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------------------------------------------------------------------------
-- The Agda standard library
--
-- A Categorical view of the Sum type (Left-biased)
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

open import Level

module Data.Sum.Categorical.Left {a} (A : Set a) (b : Level) where

open import Data.Sum.Base
open import Category.Functor
open import Category.Applicative
open import Category.Monad
import Function.Identity.Categorical as Id
open import Function

-- To minimize the universe level of the RawFunctor, we require that elements of
-- B are "lifted" to a copy of B at a higher universe level (a ⊔ b). See the
-- examples for how this is done.

------------------------------------------------------------------------
-- Left-biased monad instance for _⊎_

Sumₗ : Set (a  b)  Set (a  b)
Sumₗ B = A  B

functor : RawFunctor Sumₗ
functor = record { _<$>_ = map₂ }

applicative : RawApplicative Sumₗ
applicative = record
  { pure = inj₂
  ; _⊛_ = [ const  inj₁ , map₂ ]′
  }

-- The monad instance also requires some mucking about with universe levels.
monadT : RawMonadT (_∘′ Sumₗ)
monadT M = record
  { return = M.pure  inj₂
  ; _>>=_  = λ ma f  ma M.>>= [ M.pure ∘′ inj₁ , f ]′
  } where module M = RawMonad M

monad : RawMonad Sumₗ
monad = monadT Id.monad

------------------------------------------------------------------------
-- Get access to other monadic functions

module TraversableA {F} (App : RawApplicative {a  b} F) where

  open RawApplicative App

  sequenceA :  {A}  Sumₗ (F A)  F (Sumₗ A)
  sequenceA (inj₁ a) = pure (inj₁ a)
  sequenceA (inj₂ x) = inj₂ <$> x

  mapA :  {A B}  (A  F B)  Sumₗ A  F (Sumₗ B)
  mapA f = sequenceA  map₂ f

  forA :  {A B}  Sumₗ A  (A  F B)  F (Sumₗ B)
  forA = flip mapA

module TraversableM {M} (Mon : RawMonad {a  b} M) where

  open RawMonad Mon

  open TraversableA rawIApplicative public
    renaming
    ( sequenceA to sequenceM
    ; mapA      to mapM
    ; forA      to forM
    )