toHom g.toHom
  Dist n f g := Dist n f.toHom g.toHom
  dist_eqv := {
    refl _ := dist_eqv.refl _
    symm h := dist_eqv.symm h
    trans h1 h2 := dist_eqv.trans h1 h2
  }
  equiv_dist := equiv_dist
  dist_lt := dist_lt

instance [OFE α] [OFE β] : OFE (α × β) where
  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2
  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2
  dist_eqv := {
    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩
    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩
    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩
  }
  equiv_dist {_ _} := by admit /- proof elided -/

@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where
  hom : α -n> β
  inv : β -n> α
  hom_inv : hom (inv x) ≡ x
  inv_hom : inv (hom x) ≡ x

instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne

instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne

def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where
  hom := iso.inv
  inv := iso.hom
  hom_inv := by admit /- proof elided -/

end OFE

structure Chain (α : Type _) [OFE α] where
  chain : Nat → α
  cauchy : n ≤ i → chain i ≡{n}≡ chain n

namespace Chain

end Chain

class IsCOFE (α : Type _) [OFE α] where
  compl : Chain α → α
  conv_compl {c : Chain α} : compl c ≡{n}≡ c n

namespace COFE

end COFE

@[ext] structure LeibnizO (α : Type _) where
  car : α

section DiscreteFunOF

open COFE

end DiscreteFunOF

section Option

variable [OFE α]

def optionChain (c : Chain (Option α)) (x : α) : Chain α :=

instance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where
  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)
  conv_compl {n} c := by admit /- proof elided -/

end Option

section OptionOF

open COFE

variable (F : OFunctorPre)

end OptionOF

section Fixpoint

def Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where
  chain n := Nat.repeat f (n + 1) default
  cauchy {n} := by admit /- proof elided -/

def fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=
  COFE.compl <| Fixpoint.chain f

nonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f

-- Repository definitions used:
content: scoped notation:40 x " ≡{" n "}≡ " y:41 => OFE.Dist n x y

content: notation:50 x " ≼{" n "} " y:51 => IncludedN n x y

content: class Transitive (R : Relation α) where
  trans {x y z : α} : R x y → R y z → R x z

content: class Reflexive (R : Relation α) where
  refl {x : α} : R x x

content: instance : OFE.Discrete (DFrac F) := ⟨congrArg id⟩

content: def Agree.dist (n : Nat) (x y : Agree α) : Prop :=
  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧
  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)

content: instance : OFE (Tower F) where
  Equiv f g := ∀ k, f k ≡ g k
  Dist n f g := ∀ k, f k ≡{n}≡ g k
  dist_eqv := {
    refl _ _ := dist_eqv.refl _
    symm h _ := dist_eqv.symm (h _)
    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)
  }
  equiv_dist {_ _} := by admit /- proof elided -/

content: instance : OFE.NonExpansive (BUpd.bupd (PROP := UPred M)) := bupd_ne

content: instance bupd_ne : OFE.NonExpansive (bupd : UPred M → UPred M) where
  ne n x1 x2 Hx m y Hm Hv := by admit /- proof elided -/

content: class CMRA (α : Type _) extends OFE α where
  pcore : α → Option α
  op : α → α → α
  ValidN : Nat → α → Prop
  Valid : α → Prop

  op_ne : NonExpansive (op x)
  pcore_ne : x ≡{n}≡ y → pcore x = some cx →
    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy
  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y

  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x
  validN_succ : ValidN n.succ x → ValidN n x
  validN_op_left : ValidN n (op x y) → ValidN n x

  assoc : op x (op y z) ≡ op (op x y) z
  comm : op x y ≡ op y x

  pcore_op_left : pcore x = some cx → op cx x ≡ x
  pcore_idem : pcore x = some cx → pcore cx ≡ some cx
  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)

