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{-# OPTIONS --without-K --safe #-}
open import Algebra using (Semiring)
open import Level using (Level; 0ℓ; _⊔_)
module Data.Matrix.Semiadditive {c ℓ : Level} (R : Semiring c ℓ) where
module R = Semiring R
import Data.Nat as ℕ
import Data.Nat.Properties as ℕ-Props
import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
open import Categories.Category.Cocartesian using (Cocartesian)
open import Categories.Object.Biproduct using (Biproduct)
open import Categories.Object.Coproduct using (IsCoproduct)
open import Categories.Object.Initial using (IsInitial)
open import Categories.Object.Product using (IsProduct)
open import Categories.Object.Terminal using (IsTerminal)
open import Categories.Object.Zero using (Zero)
open import Category.Semiadditive using (Semiadditive)
open import Data.Matrix.Category R using (Mat; _·_; ≑-·; ·-Iˡ; ·-Iʳ; ·-𝟎ˡ; ·-𝟎ʳ; ·-∥; ∥-·-≑; ·-resp-≋; ·-assoc)
open import Data.Matrix.Core R.setoid using (Matrix; Matrixₛ; _≋_; module ≋; ∥-cong; ≑-cong; ᵀ-cong)
open import Data.Matrix.Monoid R.+-monoid using (𝟎; 𝟎ᵀ; 𝟎≑𝟎; 𝟎∥𝟎; _[+]_; [+]-cong; [+]-𝟎ˡ; [+]-𝟎ʳ)
open import Data.Matrix.Raw using (_ᵀ; _ᵀᵀ; mapRows; []ᵥ; []ᵥ-∥; []ₕ; []ₕ-!; []ₕ-≑; _∷ᵥ_; _∷ₕ_; ∷ᵥ-ᵀ; _∥_; _≑_; ∷ₕ-ᵀ; ∷ₕ-≑; []ᵥ-ᵀ; head-∷-tailₕ; headₕ; tailₕ; ∷ₕ-∥; ∷ᵥ-≑; []ᵥ-!)
open import Data.Matrix.Transform R 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 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.Raw using (⟨⟩)
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
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 _∷ᵥ_
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
split-≑ : (B : ℕ) (M : Matrix A (B ℕ.+ C)) → Σ[ M₁ ∈ Matrix A B ] Σ[ M₂ ∈ Matrix A C ] M₁ ≑ M₂ ≡ M
split-≑ zero M = []ₕ , M , []ₕ-≑ M
split-≑ (suc B) (M₀ ∷ M) with split-≑ B M
... | M₁ , M₂ , M₁≑M₂≡M = M₀ ∷ᵥ M₁ , M₂ , (begin
(M₀ ∷ᵥ M₁) ≑ M₂ ≡⟨ ∷ᵥ-≑ M₀ M₁ M₂ ⟨
M₀ ∷ᵥ M₁ ≑ M₂ ≡⟨ ≡.cong (M₀ ∷ᵥ_) M₁≑M₂≡M ⟩
