(** An even more general induction principle using [Z.lt]. *)

(* Specifications: *)
Lemma Zlt_lower_bound_rec :
forall P:Z -> Type, forall z:Z,
(forall x:Z, (forall y:Z, z <= y < x -> P y) -> z <= x -> P x) ->
forall x:Z, z <= x -> P x.
Lemma Zlt_lower_bound_ind :
forall P:Z -> Prop, forall z:Z,
(forall x:Z, (forall y:Z, z <= y < x -> P y) -> z <= x -> P x) ->
forall x:Z, z <= x -> P x.