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{- 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)
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