```------------------------------------------------------------------------
-- The Agda standard library
--
-- Algebraic properties of sums (disjoint unions)
------------------------------------------------------------------------

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

module Data.Sum.Algebra where

open import Algebra
open import Data.Empty.Polymorphic using (⊥)
open import Data.Product using (_,_)
open import Data.Sum.Base
open import Data.Sum.Properties
open import Data.Unit.Polymorphic using (⊤; tt)
open import Function.Base using (id; _∘_)
open import Function.Properties.Inverse using (↔-isEquivalence)
open import Function.Bundles using (_↔_; Inverse; mk↔′)
open import Level using (Level; suc)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; refl; cong; cong′)

import Function.Definitions as FuncDef

------------------------------------------------------------------------
-- Setup

private
variable
a b c d : Level
A : Set a
B : Set b
C : Set c
D : Set d

-- The module is needed because we need to pass `A` and `B` to `FuncDef`
module _ {A : Set a} {B : Set b} where
open FuncDef {A = A} {B} _≡_ _≡_

♯ : {B : ⊥ {a} → Set b} → (w : ⊥) → B w
♯ ()

------------------------------------------------------------------------
-- Algebraic properties

⊎-cong : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)
⊎-cong i j = mk↔′ (map I.to J.to) (map I.from J.from)
[ cong inj₁ ∘ I.inverseˡ , cong inj₂ ∘ J.inverseˡ ]
[ cong inj₁ ∘ I.inverseʳ , cong inj₂ ∘ J.inverseʳ ]
where module I = Inverse i; module J = Inverse j

-- ⊎ is commutative.
-- We don't use Commutative because it isn't polymorphic enough.
⊎-comm : (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)
⊎-comm _ _ = mk↔′ swap swap swap-involutive swap-involutive

module _ (ℓ : Level) where

-- ⊎ is associative
⊎-assoc : Associative {ℓ = ℓ} _↔_ _⊎_
⊎-assoc _ _ _ = mk↔′ assocʳ assocˡ
[ cong′ , [ cong′ , cong′ ] ] [ [ cong′ , cong′ ] , cong′ ]

-- ⊥ is an identity for ⊎
⊎-identityˡ : LeftIdentity {ℓ = ℓ} _↔_ ⊥ _⊎_
⊎-identityˡ A = mk↔′ [ ♯ , id ] inj₂ cong′ [ ♯ , cong′ ]

⊎-identityʳ : RightIdentity {ℓ = ℓ} _↔_ ⊥ _⊎_
⊎-identityʳ _ = mk↔′ [ id , ♯ ] inj₁ cong′ [ cong′ , ♯ ]

⊎-identity : Identity _↔_ ⊥ _⊎_
⊎-identity = ⊎-identityˡ , ⊎-identityʳ

------------------------------------------------------------------------
-- Algebraic structures

⊎-isMagma : IsMagma {ℓ = ℓ} _↔_ _⊎_
⊎-isMagma = record
{ isEquivalence = ↔-isEquivalence
; ∙-cong        = ⊎-cong
}

⊎-isSemigroup : IsSemigroup _↔_ _⊎_
⊎-isSemigroup = record
{ isMagma = ⊎-isMagma
; assoc   = ⊎-assoc
}

⊎-isMonoid : IsMonoid _↔_ _⊎_ ⊥
⊎-isMonoid = record
{ isSemigroup = ⊎-isSemigroup
; identity    = ⊎-identityˡ , ⊎-identityʳ
}

⊎-isCommutativeMonoid : IsCommutativeMonoid _↔_ _⊎_ ⊥
⊎-isCommutativeMonoid = record
{ isMonoid = ⊎-isMonoid
; comm     = ⊎-comm
}

------------------------------------------------------------------------
-- Algebraic bundles

⊎-magma : Magma (suc ℓ) ℓ
⊎-magma = record
{ isMagma = ⊎-isMagma
}

⊎-semigroup : Semigroup (suc ℓ) ℓ
⊎-semigroup = record
{ isSemigroup = ⊎-isSemigroup
}

⊎-monoid : Monoid (suc ℓ) ℓ
⊎-monoid = record
{ isMonoid = ⊎-isMonoid
}

⊎-commutativeMonoid : CommutativeMonoid (suc ℓ) ℓ
⊎-commutativeMonoid = record
{ isCommutativeMonoid = ⊎-isCommutativeMonoid
}
```