/usr/share/agda-stdlib/Algebra/Structures.agda is in agda-stdlib 0.7-2.
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-- The Agda standard library
--
-- Some algebraic structures (not packed up with sets, operations,
-- etc.)
------------------------------------------------------------------------
open import Relation.Binary
module Algebra.Structures where
import Algebra.FunctionProperties as FunctionProperties
open import Data.Product
open import Function
open import Level using (_⊔_)
import Relation.Binary.EqReasoning as EqR
open FunctionProperties using (Op₁; Op₂)
------------------------------------------------------------------------
-- One binary operation
record IsSemigroup {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(∙ : Op₂ A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isEquivalence : IsEquivalence ≈
assoc : Associative ∙
∙-cong : ∙ Preserves₂ ≈ ⟶ ≈ ⟶ ≈
open IsEquivalence isEquivalence public
record IsMonoid {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isSemigroup : IsSemigroup ≈ ∙
identity : Identity ε ∙
open IsSemigroup isSemigroup public
record IsCommutativeMonoid {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(_∙_ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isSemigroup : IsSemigroup ≈ _∙_
identityˡ : LeftIdentity ε _∙_
comm : Commutative _∙_
open IsSemigroup isSemigroup public
identity : Identity ε _∙_
identity = (identityˡ , identityʳ)
where
open EqR (record { isEquivalence = isEquivalence })
identityʳ : RightIdentity ε _∙_
identityʳ = λ x → begin
(x ∙ ε) ≈⟨ comm x ε ⟩
(ε ∙ x) ≈⟨ identityˡ x ⟩
x ∎
isMonoid : IsMonoid ≈ _∙_ ε
isMonoid = record
{ isSemigroup = isSemigroup
; identity = identity
}
record IsGroup {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
infixl 7 _-_
field
isMonoid : IsMonoid ≈ _∙_ ε
inverse : Inverse ε _⁻¹ _∙_
⁻¹-cong : _⁻¹ Preserves ≈ ⟶ ≈
open IsMonoid isMonoid public
_-_ : Op₂ A
x - y = x ∙ (y ⁻¹)
record IsAbelianGroup
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(∙ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isGroup : IsGroup ≈ ∙ ε ⁻¹
comm : Commutative ∙
open IsGroup isGroup public
isCommutativeMonoid : IsCommutativeMonoid ≈ ∙ ε
isCommutativeMonoid = record
{ isSemigroup = isSemigroup
; identityˡ = proj₁ identity
; comm = comm
}
------------------------------------------------------------------------
-- Two binary operations
record IsNearSemiring {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
+-isMonoid : IsMonoid ≈ + 0#
*-isSemigroup : IsSemigroup ≈ *
distribʳ : * DistributesOverʳ +
zeroˡ : LeftZero 0# *
open IsMonoid +-isMonoid public
renaming ( assoc to +-assoc
; ∙-cong to +-cong
; isSemigroup to +-isSemigroup
; identity to +-identity
)
open IsSemigroup *-isSemigroup public
using ()
renaming ( assoc to *-assoc
; ∙-cong to *-cong
)
record IsSemiringWithoutOne {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
+-isCommutativeMonoid : IsCommutativeMonoid ≈ + 0#
*-isSemigroup : IsSemigroup ≈ *
distrib : * DistributesOver +
zero : Zero 0# *
open IsCommutativeMonoid +-isCommutativeMonoid public
hiding (identityˡ)
renaming ( assoc to +-assoc
; ∙-cong to +-cong
; isSemigroup to +-isSemigroup
; identity to +-identity
; isMonoid to +-isMonoid
; comm to +-comm
)
open IsSemigroup *-isSemigroup public
using ()
renaming ( assoc to *-assoc
; ∙-cong to *-cong
)
isNearSemiring : IsNearSemiring ≈ + * 0#
isNearSemiring = record
{ +-isMonoid = +-isMonoid
; *-isSemigroup = *-isSemigroup
; distribʳ = proj₂ distrib
; zeroˡ = proj₁ zero
}
record IsSemiringWithoutAnnihilatingZero
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
-- Note that these structures do have an additive unit, but this
-- unit does not necessarily annihilate multiplication.
