open import Cat.Functor.Subcategory
open import Cat.Displayed.Total
open import Cat.Prelude

open import Data.Maybe.Base

open import Order.Diagram.Fixpoint
open import Order.Diagram.Bottom
open import Order.Diagram.Lub
open import Order.Base
open import Order.DCPO

import Cat.Reasoning

import Data.Nat as Nat

module Order.DCPO.Pointed where

private variable
  o ℓ : Level
  Ix : Type o

Pointed DCPOs🔗

A DCPO is pointed if it has a least element $⊥ .$ This is a property of a DCPO, as bottom elements are unique.

is-pointed-dcpo : DCPO o ℓ → Type _
is-pointed-dcpo D = Bottom (DCPO.poset D)

is-pointed-dcpo-is-prop : ∀ (D : DCPO o ℓ) → is-prop (is-pointed-dcpo D)
is-pointed-dcpo-is-prop D = Bottom-is-prop (DCPO.poset D)

A DCPO is pointed if and only if it has least upper bounds of all semidirected families.

module _ {o ℓ} (D : DCPO o ℓ) where
  open DCPO D
  open Lub

The forward direction is straightforward: bottom elements are equivalent to least upper bounds of the empty family $⊥ \to D,$ and this family is trivially semidirected.

  semidirected-lub→pointed
    : (∀ {Ix : Type o} (s : Ix → Ob) → is-semidirected-family poset s → Lub poset s)
    → is-pointed-dcpo D
  semidirected-lub→pointed lub =
    Lub→Bottom poset (lower-lub (lub (λ ()) (λ ())))

Conversely, if $D$ has a bottom element $⊥,$ then we can extend any semidirected family $I \to D$ to a directed family $Maybe (I) \to D$ by sending nothing to the bottom element.

module _ {o ℓ} (D : DCPO o ℓ) (pointed : is-pointed-dcpo D) where
  open DCPO D
  open Bottom pointed
  open is-directed-family
  open is-lub

  extend-bottom : (Ix → Ob) → Maybe Ix → Ob
  extend-bottom s nothing = bot
  extend-bottom s (just i) = s i

  extend-bottom-directed
    : (s : Ix → Ob) → is-semidirected-family poset s
    → is-directed-family poset (extend-bottom s)
  extend-bottom-directed s semidir .elt = inc nothing
  extend-bottom-directed s semidir .semidirected (just i) (just j) = do
    (k , i≤k , j≤k) ← semidir i j
    pure (just k , i≤k , j≤k)
  extend-bottom-directed s semidir .semidirected (just x) nothing =
    pure (just x , ≤-refl , ¡)
  extend-bottom-directed s semidir .semidirected nothing (just y) =
    pure (just y , ¡ , ≤-refl)
  extend-bottom-directed s semidir .semidirected nothing nothing =
   pure (nothing , ≤-refl , ≤-refl)

Furthermore, $s$ has a least upper bound only if the extended family does.

  lub→extend-bottom-lub
    : ∀ {s : Ix → Ob} {x : Ob} → is-lub poset s x → is-lub poset (extend-bottom s) x
  lub→extend-bottom-lub {s = s} {x = x} x-lub .fam≤lub (just i) = x-lub .fam≤lub i
  lub→extend-bottom-lub {s = s} {x = x} x-lub .fam≤lub nothing = ¡
  lub→extend-bottom-lub {s = s} {x = x} x-lub .least y le = x-lub .least y (le ⊙ just)

  extend-bottom-lub→lub
    : ∀ {s : Ix → Ob} {x : Ob} → is-lub poset (extend-bottom s) x → is-lub poset s x
  extend-bottom-lub→lub x-lub .fam≤lub i = x-lub .fam≤lub (just i)
  extend-bottom-lub→lub x-lub .least y le = x-lub .least y λ where
    nothing → ¡
    (just i) → le i

If we put this all together, we see that any semidirected family has a least upper bound in a pointed DCPO.

