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91 lines
No EOL
3.1 KiB
Markdown
91 lines
No EOL
3.1 KiB
Markdown
---
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title: Delay Monad
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author: Leon Vatthauer
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format: pdf
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output:
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pdf_document:
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md_extensions: +task-lists
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mainfont: DejaVu Serif
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monofont: mononoki
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geometry: margin=0.5cm
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header-includes:
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- \usepackage{fvextra}
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- \DefineVerbatimEnvironment{Highlighting}{Verbatim}{breaklines,commandchars=\\\{\}}
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---
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<!--
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```agda
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open import Level
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open import Categories.Category.Core
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open import Categories.Category.Distributive
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open import Categories.Category.Extensive.Bundle
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open import Categories.Category.Extensive
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open import Categories.Category.BinaryProducts
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open import Categories.Category.Cocartesian
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open import Categories.Category.Cartesian
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open import Categories.Object.Terminal
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open import Categories.Category.Construction.F-Coalgebras
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open import Categories.Functor.Coalgebra
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open import Categories.Functor
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open import Categories.Monad.Construction.Kleisli
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import Categories.Morphism as M
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import Categories.Morphism.Reasoning as MR
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```
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-->
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```agda
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module Monad.Instance.Delay {o ℓ e} (ED : ExtensiveDistributiveCategory o ℓ e) where
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open ExtensiveDistributiveCategory ED renaming (U to C; id to idC)
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open Cocartesian (Extensive.cocartesian extensive)
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open Cartesian (ExtensiveDistributiveCategory.cartesian ED)
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open BinaryProducts products
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open M C
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open MR C
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open Equiv
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open HomReasoning
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-- Proposition 1
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record DelayMonad (D : Endofunctor C) : Set (o ⊔ ℓ ⊔ e) where
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open Functor D using () renaming (F₀ to D₀; F₁ to D₁)
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field
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now : ∀ {X} → X ⇒ D₀ X
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later : ∀ {X} → D₀ X ⇒ D₀ X
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isIso : ∀ {X} → IsIso [ now {X} , later {X} ]
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out : ∀ {X} → D₀ X ⇒ X + D₀ X
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out {X} = IsIso.inv (isIso {X})
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field
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_* : ∀ {X Y} → X ⇒ D₀ Y → D₀ X ⇒ D₀ Y
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*-law : ∀ {X Y} {f : X ⇒ D₀ Y} → out ∘ (f *) ≈ [ out ∘ f , i₂ ∘ (f *) ] ∘ out
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*-unique : ∀ {X Y} (f : X ⇒ D₀ Y) (h : D₀ X ⇒ D₀ Y) → h ≈ f *
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*-resp-≈ : ∀ {X Y} {f h : X ⇒ D₀ Y} → f ≈ h → f * ≈ h *
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unitLaw : ∀ {X} → out {X} ∘ now {X} ≈ i₁
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unitLaw = begin
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out ∘ now ≈⟨ refl⟩∘⟨ sym inject₁ ⟩
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out ∘ [ now , later ] ∘ i₁ ≈⟨ cancelˡ (IsIso.isoˡ isIso) ⟩
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i₁ ∎
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toMonad : KleisliTriple C
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toMonad = record
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{ F₀ = D₀
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; unit = now
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; extend = _*
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; identityʳ = λ {X} {Y} {k} → begin
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k * ∘ now ≈⟨ introˡ (IsIso.isoʳ isIso) ⟩∘⟨refl ⟩
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(([ now , later ] ∘ out) ∘ k *) ∘ now ≈⟨ pullʳ *-law ⟩∘⟨refl ⟩
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([ now , later ] ∘ [ out ∘ k , i₂ ∘ (k *) ] ∘ out) ∘ now ≈⟨ pullʳ (pullʳ unitLaw) ⟩
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[ now , later ] ∘ [ out ∘ k , i₂ ∘ (k *) ] ∘ i₁ ≈⟨ refl⟩∘⟨ inject₁ ⟩
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[ now , later ] ∘ out ∘ k ≈⟨ cancelˡ (IsIso.isoʳ isIso) ⟩
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k ∎
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; identityˡ = λ {X} → sym (*-unique now idC)
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; assoc = λ {X} {Y} {Z} {f} {g} → sym (*-unique ((g *) ∘ f) ((g *) ∘ (f *)))
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; sym-assoc = λ {X} {Y} {Z} {f} {g} → *-unique ((g *) ∘ f) ((g *) ∘ (f *))
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; extend-≈ = *-resp-≈
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}
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-- record Search
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``` |