+-isCommutativeMonoid : IsCommutativeMonoid ≈ + 0#
*-isMonoid : IsMonoid ≈ * 1#
distrib : * DistributesOver +
open IsCommutativeMonoid +-isCommutativeMonoid public
hiding (identityˡ)
renaming ( assoc to +-assoc
; ∙-cong to +-cong
; isSemigroup to +-isSemigroup
; identity to +-identity
; isMonoid to +-isMonoid
; comm to +-comm
)
open IsMonoid *-isMonoid public
using ()
renaming ( assoc to *-assoc
; ∙-cong to *-cong
; isSemigroup to *-isSemigroup
; identity to *-identity
)
record IsSemiring {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isSemiringWithoutAnnihilatingZero :
IsSemiringWithoutAnnihilatingZero ≈ + * 0# 1#
zero : Zero 0# *
open IsSemiringWithoutAnnihilatingZero
isSemiringWithoutAnnihilatingZero public
isSemiringWithoutOne : IsSemiringWithoutOne ≈ + * 0#
isSemiringWithoutOne = record
{ +-isCommutativeMonoid = +-isCommutativeMonoid
; *-isSemigroup = *-isSemigroup
; distrib = distrib
; zero = zero
}
open IsSemiringWithoutOne isSemiringWithoutOne public
using (isNearSemiring)
record IsCommutativeSemiringWithoutOne
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isSemiringWithoutOne : IsSemiringWithoutOne ≈ + * 0#
*-comm : Commutative *
open IsSemiringWithoutOne isSemiringWithoutOne public
record IsCommutativeSemiring
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(_+_ _*_ : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
+-isCommutativeMonoid : IsCommutativeMonoid ≈ _+_ 0#
*-isCommutativeMonoid : IsCommutativeMonoid ≈ _*_ 1#
distribʳ : _*_ DistributesOverʳ _+_
zeroˡ : LeftZero 0# _*_
private
module +-CM = IsCommutativeMonoid +-isCommutativeMonoid
open module *-CM = IsCommutativeMonoid *-isCommutativeMonoid public
using () renaming (comm to *-comm)
open EqR (record { isEquivalence = +-CM.isEquivalence })
distrib : _*_ DistributesOver _+_
distrib = (distribˡ , distribʳ)
where
distribˡ : _*_ DistributesOverˡ _+_
distribˡ x y z = begin
(x * (y + z)) ≈⟨ *-comm x (y + z) ⟩
((y + z) * x) ≈⟨ distribʳ x y z ⟩
((y * x) + (z * x)) ≈⟨ *-comm y x ⟨ +-CM.∙-cong ⟩ *-comm z x ⟩
((x * y) + (x * z)) ∎
zero : Zero 0# _*_
zero = (zeroˡ , zeroʳ)
where
zeroʳ : RightZero 0# _*_
zeroʳ x = begin
(x * 0#) ≈⟨ *-comm x 0# ⟩
(0# * x) ≈⟨ zeroˡ x ⟩
0# ∎
isSemiring : IsSemiring ≈ _+_ _*_ 0# 1#
isSemiring = record
{ isSemiringWithoutAnnihilatingZero = record
{ +-isCommutativeMonoid = +-isCommutativeMonoid
; *-isMonoid = *-CM.isMonoid
; distrib = distrib
}
; zero = zero
}
open IsSemiring isSemiring public
hiding (distrib; zero; +-isCommutativeMonoid)
isCommutativeSemiringWithoutOne :
IsCommutativeSemiringWithoutOne ≈ _+_ _*_ 0#
isCommutativeSemiringWithoutOne = record
{ isSemiringWithoutOne = isSemiringWithoutOne
; *-comm = *-CM.comm
}
record IsRing
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(_+_ _*_ : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
+-isAbelianGroup : IsAbelianGroup ≈ _+_ 0# -_
*-isMonoid : IsMonoid ≈ _*_ 1#
distrib : _*_ DistributesOver _+_
open IsAbelianGroup +-isAbelianGroup public
renaming ( assoc to +-assoc
; ∙-cong to +-cong
; isSemigroup to +-isSemigroup
; identity to +-identity
; isMonoid to +-isMonoid
; inverse to -‿inverse
; ⁻¹-cong to -‿cong
; isGroup to +-isGroup
; comm to +-comm