  pointed→semidirected-lub
    : is-pointed-dcpo D
    → ∀ {Ix : Type o} (s : Ix → Ob) → is-semidirected-family poset s → Lub poset s
  pointed→semidirected-lub pointed {Ix = Ix} s semidir .Lub.lub =
    ⋃ (extend-bottom s) (extend-bottom-directed s semidir)
  pointed→semidirected-lub pointed {Ix = Ix} s semidir .Lub.has-lub =
    extend-bottom-lub→lub (⋃.has-lub _ _)

Fixpoints🔗

Let $D$ be a pointed DCPO. Every Scott continuous function $f : D \to D$ has a least fixed point.

module _ {o ℓ} {D : DCPO o ℓ} where
  open DCPO D

  pointed→least-fixpoint
    : is-pointed-dcpo D
    → (f : DCPOs.Hom D D)
    → Least-fixpoint poset (Scott.mono f)
  pointed→least-fixpoint pointed f = f-fix where
    open Bottom pointed
    module f = Scott f

We begin by constructing a directed family $N \to D$ that maps $n$ to $f^{n} (⊥) .$

    fⁿ : Nat → Ob → Ob
    fⁿ zero x = x
    fⁿ (suc n) x = f · (fⁿ n x)

    fⁿ-mono : ∀ {i j} → i Nat.≤ j → fⁿ i bot ≤ fⁿ j bot
    fⁿ-mono Nat.0≤x = ¡
    fⁿ-mono (Nat.s≤s p) = f.monotone (fⁿ-mono p)

    fⁿ⊥ : Lift o Nat → Ob
    fⁿ⊥ (lift n) = fⁿ n bot

    fⁿ⊥-dir : is-directed-family poset fⁿ⊥
    fⁿ⊥-dir .is-directed-family.elt = inc (lift zero)
    fⁿ⊥-dir .is-directed-family.semidirected (lift i) (lift j) =
      inc (lift (Nat.max i j) , fⁿ-mono (Nat.max-≤l i j) , fⁿ-mono (Nat.max-≤r i j))

The least upper bound of this sequence shall be our least fixpoint. We begin by proving a slightly stronger variation of the universal property; namely for any $y$ such that $f y \leq y,$ $⋃ (f^{n} (⊥)) \leq y .$ This follows from som quick induction.

    fⁿ⊥≤fix : ∀ (y : Ob) → f · y ≤ y → ∀ n → fⁿ⊥ n ≤ y
    fⁿ⊥≤fix y p (lift zero) = ¡
    fⁿ⊥≤fix y p (lift (suc n)) =
      f · (fⁿ n bot)   ≤⟨ f.monotone (fⁿ⊥≤fix y p (lift n)) ⟩≤
      f · y            ≤⟨ p ⟩≤
      y                ≤∎

    least-fix : ∀ (y : Ob) → f · y ≤ y → ⋃ fⁿ⊥ fⁿ⊥-dir ≤ y
    least-fix y p = ⋃.least _ _ _ (fⁿ⊥≤fix y p)

Now, let’s show that $⋃ (f^{n} (⊥))$ is actually a fixpoint. First, the forward direction: $⋃ (f^{n} (⊥)) \leq f (⋃ (f^{n} (⊥))) .$ This follows directly from Scott-continuity of $f .$

    roll : f · (⋃ fⁿ⊥ fⁿ⊥-dir) ≤ ⋃ fⁿ⊥ fⁿ⊥-dir
    roll =
      f · (⋃ fⁿ⊥ _)        =⟨ f.pres-⋃ fⁿ⊥ fⁿ⊥-dir ⟩=
      ⋃ (apply f ⊙ fⁿ⊥) _  ≤⟨ ⋃.least _ _ _ (λ (lift n) → ⋃.fam≤lub _ _ (lift (suc n))) ⟩≤
      ⋃ fⁿ⊥ _              ≤∎

To show the converse, we use universal property we proved earlier to re-arrange the goal to $f (f (⋃ (f^{n} (⊥)))) \leq f (⋃ (f^{n} (⊥))) .$ We can then apply monotonicity of $f,$ and then use the forward direction to finish off the proof.