M₀ ∷ᵥ 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)
proj₁ : (M : Matrix A B) (N : Matrix A C) → (I ∥ 𝟎) · (M ≑ N) ≋ M
proj₁ {A} {B} M N = begin
(I ∥ 𝟎) · (M ≑ N) ≈⟨ ∥-·-≑ I 𝟎 M N ⟩
(I · M) [+] (𝟎 · N) ≈⟨ [+]-cong ·-Iˡ (·-𝟎ˡ N) ⟩
M [+] 𝟎 ≈⟨ [+]-𝟎ʳ M ⟩
M ∎
where
open ≈-Reasoning (Matrixₛ A B)
proj₂ : (M : Matrix A B) (N : Matrix A C) → (𝟎 ∥ I) · (M ≑ N) ≋ N
proj₂ {A} {_} {C} M N = begin
(𝟎 ∥ I) · (M ≑ N) ≈⟨ ∥-·-≑ 𝟎 I M N ⟩
(𝟎 · M) [+] (I · N) ≈⟨ [+]-cong (·-𝟎ˡ M) ·-Iˡ ⟩
𝟎 [+] N ≈⟨ [+]-𝟎ˡ N ⟩
N ∎
where
open ≈-Reasoning (Matrixₛ A C)
≑-uniq
: (H : Matrix A (B ℕ.+ C))
(M : Matrix A B)
(N : Matrix A C)
→ (I ∥ 𝟎) · H ≋ M
→ (𝟎 ∥ I) · H ≋ N
→ M ≑ N ≋ H
≑-uniq {A} {B} {C} H M N eq₁ eq₂
with (H₁ , H₂ , H₁≑H₂≡H) ← split-≑ B H
rewrite ≡.sym H₁≑H₂≡H = begin
M ≑ N ≈⟨ ≑-cong eq₁ eq₂ ⟨
(I {B} ∥ 𝟎) · (H₁ ≑ H₂) ≑ (𝟎 ∥ I) · (H₁ ≑ H₂) ≈⟨ ≑-cong (proj₁ H₁ H₂) (proj₂ H₁ H₂) ⟩
H₁ ≑ H₂ ∎
where
open ≈-Reasoning (Matrixₛ A (B ℕ.+ C))
isCoproduct : IsCoproduct Mat (I {A} ≑ 𝟎) (𝟎 ≑ I {B})
isCoproduct {A} {B} = record
{ [_,_] = _∥_
; inject₁ = λ {a} {b} {c} → inj₁ b c
; inject₂ = λ {a} {b} {c} → inj₂ b c
; unique = λ eq₁ eq₂ → ∥-uniq _ _ _ eq₁ eq₂
}
isProduct : IsProduct Mat (I {A} ∥ 𝟎) (𝟎 ∥ I {B})
isProduct {A} {B} = record
{ ⟨_,_⟩ = _≑_
; project₁ = λ {a} {b} {c} → proj₁ b c
; project₂ = λ {a} {b} {c} → proj₂ b c
; unique = λ eq₁ eq₂ → ≑-uniq _ _ _ eq₁ eq₂
}
opaque
unfolding Matrix
π₁∘i₁ : (I {A} ∥ 𝟎 {B}) · (I ≑ 𝟎) ≋ I
π₁∘i₁ {A} = begin
(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 A A)
π₂∘i₂ : (𝟎 {A} {B} ∥ I) · (𝟎 ≑ I) ≋ I
π₂∘i₂ {A} {B} = begin
(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 B B)
π₁∘i₂ : (I {A} ∥ 𝟎 {B}) · (𝟎 ≑ I) ≋ 𝟎 {B} {A}
π₁∘i₂ {A} {B} = begin
(I ∥ 𝟎) · (𝟎 ≑ I) ≈⟨ ∥-·-≑ I 𝟎 𝟎 I ⟩
(I · 𝟎) [+] (𝟎 · I) ≈⟨ [+]-cong (·-𝟎ʳ I) (·-𝟎ˡ I) ⟩
𝟎 [+] 𝟎 ≈⟨ [+]-𝟎ʳ 𝟎 ⟩
𝟎 ∎
where
open ≈-Reasoning (Matrixₛ B A)
π₂∘i₁ : (𝟎 {A} {B} ∥ I) · (I ≑ 𝟎) ≋ 𝟎 {A} {B}
π₂∘i₁ {A} {B} = begin
(𝟎 ∥ I) · (I ≑ 𝟎) ≈⟨ ∥-·-≑ 𝟎 I I 𝟎 ⟩