  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →
    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂

content: instance later_contractive : OFE.Contractive UPred.later (α := UPred M) where
  distLater_dist {n x y} Hl :=
    match n with
    | 0 => by admit /- proof elided -/
    | n + 1 => fun
      | 0 => by admit /- proof elided -/
      | n' + 1 => fun x' Hn' Hx' => Hl _ Hn' _ _ (Nat.le_refl _) (CMRA.validN_succ Hx')

content: protected def later (P : UPred M) : UPred M where
  holds n x := match n with | 0 => True | Nat.succ n' => P n' x
  mono {n₁ n₂} := by admit /- proof elided -/

content: class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where
  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]
  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q
  elim_persistently {P : PROP} : ■ P ⊢ <pers> P
  idem {P : PROP} : ■ P ⊢ ■ ■ P
  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ
  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)
  emp_intro {P : PROP} : P ⊢ ■ emp
  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P
  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P

content: class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where
  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]
  intro {P : PROP} : iprop(P ⊢ |==> P)
  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)
  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)
  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))

content: @[ext]
structure UPred (M : Type _) [UCMRA M] where
  holds : Nat → M → Prop
  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2

content: class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)
    (iaction saction : ModalityAction PROP1 PROP2) where
  spec_intuitionistic : iaction.intuitionistic_action_spec M
  spec_spatial : saction.spatial_action_spec M
  emp : iprop(emp) ⊢ M iprop(emp)
  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q
  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)

content: class UCMRA (α : Type _) extends CMRA α where
  unit : α
  unit_valid : ✓ unit
  unit_left_id : unit • x ≡ x
  pcore_unit : pcore unit ≡ some unit

content: def pcore : DFrac F → Option (DFrac F)
  | own _        => none
  | .discard     => some discard
  | ownDiscard _ => some discard

content: inductive DFrac (F : Type _) where
 
| own (f : F) : DFrac F
 
| discard : DFrac F
 
| ownDiscard (f : F) : DFrac F

content: def op : DFrac F → DFrac F → DFrac F
  | .discard, .discard => discard
  | own f, .discard
  | ownDiscard f, .discard
  | .discard, own f
  | .discard, ownDiscard f => ownDiscard f
  | own f, own f' => own (f + f')
  | own f, ownDiscard f'
  | ownDiscard f, own f'
  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')

content: class Associative (R : Relation α) (f : α → α → α) where
  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))

content: class Commutative (R : Relation α) (f : β → β → α) where
  comm {x y : β} : R (f x y) (f y x)

content: def Included (x y : α) : Prop := ∃ z, y ≡ x • z

content: instance instOFE [Store T K V] [OFE V] : OFE T where
  Equiv s0 s1  := get s0 ≡ get s1
  Dist n s0 s1 := get s0 ≡{n}≡ get s1
  dist_eqv     := ⟨fun _ => .of_eq rfl, (·.symm), (·.trans ·)⟩
  equiv_dist   := equiv_dist
  dist_lt      := dist_lt

content: @[simp] def Store.Equiv [Store T K V] (t1 t2 : T) : Prop := get t1 = get t2

content: instance Store.Equiv_trans [Store T K V] : Trans Equiv (Equiv (T := T)) Equiv := ⟨by admit /- proof elided -/
⟩

content: instance [CMRA β] : OFE (α -C> β) where
  Equiv f g := f.toHom ≡ g.toHom
  Dist n f g := f.toHom ≡{n}≡ g.toHom
  dist_eqv := {
    refl _ := dist_eqv.refl _
    symm h := dist_eqv.symm h
    trans h1 h2 := dist_eqv.trans h1 h2
  }
  equiv_dist := equiv_dist
  dist_lt := dist_lt

content: instance [OFE α] [IsCOFE α] : IsCOFE (Excl α) where
  compl c := (c 0).map fun x => IsCOFE.compl (exclChain c x)
  conv_compl {n} c := by admit /- proof elided -/

content: instance (k : Nat) : NonExpansive (fun X : Tower F => X.val k) := ⟨fun _ _ _ => (· _)⟩

content: instance [OFE α] : NonExpansive excl (α := α) where
  ne _ _ _ a := a

content: instance get_ne [Store T K V] [OFE V] (k : K) : NonExpansive (get · k : T → V) where
  ne {_ _ _} Ht := Ht k

