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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; _⊔_)
open import Relation.Binary using (Setoid; Rel; IsEquivalence)
module Data.Matrix.Core {c ℓ : Level} (S : Setoid c ℓ) where
import Data.Vec.Relation.Binary.Equality.Setoid as PW-≈
import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Data.Matrix.Raw as Raw using (_∥_; _≑_; _∷ᵥ_; _∷ₕ_; headₕ; tailₕ; headᵥ; tailᵥ; _ᵀ)
open import Data.Matrix.Vec using (transpose)
open import Data.Nat using (ℕ; _+_)
open import Data.Vec as Vec using (map; zipWith; head; tail; replicate)
open import Data.Vec.Properties using (map-cong; map-id)
open import Data.Vector.Core S using (Vector; Vectorₛ; _≊_) -- ; _++_; ⟨⟩; ⟨⟩-!)
open import Data.Vector.Vec using (zipWith-map; replicate-++)
open import Function using (id)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; module ≡-Reasoning)
open Setoid S
open ℕ
private
variable
n m p : ℕ
A B C : ℕ
private
module PW-≊ {n} = PW-≈ (Vectorₛ n)
Matrix : ℕ → ℕ → Set c
Matrix = Raw.Matrix Carrier
opaque
unfolding Raw.PW
-- Pointwise equality of matrices
_≋_ : Rel (Matrix n m) (c ⊔ ℓ)
_≋_ = Raw.PW _≈_
infix 4 _≋_
-- Pointwise equivalence is an equivalence relation
≋-isEquiv : IsEquivalence (_≋_ {n} {m})
≋-isEquiv {n} {m} = PW-≊.≋-isEquivalence m
module ≋ {n} {m} = IsEquivalence (≋-isEquiv {n} {m})
Matrixₛ : ℕ → ℕ → Setoid c (c ⊔ ℓ)
Matrixₛ n m = record
{ Carrier = Matrix n m
; _≈_ = _≋_ {n} {m}
; isEquivalence = ≋-isEquiv
}
opaque
unfolding _≋_
∥-cong
: {M₁ M₂ : Matrix A C}
{N₁ N₂ : Matrix B C}
→ M₁ ≋ M₂
→ N₁ ≋ N₂
→ M₁ ∥ N₁ ≋ M₂ ∥ N₂
∥-cong = Raw.Relation.R-∥
≑-cong
: {M₁ M₂ : Matrix A B}
{N₁ N₂ : Matrix A C}
→ M₁ ≋ M₂
→ N₁ ≋ N₂
→ M₁ ≑ N₁ ≋ M₂ ≑ N₂
≑-cong = Raw.Relation.R-≑
∷ᵥ-cong
: {V₁ V₂ : Vector n}
{M₁ M₂ : Matrix n m}
→ V₁ ≊ V₂
→ M₁ ≋ M₂
→ V₁ ∷ᵥ M₁ ≋ V₂ ∷ᵥ M₂
∷ᵥ-cong = Raw.Relation.R-∷ᵥ
∷ₕ-cong
: {V₁ V₂ : Vector n}
{M₁ M₂ : Matrix m n}
→ V₁ ≊ V₂
→ M₁ ≋ M₂
→ V₁ ∷ₕ M₁ ≋ V₂ ∷ₕ M₂
∷ₕ-cong = Raw.Relation.R-∷ₕ
headₕ-cong : {M₁ M₂ : Matrix (suc n) m} → M₁ ≋ M₂ → headₕ M₁ ≊ headₕ M₂
headₕ-cong = Raw.Relation.R-headₕ
tailₕ-cong : {M₁ M₂ : Matrix (suc n) m} → M₁ ≋ M₂ → tailₕ M₁ ≋ tailₕ M₂
tailₕ-cong = Raw.Relation.R-tailₕ
headᵥ-cong : {M₁ M₂ : Matrix n (suc m)} → M₁ ≋ M₂ → headᵥ M₁ ≊ headᵥ M₂
headᵥ-cong = Raw.Relation.R-headᵥ
tailᵥ-cong : {M₁ M₂ : Matrix n (suc m)} → M₁ ≋ M₂ → tailᵥ M₁ ≋ tailᵥ M₂
tailᵥ-cong = Raw.Relation.R-tailᵥ
ᵀ-cong : {M₁ M₂ : Matrix n m} → M₁ ≋ M₂ → M₁ ᵀ ≋ M₂ ᵀ
ᵀ-cong = Raw.Relation.R-ᵀ
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