{-# OPTIONS --without-K --safe #-} open import Algebra.Bundles using (CommutativeSemiring) open import Level using (Level) module Data.Matrix.SemiadditiveDagger {c ℓ : Level} (R : CommutativeSemiring c ℓ) where module R = CommutativeSemiring R import Relation.Binary.Reasoning.Setoid as ≈-Reasoning import Data.Vec.Relation.Binary.Pointwise.Inductive as PW import Data.Nat.Properties as ℕ-Props import Data.Nat as ℕ open import Categories.Category.Cocartesian using (Cocartesian) open import Categories.Object.Coproduct using (Coproduct) open import Categories.Object.Initial using (Initial) open import Category.Dagger.Semiadditive using (DaggerCocartesianMonoidal; SemiadditiveDagger) open import Data.Matrix.Cast R.setoid using (cast₂; cast₂-∥; ∥-≑; ∥-≑⁴; ≑-sym-assoc) open import Data.Matrix.Category R.semiring using (Mat; _·_; ≑-·; ·-Iˡ; ·-Iʳ; ·-𝟎ˡ; ·-𝟎ʳ; ·-∥; ∥-·-≑) open import Data.Matrix.Core R.setoid using (Matrix; Matrixₛ; _ᵀ; _ᵀᵀ; _≋_; module ≋; mapRows; []ᵥ; []ᵥ-∥; []ₕ; []ₕ-!; []ₕ-≑; _∷ᵥ_; _∷ₕ_; ∷ᵥ-ᵀ; _∥_; _≑_; ∷ₕ-ᵀ; ∷ₕ-≑; []ᵥ-ᵀ; ∥-cong; ≑-cong; -ᵀ-cong; head-∷-tailₕ; headₕ; tailₕ; ∷ₕ-∥; []ᵥ-!) open import Data.Matrix.Monoid R.+-monoid using (𝟎; 𝟎ᵀ; 𝟎≑𝟎; 𝟎∥𝟎; _[+]_; [+]-cong; [+]-𝟎ˡ; [+]-𝟎ʳ) open import Data.Matrix.Transform R.semiring using (I; Iᵀ; [_]_; _[_]; -[-]ᵀ; [-]--cong; [-]-[]ᵥ; [⟨⟩]-[]ₕ) open import Data.Nat using (ℕ) open import Data.Product using (_,_; Σ-syntax) open import Data.Vec using (Vec; map; replicate) open import Data.Vec.Properties using (map-cong; map-const) open import Data.Vector.Bisemimodule R.semiring using (_∙_ ; ∙-cong) open import Data.Vector.Core R.setoid using (Vector; Vectorₛ; ⟨⟩; _++_; module ≊; _≊_) open import Data.Vector.Monoid R.+-monoid using () renaming (⟨ε⟩ to ⟨0⟩) open import Data.Vector.Vec using (replicate-++) open import Function using (_∘_) open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; module ≡-Reasoning) open R open Vec open ℕ.