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------------------------------------------------------------------------
-- Agda Helium
--
-- Some ordered algebraic structures (not packed up with sets,
-- operations, etc.)
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Helium.Algebra.Ordered.StrictTotal.Structures
{a ℓ₁ ℓ₂} {A : Set a} -- The underlying set
(_≈_ : Rel A ℓ₁) -- The underlying equality
(_<_ : Rel A ℓ₂) -- The underlying order
where
import Algebra.Consequences.Setoid as Consequences
open import Algebra.Core
open import Algebra.Definitions _≈_
import Algebra.Structures _≈_ as Unordered
open import Data.Product using (_,_; proj₁; proj₂)
open import Helium.Algebra.Core
open import Helium.Algebra.Definitions _≈_
open import Helium.Algebra.Structures _≈_ using (IsAlmostGroup)
open import Helium.Algebra.Ordered.Definitions _<_
open import Level using (_⊔_)
record IsMagma (∙ : Op₂ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_
∙-cong : Congruent₂ ∙
∙-invariant : Invariant ∙
open IsStrictTotalOrder isStrictTotalOrder public
strictTotalOrder : StrictTotalOrder _ _ _
strictTotalOrder = record { isStrictTotalOrder = isStrictTotalOrder }
open module strictTotalOrder = StrictTotalOrder strictTotalOrder public
using (strictPartialOrder)
∙-congˡ : LeftCongruent ∙
∙-congˡ y≈z = ∙-cong Eq.refl y≈z
∙-congʳ : RightCongruent ∙
∙-congʳ y≈z = ∙-cong y≈z Eq.refl
∙-invariantˡ : LeftInvariant ∙
∙-invariantˡ = proj₁ ∙-invariant
∙-invariantʳ : RightInvariant ∙
∙-invariantʳ = proj₂ ∙-invariant
record IsSemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isMagma : IsMagma ∙
assoc : Associative ∙
open IsMagma isMagma public
record IsMonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isSemigroup : IsSemigroup ∙
identity : Identity ε ∙
open IsSemigroup isSemigroup public
identityˡ : LeftIdentity ε ∙
identityˡ = proj₁ identity
identityʳ : RightIdentity ε ∙
identityʳ = proj₂ identity
record IsGroup (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isMonoid : IsMonoid _∙_ ε
inverse : Inverse ε _⁻¹ _∙_
⁻¹-cong : Congruent₁ _⁻¹
open IsMonoid isMonoid public
infixl 6 _-_
_-_ : Op₂ A
x - y = x ∙ (y ⁻¹)
inverseˡ : LeftInverse ε _⁻¹ _∙_
inverseˡ = proj₁ inverse
inverseʳ : RightInverse ε _⁻¹ _∙_
inverseʳ = proj₂ inverse
uniqueˡ-⁻¹ : ∀ x y → (x ∙ y) ≈ ε → x ≈ (y ⁻¹)
uniqueˡ-⁻¹ = Consequences.assoc+id+invʳ⇒invˡ-unique
strictTotalOrder.Eq.setoid ∙-cong assoc identity inverseʳ
uniqueʳ-⁻¹ : ∀ x y → (x ∙ y) ≈ ε → y ≈ (x ⁻¹)
uniqueʳ-⁻¹ = Consequences.assoc+id+invˡ⇒invʳ-unique
strictTotalOrder.Eq.setoid ∙-cong assoc identity inverseˡ
record IsAbelianGroup (∙ : Op₂ A)
(ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isGroup : IsGroup ∙ ε ⁻¹
comm : Commutative ∙
open IsGroup isGroup public
record IsRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
+-isAbelianGroup : IsAbelianGroup + 0# -_
*-isMonoid : Unordered.IsMonoid * 1#
distrib : * DistributesOver +
zero : Zero 0# *
preservesPositive : PreservesPositive 0# *
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
; comm to +-comm
; isMagma to +-isMagma
; isSemigroup to +-isSemigroup
; isMonoid to +-isMonoid
; isGroup to +-isGroup
)
open Unordered.IsMonoid *-isMonoid public
using ()
renaming
( assoc to *-assoc
; ∙-cong to *-cong
; ∙-congˡ to *-congˡ
; ∙-congʳ to *-congʳ
; identity to *-identity
; identityˡ to *-identityˡ
; identityʳ to *-identityʳ
; isMagma to *-isMagma
; isSemigroup to *-isSemigroup
)
distribˡ : * DistributesOverˡ +
distribˡ = proj₁ distrib
distribʳ : * DistributesOverʳ +
distribʳ = proj₂ distrib
zeroˡ : LeftZero 0# *
zeroˡ = proj₁ zero
zeroʳ : RightZero 0# *
zeroʳ = proj₂ zero
record IsCommutativeRing
(+ * : Op₂ A) (- : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isRing : IsRing + * - 0# 1#
*-comm : Commutative *
open IsRing isRing public
record IsDivisionRing
(+ _*_ : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
(_⁻¹ : AlmostOp₁ _≈_ 0#) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
+-isAbelianGroup : IsAbelianGroup + 0# -_
*-isAlmostGroup : IsAlmostGroup _*_ 0# 1# _⁻¹
distrib : _*_ DistributesOver +
zero : Zero 0# _*_
preservesPositive : PreservesPositive 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
; 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
; preservesPositive = preservesPositive
}
open IsRing isRing public
using (distribˡ ; distribʳ ; zeroˡ ; zeroʳ)
record IsField
(+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
(⁻¹ : AlmostOp₁ _≈_ 0#) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
field
isDivisionRing : IsDivisionRing + * -_ 0# 1# ⁻¹
*-comm : Commutative *
open IsDivisionRing isDivisionRing public
isCommutativeRing : IsCommutativeRing + * -_ 0# 1#
isCommutativeRing = record
{ isRing = isRing
; *-comm = *-comm
}
|
