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src/Monad/Instance/K/Instance/Delay.lagda.md
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src/Monad/Instance/K/Instance/Delay.lagda.md
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<!--
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```agda
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{-# OPTIONS --allow-unsolved-metas --guardedness #-}
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open import Level
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open import Category.Ambient using (Ambient)
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open import Categories.Category
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open import Categories.Category.Instance.Setoids
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open import Categories.Monad
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open import Categories.Category.Monoidal.Instance.Setoids
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open import Categories.Category.Cocartesian
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open import Categories.Object.Terminal
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open import Function.Equality as SΠ renaming (id to idₛ)
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import Categories.Morphism.Reasoning as MR
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open import Relation.Binary
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Sum.Function.Setoid
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open import Data.Sum.Relation.Binary.Pointwise
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open import Data.Unit.Polymorphic using (tt; ⊤)
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open import Data.Empty.Polymorphic using (⊥)
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open import Categories.NaturalTransformation using (ntHelper)
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open import Function.Base using (id)
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_)
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open import Data.Product using (Σ; _,_; ∃; Σ-syntax; ∃-syntax)
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```
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-->
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```agda
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module Monad.Instance.K.Instance.Delay {c ℓ} where
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```
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# Capretta's Delay Monad is an Instance of K in the Category of Setoids
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```agda
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record Delay (A : Set c) : Set c where
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coinductive
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field
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node : A ⊎ Delay A
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open Delay
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now : ∀ {A : Set c} → A → Delay A
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node (now {A} a) = inj₁ a
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later : ∀ {A : Set c} → Delay A → Delay A
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node (later {A} a) = inj₂ a
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never : ∀ {A : Set c} → Delay A
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node (never {A}) = inj₂ never
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-- first try
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delay-eq-strong : ∀ (A : Setoid c ℓ) → Delay (Setoid.Carrier A) → Delay (Setoid.Carrier A) → Set ℓ
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delay-eq-strong A x y with node x | node y
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... | inj₁ a | inj₁ b = Setoid._≈_ A a b
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... | inj₁ a | inj₂ b = ⊥
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... | inj₂ a | inj₁ b = ⊥
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... | inj₂ a | inj₂ b = {! !}
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-- second try
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record eq-strong (A : Setoid c ℓ) (x : Delay (Setoid.Carrier A)) (y : Delay (Setoid.Carrier A)) : Set c where
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field
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node-eq₁ : (node x) ≡ (node y)
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-- third try
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mutual
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node-eq : (A : Setoid c ℓ) (x y : Setoid.Carrier A ⊎ Delay (Setoid.Carrier A)) → Set ℓ
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node-eq A (inj₁ x) (inj₁ y) = Setoid._≈_ A x y
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node-eq A (inj₁ x) (inj₂ y) = ⊥
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node-eq A (inj₂ x) (inj₁ y) = ⊥
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node-eq A (inj₂ x) (inj₂ y) = delay-eq-strong' A x y
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delay-eq-strong' : ∀ (A : Setoid c ℓ) → Delay (Setoid.Carrier A) → Delay (Setoid.Carrier A) → Set ℓ
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delay-eq-strong' A x y = node-eq A {! !} {! !}
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Delay-setoid : Setoid c ℓ → Setoid c ℓ
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Delay-setoid A = record { Carrier = Delay A.Carrier ; _≈_ = {! !} ; isEquivalence = {! !} }
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where
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module A = Setoid A
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-- finally this should work
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mutual
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record eq-strong' (A : Setoid c ℓ) (x : Delay (Setoid.Carrier A)) (y : Delay (Setoid.Carrier A)) : Set ℓ where
