------------------------------------------------------------------------
-- The Agda standard library
--
-- Almost commutative rings
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Tactic.RingSolver.Core.AlmostCommutativeRing where
open import Level
open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
open import Algebra.Core using (Op₁; Op₂)
open import Algebra.Structures using (IsCommutativeSemiring)
open import Algebra.Definitions
open import Algebra.Bundles using (RawRing; CommutativeRing; CommutativeSemiring)
import Algebra.Morphism as Morphism
open import Function.Base using (id)
open import Level
open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
record IsAlmostCommutativeRing
{a ℓ} {A : Set a} (_≈_ : Rel A ℓ)
(_+_ _*_ : A → A → A) (-_ : A → 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
import Algebra.Definitions.RawSemiring as Exp
record AlmostCommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_
infixr 8 _^_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier
0# : Carrier
0≟_ : (x : Carrier) → Maybe (0# ≈ x)
1# : Carrier
isAlmostCommutativeRing :
IsAlmostCommutativeRing _≈_ _+_ _*_ -_ 0# 1#
open IsAlmostCommutativeRing isAlmostCommutativeRing hiding (refl) public
open import Data.Nat.Base as ℕ using (ℕ)
commutativeSemiring : CommutativeSemiring _ _
commutativeSemiring = record
{ isCommutativeSemiring = isCommutativeSemiring
}
open CommutativeSemiring commutativeSemiring public
using
( +-semigroup; +-monoid; +-commutativeMonoid
; *-semigroup; *-monoid; *-commutativeMonoid
; rawSemiring; semiring
)
rawRing : RawRing _ _
rawRing = record
{ _≈_ = _≈_
; _+_ = _+_
; _*_ = _*_
; -_ = -_
; 0# = 0#
; 1# = 1#
}
_^_ : Carrier → ℕ → Carrier
_^_ = Exp._^′_ rawSemiring
{-# NOINLINE _^_ #-}
_-_ : Carrier → Carrier → Carrier
_-_ x y = x + (- y)
refl : ∀ {x} → x ≈ x
refl = IsAlmostCommutativeRing.refl isAlmostCommutativeRing
record _-Raw-AlmostCommutative⟶_
{r₁ r₂ r₃ r₄}
(From : RawRing r₁ r₂)
(To : AlmostCommutativeRing r₃ r₄) : Set (r₁ ⊔ r₂ ⊔ r₃ ⊔ r₄) 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 c₃ ℓ} {Coeff : RawRing c₁ c₂} {R : AlmostCommutativeRing c₃ 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} (CR : CommutativeRing r₁ r₂) →
(open CommutativeRing CR) →
(∀ x → Maybe (0# ≈ x)) →
AlmostCommutativeRing _ _
fromCommutativeRing CR 0≟_ = record
{ isAlmostCommutativeRing = record
{ isCommutativeSemiring = isCommutativeSemiring
; -‿cong = -‿cong
; -‿*-distribˡ = λ x y → sym (-‿distribˡ-* x y)
; -‿+-comm = ⁻¹-∙-comm
}
; 0≟_ = 0≟_
}
where
open CommutativeRing CR
open import Algebra.Properties.Ring ring
open import Algebra.Properties.AbelianGroup +-abelianGroup
fromCommutativeSemiring : ∀ {r₁ r₂} (CS : CommutativeSemiring r₁ r₂)
(open CommutativeSemiring CS) →
(∀ x → Maybe (0# ≈ x)) →
AlmostCommutativeRing _ _
fromCommutativeSemiring CS 0≟_ = record
{ -_ = id
; isAlmostCommutativeRing = record
{ isCommutativeSemiring = isCommutativeSemiring
; -‿cong = id
; -‿*-distribˡ = λ _ _ → refl
; -‿+-comm = λ _ _ → refl
}
; 0≟_ = 0≟_
}
where open CommutativeSemiring CS
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