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------------------------------------------------------------------------
-- Agda Helium
--
-- Some more algebraic structures
-- (not packed up with sets, operations, etc.)
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
open import Relation.Binary.Core using (Rel)
module Helium.Algebra.Structures
{a ℓ} {A : Set a} -- The underlying set
(_≈_ : Rel A ℓ) -- The underlying equality relation
where
open import Algebra.Core
open import Algebra.Definitions _≈_
open import Algebra.Structures _≈_
open import Data.Product using (proj₁; proj₂)
open import Helium.Algebra.Core
open import Helium.Algebra.Definitions _≈_
import Helium.Algebra.Consequences.Setoid as Consequences
open import Level using (_⊔_)
open import Relation.Nullary using (¬_)
record IsAlmostGroup
(_∙_ : Op₂ A) (0# 1# : A) (_⁻¹ : AlmostOp₁ _≈_ 0#) : Set (a ⊔ ℓ) where
field
isMonoid : IsMonoid _∙_ 1#
inverse : AlmostInverse 1# _⁻¹ _∙_
⁻¹-cong : AlmostCongruent₁ _⁻¹
open IsMonoid isMonoid public
infixl 6 _-_
_-_ : AlmostOp₂ _≈_ 0#
x - y≉0 = x ∙ (y≉0 ⁻¹)
inverseˡ : AlmostLeftInverse 1# _⁻¹ _∙_
inverseˡ = proj₁ inverse
inverseʳ : AlmostRightInverse 1# _⁻¹ _∙_
inverseʳ = proj₂ inverse
uniqueˡ-⁻¹ : ∀ x {y} (y≉0 : ¬ y ≈ 0#) → (x ∙ y) ≈ 1# → x ≈ (y≉0 ⁻¹)
uniqueˡ-⁻¹ = Consequences.assoc+id+invʳ⇒invˡ-unique
setoid ∙-cong assoc identity inverseʳ
uniqueʳ-⁻¹ : ∀ {x} (x≉0 : ¬ x ≈ 0#) y → (x ∙ y) ≈ 1# → y ≈ (x≉0 ⁻¹)
uniqueʳ-⁻¹ = Consequences.assoc+id+invˡ⇒invʳ-unique
setoid ∙-cong assoc identity inverseˡ
record IsAlmostAbelianGroup
(∙ : Op₂ A) (0# 1# : A) (⁻¹ : AlmostOp₁ _≈_ 0#) : Set (a ⊔ ℓ) where
field
isAlmostGroup : IsAlmostGroup ∙ 0# 1# ⁻¹
comm : Commutative ∙
open IsAlmostGroup isAlmostGroup
isCommutativeMonoid : IsCommutativeMonoid ∙ 1#
isCommutativeMonoid = record
{ isMonoid = isMonoid
; comm = comm
}
open IsCommutativeMonoid isCommutativeMonoid public
using (isCommutativeMagma; isCommutativeSemigroup)
record IsDivisionRing
(+ _*_ : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
(_⁻¹ : AlmostOp₁ _≈_ 0#) : Set (a ⊔ ℓ) where
field
+-isAbelianGroup : IsAbelianGroup + 0# -_
*-isAlmostGroup : IsAlmostGroup _*_ 0# 1# _⁻¹
distrib : _*_ DistributesOver +
zero : Zero 0# _*_
infixl 7 _/_
_/_ : AlmostOp₂ _≈_ 0#
x / y≉0 = x * (y≉0 ⁻¹)
open IsAbelianGroup +-isAbelianGroup public
renaming
( assoc to +-assoc
; ∙-cong to +-cong
; ∙-congˡ to +-congˡ
; ∙-congʳ to +-congʳ
; identity to +-identity
; identityˡ to +-identityˡ
; identityʳ to +-identityʳ
; inverse to -‿inverse
; inverseˡ to -‿inverseˡ
; inverseʳ to -‿inverseʳ
; ⁻¹-cong to -‿cong
; uniqueˡ-⁻¹ to uniqueˡ‿-
; uniqueʳ-⁻¹ to uniqueʳ‿-
; comm to +-comm
; isMagma to +-isMagma
; isSemigroup to +-isSemigroup
; isMonoid to +-isMonoid
; isCommutativeMagma to +-isCommutativeMagma
; isCommutativeMonoid to +-isCommutativeMonoid
; isCommutativeSemigroup to +-isCommutativeSemigroup
; isGroup to +-isGroup
)
open IsAlmostGroup *-isAlmostGroup public
using (⁻¹-cong; uniqueˡ-⁻¹; uniqueʳ-⁻¹)
renaming
( assoc to *-assoc
; ∙-cong to *-cong
; ∙-congˡ to *-congˡ
; ∙-congʳ to *-congʳ
; identity to *-identity
; identityˡ to *-identityˡ
; identityʳ to *-identityʳ
; inverse to ⁻¹-inverse
; inverseˡ to ⁻¹-inverseˡ
; inverseʳ to ⁻¹-inverseʳ
; isMagma to *-isMagma
; isSemigroup to *-isSemigroup
; isMonoid to *-isMonoid
)
isRing : IsRing + _*_ -_ 0# 1#
isRing = record
{ +-isAbelianGroup = +-isAbelianGroup
; *-isMonoid = *-isMonoid
; distrib = distrib
; zero = zero
}
open IsRing isRing public
using
( distribˡ ; distribʳ ; zeroˡ ; zeroʳ
; isNearSemiring ; isSemiringWithoutOne ; isSemiringWithoutAnnihilatingZero
)
record IsField
(+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
(⁻¹ : AlmostOp₁ _≈_ 0#) : Set (a ⊔ ℓ) where
field
isDivisionRing : IsDivisionRing + * -_ 0# 1# ⁻¹
*-comm : Commutative *
open IsDivisionRing isDivisionRing public
isCommutativeRing : IsCommutativeRing + * -_ 0# 1#
isCommutativeRing = record
{ isRing = isRing
; *-comm = *-comm
}
open IsCommutativeRing isCommutativeRing public
using
( isCommutativeSemiring
; isCommutativeSemiringWithoutOne
; *-isCommutativeMagma
; *-isCommutativeSemigroup
; *-isCommutativeMonoid
)
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