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{-# OPTIONS --without-K --safe #-}
open import Algebra using (CommutativeSemiring)
open import Level using (Level; _⊔_)
module Data.Matrix.FreeSemimodule {c ℓ : Level} (R : CommutativeSemiring c ℓ) where
module R = CommutativeSemiring R
import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Algebra.Module using (Semimodule)
open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
open import Categories.Functor using (Functor)
open import Categories.Functor.Cartesian using (IsCartesianF; CartesianF)
open import Categories.Object.Product using (IsProduct)
open import Categories.Object.Terminal using (IsTerminal)
open import Category.Cartesian.Instance.Semimodules {c} {ℓ} {c} {c ⊔ ℓ} R using (Semimodules-CC)
open import Category.Instance.Semimodules {c} {ℓ} {c} {c ⊔ ℓ} R using (Semimodules; SemimoduleHomomorphism) renaming (_≈_ to _≈-SM_)
open import Data.Matrix.Category R.semiring using (Mat; _·_; ·-[])
open import Data.Matrix.Core R.setoid using (Matrix; module ≋)
open import Data.Matrix.Monoid R.+-monoid using (𝟎)
open import Data.Matrix.Raw using (_∥_; _≑_)
open import Data.Matrix.Semiadditive R.semiring using (Mat-CC)
open import Data.Matrix.Transform R.semiring using (I; _[_]; -[-]-cong; -[-]-cong₁; [_]_; -[⟨0⟩]; I[-]; -[⊕]; ∥-[++]; 𝟎-[-]; ≑--[-]; I-∥-≑)
open import Data.Nat as Nat using (ℕ)
open import Data.Vec using ([]; _++_)
open import Data.Vec using (map)
open import Data.Vec.Properties using (map-∘; map-++; zipWith-++)
open import Data.Vector.Bisemimodule R.semiring using (_⟨_⟩; ⟨_⟩_; _∙_; *-∙ˡ; *-∙ʳ; ∙-cong)
open import Data.Vector.Core R.setoid using (Vector; Vectorₛ; _≊_; module ≊)
open import Data.Vector.Monoid R.+-monoid using (_⊕_; ⊕-cong; ⟨ε⟩; ⊕-identityˡ; ⊕-identityʳ)
open import Data.Vector.Semimodule R using (Vector-Semimodule; ⟨-⟩-comm)
open import Data.Vector.Vec using (replicate-++)
open import Relation.Binary using (Setoid)
open R
open SemimoduleHomomorphism using (⟦_⟧; ⟦⟧-cong)
opaque
unfolding _[_] _⟨_⟩
-[-⟨-⟩] : {A B : ℕ} (M : Matrix A B) (r : Carrier) (V : Vector A) → M [ r ⟨ V ⟩ ] ≊ r ⟨ M [ V ] ⟩
-[-⟨-⟩] {A} M r V = begin
map (λ x → x ∙ r ⟨ V ⟩) M ≈⟨ PW.map⁺ lemma {xs = M} ≋.refl ⟩
