------------------------------------------------------------------------
-- The Agda standard library
--
-- Commutative semirings with some additional structure ("almost"
-- commutative rings), used by the ring solver
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Algebra.Solver.Ring.AlmostCommutativeRing where
 
open import Algebra
open import Algebra.Structures
open import Algebra.Definitions
import Algebra.Morphism as Morphism
import Algebra.Morphism.Definitions as MorphismDefinitions
open import Function.Base using (id)
open import Level
open import Relation.Binary.Core using (Rel)
 
 
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                : Congruent₁ _≈_ -_
    -‿*-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   -_
  infixl  _*_
  infixl  _+_
  infix   _≈_
  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
    ( +-magma; +-semigroup
    ; *-magma; *-semigroup; *-commutativeSemigroup
    ; +-monoid; +-commutativeMonoid
    ; *-monoid; *-commutativeMonoid
    ; semiring
    )
 
  rawRing : RawRing _ _
  rawRing = record
    { _≈_ = _≈_
    ; _+_ = _+_
    ; _*_ = _*_
    ; -_  = -_
    ; 0#  = 0#
    ; 1#  = 1#
    }
 
 
------------------------------------------------------------------------
-- Homomorphisms
 
infix  _-Raw-AlmostCommutative⟶_
 
record _-Raw-AlmostCommutative⟶_
  {r₁ r₂ r₃ r₄}
  (From : RawRing r₁ r₄)
  (To : AlmostCommutativeRing r₂ r₃) : Set (r₁ ⊔ r₂ ⊔ r₃) where
  private
    module F = RawRing From
    module T = AlmostCommutativeRing To
  open MorphismDefinitions 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
 
Induced-equivalence : ∀ {c₁ c₂ ℓ₁ ℓ₂} {Coeff : RawRing c₁ ℓ₁}
                      {R : AlmostCommutativeRing c₂ ℓ₂} →
                      Coeff -Raw-AlmostCommutative⟶ R →
                      Rel (RawRing.Carrier Coeff) ℓ₂
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 r₁ r₂
fromCommutativeRing CR = record
  { isAlmostCommutativeRing = record
      { isCommutativeSemiring = isCommutativeSemiring
      ; -‿cong                = -‿cong
      ; -‿*-distribˡ          = λ x y → sym (-‿distribˡ-* x y)
      ; -‿+-comm              = ⁻¹-∙-comm
      }
  }
  where
  open CommutativeRing CR
  open import Algebra.Properties.Ring ring
  open import Algebra.Properties.AbelianGroup +-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