Module NZCyclicAxiomsMod (Import Cyclic : CyclicType) <: NZAxiomsSig.
Local Open Scope Z_scope.
Local Notation wB := (base ZnZ.digits).
Local Notation "[| x |]" := (ZnZ.to_Z x) (at level 0, x at level 99).
Definition eq (n m : t) := [| n |] = [| m |].
Definition zero := ZnZ.zero.
Definition one := ZnZ.one.
Definition two := ZnZ.succ ZnZ.one.
Definition succ := ZnZ.succ.
Definition pred := ZnZ.pred.
Definition add := ZnZ.add.
Definition sub := ZnZ.sub.
Definition mul := ZnZ.mul.
Local Infix "=="  := eq (at level 70).
Local Notation "0" := zero.
Local Notation S := succ.
Local Notation P := pred.
Local Infix "+" := add.
Local Infix "-" := sub.
Local Infix "*" := mul.
Hint Rewrite ZnZ.spec_0 ZnZ.spec_1 ZnZ.spec_succ ZnZ.spec_pred
ZnZ.spec_add ZnZ.spec_mul ZnZ.spec_sub : cyclic.
Ltac zify :=
unfold eq, zero, one, two, succ, pred, add, sub, mul in *;
autorewrite with cyclic.
Ltac zcongruence := repeat red; intros; zify; congruence.
Instance eq_equiv : Equivalence eq.
Local Obligation Tactic := zcongruence.
Program Instance succ_wd : Proper (eq ==> eq) succ.
Program Instance pred_wd : Proper (eq ==> eq) pred.
Program Instance add_wd : Proper (eq ==> eq ==> eq) add.
Program Instance sub_wd : Proper (eq ==> eq ==> eq) sub.
Program Instance mul_wd : Proper (eq ==> eq ==> eq) mul.

Lemma pred_mod_wB : forall n : Z, (n - 1) mod wB = ((n mod wB) - 1) mod wB.