map (λ x → r * x ∙ V) M ≡⟨ map-∘ (r *_) (_∙ V) M ⟩
map (r *_) (map (_∙ V) M) ∎
where
lemma : {X Y : Vector A} → X ≊ Y → X ∙ r ⟨ V ⟩ ≈ r * Y ∙ V
lemma {X} {Y} X≊Y = begin
X ∙ r ⟨ V ⟩ ≈⟨ ∙-cong ≊.refl (⟨-⟩-comm r V) ⟩
X ∙ ⟨ V ⟩ r ≈⟨ *-∙ʳ X V r ⟨
X ∙ V * r ≈⟨ *-comm (X ∙ V) r ⟩
r * X ∙ V ≈⟨ *-congˡ (∙-cong X≊Y ≊.refl) ⟩
r * Y ∙ V ∎
where
open ≈-Reasoning R.setoid
open ≈-Reasoning (Vectorₛ _)
-[⟨-⟩-] : {A B : ℕ} (M : Matrix A B) (r : Carrier) (V : Vector A) → M [ ⟨ V ⟩ r ] ≊ ⟨ M [ V ] ⟩ r
-[⟨-⟩-] {A} {B} M r V = begin
map (λ x → x ∙ ⟨ V ⟩ r) M ≈⟨ PW.map⁺ (λ {W} ≊W → trans (*-∙ʳ W V r) (∙-cong ≊W ≊.refl)) {xs = M} ≋.refl ⟨
map (λ x → x ∙ V * r) M ≡⟨ map-∘ (_* r) (_∙ V) M ⟩
map (_* r) (map (_∙ V) M) ∎
where
open ≈-Reasoning (Vectorₛ _)
F₁ : {A B : ℕ}
→ Matrix A B
→ SemimoduleHomomorphism (Vector-Semimodule A) (Vector-Semimodule B)
F₁ M = record
{ ⟦_⟧ = M [_]
; isSemimoduleHomomorphism = record
{ isBisemimoduleHomomorphism = record
{ +ᴹ-isMonoidHomomorphism = record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record
{ cong = -[-]-cong M
}
; homo = -[⊕] M
}
; ε-homo = -[⟨0⟩] M
}
; *ₗ-homo = -[-⟨-⟩] M
; *ᵣ-homo = -[⟨-⟩-] M
}
}
}
Free : Functor Mat Semimodules
Free = record
{ F₀ = Vector-Semimodule
; F₁ = F₁
; identity = I[-]
; homomorphism = λ {f = M} {N} V → ·-[] M N V
; F-resp-≈ = -[-]-cong₁
}
module Free-resp-⊤ where
opaque
unfolding _⊕_ ⟨ε⟩ _⟨_⟩
! : {A : Semimodule R c (c ⊔ ℓ)} → SemimoduleHomomorphism A (Vector-Semimodule 0)
! {A} = record
{ ⟦_⟧ = λ _ → []
; isSemimoduleHomomorphism = record
{ isBisemimoduleHomomorphism = record
{ +ᴹ-isMonoidHomomorphism = record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record
{ cong = λ _ → PW.[]
}
; homo = λ _ _ → PW.[]
}
; ε-homo = PW.[]
}
; *ₗ-homo = λ _ _ → PW.[]
; *ᵣ-homo = λ _ _ → PW.[]
}
}
}
!-unique
: {A : Semimodule R c (c ⊔ ℓ)}
(f : SemimoduleHomomorphism A (Vector-Semimodule 0))
→ ! ≈-SM f
!-unique f x with [] ← ⟦ f ⟧ x = PW.[]
Free-resp-⊤ : IsTerminal Semimodules (Vector-Semimodule Mat-CC.⊤)
Free-resp-⊤ = record { Free-resp-⊤ }
⟨_,_⟩
: {A B : ℕ}
{X : Semimodule R c (c ⊔ ℓ)}
→ SemimoduleHomomorphism X (Vector-Semimodule A)
→ SemimoduleHomomorphism X (Vector-Semimodule B)