ℕ private variable A B C D E F : ℕ opaque unfolding Vector _∙_ ∙-comm : (V W : Vector A) → V ∙ W ≈ W ∙ V ∙-comm [] [] = refl ∙-comm (x ∷ V) (w ∷ W) = +-cong (*-comm x w) (∙-comm V W) opaque unfolding _[_] [_]_ _ᵀ []ᵥ ⟨⟩ _∷ₕ_ _≊_ _≋_ _∷ᵥ_ [-]-ᵀ : (M : Matrix A B) (V : Vector A) → M [ V ] ≊ [ V ] (M ᵀ) [-]-ᵀ [] V = ≊.sym (≊.reflexive ([-]-[]ᵥ V)) [-]-ᵀ (M₀ ∷ M) V = begin M₀ ∙ V ∷ map (_∙ V) M ≈⟨ ∙-comm M₀ V PW.∷ (PW.map⁺ (λ {x} ≊V → trans (∙-comm x V) (∙-cong ≊.refl ≊V)) ≋.refl) ⟩ V ∙ M₀ ∷ map (V ∙_) M ≡⟨⟩ map (V ∙_) (M₀ ∷ᵥ M) ≡⟨ ≡.cong (map (V ∙_) ∘ (M₀ ∷ᵥ_)) (M ᵀᵀ) ⟨ map (V ∙_) (M₀ ∷ᵥ M ᵀ ᵀ) ≡⟨ ≡.cong (map (V ∙_)) (∷ₕ-ᵀ M₀ (M ᵀ)) ⟨ map (V ∙_) ((M₀ ∷ₕ (M ᵀ)) ᵀ) ∎ where open ≈-Reasoning (Vectorₛ _) opaque unfolding []ᵥ mapRows ⟨⟩ _∷ₕ_ _∷ᵥ_ _ᵀ ·-ᵀ : {A B C : ℕ} (M : Matrix A B) (N : Matrix B C) → (N · M) ᵀ ≋ M ᵀ · N ᵀ ·-ᵀ {A} {B} {zero} M [] = begin []ᵥ ≡⟨ map-const (M ᵀ) ⟨⟩ ⟨ map (λ _ → ⟨⟩) (M ᵀ) ≡⟨ map-cong [-]-[]ᵥ (M ᵀ) ⟨ map ([_] []ᵥ) (M ᵀ) ∎ where open ≈-Reasoning (Matrixₛ 0 A) ·-ᵀ {A} {B} {suc C} M (N₀ ∷ N) = begin map ([_] M) (N₀ ∷ N) ᵀ ≡⟨ -[-]ᵀ (N₀ ∷ N) M ⟨ map ((N₀ ∷ N) [_]) (M ᵀ) ≈⟨ PW.map⁺ (λ {V} ≋V → ≊.trans ([-]-ᵀ (N₀ ∷ N) V) ([-]--cong {A = (N₀ ∷ᵥ N) ᵀ} ≋V ≋.refl)) ≋.refl ⟩ map ([_] ((N₀ ∷ N) ᵀ)) (M ᵀ) ≡⟨ map-cong (λ V → ≡.cong ([ V ]_) (∷ᵥ-ᵀ N₀ N)) (M ᵀ) ⟩ map ([_] (N₀ ∷ₕ N ᵀ)) (M ᵀ) ∎ where open ≈-Reasoning (Matrixₛ (suc C) A) opaque unfolding _≋_ ᵀ-involutive : (M : Matrix A B) → (M ᵀ) ᵀ ≋ M ᵀ-involutive M = ≋.reflexive (M ᵀᵀ) opaque unfolding _≋_ ≋λᵀ : ([]ᵥ ∥ I) ᵀ ≋ 𝟎 ≑ I {A} ≋λᵀ = begin ([]ᵥ ∥ I) ᵀ ≡⟨ ≡.cong (_ᵀ) ([]ᵥ-∥ I) ⟩ I ᵀ ≡⟨ Iᵀ ⟩ I ≡⟨ []ₕ-≑ I ⟨ []ₕ ≑ I ≡⟨ ≡.cong (_≑ I) ([]ₕ-! 𝟎) ⟨ 𝟎 ≑ I ∎ where open ≈-Reasoning (Matrixₛ _ _) opaque unfolding Matrix _∥_ _ᵀ _≑_ _∷ₕ_ ∥-ᵀ : (M : Matrix A C) (N : Matrix B C) → (M ∥ N) ᵀ ≡ M ᵀ ≑ N ᵀ ∥-ᵀ {A} {zero} {B} [] [] = ≡.sym (replicate-++ A B []) ∥-ᵀ (M₀ ∷ M) (N₀ ∷ N) = begin (M₀ ++ N₀) ∷ₕ ((M ∥ N) ᵀ) ≡⟨ ≡.cong ((M₀ ++ N₀) ∷ₕ_) (∥-ᵀ M N) ⟩ (M₀ ++ N₀) ∷ₕ (M ᵀ ≑ N ᵀ) ≡⟨ ∷ₕ-≑ M₀ N₀ (M ᵀ) (N ᵀ) ⟩ (M₀ ∷ₕ M ᵀ) ≑ (N₀ ∷ₕ N ᵀ) ∎ where open ≡-Reasoning ≑-ᵀ : (M : Matrix A B) (N : Matrix A C) → (M ≑ N) ᵀ ≡ M ᵀ ∥ N ᵀ ≑-ᵀ M N = begin (M ≑ N) ᵀ ≡⟨ ≡.cong₂ (λ h₁ h₂ → (h₁ ≑ h₂) ᵀ) (M ᵀᵀ) (N ᵀᵀ) ⟨ (M ᵀ ᵀ ≑ N ᵀ ᵀ ) ᵀ ≡⟨ ≡.cong (_ᵀ) (∥-ᵀ (M ᵀ) (N ᵀ)) ⟨ (M ᵀ ∥ N ᵀ ) ᵀ ᵀ ≡⟨ (M ᵀ ∥ N ᵀ ) ᵀᵀ ⟩ M ᵀ ∥ N ᵀ ∎ where open ≡-Reasoning opaque unfolding _≋_ ≋ρᵀ : (I ∥ []ᵥ) ᵀ ≋ I {A} ≑ 𝟎 ≋ρᵀ {A} = begin (I ∥ []ᵥ) ᵀ ≡⟨ ∥-ᵀ I []ᵥ ⟩ I ᵀ ≑ []ᵥ ᵀ ≡⟨ ≡.cong (I ᵀ ≑_) []ᵥ-ᵀ ⟩ I ᵀ ≑ []ₕ ≡⟨ ≡.cong (_≑ []ₕ) Iᵀ ⟩ I ≑ []ₕ ≡⟨ ≡.cong (I ≑_) ([]ₕ-! 𝟎) ⟨ I ≑ 𝟎 ∎ where open ≈-Reasoning (Matrixₛ _ _) opaque unfolding _≋_ ≋αᵀ : (((I {A} ≑ 𝟎 {A} {B ℕ.+ C}) ∥ (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎)) ∥ (𝟎 {_} {A} ≑ I {B ℕ.+ C}) · (𝟎 ≑ I {C})) ᵀ ≋ (I {A ℕ.+ B} ≑ 𝟎) · (I {A} ≑ 𝟎) ∥ (I {A ℕ.+ B} ≑ 𝟎) · (𝟎 ≑ I {B}) ∥ (𝟎 ≑ I {C}) ≋αᵀ {A} {B} {C} = begin (((I {A} ≑ 𝟎 {A} {B ℕ.+ C}) ∥ (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ∥ (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ ≡⟨ ∥-ᵀ ((I {A} ≑ 𝟎 {A} {B ℕ.+ C}) ∥ (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ⟩ ((I {A} ≑ 𝟎 {A} {B ℕ.+ C}) ∥ (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ᵀ ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ ≡⟨ ≡.cong (_≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ) (∥-ᵀ (I {A} ≑ 𝟎 {A} {B ℕ.+ C}) ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C}))) ⟩ ((I {A} ≑ 𝟎 {A} {B ℕ.+ C}) ᵀ ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ᵀ) ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ ≡⟨ ≡.cong (λ h → (h ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ᵀ) ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ) (≑-ᵀ I 𝟎) ⟩ (I {A} ᵀ ∥ 𝟎 {A} {B ℕ.+ C} ᵀ ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ᵀ) ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ ≡⟨ ≡.cong (λ h → (h ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ᵀ) ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ) (≡.cong₂ _∥_ Iᵀ 𝟎ᵀ) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (I {B} ≑ 𝟎 {B} {C})) ᵀ) ≑ ((𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) · (𝟎 {C} {B} ≑ I {C})) ᵀ ≈⟨ ≑-cong (≑-cong ≋.refl (·-ᵀ (I ≑ 𝟎) (𝟎 ≑ I))) (·-ᵀ (𝟎 ≑ I) (𝟎 ≑ I)) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ (I {B} ≑ 𝟎 {B} {C}) ᵀ · (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) ᵀ) ≑ (𝟎 {C} {B} ≑ I {C}) ᵀ · (𝟎 {B ℕ.