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{-# OPTIONS --without-K --safe #-}
open import Algebra using (Idempotent; CommutativeSemiring)
open import Level using (Level)
module Data.Matrix.Dagger-2-Poset
{c ℓ : Level}
(R : CommutativeSemiring c ℓ)
(let module R = CommutativeSemiring R)
(+-idem : Idempotent R._≈_ R._+_)
where
import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Category.Dagger.2-Poset using (Dagger-2-Poset)
open import Category.Dagger.Semiadditive using (IdempotentSemiadditiveDagger)
open import Data.Matrix.Category R.semiring using (Mat; _·_; ·-Iˡ; ·-Iʳ; ·-resp-≋; ·-assoc; ∥-·-≑; ·-∥; ·-𝟎ʳ; ≑-·)
open import Data.Matrix.Core R.setoid using (Matrix; Matrixₛ; _≋_; module ≋; ∥-cong; ≑-cong)
open import Data.Matrix.Monoid R.+-monoid using (𝟎; _[+]_; [+]-cong; [+]-𝟎ˡ; [+]-𝟎ʳ)
open import Data.Matrix.Raw using (_∥_; _≑_; _ᵀ)
open import Data.Matrix.SemiadditiveDagger R using (∥-ᵀ; Mat-SemiadditiveDagger)
open import Data.Matrix.Transform R.semiring using (I; Iᵀ)
open import Data.Nat using (ℕ)
open import Data.Vec using (Vec)
open import Data.Vector.Core R.setoid using (Vector; _≊_)
open import Data.Vector.Monoid R.+-monoid using (_⊕_)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
open Vec
private
variable
A B : ℕ
opaque
unfolding _⊕_
⊕-idem : (V : Vector A) → V ⊕ V ≊ V
⊕-idem [] = PW.[]
⊕-idem (v ∷ V) = +-idem v PW.∷ ⊕-idem V
opaque
unfolding _≋_ _[+]_
[+]-idem : (M : Matrix A B) → M [+] M ≋ M
[+]-idem [] = PW.[]
[+]-idem (M₀ ∷ M) = ⊕-idem M₀ PW.∷ [+]-idem M
Mat-IdempotentSemiadditiveDagger : IdempotentSemiadditiveDagger Mat
Mat-IdempotentSemiadditiveDagger = record
{ semiadditiveDagger = Mat-SemiadditiveDagger
; idempotent = [+]-idem _
}
open IdempotentSemiadditiveDagger Mat-IdempotentSemiadditiveDagger
Mat-Dagger-2-Poset : Dagger-2-Poset
Mat-Dagger-2-Poset = dagger-2-poset
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