DomainTheory.Basics.Pointed
Tom de Jong, May 2019.
Refactored January 2020, December 2021.
February 2022: Show that pointed dcpos have semidirected and subsingleton
suprema.
We define dcpos with a least element, typically denoted by ⊥, which are called
pointed dcpos. A map between pointed dcpos is called strict if it preserves
least elements. We show that every isomorphism of dcpos is strict.
Finally, we show that pointed dcpos have semidirected and subsingleton suprema
and that these are preserved by maps that are both strict and Scott continuous.
{-# OPTIONS --safe --without-K #-}
open import MLTT.Spartan
open import UF.FunExt
open import UF.PropTrunc
module DomainTheory.Basics.Pointed
(pt : propositional-truncations-exist)
(fe : Fun-Ext)
(𝓥 : Universe)
where
open PropositionalTruncation pt hiding (_∨_)
open import UF.Subsingletons
open import DomainTheory.Basics.Dcpo pt fe 𝓥
open import DomainTheory.Basics.Miscelanea pt fe 𝓥
module _ {𝓤 𝓣 : Universe} where
DCPO⊥ : (𝓥 ⁺) ⊔ (𝓤 ⁺) ⊔ (𝓣 ⁺) ̇
DCPO⊥ = Σ 𝓓 ꞉ DCPO {𝓤} {𝓣} , has-least (underlying-order 𝓓)
_⁻ : DCPO⊥ → DCPO
_⁻ = pr₁
⟪_⟫ : DCPO⊥ → 𝓤 ̇
⟪ 𝓓 ⟫ = ⟨ 𝓓 ⁻ ⟩
underlying-order⊥ : (𝓓 : DCPO⊥) → ⟪ 𝓓 ⟫ → ⟪ 𝓓 ⟫ → 𝓣 ̇
underlying-order⊥ 𝓓 = underlying-order (𝓓 ⁻)
syntax underlying-order⊥ 𝓓 x y = x ⊑⟪ 𝓓 ⟫ y
⊥ : (𝓓 : DCPO⊥) → ⟨ 𝓓 ⁻ ⟩
⊥ (𝓓 , x , p) = x
⊥-is-least : (𝓓 : DCPO⊥) → is-least (underlying-order (𝓓 ⁻)) (⊥ 𝓓)
⊥-is-least (𝓓 , x , p) = p
transitivity'' : (𝓓 : DCPO⊥) (x : ⟪ 𝓓 ⟫) {y z : ⟪ 𝓓 ⟫}
→ x ⊑⟪ 𝓓 ⟫ y → y ⊑⟪ 𝓓 ⟫ z → x ⊑⟪ 𝓓 ⟫ z
transitivity'' 𝓓 = transitivity' (𝓓 ⁻)
reflexivity' : (𝓓 : DCPO⊥) → reflexive (underlying-order (𝓓 ⁻))
reflexivity' (D , _) = reflexivity D
syntax transitivity'' 𝓓 x u v = x ⊑⟪ 𝓓 ⟫[ u ] v
infixr 0 transitivity''
syntax reflexivity' 𝓓 x = x ∎⟪ 𝓓 ⟫
infix 1 reflexivity'
is-a-non-trivial-pointed-dcpo : (𝓓 : DCPO⊥ {𝓤} {𝓣}) → 𝓤 ̇
is-a-non-trivial-pointed-dcpo 𝓓 = ∃ x ꞉ ⟪ 𝓓 ⟫ , x ≠ ⊥ 𝓓
＝-to-⊥-criterion : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {x : ⟪ 𝓓 ⟫} → x ⊑⟪ 𝓓 ⟫ ⊥ 𝓓 → x ＝ ⊥ 𝓓
＝-to-⊥-criterion 𝓓 {x} x-below-⊥ =
antisymmetry (𝓓 ⁻) x (⊥ 𝓓) x-below-⊥ (⊥-is-least 𝓓 x)
DCPO⊥[_,_] : DCPO⊥ {𝓤} {𝓣} → DCPO⊥ {𝓤'} {𝓣'} → (𝓥 ⁺) ⊔ 𝓤 ⊔ 𝓣 ⊔ 𝓤' ⊔ 𝓣' ̇
