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Add a few algebraic structures missing from the Algebra.Construct.Pointwise (#2555)
* add a few algebraic structures missing from the Pointwise construct * update CHANGELOG * fix ws violation * resolve conflicts
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‎CHANGELOG.md

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@@ -26,3 +26,32 @@ New modules
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Additions to existing modules
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-----------------------------
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* In `Algebra.Construct.Pointwise`:
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```agda
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isNearSemiring : IsNearSemiring _≈_ _+_ _*_ 0# →
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IsNearSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#)
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isSemiringWithoutOne : IsSemiringWithoutOne _≈_ _+_ _*_ 0# →
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IsSemiringWithoutOne (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#)
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isCommutativeSemiringWithoutOne : IsCommutativeSemiringWithoutOne _≈_ _+_ _*_ 0# →
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IsCommutativeSemiringWithoutOne (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#)
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isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1# →
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IsCommutativeSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isIdempotentSemiring : IsIdempotentSemiring _≈_ _+_ _*_ 0# 1# →
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IsIdempotentSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isKleeneAlgebra : IsKleeneAlgebra _≈_ _+_ _*_ _⋆ 0# 1# →
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IsKleeneAlgebra (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ _⋆) (lift₀ 0#) (lift₀ 1#)
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isQuasiring : IsQuasiring _≈_ _+_ _*_ 0# 1# →
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IsQuasiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isCommutativeRing : IsCommutativeRing _≈_ _+_ _*_ -_ 0# 1# →
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IsCommutativeRing (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ -_) (lift₀ 0#) (lift₀ 1#)
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commutativeMonoid : CommutativeMonoid c ℓ → CommutativeMonoid (a ⊔ c) (a ⊔ ℓ)
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nearSemiring : NearSemiring c ℓ → NearSemiring (a ⊔ c) (a ⊔ ℓ)
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semiringWithoutOne : SemiringWithoutOne c ℓ → SemiringWithoutOne (a ⊔ c) (a ⊔ ℓ)
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commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne c ℓ → CommutativeSemiringWithoutOne (a ⊔ c) (a ⊔ ℓ)
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commutativeSemiring : CommutativeSemiring c ℓ → CommutativeSemiring (a ⊔ c) (a ⊔ ℓ)
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idempotentSemiring : IdempotentSemiring c ℓ → IdempotentSemiring (a ⊔ c) (a ⊔ ℓ)
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kleeneAlgebra : KleeneAlgebra c ℓ → KleeneAlgebra (a ⊔ c) (a ⊔ ℓ)
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quasiring : Quasiring c ℓ → Quasiring (a ⊔ c) (a ⊔ ℓ)
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commutativeRing : CommutativeRing c ℓ → CommutativeRing (a ⊔ c) (a ⊔ ℓ)
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```

‎src/Algebra/Construct/Pointwise.agda

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@@ -22,15 +22,14 @@ open import Relation.Binary.Core using (Rel)
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open import Relation.Binary.Bundles using (Setoid)
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open import Relation.Binary.Structures using (IsEquivalence)
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25-
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private
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variable
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c ℓ : Level
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C : Set c
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_≈_ : Rel C ℓ
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ε 0# 1# : C
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_⁻¹ -_ : Op₁ C
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_⁻¹ -_ _⋆ : Op₁ C
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_∙_ _+_ _*_ : Op₂ C
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lift₀ : C → A → C
@@ -121,6 +120,36 @@ isAbelianGroup isAbelianGroup = record
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}
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where module M = IsAbelianGroup isAbelianGroup
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isNearSemiring : IsNearSemiring _≈_ _+_ _*_ 0# →
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IsNearSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#)
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isNearSemiring isNearSemiring = record
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{ +-isMonoid = isMonoid M.+-isMonoid
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; *-cong = λ g h _ → M.*-cong (g _) (h _)
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; *-assoc = λ f g h _ → M.*-assoc (f _) (g _) (h _)
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; distribʳ = λ f g h _ → M.distribʳ (f _) (g _) (h _)
