From 6a35dcbbf1b3859b012e5f8546e8cb353898bde2 Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Thu, 2 Apr 2026 20:27:02 -0500
Subject: Reorganize matrix code

---
 Data/Mat/Dagger-2-Poset.agda | 79 --------------------------------------------
 1 file changed, 79 deletions(-)
 delete mode 100644 Data/Mat/Dagger-2-Poset.agda

(limited to 'Data/Mat/Dagger-2-Poset.agda')

diff --git a/Data/Mat/Dagger-2-Poset.agda b/Data/Mat/Dagger-2-Poset.agda
deleted file mode 100644
index cb939d1..0000000
--- a/Data/Mat/Dagger-2-Poset.agda
+++ /dev/null
@@ -1,79 +0,0 @@
-{-# OPTIONS --without-K --safe #-}
-
-open import Algebra using (Idempotent; CommutativeSemiring)
-open import Level using (Level)
-
-module Data.Mat.Dagger-2-Poset
-    {c ℓ : Level}
-    (Rig : CommutativeSemiring c ℓ)
-    (let module Rig = CommutativeSemiring Rig)
-    (+-idem : Idempotent Rig._≈_ Rig._+_)
-  where
-
-import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
-import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
-
-open import Data.Mat.Util using (transpose-cong; replicate-++)
-open import Data.Mat.Category Rig.semiring
-  using
-    ( Mat; _ᵀ; transpose-I; I; _≋_; module ≋; _≊_; Matrix; Vector
-    ; ≋-setoid; _·_; ·-identityˡ; ·-identityʳ; ·-resp-≋; ·-assoc; _⊕_
-    )
-open import Data.Mat.Cocartesian Rig.semiring
-  using
-    ( 𝟎 ; _∥_; _≑_; ∥-cong; ≑-cong; ≑-·; ·-𝟎ˡ; ·-∥; ∥-·-≑
-    ; _[+]_; [+]-cong; [+]-𝟎ʳ; [+]-𝟎ˡ
-    )
-open import Data.Mat.SemiadditiveDagger Rig using (∥-ᵀ; Mat-SemiadditiveDagger)
-open import Category.Dagger.Semiadditive Mat using (IdempotentSemiadditiveDagger)
-open import Category.Dagger.2-Poset using (dagger-2-poset; Dagger-2-Poset)
-open import Data.Nat using (ℕ)
-open import Data.Vec using (Vec)
-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
-
-opaque
-  unfolding ≋-setoid
-  idem : (M : Matrix A B) → (I ∥ I) · (((I ≑ 𝟎) · M) ∥ ((𝟎 ≑ I) · M)) · (I ∥ I) ᵀ ≋ M
-  idem M = begin
-      (I ∥ I) · (((I ≑ 𝟎) · M) ∥ ((𝟎 ≑ I) · M)) · (I ∥ I) ᵀ     ≡⟨ ≡.cong₂ (λ h₁ h₂ → (I ∥ I) · (h₁ ∥ h₂) · (I ∥ I) ᵀ) (≑-· I 𝟎 M) (≑-· 𝟎 I M) ⟩
-      (I ∥ I) · ((I · M ≑ 𝟎 · M) ∥ (𝟎 · M ≑ I · M)) · (I ∥ I) ᵀ ≈⟨ ·-resp-≋ ≋.refl (·-resp-≋ (∥-cong (≑-cong ·-identityˡ (·-𝟎ˡ M)) (≑-cong (·-𝟎ˡ M) ·-identityˡ)) ≋.refl) ⟩
-      (I ∥ I) · ((M ≑ 𝟎) ∥ (𝟎 ≑ M)) · (I ∥ I) ᵀ                 ≡⟨ ≡.cong (λ h → (I ∥ I) · ((M ≑ 𝟎) ∥ (𝟎 ≑ M)) · h) (∥-ᵀ I I) ⟩
-      (I ∥ I) · ((M ≑ 𝟎) ∥ (𝟎 ≑ M)) · (I ᵀ ≑ I ᵀ)               ≡⟨ ≡.cong₂ (λ h₁ h₂ → (I ∥ I) · ((M ≑ 𝟎) ∥ (𝟎 ≑ M)) · (h₁ ≑ h₂)) transpose-I transpose-I ⟩
-      (I ∥ I) · ((M ≑ 𝟎) ∥ (𝟎 ≑ M)) · (I ≑ I)                   ≈⟨ ·-assoc ⟨
-      ((I ∥ I) · ((M ≑ 𝟎) ∥ (𝟎 ≑ M))) · (I ≑ I)                 ≡⟨ ≡.cong (_· (I ≑ I)) (·-∥ (I ∥ I) (M ≑ 𝟎) (𝟎 ≑ M)) ⟩
-      (((I ∥ I) · (M ≑ 𝟎)) ∥ ((I ∥ I) · (𝟎 ≑ M))) · (I ≑ I)     ≈⟨ ∥-·-≑ ((I ∥ I) · (M ≑ 𝟎)) ((I ∥ I) · (𝟎 ≑ M)) I I ⟩
-      (((I ∥ I) · (M ≑ 𝟎)) · I) [+] (((I ∥ I) · (𝟎 ≑ M)) · I)   ≈⟨ [+]-cong ·-identityʳ ·-identityʳ ⟩
-      ((I ∥ I) · (M ≑ 𝟎)) [+] ((I ∥ I) · (𝟎 ≑ M))               ≈⟨ [+]-cong (∥-·-≑ I I M 𝟎) (∥-·-≑ I I 𝟎 M) ⟩
-      ((I · M) [+] (I · 𝟎)) [+] ((I · 𝟎) [+] (I · M))           ≈⟨ [+]-cong ([+]-cong ·-identityˡ ·-identityˡ) ([+]-cong ·-identityˡ ·-identityˡ) ⟩
-      (M [+] 𝟎) [+] (𝟎 [+] M)                                   ≈⟨ [+]-cong ([+]-𝟎ʳ M) ([+]-𝟎ˡ M) ⟩
-      M [+] M                                                   ≈⟨ [+]-idem M ⟩
-      M ∎
-    where
-      open ≈-Reasoning (≋-setoid _ _)
-
-Mat-IdempotentSemiadditiveDagger : IdempotentSemiadditiveDagger
-Mat-IdempotentSemiadditiveDagger = record
-    { semiadditiveDagger = Mat-SemiadditiveDagger
-    ; idempotent = idem _
-    }
-
-Mat-Dagger-2-Poset : Dagger-2-Poset
-Mat-Dagger-2-Poset = dagger-2-poset Mat-IdempotentSemiadditiveDagger
-- 
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