------------------------------------------------------------------------
-- Some defined operations (multiplication by natural number and
-- exponentiation)
------------------------------------------------------------------------

{-# OPTIONS --universe-polymorphism #-}

open import Algebra

module Algebra.Operations {s₁ s₂} (S : Semiring s₁ s₂) where

open Semiring S hiding (zero)
open import Data.Nat using (zero; suc;)
open import Function
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as PropEq using (_≡_)
import Relation.Binary.EqReasoning as EqR
open EqR setoid

------------------------------------------------------------------------
-- Operations

-- Multiplication by natural number.

infixr 7 _×_

_×_ :   Carrier  Carrier
zero  × x = 0#
suc n × x = x + n × x

-- Exponentiation.

infixr 8 _^_

_^_ : Carrier    Carrier
x ^ zero  = 1#
x ^ suc n = x * x ^ n

------------------------------------------------------------------------
-- Some properties

×-cong : _×_ Preserves₂ _≡_  _≈_  _≈_
×-cong {n} {n'} {x} {x'} n≡n' x≈x' = begin
  n  × x   ≈⟨ reflexive $ PropEq.cong  n  n × x) n≡n' 
  n' × x   ≈⟨ ×-congʳ n' x≈x' 
  n' × x'  
  where
  ×-congʳ :  n  (_×_ n) Preserves _≈_  _≈_
  ×-congʳ zero    x≈x' = refl
  ×-congʳ (suc n) x≈x' = x≈x'  +-cong  ×-congʳ n x≈x'

^-cong : _^_ Preserves₂ _≈_  _≡_  _≈_
^-cong {x} {x'} {n} {n'} x≈x' n≡n' = begin
  x  ^ n   ≈⟨ reflexive $ PropEq.cong (_^_ x) n≡n' 
  x  ^ n'  ≈⟨ ^-congˡ n' x≈x' 
  x' ^ n'  
  where
  ^-congˡ :  n   x  x ^ n) Preserves _≈_  _≈_
  ^-congˡ zero    x≈x' = refl
  ^-congˡ (suc n) x≈x' = x≈x'  *-cong  ^-congˡ n x≈x'