+ C} {A} ≑ I {B ℕ.+ C}) ᵀ ≡⟨ ≡.cong₂ _≑_ (≡.cong₂ (λ h₁ h₂ → I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ h₁ · h₂) (≑-ᵀ I 𝟎) (≑-ᵀ 𝟎 I)) (≡.cong₂ _·_ (≑-ᵀ 𝟎 I) (≑-ᵀ 𝟎 I)) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ (I {B} ᵀ ∥ 𝟎 {B} {C} ᵀ) · (𝟎 {B ℕ.+ C} {A} ᵀ ∥ I {B ℕ.+ C} ᵀ)) ≑ (𝟎 {C} {B} ᵀ ∥ I {C} ᵀ) · (𝟎 {B ℕ.+ C} {A} ᵀ ∥ I {B ℕ.+ C} ᵀ) ≡⟨ ≡.cong₂ _≑_ (≡.cong₂ (λ h₁ h₂ → I {A} ∥ 𝟎 ≑ h₁ · h₂) (≡.cong₂ _∥_ Iᵀ 𝟎ᵀ) (≡.cong₂ _∥_ 𝟎ᵀ Iᵀ)) (≡.cong₂ _·_ (≡.cong₂ _∥_ 𝟎ᵀ Iᵀ) (≡.cong₂ _∥_ 𝟎ᵀ Iᵀ)) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ (I {B} ∥ 𝟎 {C} {B}) · (𝟎 {A} {B ℕ.+ C} ∥ I {B ℕ.+ C})) ≑ (𝟎 {B} {C} ∥ I {C}) · (𝟎 {A} {B ℕ.+ C} ∥ I {B ℕ.+ C}) ≡⟨ ≡.cong (λ h → (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ h) ≑ (𝟎 {B} {C} ∥ I {C}) · (𝟎 {A} {B ℕ.+ C} ∥ I {B ℕ.+ C})) (·-∥ (I ∥ 𝟎) 𝟎 I) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ (I {B} ∥ 𝟎 {C} {B}) · 𝟎 {A} {B ℕ.+ C} ∥ (I {B} ∥ 𝟎 {C} {B}) · I {B ℕ.+ C}) ≑ (𝟎 {B} {C} ∥ I {C}) · (𝟎 {A} {B ℕ.+ C} ∥ I {B ℕ.+ C}) ≈⟨ ≑-cong (≑-cong ≋.refl (∥-cong (·-𝟎ʳ (I ∥ 𝟎)) ·-Iʳ)) (≋.refl {x = (𝟎 {B} {C} ∥ I {C}) · (𝟎 {A} {B ℕ.+ C} ∥ I {B ℕ.+ C})}) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ 𝟎 {A} {B} ∥ I {B} ∥ 𝟎 {C} {B}) ≑ (𝟎 {B} {C} ∥ I {C}) · (𝟎 {A} {B ℕ.+ C} ∥ I {B ℕ.+ C}) ≡⟨ ≡.cong ((I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ 𝟎 {A} {B} ∥ I {B} ∥ 𝟎 {C} {B}) ≑_) (·-∥ (𝟎 ∥ I) 𝟎 I) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ 𝟎 {A} {B} ∥ I {B} ∥ 𝟎 {C} {B}) ≑ (𝟎 {B} {C} ∥ I {C}) · (𝟎 {A} {B ℕ.+ C}) ∥ (𝟎 {B} {C} ∥ I {C}) · I {B ℕ.+ C} ≈⟨ ≑-cong ≋.refl (∥-cong (·-𝟎ʳ (𝟎 ∥ I)) ·-Iʳ) ⟩ (I {A} ∥ 𝟎 {B ℕ.