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{-# OPTIONS --without-K --safe #-}
open import Level using (Level)
open import Algebra using (Semiring)
module Data.Matrix.Functional {c ℓ : Level} (R : Semiring c ℓ) where
open import Data.Bool using (if_then_else_)
open import Data.Fin using (Fin; _≟_)
open import Data.Nat using (ℕ)
open import Data.Vec.Functional using (Vector; head; tail)
open import Function using (flip)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_)
open import Relation.Nullary.Decidable using (⌊_⌋)
open Semiring R
open ℕ
Matrix : ℕ → ℕ → Set c
Matrix n m = Vector (Vector Carrier m) n
sum : {n : ℕ} → Vector Carrier n → Carrier
sum {zero} _ = 0#
sum {suc n} v = head v + sum (tail v)
sum-cong : {n : ℕ} {V W : Vector Carrier n} → V ≗ W → sum V ≡ sum W
sum-cong {zero} V≗W = ≡.refl
sum-cong {suc n} {V} {W} V≗W = ≡.cong₂ _+_ (V≗W Fin.zero) (sum-cong (λ i → V≗W (Fin.suc i)))
_⟨*⟩_ : {n : ℕ} → Vector Carrier n → Vector Carrier n → Vector Carrier n
_⟨*⟩_ v w i = v i * w i
_⟨+⟩_ : {n : ℕ} → Vector Carrier n → Vector Carrier n → Vector Carrier n
_⟨+⟩_ v w i = v i + w i
_[+]_ : {n m : ℕ} → Matrix n m → Matrix n m → Matrix n m
_[+]_ v w i = v i ⟨+⟩ w i
_∙_ : {n : ℕ} → Vector Carrier n → Vector Carrier n → Carrier
_∙_ v w = sum (v ⟨*⟩ w)
_·_ : {n m o : ℕ} → Matrix m o → Matrix n m → Matrix n o
_·_ A B i j = flip A j ∙ B i
identity : {n : ℕ} → Matrix n n
identity {n} i j = if ⌊ i ≟ j ⌋ then 1# else 0#
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