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Definitons of domain theory
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‎CHANGELOG.md

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* `Data.Sign.Show` to show a sign
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* Added a new domain theory section to the library under `Relation.Binary.Domain.*`:
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- Introduced new modules and bundles for domain theory, including `DCPO`, `Lub`, and `ScottContinuous`.
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- All files for domain theory are now available in `src/Relation/Binary/Domain/`.
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Additions to existing modules
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-----------------------------
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‎src/Relation/Binary/Domain.agda

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------------------------------------------------------------------------
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-- The Agda standard library
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--
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-- Order-theoretic Domains
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------------------------------------------------------------------------
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{-# OPTIONS --cubical-compatible --safe #-}
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module Relation.Binary.Domain where
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------------------------------------------------------------------------
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-- Re-export various components of the Domain hierarchy
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open import Relation.Binary.Domain.Definitions public
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open import Relation.Binary.Domain.Structures public
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open import Relation.Binary.Domain.Bundles public

‎src/Relation/Binary/Domain/Bundles.agda

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------------------------------------------------------------------------
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-- The Agda standard library
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--
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-- Bundles for domain theory
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------------------------------------------------------------------------
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{-# OPTIONS --cubical-compatible --safe #-}
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module Relation.Binary.Domain.Bundles where
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open import Level using (Level; _⊔_; suc)
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open import Relation.Binary.Bundles using (Poset)
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open import Relation.Binary.Domain.Structures
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open import Relation.Binary.Domain.Definitions
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private
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variable
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o ℓ e o' ℓ' e' ℓ₂ : Level
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Ix A B : Set o
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------------------------------------------------------------------------
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-- DCPOs
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------------------------------------------------------------------------
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record DCPO (c ℓ₁ ℓ₂ : Level) : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
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field
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poset : Poset c ℓ₁ ℓ₂
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DcpoStr : IsDCPO poset
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open Poset poset public
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open IsDCPO DcpoStr public
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------------------------------------------------------------------------
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-- Scott-continuous functions
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------------------------------------------------------------------------
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record ScottContinuous {c ℓ₁ ℓ₂ : Level} {P : Poset c ℓ₁ ℓ₂} {Q : Poset c ℓ₁ ℓ₂} :
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Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
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field
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f : Poset.Carrier P → Poset.Carrier Q
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Scottfunction : IsScottContinuous {P = P} {Q = Q} f
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------------------------------------------------------------------------
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-- Lubs
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------------------------------------------------------------------------
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record Lub {c ℓ₁ ℓ₂ : Level} {P : Poset c ℓ₁ ℓ₂} {Ix : Set c} (s : Ix → Poset.Carrier P) : Set (c ⊔ ℓ₁ ⊔ ℓ₂) where
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private
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module P = Poset P
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field
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lub : P.Carrier
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is-upperbound : ∀ i → P._≤_ (s i) lub
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is-least : ∀ y → (∀ i → P._≤_ (s i) y) → P._≤_ lub y

‎src/Relation/Binary/Domain/Definitions.agda

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------------------------------------------------------------------------
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-- The Agda standard library
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--
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-- Definitions for domain theory
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------------------------------------------------------------------------
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{-# OPTIONS --cubical-compatible --safe #-}
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module Relation.Binary.Domain.Definitions where
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open import Data.Product using (∃-syntax; _×_; _,_)
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open import Level using (Level; _⊔_)
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open import Relation.Binary.Bundles using (Poset)
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open import Relation.Binary.Morphism.Structures using (IsOrderHomomorphism)
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private
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variable
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c o ℓ e o' ℓ' e' ℓ₁ ℓ₂ : Level
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Ix A B : Set o
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P : Poset c ℓ e
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------------------------------------------------------------------------
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-- Directed families
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------------------------------------------------------------------------
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IsSemidirectedFamily : (P : Poset c ℓ₁ ℓ₂) → ∀ {Ix : Set c} → (s : Ix → Poset.Carrier P) → Set _
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IsSemidirectedFamily P {Ix} s = ∀ i j → ∃[ k ] (Poset._≤_ P (s i) (s k) × Poset._≤_ P (s j) (s k))
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‎src/Relation/Binary/Domain/Structures.agda

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------------------------------------------------------------------------
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-- The Agda standard library
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--
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-- Structures for domain theory
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------------------------------------------------------------------------
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{-# OPTIONS --cubical-compatible --safe #-}
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module Relation.Binary.Domain.Structures where
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open import Data.Product using (_×_; _,_)
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open import Data.Nat.Properties using (≤-trans)
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open import Function using (_∘_)
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open import Level using (Level; _⊔_; suc)
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open import Relation.Binary.Bundles using (Poset)
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open import Relation.Binary.Domain.Definitions
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open import Relation.Binary.Morphism.Structures using (IsOrderHomomorphism)
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private variable
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o ℓ e o' ℓ' e' ℓ₂ : Level
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Ix A B : Set o
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module _ {c ℓ₁ ℓ₂ : Level} (P : Poset c ℓ₁ ℓ₂) where
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open Poset P
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record IsDirectedFamily {Ix : Set c} (s : Ix → Carrier) : Set (c ⊔ ℓ₁ ⊔ ℓ₂) where
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no-eta-equality
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field
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elt : Ix
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SemiDirected : IsSemidirectedFamily P s
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record IsLub {Ix : Set c} (s : Ix → Carrier) (lub : Carrier) : Set (c ⊔ ℓ₁ ⊔ ℓ₂) where
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field
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is-upperbound : ∀ i → s i ≤ lub
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is-least : ∀ y → (∀ i → s i ≤ y) → lub ≤ y
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record IsDCPO : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
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field
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⋁ : ∀ {Ix : Set c}
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→ (s : Ix → Carrier)
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→ IsDirectedFamily s
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→ Carrier
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⋁-isLub : ∀ {Ix : Set c}
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→ (s : Ix → Carrier)
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→ (dir : IsDirectedFamily s)
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→ IsLub s (⋁ s dir)
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module _ {Ix : Set c} {s : Ix → Carrier} {dir : IsDirectedFamily s} where
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open IsLub (⋁-isLub s dir)
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renaming (is-upperbound to ⋁-≤; is-least to ⋁-least)
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public
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module _ {c ℓ₁ ℓ₂ : Level} {P : Poset c ℓ₁ ℓ₂} {Q : Poset c ℓ₁ ℓ₂} where
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private
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module P = Poset P
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module Q = Poset Q
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record IsScottContinuous (f : P.Carrier → Q.Carrier) : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
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field
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PreserveLub : ∀ {Ix : Set c} {s : Ix → P.Carrier}
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→ (dir : IsDirectedFamily P s)
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→ (lub : P.Carrier)
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→ IsLub P s lub
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→ IsLub Q (f ∘ s) (f lub)
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PreserveEquality : ∀ {x y} → x P.≈ y → f x Q.≈ f y

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