+ C} {A} ≑ 𝟎 {A} {B} ∥ I {B} ∥ 𝟎 {C} {B}) ≑ 𝟎 {A} {C} ∥ 𝟎 {B} {C} ∥ I {C} ≡⟨ ≡.cong (λ h → (I {A} ∥ h ≑ 𝟎 {A} {B} ∥ I {B} ∥ 𝟎 {C} {B}) ≑ 𝟎 {A} {C} ∥ 𝟎 {B} {C} ∥ I {C}) 𝟎∥𝟎 ⟨ (I {A} ∥ 𝟎 {B} ∥ 𝟎 {C} ≑ 𝟎 {A} ∥ I {B} ∥ 𝟎 {C}) ≑ 𝟎 {A} ∥ 𝟎 {B} ∥ I {C} ≡⟨ ≑-sym-assoc (I {A} ∥ 𝟎 {B} ∥ 𝟎 {C}) (𝟎 {A} ∥ I {B} ∥ 𝟎 {C}) (𝟎 {A} ∥ 𝟎 {B} ∥ I {C}) ⟨ cast₂ _ (I {A} ∥ 𝟎 {B} ∥ 𝟎 {C} ≑ 𝟎 {A} ∥ I {B} ∥ 𝟎 {C} ≑ 𝟎 {A} ∥ 𝟎 {B} ∥ I {C}) ≡⟨ ≡.cong (cast₂ _) (∥-≑⁴ I 𝟎 𝟎 𝟎 I 𝟎 𝟎 𝟎 I) ⟩ cast₂ (≡.sym assoc) ((I {A} ≑ 𝟎 {A} {B} ≑ (𝟎 {A} {C})) ∥ (𝟎 {B} {A} ≑ I {B} ≑ 𝟎 {B} {C}) ∥ ((𝟎 {C} {A} ≑ 𝟎 {C} {B} ≑ I {C}))) ≡⟨ cast₂-∥ (≡.sym assoc) ((I {A} ≑ 𝟎 {A} {B} ≑ (𝟎 {A} {C}))) ((𝟎 {B} {A} ≑ I {B} ≑ 𝟎 {B} {C}) ∥ ((𝟎 {C} {A} ≑ 𝟎 {C} {B} ≑ I {C}))) ⟨ (cast₂ (≡.sym assoc) (I {A} ≑ 𝟎 {A} {B} ≑ (𝟎 {A} {C}))) ∥ cast₂ (≡.sym assoc) ((𝟎 {B} {A} ≑ I {B} ≑ 𝟎 {B} {C}) ∥ ((𝟎 {C} {A} ≑ 𝟎 {C} {B} ≑ I {C}))) ≡⟨ ≡.cong (cast₂ (≡.sym assoc) (I {A} ≑ 𝟎 {A} {B} ≑ (𝟎 {A} {C})) ∥_) (cast₂-∥ (≡.sym assoc) (𝟎 {B} {A} ≑ I {B} ≑ 𝟎 {B} {C}) (𝟎 {C} {A} ≑ 𝟎 {C} {B} ≑ I {C})) ⟨ cast₂ (≡.sym assoc) (I {A} ≑ 𝟎 {A} {B} ≑ (𝟎 {A} {C})) ∥ cast₂ (≡.sym assoc) (𝟎 {B} {A} ≑ I {B} ≑ 𝟎 {B} {C}) ∥ cast₂ (≡.sym assoc) (𝟎 {C} {A} ≑ 𝟎 {C} {B} ≑ I {C}) ≡⟨ ≡.cong₂ _∥_ (≑-sym-assoc I 𝟎 𝟎) (≡.cong₂ _∥_ (≑-sym-assoc 𝟎 I 𝟎) (≑-sym-assoc 𝟎 𝟎 I)) ⟩ ((I {A} ≑ 𝟎 {A} {B}) ≑ (𝟎 {A} {C})) ∥ ((𝟎 {B} {A} ≑ I {B}) ≑ 𝟎 {B} {C}) ∥ ((𝟎 {C} {A} ≑ 𝟎 {C} {B}) ≑ I {C}) ≡⟨ ≡.cong (λ h → ((I {A} ≑ 𝟎 {A} {B}) ≑ (𝟎 {A} {C})) ∥ ((𝟎 {B} {A} ≑ I {B}) ≑ 𝟎 {B} {C}) ∥ (h ≑ I {C})) 𝟎≑𝟎 ⟩ ((I {A} ≑ 𝟎 {A} {B}) ≑ (𝟎 {A} {C})) ∥ ((𝟎 {B} {A} ≑ I {B}) ≑ 𝟎 {B} {C}) ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C}) ≈⟨ ∥-cong ≋.refl (∥-cong (≑-cong ·-Iˡ (·-𝟎ˡ (𝟎 ≑ I))) ≋.refl) ⟨ ((I {A} ≑ 𝟎 {A} {B}) ≑ (𝟎 {A} {C})) ∥ (((I {A ℕ.+ B} · (𝟎 {B} {A} ≑ I {B})) ≑ (𝟎 {A ℕ.+ B} {C} · (𝟎 {B} {A} ≑ I {B})))) ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C}) ≡⟨ ≡.cong (λ h → ((I {A} ≑ 𝟎 {A} {B}) ≑ (𝟎 {A} {C})) ∥ h ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C})) (≑-· I 𝟎 (𝟎 ≑ I)) ⟨ ((I {A} ≑ 𝟎 {A} {B}) ≑ (𝟎 {A} {C})) ∥ ((I {A ℕ.+ B} ≑ 𝟎 {A ℕ.