→ SemimoduleHomomorphism X (Vector-Semimodule (A Nat.+ B))
⟨_,_⟩ {A} {B} {X} f g = record
{ ⟦_⟧ = λ x → ⟦ f ⟧ x ++ ⟦ g ⟧ x
; isSemimoduleHomomorphism = record
{ isBisemimoduleHomomorphism = record
{ +ᴹ-isMonoidHomomorphism = record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record
{ cong = λ ≈x → PW.++⁺ (⟦⟧-cong f ≈x) (⟦⟧-cong g ≈x)
}
; homo = homo
}
; ε-homo = ε-homo
}
; *ₗ-homo = *ₗ-homo
; *ᵣ-homo = *ᵣ-homo
}
}
}
where
open ≈-Reasoning (PW.setoid setoid (A Nat.+ B))
module f = SemimoduleHomomorphism f
module g = SemimoduleHomomorphism g
open Semimodule X
opaque
unfolding ⟨ε⟩
ε-homo : PW.Pointwise {c} {c} {ℓ} {Carrier} {Carrier} _≈_ {A Nat.+ B} {A Nat.+ B} (⟦ f ⟧ 0ᴹ ++ ⟦ g ⟧ 0ᴹ) ⟨ε⟩
ε-homo = begin
⟦ f ⟧ 0ᴹ ++ ⟦ g ⟧ 0ᴹ ≈⟨ PW.++⁺ f.0ᴹ-homo g.0ᴹ-homo ⟩
⟨ε⟩ {A} ++ ⟨ε⟩ {B} ≡⟨ replicate-++ A B 0# ⟩
⟨ε⟩ ∎
opaque
unfolding _⊕_
homo : (x y : Carrierᴹ) → PW.Pointwise _≈_ (f.⟦ x +ᴹ y ⟧ ++ g.⟦ x +ᴹ y ⟧) ((f.⟦ x ⟧ ++ g.⟦ x ⟧) ⊕ (f.⟦ y ⟧ ++ g.⟦ y ⟧))
homo x y = begin
f.⟦ x +ᴹ y ⟧ ++ g.⟦ x +ᴹ y ⟧ ≈⟨ PW.++⁺ (f.+ᴹ-homo x y) (g.+ᴹ-homo x y) ⟩
(f.⟦ x ⟧ ⊕ f.⟦ y ⟧) ++ (g.⟦ x ⟧ ⊕ g.⟦ y ⟧) ≡⟨ zipWith-++ _+_ f.⟦ x ⟧ g.⟦ x ⟧ f.⟦ y ⟧ g.⟦ y ⟧ ⟨
(f.⟦ x ⟧ ++ g.⟦ x ⟧) ⊕ (f.⟦ y ⟧ ++ g.⟦ y ⟧) ∎
opaque
unfolding _⟨_⟩
*ₗ-homo : (r : Carrier) (x : Carrierᴹ) → PW.Pointwise _≈_ (f.⟦ r *ₗ x ⟧ ++ g.⟦ r *ₗ x ⟧) (r ⟨ f.⟦ x ⟧ ++ g.⟦ x ⟧ ⟩)
*ₗ-homo r x = begin
f.⟦ r *ₗ x ⟧ ++ g.⟦ r *ₗ x ⟧ ≈⟨ PW.++⁺ (f.*ₗ-homo r x) (g.*ₗ-homo r x) ⟩
r ⟨ f.⟦ x ⟧ ⟩ ++ r ⟨ g.⟦ x ⟧ ⟩ ≡⟨ map-++ (r *_) f.⟦ x ⟧ g.⟦ x ⟧ ⟨
r ⟨ f.⟦ x ⟧ ++ g.⟦ x ⟧ ⟩ ∎
opaque
unfolding ⟨_⟩_
*ᵣ-homo : (r : Carrier) (x : Carrierᴹ) → PW.Pointwise _≈_ (f.⟦ x *ᵣ r ⟧ ++ g.⟦ x *ᵣ r ⟧) (⟨ f.⟦ x ⟧ ++ g.⟦ x ⟧ ⟩ r)
*ᵣ-homo r x = begin
f.⟦ x *ᵣ r ⟧ ++ g.⟦ x *ᵣ r ⟧ ≈⟨ PW.++⁺ (f.*ᵣ-homo r x) (g.*ᵣ-homo r x) ⟩
⟨ f.⟦ x ⟧ ⟩ r ++ ⟨ g.⟦ x ⟧ ⟩ r ≡⟨ map-++ (_* r) f.⟦ x ⟧ g.⟦ x ⟧ ⟨
⟨ f.⟦ x ⟧ ++ g.⟦ x ⟧ ⟩ r ∎
module Project
{A B : ℕ}
{X : Semimodule R c (c ⊔ ℓ)}
{f : SemimoduleHomomorphism X (Vector-Semimodule A)}
{g : SemimoduleHomomorphism X (Vector-Semimodule B)}
where
module f = SemimoduleHomomorphism f
module g = SemimoduleHomomorphism g
open Semimodule X
project₁ : (x : Carrierᴹ) → PW.Pointwise _≈_ ((I {A} ∥ 𝟎 ) [ f.⟦ x ⟧ ++ g.⟦ x ⟧ ]) f.⟦ x ⟧
project₁ x = begin
