```------------------------------------------------------------------------
-- The Agda standard library
--
-- Commutative semirings with some additional structure ("almost"
-- commutative rings), used by the ring solver
------------------------------------------------------------------------

module Algebra.RingSolver.AlmostCommutativeRing where

open import Relation.Binary
open import Algebra
open import Algebra.Structures
open import Algebra.FunctionProperties
import Algebra.Morphism as Morphism
open import Function
open import Level

------------------------------------------------------------------------
-- Definitions

record IsAlmostCommutativeRing
{a ℓ} {A : Set a} (_≈_ : Rel A ℓ)
(_+_ _*_ : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
field
isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1#
-‿cong                : -_ Preserves _≈_ ⟶ _≈_
-‿*-distribˡ          : ∀ x y → ((- x) * y)     ≈ (- (x * y))
-‿+-comm              : ∀ x y → ((- x) + (- y)) ≈ (- (x + y))

open IsCommutativeSemiring isCommutativeSemiring public

record AlmostCommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
infix  8 -_
infixl 7 _*_
infixl 6 _+_
infix  4 _≈_
field
Carrier                 : Set c
_≈_                     : Rel Carrier ℓ
_+_                     : Op₂ Carrier
_*_                     : Op₂ Carrier
-_                      : Op₁ Carrier
0#                      : Carrier
1#                      : Carrier
isAlmostCommutativeRing :
IsAlmostCommutativeRing _≈_ _+_ _*_ -_ 0# 1#

open IsAlmostCommutativeRing isAlmostCommutativeRing public

commutativeSemiring : CommutativeSemiring _ _
commutativeSemiring =
record { isCommutativeSemiring = isCommutativeSemiring }

open CommutativeSemiring commutativeSemiring public
using ( setoid
; +-semigroup; +-monoid; +-commutativeMonoid
; *-semigroup; *-monoid; *-commutativeMonoid
; semiring
)

rawRing : RawRing _
rawRing = record
{ _+_ = _+_
; _*_ = _*_
; -_  = -_
; 0#  = 0#
; 1#  = 1#
}

------------------------------------------------------------------------
-- Homomorphisms

record _-Raw-AlmostCommutative⟶_
{r₁ r₂ r₃}
(From : RawRing r₁)
(To : AlmostCommutativeRing r₂ r₃) : Set (r₁ ⊔ r₂ ⊔ r₃) where
private
module F = RawRing From
module T = AlmostCommutativeRing To
open Morphism.Definitions F.Carrier T.Carrier T._≈_
field
⟦_⟧    : Morphism
+-homo : Homomorphic₂ ⟦_⟧ F._+_ T._+_
*-homo : Homomorphic₂ ⟦_⟧ F._*_ T._*_
-‿homo : Homomorphic₁ ⟦_⟧ F.-_  T.-_
0-homo : Homomorphic₀ ⟦_⟧ F.0#  T.0#
1-homo : Homomorphic₀ ⟦_⟧ F.1#  T.1#

-raw-almostCommutative⟶
: ∀ {r₁ r₂} (R : AlmostCommutativeRing r₁ r₂) →
AlmostCommutativeRing.rawRing R -Raw-AlmostCommutative⟶ R
-raw-almostCommutative⟶ R = record
{ ⟦_⟧    = id
; +-homo = λ _ _ → refl
; *-homo = λ _ _ → refl
; -‿homo = λ _ → refl
; 0-homo = refl
; 1-homo = refl
}
where open AlmostCommutativeRing R

-- A homomorphism induces a notion of equivalence on the raw ring.

Induced-equivalence :
∀ {c₁ c₂ ℓ} {Coeff : RawRing c₁} {R : AlmostCommutativeRing c₂ ℓ} →
Coeff -Raw-AlmostCommutative⟶ R → Rel (RawRing.Carrier Coeff) ℓ
Induced-equivalence {R = R} morphism a b = ⟦ a ⟧ ≈ ⟦ b ⟧
where
open AlmostCommutativeRing R
open _-Raw-AlmostCommutative⟶_ morphism

------------------------------------------------------------------------
-- Conversions

-- Commutative rings are almost commutative rings.

fromCommutativeRing :
∀ {r₁ r₂} → CommutativeRing r₁ r₂ → AlmostCommutativeRing _ _
fromCommutativeRing CR = record
{ isAlmostCommutativeRing = record
{ isCommutativeSemiring = isCommutativeSemiring
; -‿cong                = -‿cong
; -‿*-distribˡ          = -‿*-distribˡ
; -‿+-comm              = ⁻¹-∙-comm
}
}
where
open CommutativeRing CR
import Algebra.Props.Ring as R; open R ring
import Algebra.Props.AbelianGroup as AG; open AG +-abelianGroup

-- Commutative semirings can be viewed as almost commutative rings by
-- using identity as the "almost negation".

fromCommutativeSemiring :
∀ {r₁ r₂} → CommutativeSemiring r₁ r₂ → AlmostCommutativeRing _ _
fromCommutativeSemiring CS = record
{ -_                      = id
; isAlmostCommutativeRing = record
{ isCommutativeSemiring = isCommutativeSemiring
; -‿cong                = id
; -‿*-distribˡ          = λ _ _ → refl
; -‿+-comm              = λ _ _ → refl
}
}
where open CommutativeSemiring CS
```