------------------------------------------------------------------------
-- Sums (disjoint unions)
------------------------------------------------------------------------

{-# OPTIONS --universe-polymorphism #-}

module Data.Sum where

open import Data.Function
open import Data.Maybe.Core
open import Level

------------------------------------------------------------------------
-- Definition

infixr 1 _⊎_

data _⊎_ {a b} (A : Set a) (B : Set b) : Set (a  b) where
  inj₁ : (x : A)  A  B
  inj₂ : (y : B)  A  B

------------------------------------------------------------------------
-- Functions

[_,_] :  {a b c} {A : Set a} {B : Set b} {C : A  B  Set c} 
        ((x : A)  C (inj₁ x))  ((x : B)  C (inj₂ x)) 
        ((x : A  B)  C x)
[ f , g ] (inj₁ x) = f x
[ f , g ] (inj₂ y) = g y

[_,_]′ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
         (A  C)  (B  C)  (A  B  C)
[_,_]′ = [_,_]

map :  {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} 
      (A  C)  (B  D)  (A  B  C  D)
map f g = [ inj₁  f , inj₂  g ]

infixr 1 _-⊎-_

_-⊎-_ :  {a b c d} {A : Set a} {B : Set b} 
        (A  B  Set c)  (A  B  Set d)  (A  B  Set (c  d))
f -⊎- g = f -[ _⊎_ ]- g

isInj₁ : {A B : Set}  A  B  Maybe A
isInj₁ (inj₁ x) = just x
isInj₁ (inj₂ y) = nothing

isInj₂ : {A B : Set}  A  B  Maybe B
isInj₂ (inj₁ x) = nothing
isInj₂ (inj₂ y) = just y