DCPO⊥[ 𝓓 , 𝓔 ] = DCPO[ 𝓓 ⁻ , 𝓔 ⁻ ]
is-strict : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
→ (⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫)
→ 𝓤' ̇
is-strict 𝓓 𝓔 f = f (⊥ 𝓓) ＝ ⊥ 𝓔
∘-is-strict : {𝓤'' 𝓣'' : Universe}
(𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
(𝓔' : DCPO⊥ {𝓤''} {𝓣''})
(f : ⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫) (g : ⟪ 𝓔 ⟫ → ⟪ 𝓔' ⟫)
→ is-strict 𝓓 𝓔 f
→ is-strict 𝓔 𝓔' g
→ is-strict 𝓓 𝓔' (g ∘ f)
∘-is-strict 𝓓 𝓔 𝓔' f g sf sg = ap g sf ∙ sg
being-strict-is-prop : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
(f : ⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫)
→ is-prop (is-strict 𝓓 𝓔 f)
being-strict-is-prop 𝓓 𝓔 f = sethood (𝓔 ⁻)
strictness-criterion : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
(f : ⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫)
→ f (⊥ 𝓓) ⊑⟪ 𝓔 ⟫ ⊥ 𝓔
→ is-strict 𝓓 𝓔 f
strictness-criterion 𝓓 𝓔 f crit =
antisymmetry (𝓔 ⁻) (f (⊥ 𝓓)) (⊥ 𝓔) crit (⊥-is-least 𝓔 (f (⊥ 𝓓)))
Defining isomorphisms of pointed dcpos and showing that every isomorphism of
dcpos is automatically strict.
_≃ᵈᶜᵖᵒ⊥_ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'}) → 𝓥 ⁺ ⊔ 𝓤 ⊔ 𝓣 ⊔ 𝓤' ⊔ 𝓣' ̇
𝓓 ≃ᵈᶜᵖᵒ⊥ 𝓔 = Σ f ꞉ (⟨ 𝓓 ⁻ ⟩ → ⟨ 𝓔 ⁻ ⟩) , Σ g ꞉ (⟨ 𝓔 ⁻ ⟩ → ⟨ 𝓓 ⁻ ⟩) ,
((d : ⟨ 𝓓 ⁻ ⟩) → g (f d) ＝ d)
× ((e : ⟨ 𝓔 ⁻ ⟩) → f (g e) ＝ e)
× is-continuous (𝓓 ⁻) (𝓔 ⁻) f
× is-continuous (𝓔 ⁻) (𝓓 ⁻) g
× is-strict 𝓓 𝓔 f
× is-strict 𝓔 𝓓 g
≃ᵈᶜᵖᵒ⊥-to-≃ᵈᶜᵖᵒ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
→ 𝓓 ≃ᵈᶜᵖᵒ⊥ 𝓔 → (𝓓 ⁻) ≃ᵈᶜᵖᵒ (𝓔 ⁻)
≃ᵈᶜᵖᵒ⊥-to-≃ᵈᶜᵖᵒ 𝓓 𝓔 (f , g , s , r , cf , cg , sf , sg) =
f , g , s , r , cf , cg
≃ᵈᶜᵖᵒ-to-≃ᵈᶜᵖᵒ⊥ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
→ (𝓓 ⁻) ≃ᵈᶜᵖᵒ (𝓔 ⁻) → 𝓓 ≃ᵈᶜᵖᵒ⊥ 𝓔
≃ᵈᶜᵖᵒ-to-≃ᵈᶜᵖᵒ⊥ 𝓓 𝓔 (f , g , gf , fg , cf , cg) =
f , g , gf , fg , cf , cg , sf , sg
where
sf : is-strict 𝓓 𝓔 f
sf = antisymmetry (𝓔 ⁻) (f (⊥ 𝓓)) (⊥ 𝓔) γ (⊥-is-least 𝓔 (f (⊥ 𝓓)))
where
γ = f (⊥ 𝓓) ⊑⟨ 𝓔 ⁻ ⟩[ l₁ ]
f (g (⊥ 𝓔)) ⊑⟨ 𝓔 ⁻ ⟩[ l₂ ]
⊥ 𝓔 ∎⟨ 𝓔 ⁻ ⟩
where
l₁ = monotone-if-continuous (𝓓 ⁻) (𝓔 ⁻) (f , cf) (⊥ 𝓓) (g (⊥ 𝓔))
(⊥-is-least 𝓓 (g (⊥ 𝓔)))
l₂ = ＝-to-⊑ (𝓔 ⁻) (fg (⊥ 𝓔))
sg : is-strict 𝓔 𝓓 g
sg = antisymmetry (𝓓 ⁻) (g (⊥ 𝓔)) (⊥ 𝓓) γ (⊥-is-least 𝓓 (g (⊥ 𝓔)))