+ B} {C}) · (𝟎 {B} {A} ≑ I {B})) ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C}) ≈⟨ ∥-cong (≑-cong ·-Iˡ (·-𝟎ˡ (I ≑ 𝟎))) ≋.refl ⟨ ((I {A ℕ.+ B} · (I {A} ≑ 𝟎 {A} {B})) ≑ (𝟎 {A ℕ.+ B} {C} · (I {A} ≑ 𝟎 {A} {B}))) ∥ ((I {A ℕ.+ B} ≑ 𝟎 {A ℕ.+ B} {C}) · (𝟎 {B} {A} ≑ I {B})) ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C}) ≡⟨ ≡.cong (λ h → h ∥ ((I {A ℕ.+ B} ≑ 𝟎 {A ℕ.+ B} {C}) · (𝟎 {B} {A} ≑ I {B})) ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C})) (≑-· I 𝟎 (I ≑ 𝟎)) ⟨ (I {A ℕ.+ B} ≑ 𝟎 {A ℕ.+ B} {C}) · (I {A} ≑ 𝟎 {A} {B}) ∥ ((I {A ℕ.+ B} ≑ 𝟎 {A ℕ.+ B} {C}) · (𝟎 {B} {A} ≑ I {B})) ∥ (𝟎 {C} {A ℕ.+ B} ≑ I {C}) ∎ where assoc : A ℕ.+ B ℕ.+ C ≡ A ℕ.+ (B ℕ.+ C) assoc = ℕ-Props.+-assoc A B C open ≈-Reasoning (Matrixₛ _ _) ≋σᵀ : ((𝟎 ≑ I {A}) ∥ (I {B} ≑ 𝟎)) ᵀ ≋ (𝟎 ≑ I {B}) ∥ (I {A} ≑ 𝟎) ≋σᵀ {A} {B} = begin ((𝟎 ≑ I) ∥ (I ≑ 𝟎)) ᵀ ≡⟨ ∥-ᵀ (𝟎 ≑ I) (I ≑ 𝟎) ⟩ (𝟎 ≑ I {A}) ᵀ ≑ (I ≑ 𝟎) ᵀ ≡⟨ ≡.cong₂ _≑_ (≑-ᵀ 𝟎 I) (≑-ᵀ I 𝟎) ⟩ 𝟎 ᵀ ∥ (I {A}) ᵀ ≑ I ᵀ ∥ 𝟎 ᵀ ≡⟨ ≡.cong₂ _≑_ (≡.cong₂ _∥_ 𝟎ᵀ Iᵀ) (≡.cong₂ _∥_ Iᵀ 𝟎ᵀ) ⟩ 𝟎 ∥ I {A} ≑ I ∥ 𝟎 ≡⟨ ∥-≑ 𝟎 I I 𝟎 ⟩ (𝟎 ≑ I {B}) ∥ (I ≑ 𝟎) ∎ where open ≈-Reasoning (Matrixₛ _ _) ≋⊗ : (M : Matrix A B) (N : Matrix C D) → (I ≑ 𝟎) · M ∥ (𝟎 ≑ I) · N ≋ (M ≑ 𝟎) ∥ (𝟎 ≑ N) ≋⊗ M N = begin (I ≑ 𝟎) · M ∥ (𝟎 ≑ I) · N ≡⟨ ≡.cong₂ _∥_ (≑-· I 𝟎 M) (≑-· 𝟎 I N) ⟩ (I · M ≑ 𝟎 · M) ∥ (𝟎 · N ≑ I · N) ≈⟨ ∥-cong (≑-cong ·-Iˡ (·-𝟎ˡ M)) (≑-cong (·-𝟎ˡ N) ·-Iˡ) ⟩ (M ≑ 𝟎) ∥ (𝟎 ≑ N) ∎ where open ≈-Reasoning (Matrixₛ _ _) ᵀ-resp-⊗ : {M : Matrix A B} {N : Matrix C D} → ((I ≑ 𝟎) · M ∥ (𝟎 ≑ I) · N) ᵀ ≋ (I ≑ 𝟎) · M ᵀ ∥ (𝟎 ≑ I) · N ᵀ ᵀ-resp-⊗ {M = M} {N = N} = begin ((I ≑ 𝟎) · M ∥ (𝟎 ≑ I) · N) ᵀ ≈⟨ -ᵀ-cong (≋⊗ M N) ⟩ ((M ≑ 𝟎) ∥ (𝟎 ≑ N)) ᵀ ≡⟨ ≡.cong (_ᵀ) (∥-≑ M 𝟎 𝟎 N) ⟨ ((M ∥ 𝟎) ≑ (𝟎 ∥ N)) ᵀ ≡⟨ ≑-ᵀ (M ∥ 𝟎) (𝟎 ∥ N) ⟩ (M ∥ 𝟎) ᵀ ∥ (𝟎 ∥ N) ᵀ ≡⟨ ≡.cong₂ _∥_ (∥-ᵀ M 𝟎) (∥-ᵀ 𝟎 N) ⟩ (M ᵀ ≑ 𝟎 ᵀ) ∥ (𝟎 ᵀ ≑ N ᵀ) ≡⟨ ≡.cong₂ (λ h₁ h₂ → (M ᵀ ≑ h₁) ∥ (h₂ ≑ N ᵀ)) 𝟎ᵀ 𝟎ᵀ ⟩ (M ᵀ ≑ 𝟎) ∥ (𝟎 ≑ N ᵀ) ≈⟨ ≋⊗ (M ᵀ) (N ᵀ) ⟨ (I ≑ 𝟎) · M ᵀ ∥ (𝟎 ≑ I) · N ᵀ ∎ where open ≈-Reasoning (Matrixₛ _ _) inj₁ : (M : Matrix A C) (N : Matrix B C) → (M ∥ N) · (I ≑ 𝟎) ≋ M inj₁ {A} {C} M N = begin (M ∥ N) · (I ≑ 𝟎) ≈⟨ ∥-·-≑ M N I 𝟎 ⟩ (M · I) [+] (N · 𝟎) ≈⟨ [+]-cong ·-Iʳ (·-𝟎ʳ N) ⟩ M [+] 𝟎 ≈⟨ [+]-𝟎ʳ M ⟩ M ∎ where open ≈-Reasoning (Matrixₛ A C) inj₂ : (M : Matrix A C) (N : Matrix B C) → (M ∥ N) · (𝟎 ≑ I) ≋ N inj₂ {A} {C} {B} M N = begin (M ∥ N) · (𝟎 ≑ I) ≈⟨ ∥-·-≑ M N 𝟎 I ⟩ (M · 𝟎) [+] (N · I) ≈⟨ [+]-cong (·-𝟎ʳ M) ·-Iʳ ⟩ 𝟎 [+] N ≈⟨ [+]-𝟎ˡ N ⟩ N ∎ where open ≈-Reasoning (Matrixₛ B C) opaque unfolding Matrix split-∥ : (A : ℕ) → (M : Matrix (A ℕ.+ B) C) → Σ[ M₁ ∈ Matrix A C ] Σ[ M₂ ∈ Matrix B C ] M₁ ∥ M₂ ≡ M split-∥ zero M = []ᵥ , M , []ᵥ-∥ M split-∥ (suc A) M′ rewrite ≡.sym (head-∷-tailₕ M′) using M₀ ← headₕ M′ using M ← tailₕ M′ with split-∥ A M ... | M₁ , M₂ , M₁∥M₂≡M = M₀ ∷ₕ M₁ , M₂ , (begin (M₀ ∷ₕ M₁) ∥ M₂ ≡⟨ ∷ₕ-∥ M₀ M₁ M₂ ⟨ M₀ ∷ₕ M₁ ∥ M₂ ≡⟨ ≡.cong (M₀ ∷ₕ_) M₁∥M₂≡M ⟩ M₀ ∷ₕ M ∎) where open ≡-Reasoning uniq : (H : Matrix (A ℕ.+ B) C) (M : Matrix A C) (N : Matrix B C) → H · (I ≑ 𝟎) ≋ M → H · (𝟎 ≑ I) ≋ N → M ∥ N ≋ H uniq {A} {B} {C} H M N eq₁ eq₂ with (H₁ , H₂ , H₁∥H₂≡H) ← split-∥ A H rewrite ≡.sym H₁∥H₂≡H = begin M ∥ N ≈⟨ ∥-cong eq₁ eq₂ ⟨ (H₁ ∥ H₂) · (I {A} ≑ 𝟎) ∥ (H₁ ∥ H₂) · (𝟎 ≑ I) ≈⟨ ∥-cong (inj₁ H₁ H₂) (inj₂ H₁ H₂) ⟩ (H₁ ∥ H₂) ∎ where open ≈-Reasoning (Matrixₛ (A ℕ.+ B) C) coproduct : Coproduct Mat A B coproduct {A} {B} = record { A+B = A ℕ.+ B ; i₁ = I ≑ 𝟎 ; i₂ = 𝟎 ≑ I ; [_,_] = _∥_ ; inject₁ = λ {a} {b} {c} → inj₁ b c ; inject₂ = λ {a} {b} {c} → inj₂ b c ; unique = λ eq₁ eq₂ → uniq _ _ _ eq₁ eq₂ } opaque unfolding _≋_ !-unique : (E : Matrix 0 B) → []ᵥ ≋ E !-unique E = ≋.reflexive (≡.sym ([]ᵥ-! E)) initial : Initial Mat initial = record { ⊥ = 0 ; ⊥-is-initial = record { ! = []ᵥ ; !-unique = !-unique } } Mat-Cocartesian : Cocartesian Mat Mat-Cocartesian = record { initial = initial ; coproducts = record { coproduct = coproduct } } Mat-DaggerCocartesian : DaggerCocartesianMonoidal Mat Mat-DaggerCocartesian = record { cocartesian = Mat-Cocartesian ; dagger = record { _† = λ M → M ᵀ ; †-identity = ≋.reflexive Iᵀ ; †-homomorphism = λ {f = f} {g} → ·-ᵀ f g ; †-resp-≈ = -ᵀ-cong ; †-involutive = ᵀ-involutive } ; λ≅† = ≋λᵀ ; ρ≅† = ≋ρᵀ ; α≅† = ≋αᵀ ; σ≅† = ≋σᵀ ; †-resp-⊗ = ᵀ-resp-⊗ } p₁-i₁ : (I ≑ 𝟎) ᵀ · (I ≑ 𝟎 {A} {B}) ≋ I p₁-i₁ = begin (I ≑ 𝟎) ᵀ · (I ≑ 𝟎) ≡⟨ ≡.cong (_· (I ≑ 𝟎)) (≑-ᵀ I 𝟎) ⟩ (I ᵀ ∥ 𝟎 ᵀ) · (I ≑ 𝟎) ≡⟨ ≡.cong₂ (λ h₁ h₂ → (h₁ ∥ h₂) · (I ≑ 𝟎)) Iᵀ 𝟎ᵀ ⟩ (I ∥ 𝟎) · (I ≑ 𝟎) ≈⟨ ∥-·-≑ I 𝟎 I 𝟎 ⟩ (I · I) [+] (𝟎 · 𝟎) ≈⟨ [+]-cong ·-Iˡ (·-𝟎ˡ 𝟎) ⟩ I [+] 𝟎 ≈⟨ [+]-𝟎ʳ I ⟩ I ∎ where open ≈-Reasoning (Matrixₛ _ _) p₂-i₂ : (𝟎 {A} {B} ≑ I) ᵀ · (𝟎 ≑ I) ≋ I p₂-i₂ = begin (𝟎 ≑ I) ᵀ · (𝟎 ≑ I) ≡⟨ ≡.cong (_· (𝟎 ≑ I)) (≑-ᵀ 𝟎 I) ⟩ (𝟎 ᵀ ∥ I ᵀ) · (𝟎 ≑ I) ≡⟨ ≡.cong₂ (λ h₁ h₂ → (h₁ ∥ h₂) · (𝟎 ≑ I)) 𝟎ᵀ Iᵀ ⟩ (𝟎 ∥ I) · (𝟎 ≑ I) ≈⟨ ∥-·-≑ 𝟎 I 𝟎 I ⟩ (𝟎 · 𝟎) [+] (I · I) ≈⟨ [+]-cong (·-𝟎ˡ 𝟎) ·-Iˡ ⟩ 𝟎 [+] I ≈⟨ [+]-𝟎ˡ I ⟩ I ∎ where open ≈-Reasoning (Matrixₛ _ _) opaque unfolding 𝟎 ⟨⟩ []ᵥ·[]ₕ : []ᵥ · []ₕ ≡ 𝟎 {A} {B} []ᵥ·[]ₕ {A} {B} = begin map ([_] []ₕ) []ᵥ ≡⟨ map-cong (λ { [] → [⟨⟩]-[]ₕ }) []ᵥ ⟩ map (λ _ → ⟨0⟩) []ᵥ ≡⟨ map-const []ᵥ ⟨0⟩ ⟩ 𝟎 ∎ where open ≡-Reasoning p₂-i₁ : (𝟎 {A} ≑ I) ᵀ · (I ≑ 𝟎 {B}) ≋ []ᵥ · []ᵥ ᵀ p₂-i₁ = begin (𝟎 ≑ I) ᵀ · (I ≑ 𝟎) ≡⟨ ≡.cong (_· (I ≑ 𝟎)) (≑-ᵀ 𝟎 I) ⟩ (𝟎 ᵀ ∥ I ᵀ) · (I ≑ 𝟎) ≡⟨ ≡.cong₂ (λ h₁ h₂ → (h₁ ∥ h₂) · (I ≑ 𝟎)) 𝟎ᵀ Iᵀ ⟩ (𝟎 ∥ I) · (I ≑ 𝟎) ≈⟨ ∥-·-≑ 𝟎 I I 𝟎 ⟩ (𝟎 · I) [+] (I · 𝟎) ≈⟨ [+]-cong (·-𝟎ˡ I) (·-𝟎ʳ I) ⟩ 𝟎 [+] 𝟎 ≈⟨ [+]-𝟎ˡ 𝟎 ⟩ 𝟎 ≡⟨ []ᵥ·[]ₕ ⟨ []ᵥ · []ₕ ≡⟨ ≡.cong ([]ᵥ ·_) []ᵥ-ᵀ ⟨ []ᵥ · []ᵥ ᵀ ∎ where open ≈-Reasoning (Matrixₛ _ _) p₁-i₂ : (I ≑ 𝟎 {A}) ᵀ · (𝟎 {B} ≑ I) ≋ []ᵥ · []ᵥ ᵀ p₁-i₂ = begin (I ≑ 𝟎) ᵀ · (𝟎 ≑ I) ≡⟨ ≡.cong (_· (𝟎 ≑ I)) (≑-ᵀ I 𝟎) ⟩ (I ᵀ ∥ 𝟎 ᵀ) · (𝟎 ≑ I) ≡⟨ ≡.cong₂ (λ h₁ h₂ → (h₁ ∥ h₂) · (𝟎 ≑ I)) Iᵀ 𝟎ᵀ ⟩ (I ∥ 𝟎) · (𝟎 ≑ I) ≈⟨ ∥-·-≑ I 𝟎 𝟎 I ⟩ (I · 𝟎) [+] (𝟎 · I) ≈⟨ [+]-cong (·-𝟎ʳ I) (·-𝟎ˡ I) ⟩ 𝟎 [+] 𝟎 ≈⟨ [+]-𝟎ˡ 𝟎 ⟩ 𝟎 ≡⟨ []ᵥ·[]ₕ ⟨ []ᵥ · []ₕ ≡⟨ ≡.cong ([]ᵥ ·_) []ᵥ-ᵀ ⟨ []ᵥ · []ᵥ ᵀ ∎ where open ≈-Reasoning (Matrixₛ _ _) Mat-SemiadditiveDagger : SemiadditiveDagger Mat Mat-SemiadditiveDagger = record { daggerCocartesianMonoidal = Mat-DaggerCocartesian ; p₁-i₁ = p₁-i₁ ; p₂-i₂ = p₂-i₂ ; p₂-i₁ = p₂-i₁ ; p₁-i₂ = p₁-i₂ }