    unroll : ⋃ fⁿ⊥ fⁿ⊥-dir ≤ f · (⋃ fⁿ⊥ fⁿ⊥-dir)
    unroll = least-fix (f · (⋃ fⁿ⊥ fⁿ⊥-dir)) $
      f.monotone roll

All that remains is to package up the data.

    f-fix : Least-fixpoint poset f.mono
    f-fix .Least-fixpoint.fixpoint = ⋃ fⁿ⊥ fⁿ⊥-dir
    f-fix .Least-fixpoint.has-least-fixpoint .is-least-fixpoint.fixed =
      ≤-antisym roll unroll
    f-fix .Least-fixpoint.has-least-fixpoint .is-least-fixpoint.least y y-fix =
      least-fix y (≤-refl' y-fix)

Strictly Scott-continuous maps🔗

A Scott-continuous map is strictly continuous if it preserves bottoms.

module _ {o ℓ} {D E : DCPO o ℓ} where
  private
    module D = DCPO D
    module E = DCPO E

  is-strictly-scott-continuous : (f : DCPOs.Hom D E) → Type _
  is-strictly-scott-continuous f =
    ∀ (x : D.Ob) → is-bottom D.poset x → is-bottom E.poset (f · x)

  is-strictly-scott-is-prop
    : (f : DCPOs.Hom D E) → is-prop (is-strictly-scott-continuous f)
  is-strictly-scott-is-prop f = Π-is-hlevel² 1 λ x _ →
    is-bottom-is-prop E.poset (f · x)

Strictly Scott-continuous functions are closed under identities and composites.

strict-scott-id
  : ∀ {D : DCPO o ℓ}
  → is-strictly-scott-continuous (DCPOs.id {x = D})
strict-scott-id x x-bot = x-bot

strict-scott-∘
  : ∀ {D E F : DCPO o ℓ}
  → (f : DCPOs.Hom E F) (g : DCPOs.Hom D E)
  → is-strictly-scott-continuous f → is-strictly-scott-continuous g
  → is-strictly-scott-continuous (f DCPOs.∘ g)
strict-scott-∘ f g f-strict g-strict x x-bot =
  f-strict (g · x) (g-strict x x-bot)

Pointed DCPOs and strictly Scott-continuous functions form a subcategory of the category of DCPOs.

Pointed-DCPOs-subcat : ∀ o ℓ → Subcat (DCPOs o ℓ) (o ⊔ ℓ) (o ⊔ ℓ)
Pointed-DCPOs-subcat o ℓ .Subcat.is-ob = is-pointed-dcpo
Pointed-DCPOs-subcat o ℓ .Subcat.is-hom f _ _ = is-strictly-scott-continuous f
Pointed-DCPOs-subcat o ℓ .Subcat.is-hom-prop f _ _ =
  is-strictly-scott-is-prop f
Pointed-DCPOs-subcat o ℓ .Subcat.is-hom-id {D} _ = strict-scott-id {D = D}
Pointed-DCPOs-subcat o ℓ .Subcat.is-hom-∘ {f = f} {g = g} f-strict g-strict =
  strict-scott-∘ f g f-strict g-strict

Pointed-DCPOs : ∀ o ℓ → Precategory (lsuc (o ⊔ ℓ)) (lsuc o ⊔ ℓ)
Pointed-DCPOs o ℓ = Subcategory (Pointed-DCPOs-subcat o ℓ)

Pointed-DCPOs-is-category : is-category (Pointed-DCPOs o ℓ)
Pointed-DCPOs-is-category =
  subcat-is-category DCPOs-is-category is-pointed-dcpo-is-prop

Reasoning with pointed DCPOs🔗

The following module re-exports facts about pointed DCPOs, and also proves a bunch of useful lemma.s

module Pointed-DCPOs {o ℓ : Level} = Cat.Reasoning (Pointed-DCPOs o ℓ)

Pointed-dcpo : ∀ o ℓ → Type _
Pointed-dcpo o ℓ = Pointed-DCPOs.Ob {o} {ℓ}

Pointed-DCPOs↪DCPOs : Functor (Pointed-DCPOs o ℓ) (DCPOs o ℓ)
Pointed-DCPOs↪DCPOs = Forget-subcat

