{-# OPTIONS --safe --warning=error #-}

open import LogicalFormulae
open import Groups.Groups
open import Functions
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Numbers.Modulo.IntegersModN
open import Rings.Examples.Proofs
open import Numbers.Primes.PrimeNumbers

module Rings.Examples.Examples where

  nToZn : (n : ℕ) (pr : 0 <N n) (x : ℕ) → ℤn n pr
  nToZn n pr x = nToZn' n pr x

  mod : (n : ℕ) → (pr : 0 <N n) → ℤ → ℤn n pr
  mod n pr a = mod' n pr a

  modNExampleSurjective : (n : ℕ) → (pr : 0 <N n) → Surjection (mod n pr)
  modNExampleSurjective n pr = modNExampleSurjective' n pr

{-
  modNExampleGroupHom : (n : ℕ) → (pr : 0 <N n) → GroupHom ℤGroup (ℤnGroup n pr) (mod n pr)
  modNExampleGroupHom n pr = modNExampleGroupHom' n pr

  embedZnInZ : {n : ℕ} {pr : 0 <N n} → (a : ℤn n pr) → ℤ
  embedZnInZ record { x = x } = nonneg x

  modNRoundTrip : (n : ℕ) → (pr : 0 <N n) → (a : ℤn n pr) → mod n pr (embedZnInZ a) ≡ a
  modNRoundTrip zero ()
  modNRoundTrip (succ n) pr record { x = x ; xLess = xLess } with divisionAlg (succ n) x
  modNRoundTrip (succ n) _ record { x = x ; xLess = xLess } | record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inl remIsSmall ; quotSmall = quotSmall } = equalityZn _ _ p
    where
      p : rem ≡ x
      p = modIsUnique record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inl remIsSmall ; quotSmall = quotSmall } record { quot = 0 ; rem = x ; pr = identityOfIndiscernablesLeft _ _ _ _≡_ refl (applyEquality (λ i → i +N x) (multiplicationNIsCommutative 0 n)) ; remIsSmall = inl xLess ; quotSmall = inl (succIsPositive n) }
  modNRoundTrip (succ n) _ record { x = x ; xLess = xLess } | record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inr () }

-}