(*********************************************************)
(*           first and last                              *)
(*********************************************************)
(**********)
(**********)
Definition ordre_total (n m : nat) :=
(0 < alpha)%R /\ (alpha < 1)%R ->
(frac_part_n_alpha n <= frac_part_n_alpha m)%R.

(* Specifications: *)
(**********)
Lemma N_classic : forall N : nat, {N = 0} + {N = 1} + {N >= 2}.
simple induction N; auto with arith real.