Source code on Github
------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of sums (disjoint unions)
------------------------------------------------------------------------

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

module Data.Sum.Properties where

open import Level
open import Data.Sum.Base
open import Function
open import Relation.Binary using (Decidable)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary using (yes; no)
open import Relation.Nullary.Decidable using (map′)

private
  variable
    a b c d e f : Level
    A : Set a
    B : Set b
    C : Set c
    D : Set d
    E : Set e
    F : Set f

inj₁-injective :  {x y}  (A  B  inj₁ x)  inj₁ y  x  y
inj₁-injective refl = refl

inj₂-injective :  {x y}  (A  B  inj₂ x)  inj₂ y  x  y
inj₂-injective refl = refl

module _ (dec₁ : Decidable {A = A} {B = A} _≡_)
         (dec₂ : Decidable {A = B} {B = B} _≡_) where

  ≡-dec : Decidable {A = A  B} _≡_
  ≡-dec (inj₁ x) (inj₁ y) = map′ (cong inj₁) inj₁-injective (dec₁ x y)
  ≡-dec (inj₁ x) (inj₂ y) = no λ()
  ≡-dec (inj₂ x) (inj₁ y) = no λ()
  ≡-dec (inj₂ x) (inj₂ y) = map′ (cong inj₂) inj₂-injective (dec₂ x y)

swap-involutive : swap {A = A} {B = B}  swap  id
swap-involutive = [  _  refl) ,  _  refl) ]

map-id : map {A = A} {B = B} id id  id
map-id (inj₁ _) = refl
map-id (inj₂ _) = refl

[,]-∘-distr : (f : A  B)
              {g : C  A} {h : D  A} 
              f  [ g , h ]  [ f  g , f  h ]
[,]-∘-distr _ (inj₁ _) = refl
[,]-∘-distr _ (inj₂ _) = refl

[,]-map-commute : {f : A  B}  {g : C  D}
                  {f′ : B  E} {g′ : D  E} 
                  [ f′ , g′ ]  map f g  [ f′  f , g′  g ]
[,]-map-commute (inj₁ _) = refl
[,]-map-commute (inj₂ _) = refl

map-commute : {f : A  B}  {g : C  D}
              {f′ : B  E} {g′ : D  F} 
              map f′ g′  map f g  map (f′  f) (g′  g)
map-commute (inj₁ _) = refl
map-commute (inj₂ _) = refl

map₁₂-commute : {f : A  B} {g : C  D} 
                map₁ f  map₂ g  map₂ g  map₁ f
map₁₂-commute (inj₁ _) = refl
map₁₂-commute (inj₂ _) = refl

[,]-cong : {f f′ : A  B} {g g′ : C  B} 
           f  f′  g  g′ 
           [ f , g ]  [ f′ , g′ ]
[,]-cong = [_,_]

[-,]-cong : {f f′ : A  B} {g : C  B} 
            f  f′ 
            [ f , g ]  [ f′ , g ]
[-,]-cong = [_,  _  refl) ]

[,-]-cong : {f : A  B} {g g′ : C  B} 
            g  g′ 
            [ f , g ]  [ f , g′ ]
[,-]-cong = [  _  refl) ,_]

map-cong : {f f′ : A  B} {g g′ : C  D} 
           f  f′  g  g′ 
           map f g  map f′ g′
map-cong f≗f′ g≗g′ (inj₁ x) = cong inj₁ (f≗f′ x)
map-cong f≗f′ g≗g′ (inj₂ x) = cong inj₂ (g≗g′ x)

map₁-cong : {f f′ : A  B} 
            f  f′ 
            map₁ {B = C} f  map₁ f′
map₁-cong f≗f′ = [-,]-cong ((cong inj₁)  f≗f′)

map₂-cong : {g g′ : C  D} 
            g  g′ 
            map₂ {A = A} g  map₂ g′
map₂-cong g≗g′ = [,-]-cong ((cong inj₂)  g≗g′)