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

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