; isCommutativeMonoid to +-isCommutativeMonoid
)
open IsMonoid *-isMonoid public
using ()
renaming ( assoc to *-assoc
; ∙-cong to *-cong
; isSemigroup to *-isSemigroup
; identity to *-identity
)
zero : Zero 0# _*_
zero = (zeroˡ , zeroʳ)
where
open EqR (record { isEquivalence = isEquivalence })
zeroˡ : LeftZero 0# _*_
zeroˡ x = begin
0# * x ≈⟨ sym $ proj₂ +-identity _ ⟩
(0# * x) + 0# ≈⟨ refl ⟨ +-cong ⟩ sym (proj₂ -‿inverse _) ⟩
(0# * x) + ((0# * x) + - (0# * x)) ≈⟨ sym $ +-assoc _ _ _ ⟩
((0# * x) + (0# * x)) + - (0# * x) ≈⟨ sym (proj₂ distrib _ _ _) ⟨ +-cong ⟩ refl ⟩
((0# + 0#) * x) + - (0# * x) ≈⟨ (proj₂ +-identity _ ⟨ *-cong ⟩ refl)
⟨ +-cong ⟩
refl ⟩
(0# * x) + - (0# * x) ≈⟨ proj₂ -‿inverse _ ⟩
0# ∎
zeroʳ : RightZero 0# _*_
zeroʳ x = begin
x * 0# ≈⟨ sym $ proj₂ +-identity _ ⟩
(x * 0#) + 0# ≈⟨ refl ⟨ +-cong ⟩ sym (proj₂ -‿inverse _) ⟩
(x * 0#) + ((x * 0#) + - (x * 0#)) ≈⟨ sym $ +-assoc _ _ _ ⟩
((x * 0#) + (x * 0#)) + - (x * 0#) ≈⟨ sym (proj₁ distrib _ _ _) ⟨ +-cong ⟩ refl ⟩
(x * (0# + 0#)) + - (x * 0#) ≈⟨ (refl ⟨ *-cong ⟩ proj₂ +-identity _)
⟨ +-cong ⟩
refl ⟩
(x * 0#) + - (x * 0#) ≈⟨ proj₂ -‿inverse _ ⟩
0# ∎
isSemiringWithoutAnnihilatingZero
: IsSemiringWithoutAnnihilatingZero ≈ _+_ _*_ 0# 1#
isSemiringWithoutAnnihilatingZero = record
{ +-isCommutativeMonoid = +-isCommutativeMonoid
; *-isMonoid = *-isMonoid
; distrib = distrib
}
isSemiring : IsSemiring ≈ _+_ _*_ 0# 1#
isSemiring = record
{ isSemiringWithoutAnnihilatingZero =
isSemiringWithoutAnnihilatingZero
; zero = zero
}
open IsSemiring isSemiring public
using (isNearSemiring; isSemiringWithoutOne)
record IsCommutativeRing
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(+ * : Op₂ A) (- : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isRing : IsRing ≈ + * - 0# 1#
*-comm : Commutative *
open IsRing isRing public
isCommutativeSemiring : IsCommutativeSemiring ≈ + * 0# 1#
isCommutativeSemiring = record
{ +-isCommutativeMonoid = +-isCommutativeMonoid
; *-isCommutativeMonoid = record
{ isSemigroup = *-isSemigroup
; identityˡ = proj₁ *-identity
; comm = *-comm
}
; distribʳ = proj₂ distrib
; zeroˡ = proj₁ zero
}
open IsCommutativeSemiring isCommutativeSemiring public
using ( *-isCommutativeMonoid
; isCommutativeSemiringWithoutOne
)
record IsLattice {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isEquivalence : IsEquivalence ≈
∨-comm : Commutative ∨
∨-assoc : Associative ∨
∨-cong : ∨ Preserves₂ ≈ ⟶ ≈ ⟶ ≈
∧-comm : Commutative ∧
∧-assoc : Associative ∧
∧-cong : ∧ Preserves₂ ≈ ⟶ ≈ ⟶ ≈
absorptive : Absorptive ∨ ∧
open IsEquivalence isEquivalence public
record IsDistributiveLattice {a ℓ} {A : Set a} (≈ : Rel A ℓ)
(∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isLattice : IsLattice ≈ ∨ ∧
∨-∧-distribʳ : ∨ DistributesOverʳ ∧
open IsLattice isLattice public
record IsBooleanAlgebra
{a ℓ} {A : Set a} (≈ : Rel A ℓ)
(∨ ∧ : Op₂ A) (¬ : Op₁ A) (⊤ ⊥ : A) : Set (a ⊔ ℓ) where
open FunctionProperties ≈
field
isDistributiveLattice : IsDistributiveLattice ≈ ∨ ∧
∨-complementʳ : RightInverse ⊤ ¬ ∨
∧-complementʳ : RightInverse ⊥ ¬ ∧
¬-cong : ¬ Preserves ≈ ⟶ ≈
open IsDistributiveLattice isDistributiveLattice public
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