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authorBenjamin Barenblat <bbaren@mit.edu>2015-05-23 16:10:16 -0400
committerBenjamin Barenblat <bbaren@mit.edu>2015-05-23 16:10:16 -0400
commitc1fd2daa41aa1b915f74b4c09c6e62c79320e8ec (patch)
treef0ce08eaa83bd5a47bf4bb7e030a48864cc1cf20
parent374045f4a0c4aab339a49c4c4bd3dad7c813bf69 (diff)
Start implementing a Number moduleHEADmaster
The nascent Number module also serves as a proof of concept for dealing with records in the Agda standard library that are parameterized by levels instead of sets, like those in the Algebra hierarchy. These records are difficult to use as type classes, because Agda’s normal mechanism for generating type classes ('open SomeRecord ⦃...⦄') will make the levels the instance arguments, rather than the actual set the record’s functions operate on. To work around this, I defined wrapper records (effectively just named dependent pairs) parameterized in the way I desire. Each contains an underlying algebraic structure and a proof of type equality (usually just 'refl') between the parameter and the structure’s carrier. Then, I define functions over my wrapper record to rewrite with the type equality and use the underlying structure’s code. This work is related to standard library bug #35 <https://github.com/agda/agda-stdlib/issues/35>;; fixing this bug would obviate the need for my workaround.
-rw-r--r--B.agda1
-rw-r--r--Prelude/Number.agda60
2 files changed, 61 insertions, 0 deletions
diff --git a/B.agda b/B.agda
index d6087d8..3e28f35 100644
--- a/B.agda
+++ b/B.agda
@@ -20,6 +20,7 @@ open import B.Prelude.Eq public
open import B.Prelude.Functor public
open import B.Prelude.List public
open import B.Prelude.Maybe public
+open import B.Prelude.Number public
open import B.Prelude.Product public
open import B.Prelude.String public
open import B.Prelude.Sum public
diff --git a/Prelude/Number.agda b/Prelude/Number.agda
new file mode 100644
index 0000000..02024d0
--- /dev/null
+++ b/Prelude/Number.agda
@@ -0,0 +1,60 @@
+{- 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)