where
γ = g (⊥ 𝓔) ⊑⟨ 𝓓 ⁻ ⟩[ l₁ ]
g (f (⊥ 𝓓)) ⊑⟨ 𝓓 ⁻ ⟩[ l₂ ]
⊥ 𝓓 ∎⟨ 𝓓 ⁻ ⟩
where
l₁ = monotone-if-continuous (𝓔 ⁻) (𝓓 ⁻) (g , cg) (⊥ 𝓔) (f (⊥ 𝓓))
(⊥-is-least 𝓔 (f (⊥ 𝓓)))
l₂ = ＝-to-⊑ (𝓓 ⁻) (gf (⊥ 𝓓))
Pointed dcpos have semidirected & subsingleton suprema and these are preserved
by maps that are both strict and continuous.
This is used to prove (in DomainTheroy.Lifting.LiftingSet.lagda) that the
lifting yields the free pointed dcpo on a set.
add-⊥ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
→ (𝟙{𝓥} + I) → ⟪ 𝓓 ⟫
add-⊥ 𝓓 α (inl ⋆) = ⊥ 𝓓
add-⊥ 𝓓 α (inr i) = α i
add-⊥-is-directed : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } {α : I → ⟪ 𝓓 ⟫}
→ is-semidirected (underlying-order (𝓓 ⁻)) α
→ is-Directed (𝓓 ⁻) (add-⊥ 𝓓 α)
add-⊥-is-directed 𝓓 {I} {α} σ = ∣ inl ⋆ ∣ , δ
where
δ : is-semidirected (underlying-order (𝓓 ⁻)) (add-⊥ 𝓓 α)
δ (inl ⋆) a = ∣ a , ⊥-is-least 𝓓 (add-⊥ 𝓓 α a) ,
reflexivity (𝓓 ⁻) (add-⊥ 𝓓 α a) ∣
δ (inr i) (inl ⋆) = ∣ (inr i) , reflexivity (𝓓 ⁻) (α i)
, ⊥-is-least 𝓓 (α i) ∣
δ (inr i) (inr j) = ∥∥-functor (λ (k , u , v) → (inr k , u , v)) (σ i j)
adding-⊥-preserves-sup : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ }
(α : I → ⟪ 𝓓 ⟫) (x : ⟪ 𝓓 ⟫)
→ is-sup (underlying-order (𝓓 ⁻)) x α
→ is-sup (underlying-order (𝓓 ⁻)) x (add-⊥ 𝓓 α)
adding-⊥-preserves-sup 𝓓 {I} α x x-is-sup = x-is-ub , x-is-lb-of-ubs
where
x-is-ub : (i : 𝟙 + I) → add-⊥ 𝓓 α i ⊑⟪ 𝓓 ⟫ x
x-is-ub (inl ⋆) = ⊥-is-least 𝓓 x
x-is-ub (inr i) = sup-is-upperbound (underlying-order (𝓓 ⁻)) x-is-sup i
x-is-lb-of-ubs : is-lowerbound-of-upperbounds (underlying-order (𝓓 ⁻))
x (add-⊥ 𝓓 α)
x-is-lb-of-ubs y y-is-ub = sup-is-lowerbound-of-upperbounds
(underlying-order (𝓓 ⁻)) x-is-sup y
(λ i → y-is-ub (inr i))
adding-⊥-reflects-sup : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ }
(α : I → ⟪ 𝓓 ⟫) (x : ⟪ 𝓓 ⟫)
→ is-sup (underlying-order (𝓓 ⁻)) x (add-⊥ 𝓓 α)
→ is-sup (underlying-order (𝓓 ⁻)) x α
adding-⊥-reflects-sup 𝓓 {I} α x x-is-sup = x-is-ub , x-is-lb-of-ubs
where
x-is-ub : (i : I) → α i ⊑⟪ 𝓓 ⟫ x
x-is-ub i = sup-is-upperbound (underlying-order (𝓓 ⁻)) x-is-sup (inr i)
x-is-lb-of-ubs : is-lowerbound-of-upperbounds (underlying-order (𝓓 ⁻)) x α
