Open Scope R_scope.
(** * Integers *)
(** Integer part in Z *)

(* Specifications: *)
Lemma floor_ex : forall x : R, {n : Z | IZR n <= x < IZR n + 1}.
Definition floor x := proj1_sig (floor_ex x).
Lemma floor1_ex : forall x : R, {n : Z | IZR n < x <= IZR n + 1}.
Definition floor1 x := proj1_sig (floor1_ex x).