Section Induction.
Variable A : t -> Prop.
Hypothesis A_wd : Proper (eq ==> iff) A.
Hypothesis A0 : A 0.
Hypothesis AS : forall n, A n <-> A (S n).
(* Below, we use only -> direction *)
Let B (n : Z) := A (ZnZ.of_Z n).

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.