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; zeroˡ = λ f _ → M.zeroˡ (f _)
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}
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where module M = IsNearSemiring isNearSemiring
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isSemiringWithoutOne : IsSemiringWithoutOne _≈_ _+_ _*_ 0# →
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IsSemiringWithoutOne (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#)
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isSemiringWithoutOne isSemiringWithoutOne = record
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{ +-isCommutativeMonoid = isCommutativeMonoid M.+-isCommutativeMonoid
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; *-cong = λ g h _ → M.*-cong (g _) (h _)
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; *-assoc = λ f g h _ → M.*-assoc (f _) (g _) (h _)
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; distrib = (λ f g h _ → M.distribˡ (f _) (g _) (h _)) , (λ f g h _ → M.distribʳ (f _) (g _) (h _))
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; zero = (λ f _ → M.zeroˡ (f _)) , (λ f _ → M.zeroʳ (f _))
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}
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where module M = IsSemiringWithoutOne isSemiringWithoutOne
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isCommutativeSemiringWithoutOne : IsCommutativeSemiringWithoutOne _≈_ _+_ _*_ 0# →
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IsCommutativeSemiringWithoutOne (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#)
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isCommutativeSemiringWithoutOne isCommutativeSemiringWithoutOne = record
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{ isSemiringWithoutOne = isSemiringWithoutOne M.isSemiringWithoutOne
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; *-comm = λ f g _ → M.*-comm (f _) (g _)
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}
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where module M = IsCommutativeSemiringWithoutOne isCommutativeSemiringWithoutOne
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isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _≈_ _+_ _*_ 0# 1# →
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IsSemiringWithoutAnnihilatingZero (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isSemiringWithoutAnnihilatingZero isSemiringWithoutAnnihilatingZero = record
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}
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where module M = IsSemiring isSemiring
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isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1# →
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IsCommutativeSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isCommutativeSemiring isCommutativeSemiring = record
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{ isSemiring = isSemiring M.isSemiring
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; *-comm = λ f g _ → M.*-comm (f _) (g _)
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}
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where module M = IsCommutativeSemiring isCommutativeSemiring
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isIdempotentSemiring : IsIdempotentSemiring _≈_ _+_ _*_ 0# 1# →
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IsIdempotentSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isIdempotentSemiring isIdempotentSemiring = record
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{ isSemiring = isSemiring M.isSemiring
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; +-idem = λ f _ → M.+-idem (f _)
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}
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where module M = IsIdempotentSemiring isIdempotentSemiring
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isKleeneAlgebra : IsKleeneAlgebra _≈_ _+_ _*_ _⋆ 0# 1# →
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IsKleeneAlgebra (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ _⋆) (lift₀ 0#) (lift₀ 1#)
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isKleeneAlgebra isKleeneAlgebra = record
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{ isIdempotentSemiring = isIdempotentSemiring M.isIdempotentSemiring
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; starExpansive = (λ f _ → M.starExpansiveˡ (f _)) , λ f _ → M.starExpansiveʳ (f _)
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; starDestructive = (λ f g h i _ → M.starDestructiveˡ (f _) (g _) (h _) (i _))
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, (λ f g h i _ → M.starDestructiveʳ (f _) (g _) (h _) (i _))
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}
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where module M = IsKleeneAlgebra isKleeneAlgebra
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isQuasiring : IsQuasiring _≈_ _+_ _*_ 0# 1# →
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IsQuasiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
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isQuasiring isQuasiring = record
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{ +-isMonoid = isMonoid M.+-isMonoid
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; *-cong = λ g h _ → M.*-cong (g _) (h _)
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; *-assoc = λ f g h _ → M.*-assoc (f _) (g _) (h _)
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; *-identity = (λ f _ → M.*-identityˡ (f _)) , λ f _ → M.*-identityʳ (f _)
