```------------------------------------------------------------------------
-- The Agda standard library
--
------------------------------------------------------------------------

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

module Data.Sign.Properties where

open import Algebra.Bundles
open import Data.Empty
open import Data.Sign.Base
open import Data.Product using (_,_)
open import Function
open import Level using (0ℓ)
open import Relation.Binary using (Decidable; Setoid; DecSetoid)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary using (yes; no)

open import Algebra.Structures {A = Sign} _≡_
open import Algebra.Definitions {A = Sign} _≡_

------------------------------------------------------------------------
-- Equality

infix 4 _≟_

_≟_ : Decidable {A = Sign} _≡_
- ≟ - = yes refl
- ≟ + = no λ()
+ ≟ - = no λ()
+ ≟ + = yes refl

≡-setoid : Setoid 0ℓ 0ℓ
≡-setoid = setoid Sign

≡-decSetoid : DecSetoid 0ℓ 0ℓ
≡-decSetoid = decSetoid _≟_

------------------------------------------------------------------------
-- opposite

s≢opposite[s] : ∀ s → s ≢ opposite s
s≢opposite[s] - ()
s≢opposite[s] + ()

opposite-injective : ∀ {s t} → opposite s ≡ opposite t → s ≡ t
opposite-injective { - } { - } refl = refl
opposite-injective { + } { + } refl = refl

------------------------------------------------------------------------
-- _*_

-- Algebraic properties of _*_

*-identityˡ : LeftIdentity + _*_
*-identityˡ _ = refl

*-identityʳ : RightIdentity + _*_
*-identityʳ - = refl
*-identityʳ + = refl

*-identity : Identity + _*_
*-identity = *-identityˡ  , *-identityʳ

*-comm : Commutative _*_
*-comm + + = refl
*-comm + - = refl
*-comm - + = refl
*-comm - - = refl

*-assoc : Associative _*_
*-assoc + + _ = refl
*-assoc + - _ = refl
*-assoc - + _ = refl
*-assoc - - + = refl
*-assoc - - - = refl

*-cancelʳ-≡ : RightCancellative _*_
*-cancelʳ-≡ - - _  = refl
*-cancelʳ-≡ - + eq = ⊥-elim (s≢opposite[s] _ \$ sym eq)
*-cancelʳ-≡ + - eq = ⊥-elim (s≢opposite[s] _ eq)
*-cancelʳ-≡ + + _  = refl

*-cancelˡ-≡ : LeftCancellative _*_
*-cancelˡ-≡ - eq = opposite-injective eq
*-cancelˡ-≡ + eq = eq

*-cancel-≡ : Cancellative _*_
*-cancel-≡ = *-cancelˡ-≡ , *-cancelʳ-≡

*-isMagma : IsMagma _*_
*-isMagma = record
{ isEquivalence = isEquivalence
; ∙-cong        = cong₂ _*_
}

*-magma : Magma 0ℓ 0ℓ
*-magma = record
{ isMagma = *-isMagma
}

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

*-semigroup : Semigroup 0ℓ 0ℓ
*-semigroup = record
{ isSemigroup = *-isSemigroup
}

*-isMonoid : IsMonoid _*_ +
*-isMonoid = record
{ isSemigroup = *-isSemigroup
; identity    = *-identity
}

*-monoid : Monoid 0ℓ 0ℓ
*-monoid = record
{ isMonoid = *-isMonoid
}

-- Other properties of _*_

s*s≡+ : ∀ s → s * s ≡ +
s*s≡+ + = refl
s*s≡+ - = refl

s*opposite[s]≡- : ∀ s → s * opposite s ≡ -
s*opposite[s]≡- + = refl
s*opposite[s]≡- - = refl

opposite[s]*s≡- : ∀ s → opposite s * s ≡ -
opposite[s]*s≡- + = refl
opposite[s]*s≡- - = refl

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

opposite-not-equal = s≢opposite[s]
{-# WARNING_ON_USAGE opposite-not-equal
"Warning: opposite-not-equal was deprecated in v0.15.
#-}
opposite-cong = opposite-injective
{-# WARNING_ON_USAGE opposite-cong
"Warning: opposite-cong was deprecated in v0.15.
#-}
cancel-*-left = *-cancelˡ-≡
{-# WARNING_ON_USAGE cancel-*-left
"Warning: cancel-*-left was deprecated in v0.15.