content: instance [Store T1 K V1] [Store T2 K V2] [OFE V1] [OFE V2] (f : K → V1 → V2)
  [∀ k, NonExpansive (f k)] [HasStoreMap T1 T2 K V1 V2] : NonExpansive (dmap f : T1 → T2) where
  ne _ {_ _} H k := by admit /- proof elided -/

content: class HasStoreMap (T1 T2 : Type _) (K V1 V2 : outParam (Type _)) [Store T1 K V1] [Store T2 K V2] where
   
  dmap (f : K → V1 → V2) : T1 → T2
  get_dmap : get (dmap f t) k = f k (get t k)

content: instance : NonExpansive (pcore (α := α)) where
  ne n x {y} e := by admit /- proof elided -/

content: instance [Heap T K V] [OFE V] (op : V → V → V) [NonExpansive₂ op] :
    NonExpansive₂ (merge (T := T) op) where
  ne _ {_ _} Ht {_ _} Hs k := by admit /- proof elided -/

content: instance Heap.instCOFE [Heap T K V] [COFE V] : COFE T where
  compl c := hmap (fun _ => COFE.compl <| c.map ⟨_, Store.get_ne ·⟩) (c 0)
  conv_compl {_ c} k := by admit /- proof elided -/

content: infix:60 " • " => op

content: infix:50 " ≼ " => Included

content: @[inherit_doc]
infixr:25 " -C> " => Hom

-- Repository lemmas used:
content: theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]

-- Local definitions used:
content: class OFE (α : Type _) where
  Equiv : α → α → Prop
  Dist : Nat → α → α → Prop
  dist_eqv : Equivalence (Dist n)
  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y
  dist_lt : Dist n x y → m < n → Dist m x y

content: class NonExpansive [OFE α] [OFE β] (f : α → β) where
  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂

content: instance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩

content: class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where
  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂

content: def DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y

content: class Contractive [OFE α] [OFE β] (f : α → β) where
  distLater_dist : DistLater n x y → f x ≡{n}≡ f y

content: instance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where
  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)

content: instance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where
  distLater_dist := fun _ => Dist.rfl

content: @[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where
  f : α → β
  ne : NonExpansive f

content: instance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne

content: @[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where
  [contractive : Contractive f]
  ne := ne_of_contractive f

content: instance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive

content: instance : OFE Unit where
  Equiv _ _ := True
  Dist _ _ _ := True
  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩
  equiv_dist := by admit /- proof elided -/

content: instance [OFE α] : OFE (ULift α) where
  Equiv x y := x.down ≡ y.down
  Dist n x y := x.down ≡{n}≡ y.down
  dist_eqv := InvImage.equivalence dist_eqv
  equiv_dist := equiv_dist
  dist_lt := dist_lt

content: instance [OFE α] : OFE (Option α) where
  Equiv := Option.Forall₂ Equiv
  Dist n := Option.Forall₂ (Dist n)
  dist_eqv := Option.Forall₂.equivalence dist_eqv
  equiv_dist {x y} := by admit /- proof elided -/

content: instance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where
  discrete_0 {mx my} e :=
    match mx, my with
    | none,   none   => e
    | none,   some _ => e
    | some _, none   => e
    | some x, some y => show x ≡ y from discrete_0 e

content: instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩

content: instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :
    NonExpansive₂ (Option.merge op) where
  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/

content: instance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where
  Equiv f g := ∀ x, f x ≡ g x
  Dist n f g := ∀ x, f x ≡{n}≡ g x
  dist_eqv := {
    refl _ _ := dist_eqv.refl _
    symm h _ := dist_eqv.symm (h _)
    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)
  }
  equiv_dist {_ _} := by admit /- proof elided -/

