{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Groups.Groups
open import Functions
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Numbers.Integers.Integers
open import Numbers.Modulo.Group
open import Numbers.Modulo.Definition
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 () }

-}