Pointed-DCPOs↪Sets : Functor (Pointed-DCPOs o ℓ) (Sets o)
Pointed-DCPOs↪Sets = DCPOs↪Sets F∘ Pointed-DCPOs↪DCPOs

module Pointed-dcpo {o ℓ} (D : Pointed-dcpo o ℓ) where

These proofs are all quite straightforward, so we omit them.

  open is-directed-family

  dcpo : DCPO o ℓ
  dcpo = D .fst

  has-pointed : is-pointed-dcpo dcpo
  has-pointed = D .snd

  open DCPO dcpo public

  bottom : Ob
  bottom = Bottom.bot (D .snd)

  bottom≤x : ∀ x → bottom ≤ x
  bottom≤x = Bottom.has-bottom (D .snd)

  adjoin : ∀ {Ix : Type o} → (Ix → Ob) → Maybe Ix → Ob
  adjoin = extend-bottom dcpo has-pointed

  adjoin-directed
    : ∀ (s : Ix → Ob) → is-semidirected-family poset s
    → is-directed-family poset (adjoin s)
  adjoin-directed = extend-bottom-directed dcpo has-pointed

  lub→adjoin-lub : ∀ {s : Ix → Ob} {x : Ob} → is-lub poset s x → is-lub poset (adjoin s) x
  lub→adjoin-lub = lub→extend-bottom-lub dcpo has-pointed

  adjoin-lub→lub : ∀ {s : Ix → Ob} {x : Ob} → is-lub poset (adjoin s) x → is-lub poset s x
  adjoin-lub→lub = extend-bottom-lub→lub dcpo has-pointed

  -- We put these behind 'opaque' to prevent blow ups in goals.
  opaque
    ⋃-semi : (s : Ix → Ob) → is-semidirected-family poset s → Ob
    ⋃-semi s semidir = ⋃ (adjoin s) (adjoin-directed s semidir)

    ⋃-semi-lub
      : ∀ (s : Ix → Ob) (dir : is-semidirected-family poset s)
      → is-lub poset s (⋃-semi s dir)
    ⋃-semi-lub s dir = adjoin-lub→lub (⋃.has-lub (adjoin s) (adjoin-directed s dir))

    ⋃-semi-le
      : ∀ (s : Ix → Ob) (dir : is-semidirected-family poset s)
      → ∀ i → s i ≤ ⋃-semi s dir
    ⋃-semi-le s dir = is-lub.fam≤lub (⋃-semi-lub s dir)

    ⋃-semi-least
      : ∀ (s : Ix → Ob) (dir : is-semidirected-family poset s)
      → ∀ x → (∀ i → s i ≤ x) → ⋃-semi s dir ≤ x
    ⋃-semi-least s dir x le = is-lub.least (⋃-semi-lub s dir) x le

    ⋃-semi-pointwise
      : ∀ {Ix} {s s' : Ix → Ob}
      → {fam : is-semidirected-family poset s} {fam' : is-semidirected-family poset s'}
      → (∀ ix → s ix ≤ s' ix)
      → ⋃-semi s fam ≤ ⋃-semi s' fam'
    ⋃-semi-pointwise le = ⋃-pointwise λ where
      (just i) → le i
      nothing → bottom≤x _

However, we do call attention to one extremely useful fact: if $D$ is a pointed DCPO, then it has least upper bounds of families indexed by propositions.

  opaque
    ⋃-prop : ∀ {Ix : Type o} → (Ix → Ob) → is-prop Ix → Ob
    ⋃-prop s ix-prop = ⋃-semi s (prop-indexed→semidirected poset s ix-prop)

    ⋃-prop-le
      : ∀ (s : Ix → Ob) (p : is-prop Ix)
      → ∀ i → s i ≤ ⋃-prop s p
    ⋃-prop-le s p i = ⋃-semi-le _ _ i