x-is-lb-of-ubs y y-is-ub = sup-is-lowerbound-of-upperbounds
(underlying-order (𝓓 ⁻)) x-is-sup y
h
where
h : is-upperbound (underlying-order (𝓓 ⁻)) y (add-⊥ 𝓓 α)
h (inl ⋆) = ⊥-is-least 𝓓 y
h (inr i) = y-is-ub i
semidirected-complete-if-pointed : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } {α : I → ⟪ 𝓓 ⟫}
→ is-semidirected (underlying-order (𝓓 ⁻)) α
→ has-sup (underlying-order (𝓓 ⁻)) α
semidirected-complete-if-pointed 𝓓 {I} {α} σ = x , x-is-sup
where
δ : is-Directed (𝓓 ⁻) (add-⊥ 𝓓 α)
δ = add-⊥-is-directed 𝓓 σ
x : ⟪ 𝓓 ⟫
x = ∐ (𝓓 ⁻) δ
x-is-sup : is-sup (underlying-order (𝓓 ⁻)) x α
x-is-sup = adding-⊥-reflects-sup 𝓓 α x (∐-is-sup (𝓓 ⁻) δ)
pointed-if-semidirected-complete : (𝓓 : DCPO {𝓤} {𝓣})
→ ({I : 𝓥 ̇ } {α : I → ⟨ 𝓓 ⟩}
→ is-semidirected (underlying-order 𝓓) α
→ has-sup (underlying-order 𝓓) α)
→ has-least (underlying-order 𝓓)
pointed-if-semidirected-complete 𝓓 c = x , x-is-least
where
α : 𝟘 → ⟨ 𝓓 ⟩
α = 𝟘-elim
σ : is-semidirected (underlying-order 𝓓) α
σ = 𝟘-induction
x : ⟨ 𝓓 ⟩
x = the-sup (underlying-order 𝓓) (c σ)
x-is-least : is-least (underlying-order 𝓓) x
x-is-least y =
sup-is-lowerbound-of-upperbounds
(underlying-order 𝓓)
(sup-property (underlying-order 𝓓) (c σ))
y 𝟘-induction
∐ˢᵈ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } {α : I → ⟪ 𝓓 ⟫}
→ is-semidirected (underlying-order (𝓓 ⁻)) α → ⟪ 𝓓 ⟫
∐ˢᵈ 𝓓 {I} {α} σ = pr₁ (semidirected-complete-if-pointed 𝓓 σ)
∐ˢᵈ-in-terms-of-∐ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } {α : I → ⟪ 𝓓 ⟫}
(σ : is-semidirected (underlying-order (𝓓 ⁻)) α)
→ ∐ˢᵈ 𝓓 σ ＝ ∐ (𝓓 ⁻) (add-⊥-is-directed 𝓓 σ)
∐ˢᵈ-in-terms-of-∐ 𝓓 {I} {α} σ = refl
preserves-semidirected-sups-if-continuous-and-strict :
(𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
(f : ⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫)
→ is-continuous (𝓓 ⁻) (𝓔 ⁻) f
→ is-strict 𝓓 𝓔 f
→ {I : 𝓥 ̇ } {α : I → ⟪ 𝓓 ⟫}
→ (σ : is-semidirected (underlying-order (𝓓 ⁻)) α)
→ is-sup (underlying-order (𝓔 ⁻)) (f (∐ˢᵈ 𝓓 σ)) (f ∘ α)
preserves-semidirected-sups-if-continuous-and-strict 𝓓 𝓔 f con str {I} {α} σ =
ub , lb-of-ubs
where
δ : is-Directed (𝓓 ⁻) (add-⊥ 𝓓 α)
δ = add-⊥-is-directed 𝓓 σ
