MathClasses.implementations.modular_ring

Require Import
  Ring abstract_algebra interfaces.integers
  theory.integers theory.ring_ideals.

Definition is_multiple `{Equiv Z} `{Mult Z} (b x : Z) := ∃ k, x = b × k.
Notation Mod b := (Factor _ (is_multiple b)).

Section modular_ring.
  Context `{Ring Z} {b : Z}.
  Add Ring R : (rings.stdlib_ring_theory Z).

  Global Instance: RingIdeal Z (is_multiple b).
  Proof.
    unfold is_multiple. split.
        solve_proper.
       split. ∃ 0 0. ring.
      intros x y [k1 E1] [k2 E2]. ∃ (k1 - k2). rewrite E1, E2. ring.
     intros x y [k E]. ∃ (y × k). rewrite E. ring.
    intros x y [k E]. ∃ (x × k). rewrite E. ring.
  Qed.

  Lemma modular_ring_eq (x y : Mod b) : x = y ↔ ∃ k, 'x = 'y + b × k.
  Proof. split; intros [k E]; ∃ k. rewrite <-E. ring. rewrite E. ring. Qed.
End modular_ring.