    ⋃-prop-least
      : ∀ (s : Ix → Ob) (p : is-prop Ix)
      → ∀ x → (∀ i → s i ≤ x) → ⋃-prop s p ≤ x
    ⋃-prop-least s p = ⋃-semi-least _ _

    ⋃-prop-lub
      : ∀ (s : Ix → Ob) (p : is-prop Ix)
      → is-lub poset s (⋃-prop s p)
    ⋃-prop-lub s p = ⋃-semi-lub _ _

This allows us to reflect the truth value of a proposition into $D .$

  opaque
    ⋃-prop-false
      : ∀ (s : Ix → Ob) (p : is-prop Ix)
      → ¬ Ix → ⋃-prop s p ≡ bottom
    ⋃-prop-false s p ¬i =
      ≤-antisym
        (⋃-prop-least s p bottom (λ x → absurd (¬i x)))
        (bottom≤x _)

    ⋃-prop-true
      : ∀ (s : Ix → Ob) (p : is-prop Ix)
      → (i : Ix) → ⋃-prop s p ≡ s i
    ⋃-prop-true s p i =
      sym $ lub-of-const-fam (λ i j → ap s (p i j)) (⋃-prop-lub s p) i

We define a similar module for strictly Scott-continuous maps.

module Strict-scott {D E : Pointed-dcpo o ℓ} (f : Pointed-DCPOs.Hom D E) where

These proofs are all quite straightforward, so we omit them.

  private
    module D = Pointed-dcpo D
    module E = Pointed-dcpo E

  scott : DCPOs.Hom D.dcpo E.dcpo
  scott = Subcat-hom.hom f

  open Scott scott public

  opaque
    pres-bottoms : ∀ x → is-bottom D.poset x → is-bottom E.poset (f · x)
    pres-bottoms = Subcat-hom.witness f

    pres-⊥ : f · D.bottom ≡ E.bottom
    pres-⊥ = bottom-unique E.poset (pres-bottoms D.bottom D.bottom≤x) E.bottom≤x

    pres-adjoin-lub
      : ∀ {s : Ix → D.Ob} {x : D.Ob}
      → is-semidirected-family D.poset s
      → is-lub D.poset (D.adjoin s) x → is-lub E.poset (E.adjoin (apply f ⊙ s)) (f · x)
    pres-adjoin-lub {s = s} {x = x} sdir x-lub .is-lub.fam≤lub (just i) =
      monotone (is-lub.fam≤lub x-lub (just i))
    pres-adjoin-lub {s = s} {x = x} sdir x-lub .is-lub.fam≤lub nothing =
      E.bottom≤x (f · x)
    pres-adjoin-lub {s = s} {x = x} sdir x-lub .is-lub.least y le =
      is-lub.least
        (pres-directed-lub (D.adjoin s) (D.adjoin-directed s sdir) x x-lub) y λ where
          (just i) → le (just i)
          nothing → pres-bottoms _ D.bottom≤x y

    pres-semidirected-lub
      : ∀ {Ix} (s : Ix → D.Ob) → is-semidirected-family D.poset s
      → ∀ x → is-lub D.poset s x → is-lub E.poset (apply f ⊙ s) (f · x)
    pres-semidirected-lub s sdir x x-lub =
      E.adjoin-lub→lub (pres-adjoin-lub sdir (D.lub→adjoin-lub x-lub))

    pres-⋃-prop
      : ∀ {Ix} (s : Ix → D.Ob) (p q : is-prop Ix)
      → f · (D.⋃-prop s p) ≡ E.⋃-prop (apply f ⊙ s) q
    pres-⋃-prop s p q =
      lub-unique
        (pres-semidirected-lub _
          (prop-indexed→semidirected D.poset s p) (D.⋃-prop s p) (D.⋃-prop-lub s p))
        (E.⋃-prop-lub _ _)

    bottom-bounded : ∀ {x y} → x D.≤ y → f · y ≡ E.bottom → f · x ≡ E.bottom
    bottom-bounded {x = x} {y = y} p y-bot =
      E.≤-antisym
        (f · x    E.≤⟨ monotone p ⟩E.≤
         f · y    E.=⟨ y-bot ⟩E.=
         E.bottom E.≤∎)
        (E.bottom≤x _)

module _ {o ℓ} {D E : Pointed-dcpo o ℓ} where
  private
    module D = Pointed-dcpo D
    module E = Pointed-dcpo E
  open Total-hom
  open is-directed-family

