The library has been tested using Agda 2.7.0 and 2.7.0.1.
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In
Algebra.Definitions.RawMagma
:_∣∣_ ↦ _∥_ _∤∤_ ↦ _∦_
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In
Algebra.Properties.Magma.Divisibility
:∣∣-sym ↦ ∥-sym ∣∣-respˡ-≈ ↦ ∥-respˡ-≈ ∣∣-respʳ-≈ ↦ ∥-respʳ-≈ ∣∣-resp-≈ ↦ ∥-resp-≈ ∤∤-sym -≈ ↦ ∦-sym ∤∤-respˡ-≈ ↦ ∦-respˡ-≈ ∤∤-respʳ-≈ ↦ ∦-respʳ-≈ ∤∤-resp-≈ ↦ ∦-resp-≈
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In
Algebra.Properties.Monoid.Divisibility
:∣∣-refl ↦ ∥-refl ∣∣-reflexive ↦ ∥-reflexive ∣∣-isEquivalence ↦ ∥-isEquivalence
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In
Algebra.Properties.Semigroup.Divisibility
:∣∣-trans ↦ ∥-trans
- In
Algebra.Construct.Pointwise
:isNearSemiring : IsNearSemiring _≈_ _+_ _*_ 0# → IsNearSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) isSemiringWithoutOne : IsSemiringWithoutOne _≈_ _+_ _*_ 0# → IsSemiringWithoutOne (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) isCommutativeSemiringWithoutOne : IsCommutativeSemiringWithoutOne _≈_ _+_ _*_ 0# → IsCommutativeSemiringWithoutOne (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) isCommutativeSemiring : IsCommutativeSemiring _≈_ _+_ _*_ 0# 1# → IsCommutativeSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#) isIdempotentSemiring : IsIdempotentSemiring _≈_ _+_ _*_ 0# 1# → IsIdempotentSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#) isKleeneAlgebra : IsKleeneAlgebra _≈_ _+_ _*_ _⋆ 0# 1# → IsKleeneAlgebra (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ _⋆) (lift₀ 0#) (lift₀ 1#) isQuasiring : IsQuasiring _≈_ _+_ _*_ 0# 1# → IsQuasiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#) isCommutativeRing : IsCommutativeRing _≈_ _+_ _*_ -_ 0# 1# → IsCommutativeRing (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ -_) (lift₀ 0#) (lift₀ 1#) commutativeMonoid : CommutativeMonoid c ℓ → CommutativeMonoid (a ⊔ c) (a ⊔ ℓ) nearSemiring : NearSemiring c ℓ → NearSemiring (a ⊔ c) (a ⊔ ℓ) semiringWithoutOne : SemiringWithoutOne c ℓ → SemiringWithoutOne (a ⊔ c) (a ⊔ ℓ) commutativeSemiringWithoutOne : CommutativeSemiringWithoutOne c ℓ → CommutativeSemiringWithoutOne (a ⊔ c) (a ⊔ ℓ) commutativeSemiring : CommutativeSemiring c ℓ → CommutativeSemiring (a ⊔ c) (a ⊔ ℓ) idempotentSemiring : IdempotentSemiring c ℓ → IdempotentSemiring (a ⊔ c) (a ⊔ ℓ) kleeneAlgebra : KleeneAlgebra c ℓ → KleeneAlgebra (a ⊔ c) (a ⊔ ℓ) quasiring : Quasiring c ℓ → Quasiring (a ⊔ c) (a ⊔ ℓ) commutativeRing : CommutativeRing c ℓ → CommutativeRing (a ⊔ c) (a ⊔ ℓ)