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; distrib = (λ f g h _ → M.distribˡ (f _) (g _) (h _)) , λ f g h _ → M.distribʳ (f _) (g _) (h _)
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; zero = (λ f _ → M.zeroˡ (f _)) , λ f _ → M.zeroʳ (f _)
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}
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where module M = IsQuasiring isQuasiring
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isRing : IsRing _≈_ _+_ _*_ -_ 0# 1# →
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IsRing (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ -_) (lift₀ 0#) (lift₀ 1#)
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isRing isRing = record
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}
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where module M = IsRing isRing
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isCommutativeRing : IsCommutativeRing _≈_ _+_ _*_ -_ 0# 1# →
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IsCommutativeRing (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ -_) (lift₀ 0#) (lift₀ 1#)
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isCommutativeRing isCommutativeRing = record
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{ isRing = isRing M.isRing
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; *-comm = λ f g _ → M.*-comm (f _) (g _)
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}
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where module M = IsCommutativeRing isCommutativeRing
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------------------------------------------------------------------------
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-- Bundles
@@ -170,15 +244,42 @@ commutativeSemigroup m = record { isCommutativeSemigroup = isCommutativeSemigrou
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monoid : Monoid c ℓ → Monoid (a ⊔ c) (a ⊔ ℓ)
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monoid m = record { isMonoid = isMonoid (Monoid.isMonoid m) }
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commutativeMonoid : CommutativeMonoid c ℓ → CommutativeMonoid (a ⊔ c) (a ⊔ ℓ)
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commutativeMonoid m = record { isCommutativeMonoid = isCommutativeMonoid (CommutativeMonoid.isCommutativeMonoid m) }
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group : Group c ℓ → Group (a ⊔ c) (a ⊔ ℓ)
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group m = record { isGroup = isGroup (Group.isGroup m) }
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abelianGroup : AbelianGroup c ℓ → AbelianGroup (a ⊔ c) (a ⊔ ℓ)
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abelianGroup m = record { isAbelianGroup = isAbelianGroup (AbelianGroup.isAbelianGroup m) }
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nearSemiring : NearSemiring c ℓ → NearSemiring (a ⊔ c) (a ⊔ ℓ)
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nearSemiring m = record { isNearSemiring = isNearSemiring (NearSemiring.isNearSemiring m) }
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semiringWithoutOne : SemiringWithoutOne c ℓ → SemiringWithoutOne (a ⊔ c) (a ⊔ ℓ)
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semiringWithoutOne m = record { isSemiringWithoutOne = isSemiringWithoutOne (SemiringWithoutOne.isSemiringWithoutOne m) }
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commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne c ℓ → CommutativeSemiringWithoutOne (a ⊔ c) (a ⊔ ℓ)
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commutativeSemiringWithoutOne m = record
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{ isCommutativeSemiringWithoutOne = isCommutativeSemiringWithoutOne (CommutativeSemiringWithoutOne.isCommutativeSemiringWithoutOne m) }
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semiring : Semiring c ℓ → Semiring (a ⊔ c) (a ⊔ ℓ)
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semiring m = record { isSemiring = isSemiring (Semiring.isSemiring m) }
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commutativeSemiring : CommutativeSemiring c ℓ → CommutativeSemiring (a ⊔ c) (a ⊔ ℓ)
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commutativeSemiring m = record { isCommutativeSemiring = isCommutativeSemiring (CommutativeSemiring.isCommutativeSemiring m) }
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idempotentSemiring : IdempotentSemiring c ℓ → IdempotentSemiring (a ⊔ c) (a ⊔ ℓ)
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idempotentSemiring m = record { isIdempotentSemiring = isIdempotentSemiring (IdempotentSemiring.isIdempotentSemiring m) }
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kleeneAlgebra : KleeneAlgebra c ℓ → KleeneAlgebra (a ⊔ c) (a ⊔ ℓ)
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kleeneAlgebra m = record { isKleeneAlgebra = isKleeneAlgebra (KleeneAlgebra.isKleeneAlgebra m) }
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quasiring : Quasiring c ℓ → Quasiring (a ⊔ c) (a ⊔ ℓ)
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quasiring m = record { isQuasiring = isQuasiring (Quasiring.isQuasiring m) }
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ring : Ring c ℓ → Ring (a ⊔ c) (a ⊔ ℓ)
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ring m = record { isRing = isRing (Ring.isRing m) }
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commutativeRing : CommutativeRing c ℓ → CommutativeRing (a ⊔ c) (a ⊔ ℓ)
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commutativeRing m = record { isCommutativeRing = isCommutativeRing (CommutativeRing.isCommutativeRing m) }

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