claim₁ : is-sup (underlying-order (𝓔 ⁻)) (f (∐ (𝓓 ⁻) δ))
(f ∘ add-⊥ 𝓓 α)
claim₁ = con (𝟙 + I) (add-⊥ 𝓓 α) (add-⊥-is-directed 𝓓 σ)
claim₂ : is-sup (underlying-order (𝓔 ⁻)) (f (∐ˢᵈ 𝓓 σ))
(f ∘ add-⊥ 𝓓 α)
claim₂ = transport⁻¹
(λ - → is-sup (underlying-order (𝓔 ⁻)) (f -) (f ∘ (add-⊥ 𝓓 α)))
(∐ˢᵈ-in-terms-of-∐ 𝓓 σ) claim₁
ub : (i : I) → f (α i) ⊑⟪ 𝓔 ⟫ f (∐ˢᵈ 𝓓 σ)
ub i = sup-is-upperbound (underlying-order (𝓔 ⁻)) claim₂ (inr i)
lb-of-ubs : is-lowerbound-of-upperbounds (underlying-order (𝓔 ⁻)) (f (∐ˢᵈ 𝓓 σ))
(f ∘ α)
lb-of-ubs y y-is-ub = sup-is-lowerbound-of-upperbounds (underlying-order (𝓔 ⁻))
claim₂ y h
where
h : is-upperbound (underlying-order (𝓔 ⁻)) y (f ∘ add-⊥ 𝓓 α)
h (inl ⋆) = f (⊥ 𝓓) ⊑⟪ 𝓔 ⟫[ ＝-to-⊑ (𝓔 ⁻) str ]
⊥ 𝓔 ⊑⟪ 𝓔 ⟫[ ⊥-is-least 𝓔 y ]
y ∎⟪ 𝓔 ⟫
h (inr i) = y-is-ub i
subsingleton-indexed-is-semidirected : (𝓓 : DCPO {𝓤} {𝓣})
{I : 𝓥 ̇ } (α : I → ⟨ 𝓓 ⟩)
→ is-prop I
→ is-semidirected (underlying-order 𝓓) α
subsingleton-indexed-is-semidirected 𝓓 α ρ i j = ∣ i , r , r' ∣
where
r : α i ⊑⟨ 𝓓 ⟩ α i
r = reflexivity 𝓓 (α i)
r' : α j ⊑⟨ 𝓓 ⟩ α i
r' = transport (λ k → α k ⊑⟨ 𝓓 ⟩ α i) (ρ i j) r
subsingleton-complete-if-pointed : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
→ is-prop I
→ has-sup (underlying-order (𝓓 ⁻)) α
subsingleton-complete-if-pointed 𝓓 α ρ =
semidirected-complete-if-pointed 𝓓 σ
where
σ : is-semidirected (underlying-order (𝓓 ⁻)) α
σ = subsingleton-indexed-is-semidirected (𝓓 ⁻) α ρ
∐ˢˢ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
→ is-prop I → ⟪ 𝓓 ⟫
∐ˢˢ 𝓓 {I} α ρ = pr₁ (subsingleton-complete-if-pointed 𝓓 α ρ)
∐ˢˢ-in-terms-of-∐ˢᵈ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } {α : I → ⟪ 𝓓 ⟫}
(ρ : is-prop I)
→ ∐ˢˢ 𝓓 α ρ
＝ ∐ˢᵈ 𝓓
(subsingleton-indexed-is-semidirected (𝓓 ⁻) α ρ)
∐ˢˢ-in-terms-of-∐ˢᵈ 𝓓 {I} {α} σ = refl
preserves-subsingleton-sups-if-continuous-and-strict :
(𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
(f : ⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫)
→ is-continuous (𝓓 ⁻) (𝓔 ⁻) f
→ is-strict 𝓓 𝓔 f
→ {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
→ (ρ : is-prop I)
→ is-sup (underlying-order (𝓔 ⁻)) (f (∐ˢˢ 𝓓 α ρ)) (f ∘ α)
preserves-subsingleton-sups-if-continuous-and-strict 𝓓 𝓔 f con str α ρ =