We also provide a handful of ways of constructing strictly Scott-continuous maps. First, we note that if $f$ is monotonic and preserves the chosen least upper bound of semidirected families, then $f$ is strictly Scott-continuous.

  to-strict-scott-⋃-semi
    : (f : D.Ob → E.Ob)
    → (∀ {x y} → x D.≤ y → f x E.≤ f y)
    → (∀ {Ix} (s : Ix → D.Ob) → (dir : is-semidirected-family D.poset s)
       → is-lub E.poset (f ⊙ s) (f (D.⋃-semi s dir)))
    → Pointed-DCPOs.Hom D E
  to-strict-scott-⋃-semi f monotone pres-⋃-semi =
    sub-hom (to-scott f monotone pres-⋃) pres-bot
    where
      pres-⋃
        : ∀ {Ix} (s : Ix → D.Ob) → (dir : is-directed-family D.poset s)
        → is-lub E.poset (f ⊙ s) (f (D.⋃ s dir))
      pres-⋃ s dir .is-lub.fam≤lub i =
        monotone $ D.⋃.fam≤lub _ _ i
      pres-⋃ s dir .is-lub.least y le =
        f (D.⋃ s dir)                      E.=⟨ ap f (lub-unique (D.⋃.has-lub _ _) (D.⋃-semi-lub s (dir .semidirected))) ⟩E.=
        f (D.⋃-semi s (dir .semidirected)) E.≤⟨ pres-⋃-semi _ _ .is-lub.least y le ⟩E.≤
        y E.≤∎

      pres-bot : ∀ x → is-bottom D.poset x → is-bottom E.poset (f x)
      pres-bot x x-bot y =
        f x              E.≤⟨ monotone (x-bot _) ⟩E.≤
        f (D.⋃-semi _ _) E.≤⟨ is-lub.least (pres-⋃-semi (λ x → absurd (x .lower)) (λ ())) y (λ ()) ⟩E.≤
        y                E.≤∎

Next, if $f$ is monotonic, preserves chosen least upper bounds of directed families, and preserves chosen bottoms, then $f$ is strictly Scott-continuous.

  to-strict-scott-bottom
    : (f : D.Ob → E.Ob)
    → (∀ {x y} → x D.≤ y → f x E.≤ f y)
    → (∀ {Ix} (s : Ix → D.Ob) → (dir : is-directed-family D.poset s)
       → is-lub E.poset (f ⊙ s) (f (D.⋃ s dir)))
    → is-bottom E.poset (f D.bottom)
    → Pointed-DCPOs.Hom D E
  to-strict-scott-bottom f monotone pres-⋃ pres-bot =
    sub-hom (to-scott f monotone pres-⋃) λ x x-bot y →
      f x        E.≤⟨ monotone (x-bot _) ⟩E.≤
      f D.bottom E.≤⟨ pres-bot y ⟩E.≤
      y          E.≤∎

Finally, if $f$ preserves arbitrary least upper bounds of semidirected families, then $f$ must be monotonic, and thus strictly Scott-continuous.

  to-strict-scott-semidirected
    : (f : D.Ob → E.Ob)
    → (∀ {Ix} (s : Ix → D.Ob) → (dir : is-semidirected-family D.poset s)
       → ∀ x → is-lub D.poset s x → is-lub E.poset (f ⊙ s) (f x))
    → Pointed-DCPOs.Hom D E
  to-strict-scott-semidirected f pres =
    sub-hom
      (to-scott-directed f
        (λ s dir x lub → pres s (is-directed-family.semidirected dir) x lub))
      (λ x x-bot y → is-lub.least
          (pres _ (λ x → absurd (x .lower)) x (lift-is-lub (is-bottom→is-lub D.poset {f = λ ()} x-bot)))
          y (λ ()))