(𝟎 · I) [+] (I · 𝟎) ≈⟨ [+]-cong (·-𝟎ˡ I) (·-𝟎ʳ I) ⟩
𝟎 [+] 𝟎 ≈⟨ [+]-𝟎ʳ 𝟎 ⟩
𝟎 ∎
where
open ≈-Reasoning (Matrixₛ A B)
permute
: (I ≑ 𝟎 {A} {B}) · (I ∥ 𝟎 {B} {A}) · (𝟎 {B} {A} ≑ I) · (𝟎 {A} {B} ∥ I)
≋ (𝟎 {B} {A} ≑ I) · (𝟎 {A} {B} ∥ I) · (I ≑ 𝟎 {A} {B}) · (I ∥ 𝟎 {B} {A})
permute {A} {B} = begin
(I ≑ 𝟎) · (I ∥ 𝟎 {B} {A}) · (𝟎 {B} {A} ≑ I) · (𝟎 {A} {B} ∥ I) ≈⟨ ·-resp-≋ ≋.refl ·-assoc ⟨
(I ≑ 𝟎) · ((I ∥ 𝟎 {B} {A}) · (𝟎 {B} {A} ≑ I)) · (𝟎 {A} {B} ∥ I) ≈⟨ ·-resp-≋ ≋.refl (·-resp-≋ π₁∘i₂ ≋.refl) ⟩
(I ≑ 𝟎) · 𝟎 {B} {A} · (𝟎 {A} {B} ∥ I) ≈⟨ ·-resp-≋ ≋.refl (·-𝟎ˡ (𝟎 ∥ I)) ⟩
(I ≑ 𝟎 {A} {B}) · 𝟎 ≈⟨ ·-𝟎ʳ (I ≑ 𝟎) ⟩
𝟎 ≈⟨ ·-𝟎ʳ (𝟎 ≑ I) ⟨
(𝟎 {B} {A} ≑ I) · 𝟎 ≈⟨ ·-resp-≋ ≋.refl (·-𝟎ˡ (I ∥ 𝟎)) ⟨
(𝟎 ≑ I) · 𝟎 {A} {B} · (I ∥ 𝟎 {B} {A}) ≈⟨ ·-resp-≋ ≋.refl (·-resp-≋ π₂∘i₁ ≋.refl) ⟨
(𝟎 {B} {A} ≑ I) · ((𝟎 ∥ I) · (I ≑ 𝟎 {A} {B})) · (I ∥ 𝟎 {B} {A}) ≈⟨ ·-resp-≋ ≋.refl ·-assoc ⟩
(𝟎 {B} {A} ≑ I) · (𝟎 {A} {B} ∥ I) · (I ≑ 𝟎 {A} {B}) · (I ∥ 𝟎) ∎
where
open ≈-Reasoning (Matrixₛ (A ℕ.+ B) (A ℕ.+ B))
biproduct : Biproduct Mat A B
biproduct {A} {B} = record
{ A⊕B = A ℕ.+ B
; π₁ = I ∥ 𝟎
; π₂ = 𝟎 ∥ I
; i₁ = I ≑ 𝟎
; i₂ = 𝟎 ≑ I
; isBiproduct = record
{ isCoproduct = isCoproduct
; isProduct = isProduct
; π₁∘i₁≈id = π₁∘i₁
; π₂∘i₂≈id = π₂∘i₂
; permute = permute
}
}
opaque
unfolding _≋_
¡-unique : (E : Matrix 0 B) → []ᵥ ≋ E
¡-unique E = ≋.reflexive (≡.sym ([]ᵥ-! E))
!-unique : (E : Matrix A 0) → []ₕ ≋ E
!-unique E = ≋.reflexive (≡.sym ([]ₕ-! E))
isInitial : IsInitial Mat 0
isInitial = record
{ ¡ = []ᵥ
; ¡-unique = ¡-unique
}
isTerminal : IsTerminal Mat 0
isTerminal = record
{ ! = []ₕ
; !-unique = !-unique
}
zeroObj : Zero Mat
zeroObj = record
{ 𝟘 = 0
; isZero = record
{ isInitial = isInitial
; isTerminal = isTerminal
}
}
Mat-Semiadditive : Semiadditive Mat
Mat-Semiadditive = record
{ zero = zeroObj
; biproducts = record
{ biproduct = biproduct
}
}
open Semiadditive Mat-Semiadditive using (cartesian)
Mat-CC : CartesianCategory 0ℓ c (c ⊔ ℓ)
Mat-CC = record
{ U = Mat
; cartesian = cartesian
}
|