(I ∥ 𝟎) [ f.⟦ x ⟧ ++ g.⟦ x ⟧ ] ≈⟨ ∥-[++] f.⟦ x ⟧ g.⟦ x ⟧ I 𝟎 ⟩
I [ f.⟦ x ⟧ ] ⊕ 𝟎 [ g.⟦ x ⟧ ] ≈⟨ ⊕-cong (I[-] f.⟦ x ⟧) (𝟎-[-] g.⟦ x ⟧) ⟩
f.⟦ x ⟧ ⊕ ⟨ε⟩ ≈⟨ ⊕-identityʳ f.⟦ x ⟧ ⟩
f.⟦ x ⟧ ∎
where
open ≈-Reasoning (PW.setoid setoid A)
project₂ : (x : Carrierᴹ) → PW.Pointwise _≈_ ((𝟎 ∥ I {B} ) [ f.⟦ x ⟧ ++ g.⟦ x ⟧ ]) g.⟦ x ⟧
project₂ x = begin
(𝟎 ∥ I) [ f.⟦ x ⟧ ++ g.⟦ x ⟧ ] ≈⟨ ∥-[++] f.⟦ x ⟧ g.⟦ x ⟧ 𝟎 I ⟩
𝟎 [ f.⟦ x ⟧ ] ⊕ I [ g.⟦ x ⟧ ] ≈⟨ ⊕-cong (𝟎-[-] f.⟦ x ⟧) (I[-] g.⟦ x ⟧) ⟩
⟨ε⟩ ⊕ g.⟦ x ⟧ ≈⟨ ⊕-identityˡ g.⟦ x ⟧ ⟩
g.⟦ x ⟧ ∎
where
open ≈-Reasoning (PW.setoid setoid B)
module Unique
{A B : ℕ}
{X : Semimodule R c (c ⊔ ℓ)}
{f : SemimoduleHomomorphism X (Vector-Semimodule A)}
{g : SemimoduleHomomorphism X (Vector-Semimodule B)}
{h : SemimoduleHomomorphism X (Vector-Semimodule (A Nat.+ B))}
where
open Semimodule X
module f = SemimoduleHomomorphism f
module g = SemimoduleHomomorphism g
module h = SemimoduleHomomorphism h
unique
: (eq₁ : (x : Carrierᴹ) → PW.Pointwise _≈_ ((I {A} ∥ 𝟎) [ h.⟦ x ⟧ ]) (f.⟦ x ⟧))
→ (eq₂ : (x : Carrierᴹ) → PW.Pointwise _≈_ ((𝟎 ∥ I {B}) [ h.⟦ x ⟧ ]) (g.⟦ x ⟧))
→ (x : Carrierᴹ)
→ PW.Pointwise _≈_ (f.⟦ x ⟧ ++ g.⟦ x ⟧) h.⟦ x ⟧
unique eq₁ eq₂ x = begin
f.⟦ x ⟧ ++ g.⟦ x ⟧ ≈⟨ PW.++⁺ (eq₁ x) (eq₂ x) ⟨
((I {A} ∥ 𝟎) [ h.⟦ x ⟧ ]) ++ ((𝟎 ∥ I) [ h.⟦ x ⟧ ]) ≡⟨ ≑--[-] h.⟦ x ⟧ (I ∥ 𝟎) (𝟎 ∥ I) ⟨
((I {A} ∥ 𝟎) ≑ (𝟎 ∥ I)) [ h.⟦ x ⟧ ] ≡⟨ ≡.cong (_[ h.⟦ x ⟧ ]) I-∥-≑ ⟩
I [ h.⟦ x ⟧ ] ≈⟨ I[-] h.⟦ x ⟧ ⟩
h.⟦ x ⟧ ∎
where
open ≈-Reasoning (PW.setoid setoid (A Nat.+ B))
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
Free-resp-× : {A B : ℕ} → IsProduct Semimodules (F₁ (Mat-CC.π₁ {A} {B})) (F₁ (Mat-CC.π₂ {A} {B}))
Free-resp-× {A} {B} = record
{ ⟨_,_⟩ = ⟨_,_⟩
; project₁ = λ {X f g} → Project.project₁ {A} {B} {X} {f} {g}
; project₂ = λ {X f g} → Project.project₂ {A} {B} {X} {f} {g}
; unique = λ {X h f g} eq₁ eq₂ x → Unique.unique {A} {B} {X} {f} {g} {h} eq₁ eq₂ x
}
Free-IsCartesianF : IsCartesianF Mat-CC Semimodules-CC Free
Free-IsCartesianF = record
{ F-resp-⊤ = Free-resp-⊤
; F-resp-× = Free-resp-×
}
Free-IsCC : CartesianF Mat-CC Semimodules-CC
Free-IsCC = record
{ F = Free
; isCartesian = Free-IsCartesianF
}
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