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53 lines
1.9 KiB
Markdown
53 lines
1.9 KiB
Markdown
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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.Cocartesian
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open import Categories.Category.Cartesian
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open import Categories.Category.Distributive
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open import Categories.Functor hiding (id)
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open import Categories.Object.Terminal
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open import Categories.Monad
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open import Categories.NaturalTransformation hiding (id)
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import Categories.Morphism.Reasoning as MR
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```
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-->
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# The maybe monad
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```agda
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module Monad.Instance.Maybe {o ℓ e} {C : Category o ℓ e} (distributive : Distributive C) where
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open Category C
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open MR C
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open HomReasoning
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open Equiv
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open Distributive distributive
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open Cocartesian cocartesian
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open Cartesian cartesian using (terminal)
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open Terminal terminal
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maybeFunctor : Endofunctor C
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maybeFunctor = record
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{ F₀ = λ X → X + ⊤
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; F₁ = λ f → f +₁ id
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; identity = +-unique id-comm-sym id-comm-sym
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; homomorphism = λ {X} {Y} {Z} {f} {g} → sym (+₁∘+₁ ○ +₁-cong₂ refl identity²)
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; F-resp-≈ = λ eq → +₁-cong₂ eq refl
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}
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open Monad renaming (identityˡ to m-identityˡ; identityʳ to m-identityʳ; assoc to m-assoc; sym-assoc to m-sym-assoc)
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maybeMonad : Monad C
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maybeMonad .F = maybeFunctor
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maybeMonad .η = ntHelper (record { η = λ X → i₁ ; commute = λ f → sym inject₁ })
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maybeMonad .μ = ntHelper (record { η = λ X → [ id , i₂ ] ; commute = λ f → []∘+₁ ○ []-cong₂ id-comm-sym (sym inject₂) ○ sym ∘[]})
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maybeMonad .m-assoc = begin
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[ id , i₂ ] ∘ ([ id , i₂ ] +₁ id) ≈⟨ []∘+₁ ⟩
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[ id ∘ [ id , i₂ ] , i₂ ∘ id ] ≈˘⟨ []-cong₂ id-comm (inject₂ ○ introʳ refl) ⟩
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[ [ id , i₂ ] ∘ id , [ id , i₂ ] ∘ i₂ ] ≈˘⟨ ∘[] ⟩
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[ id , i₂ ] ∘ [ id , i₂ ] ∎
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maybeMonad .m-sym-assoc = sym (m-assoc maybeMonad)
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maybeMonad .m-identityˡ = []∘+₁ ○ +-unique refl id-comm-sym
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maybeMonad .m-identityʳ = inject₁
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```
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