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coinductive
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field
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eq : eq-strong'' A (node x) (node y)
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eq-strong'' : (A : Setoid c ℓ) → (Setoid.Carrier A ⊎ Delay (Setoid.Carrier A)) → (Setoid.Carrier A ⊎ Delay (Setoid.Carrier A)) → Set ℓ
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eq-strong'' A (inj₁ x) (inj₁ y) = Setoid._≈_ A x y
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eq-strong'' A (inj₁ x) (inj₂ y) = ⊥
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eq-strong'' A (inj₂ x) (inj₁ y) = ⊥
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eq-strong'' A (inj₂ x) (inj₂ y) = eq-strong' A x y
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-- this should also work
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mutual
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record eq-weak (A : Setoid c ℓ) (x : Delay (Setoid.Carrier A)) (y : Delay (Setoid.Carrier A)) : Set ℓ where
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coinductive
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field
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eq : eq-weak' A (node x) (node y)
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eq-weak' : (A : Setoid c ℓ) → (Setoid.Carrier A ⊎ Delay (Setoid.Carrier A)) → (Setoid.Carrier A ⊎ Delay (Setoid.Carrier A)) → Set ℓ
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eq-weak' A (inj₁ x) (inj₁ y) = Setoid._≈_ A x y
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eq-weak' A (inj₁ x) (inj₂ y) = eq-weak A (now x) y
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eq-weak' A (inj₂ x) (inj₁ y) = eq-weak A x (now y)
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eq-weak' A (inj₂ x) (inj₂ y) = eq-weak A x y
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open eq-weak
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now-cong : ∀ {A : Setoid c ℓ} {a b : Setoid.Carrier A} → Setoid._≈_ A a b → eq-weak A (now a) (now b)
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eq (now-cong {A} {a} {b} a≈b) = a≈b
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now-inj : ∀ {A : Setoid c ℓ} {a b : Setoid.Carrier A} → eq-weak A (now a) (now b) → Setoid._≈_ A a b
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now-inj {A} {a} {b} na≈nb with (eq na≈nb)
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... | a≈b = a≈b
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now-weak-eq : ∀ {A : Setoid c ℓ} {a : Setoid.Carrier A} {b : Delay (Setoid.Carrier A)} → eq-weak A (now a) b → Σ (Setoid.Carrier A) (λ c → Setoid._≈_ A a c)
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now-weak-eq {A} {a} {b} na≈b with node b | eq na≈b
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... | inj₁ x | a≈b = x , a≈b
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... | inj₂ x | a≈b = {! !}
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weak-setoid : Setoid c ℓ → Setoid c ℓ
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weak-setoid A = record { Carrier = Delay (Setoid.Carrier A) ; _≈_ = eq-weak A ; isEquivalence = record { refl = refl' ; sym = sym' ; trans = trans' } }
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where
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refl' : Reflexive (eq-weak A)
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eq (refl' {a}) with node a
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... | inj₁ x = IsEquivalence.refl (Setoid.isEquivalence A)
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... | inj₂ x = refl'
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sym' : Symmetric (eq-weak A)
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eq (sym' {a} {b} a≈b) with node a | node b | eq a≈b
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... | inj₁ x | inj₁ y | z = IsEquivalence.sym (Setoid.isEquivalence A) z
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... | inj₂ x | inj₁ y | z = sym' z
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... | inj₁ x | inj₂ y | z = sym' z
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... | inj₂ x | inj₂ y | z = sym' z
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trans' : Transitive (eq-weak A)
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eq (trans' {a} {b} {c} a≈b b≈c) with node a | node b | node c | eq a≈b | eq b≈c
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... | inj₂ x | inj₂ y | inj₂ z | eq₁ | eq₂ = trans' eq₁ eq₂
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... | inj₁ x | inj₂ y | inj₂ z | eq₁ | eq₂ = trans' eq₁ eq₂
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... | inj₂ x | inj₁ y | inj₂ z | eq₁ | eq₂ = trans' eq₁ eq₂
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... | inj₁ x | inj₁ y | inj₂ z | eq₁ | eq₂ = trans' (now-cong eq₁) eq₂ -- trans' (now-cong eq₁) eq₂
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... | inj₂ x | inj₂ y | inj₁ z | eq₁ | eq₂ = trans' eq₁ eq₂
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... | inj₁ x | inj₂ y | inj₁ z | eq₁ | eq₂ = {! trans' eq₁ eq₂ !} -- now-inj {A} {x} {z} (trans' eq₁ eq₂)
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... | inj₂ x | inj₁ y | inj₁ z | eq₁ | eq₂ = trans' eq₁ (now-cong eq₂)
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... | inj₁ x | inj₁ y | inj₁ z | eq₁ | eq₂ = IsEquivalence.trans (Setoid.isEquivalence A) eq₁ eq₂
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```
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@ -147,6 +147,4 @@ Assuming the axiom of choice, the maybe monad is an instance of K in the categor
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; identityˡ = λ {A} {i} {j} → identityˡ {A} {i} {j}
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; identityʳ = λ {A} {i} {j} → identityʳ {A} {i} {j}
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}
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```
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