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(* 02110-1301 USA                                                     *)
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(*                                                                          *)
(*  Proof of the three gap theorem.                                         *)
(*                                                                          *)
(*  Micaela Mayero (INRIA-Rocquencourt)                                     *)
(*  September 1998                                                          *)
(*                                                                          *)
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(*                               tools.v                                    *)
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(*********************************************************)
(*            Tools for The three gap theorem            *)
(*                                                       *)
(*********************************************************)
Require Export Reals.
Require Export Nat_compl.
Unset Standard Proposition Elimination Names.
Parameter alpha : R.
(*********************************************************)
(*********************************************************)
(*           first and last                              *)
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(**********)
Definition frac_part_n_alpha (n : nat) : R := frac_part (INR n * alpha).
(**********)
Definition ordre_total (n m : nat) :=
(0 < alpha)%R /\ (alpha < 1)%R ->
(frac_part_n_alpha n <= frac_part_n_alpha m)%R.

(**********)
Lemma N_classic : forall N : nat, {N = 0} + {N = 1} + {N >= 2}.