preserves-semidirected-sups-if-continuous-and-strict 𝓓 𝓔 f con str
(subsingleton-indexed-is-semidirected (𝓓 ⁻) α ρ)
∐ˢˢ-is-upperbound : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
(ρ : is-prop I)
→ is-upperbound
(underlying-order (𝓓 ⁻)) (∐ˢˢ 𝓓 α ρ) α
∐ˢˢ-is-upperbound 𝓓 {I} α ρ i = ∐-is-upperbound (𝓓 ⁻) δ (inr i)
where
δ : is-Directed (𝓓 ⁻) (add-⊥ 𝓓 α)
δ = add-⊥-is-directed 𝓓 (subsingleton-indexed-is-semidirected (𝓓 ⁻) α ρ)
∐ˢˢ-is-lowerbound-of-upperbounds : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
(ρ : is-prop I)
→ is-lowerbound-of-upperbounds
(underlying-order (𝓓 ⁻)) (∐ˢˢ 𝓓 α ρ) α
∐ˢˢ-is-lowerbound-of-upperbounds 𝓓 {I} α ρ y y-is-ub =
∐-is-lowerbound-of-upperbounds (𝓓 ⁻) δ y h
where
δ : is-Directed (𝓓 ⁻) (add-⊥ 𝓓 α)
δ = add-⊥-is-directed 𝓓 (subsingleton-indexed-is-semidirected (𝓓 ⁻) α ρ)
h : (i : 𝟙 + I) → add-⊥ 𝓓 α i ⊑⟪ 𝓓 ⟫ y
h (inl ⋆) = ⊥-is-least 𝓓 y
h (inr i) = y-is-ub i
∐ˢˢ-is-sup : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫) (ρ : is-prop I)
→ is-sup (underlying-order (𝓓 ⁻)) (∐ˢˢ 𝓓 α ρ) α
∐ˢˢ-is-sup 𝓓 α ρ = ∐ˢˢ-is-upperbound 𝓓 α ρ
, ∐ˢˢ-is-lowerbound-of-upperbounds 𝓓 α ρ
∐ˢˢ-＝-if-continuous-and-strict : (𝓓 : DCPO⊥ {𝓤} {𝓣}) (𝓔 : DCPO⊥ {𝓤'} {𝓣'})
(f : ⟪ 𝓓 ⟫ → ⟪ 𝓔 ⟫)
→ is-continuous (𝓓 ⁻) (𝓔 ⁻) f
→ is-strict 𝓓 𝓔 f
→ {I : 𝓥 ̇ } (α : I → ⟪ 𝓓 ⟫)
→ (ρ : is-prop I)
→ f (∐ˢˢ 𝓓 α ρ) ＝ ∐ˢˢ 𝓔 (f ∘ α) ρ
∐ˢˢ-＝-if-continuous-and-strict 𝓓 𝓔 f con str α ρ =
sups-are-unique
(underlying-order (𝓔 ⁻))
(pr₁ (axioms-of-dcpo (𝓔 ⁻))) (f ∘ α)
(preserves-subsingleton-sups-if-continuous-and-strict 𝓓 𝓔 f con str α ρ)
(∐ˢˢ-is-sup 𝓔 (f ∘ α) ρ)
∐ˢˢ-family-＝ : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ } {α β : I → ⟪ 𝓓 ⟫} (ρ : is-prop I)
→ α ＝ β
→ ∐ˢˢ 𝓓 α ρ ＝ ∐ˢˢ 𝓓 β ρ
∐ˢˢ-family-＝ 𝓓 ρ refl = refl
∐ˢˢ-＝-if-domain-holds : (𝓓 : DCPO⊥ {𝓤} {𝓣}) {I : 𝓥 ̇ }
{α : I → ⟪ 𝓓 ⟫} (ρ : is-prop I)
→ (i : I) → ∐ˢˢ 𝓓 α ρ ＝ α i
∐ˢˢ-＝-if-domain-holds 𝓓 {I} {α} ρ i =
antisymmetry (𝓓 ⁻) (∐ˢˢ 𝓓 α ρ) (α i)
(∐ˢˢ-is-lowerbound-of-upperbounds 𝓓 α ρ (α i) l)
(∐ˢˢ-is-upperbound 𝓓 α ρ i)
where
l : (j : I) → α j ⊑⟪ 𝓓 ⟫ α i
l j = transport (λ - → α - ⊑⟪ 𝓓 ⟫ α i) (ρ i j) (reflexivity (𝓓 ⁻) (α i))