{- Copyright © 2015 Benjamin Barenblat Licensed under the Apache License, Version 2.0 (the ‘License’); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an ‘AS IS’ BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -} module B.Prelude.Number where import Algebra import Data.Integer import Data.Integer.Properties import Data.Nat import Data.Nat.Properties open import Level using (_⊔_) open import Relation.Binary.PropositionalEquality using (_≡_; refl) open Data.Integer using (ℤ) public open Data.Nat using (ℕ) public record NearSemiring {c} {ℓ} (A : Set c) : Set (Level.suc (c ⊔ ℓ)) where field structure : Algebra.NearSemiring c ℓ typeEquality : A ≡ Algebra.NearSemiring.Carrier structure NearSemiringInstance : ∀ {c ℓ} → (structure : Algebra.NearSemiring c ℓ) → NearSemiring (Algebra.NearSemiring.Carrier structure) NearSemiringInstance structure = record { structure = structure; typeEquality = refl } private change₂ : ∀ {c ℓ A′} → ⦃ witness : NearSemiring {c} {ℓ} A′ ⦄ → ((r : Algebra.NearSemiring c ℓ) → let A = Algebra.NearSemiring.Carrier r in A → A → A) → A′ → A′ → A′ change₂ ⦃ witness ⦄ f rewrite NearSemiring.typeEquality witness = f (NearSemiring.structure witness) _*_ : ∀ {c ℓ A} → ⦃ _ : NearSemiring {c} {ℓ} A ⦄ → A → A → A _*_ = change₂ Algebra.NearSemiring._*_ instance NearSemiring-ℕ : NearSemiring ℕ NearSemiring-ℕ = NearSemiringInstance (Algebra.CommutativeSemiring.nearSemiring Data.Nat.Properties.commutativeSemiring) NearSemiring-ℤ : NearSemiring ℤ NearSemiring-ℤ = NearSemiringInstance (Algebra.CommutativeRing.nearSemiring Data.Integer.Properties.commutativeRing)