content: instance [OFE α] [OFE β] : OFE (α -n> β) where
  Equiv f g := f.f ≡ g.f
  Dist n f g := f.f ≡{n}≡ g.f
  dist_eqv := {
    refl _ := dist_eqv.refl _
    symm h := dist_eqv.symm h
    trans h1 h2 := dist_eqv.trans h1 h2
  }
  equiv_dist := equiv_dist
  dist_lt := dist_lt

content: instance [OFE α] [OFE β] : OFE (α -c> β) where
  Equiv f g := Equiv f.toHom g.toHom
  Dist n f g := Dist n f.toHom g.toHom
  dist_eqv := {
    refl _ := dist_eqv.refl _
    symm h := dist_eqv.symm h
    trans h1 h2 := dist_eqv.trans h1 h2
  }
  equiv_dist := equiv_dist
  dist_lt := dist_lt

content: instance [OFE α] [OFE β] : OFE (α × β) where
  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2
  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2
  dist_eqv := {
    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩
    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩
    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩
  }
  equiv_dist {_ _} := by admit /- proof elided -/

content: @[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where
  hom : α -n> β
  inv : β -n> α
  hom_inv : hom (inv x) ≡ x
  inv_hom : inv (hom x) ≡ x

content: instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne

content: instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne

content: def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where
  hom := iso.inv
  inv := iso.hom
  hom_inv := by admit /- proof elided -/

content: structure Chain (α : Type _) [OFE α] where
  chain : Nat → α
  cauchy : n ≤ i → chain i ≡{n}≡ chain n

content: class IsCOFE (α : Type _) [OFE α] where
  compl : Chain α → α
  conv_compl {c : Chain α} : compl c ≡{n}≡ c n

content: @[ext] structure LeibnizO (α : Type _) where
  car : α

content: def optionChain (c : Chain (Option α)) (x : α) : Chain α :=

content: instance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where
  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)
  conv_compl {n} c := by admit /- proof elided -/

content: def Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where
  chain n := Nat.repeat f (n + 1) default
  cauchy {n} := by admit /- proof elided -/

content: def fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=
  COFE.compl <| Fixpoint.chain f

nonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f

-- Local lemmas used:
content: theorem Dist.lt [OFE α] {m n} {x y : α} : x ≡{n}≡ y → m < n → x ≡{m}≡ y

content: theorem Dist.le [OFE α] {m n} {x y : α} (h : x ≡{n}≡ y) (h' : m ≤ n) : x ≡{m}≡ y

content: @[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x

content: @[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x

content: theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z

content: theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv

content: @[symm] theorem Equiv.symm [OFE α] {x : α} : x ≡ y → y ≡ x

content: theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z

content: theorem Equiv.dist [OFE α] {x : α} : x ≡ y → x ≡{n}≡ y

content: @[symm] theorem DistLater.symm [OFE α] {n} {x : α} (h : DistLater n x y) : DistLater n y x

content: theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :
    DistLater n x z

content: @[simp] theorem distLater_zero [OFE α] {x y : α} : DistLater 0 x y

content: theorem distLater_succ [OFE α] {n} {x y : α} : DistLater n.succ x y ↔ x ≡{n}≡ y

content: @[simp] theorem Contractive.zero [OFE α] [OFE β] (f : α → β) [Contractive f] {x y} :
    f x ≡{0}≡ f y

content: theorem Contractive.succ [OFE α] [OFE β] (f : α → β) [Contractive f] {n x y}
    (h : x ≡{n}≡ y) : f x ≡{n.succ}≡ f y

content: theorem fixpoint_unfold [COFE α] [Inhabited α] (f : α -c> α) :
    fixpoint f ≡ f (fixpoint f)

-- Target theorem and proof:
theorem fixpoint_unique [COFE α] [Inhabited α] {f : α -c> α} {x : α} (H : x ≡ f x) :
    x ≡ fixpoint f
:= by
  refine equiv_dist.mpr fun n => ?_
  induction n with refine H.dist.trans <| .trans ?_ (fixpoint_unfold f).dist.symm
  | zero => exact Contractive.zero f.f
  | succ _ IH => exact Contractive.succ f.f IH