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.Ltac zify :=
 unfold eq, zero, one, two, succ, pred, add, sub, mul in *;
 autorewrite with cyclic.Ltac zcongruence := repeat red; intros; zify; congruence.#[global]
Instance eq_equiv : Equivalence eq.#[global]
Program Instance succ_wd : Proper (eq ==> eq) succ.#[global]
Program Instance pred_wd : Proper (eq ==> eq) pred.#[global]
Program Instance add_wd : Proper (eq ==> eq ==> eq) add.#[global]
Program Instance sub_wd : Proper (eq ==> eq ==> eq) sub.#[global]
Program Instance mul_wd : Proper (eq ==> eq ==> eq) mul.

Theorem Zbounded_induction : (forall Q : Z -> Prop, forall b : Z, Q 0 -> (forall n, 0 <= n -> n < b - 1 -> Q n -> Q (n + 1)) -> forall n, 0 <= n -> n < b -> Q n)%Z.