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(* 02110-1301 USA                                                     *)
(****************************************************************************)
(*                                                                          *)
(*  Proof of the three gap theorem.                                         *)
(*                                                                          *)
(*  Micaela Mayero (INRIA-Rocquencourt)                                     *)
(*  September 1998                                                          *)
(*                                                                          *)
(****************************************************************************)
(****************************************************************************)
(*                               tools.v                                    *)
(****************************************************************************)
(*********************************************************)
(*            Tools for The three gap theorem            *)
(*                                                       *)
(*********************************************************)
Require Export Reals.
Require Export Nat_compl.
Unset Standard Proposition Elimination Names.
Parameter alpha : R.
(*********************************************************)
(*********************************************************)
(*           first and last                              *)
(*********************************************************)
(**********)
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.
simple induction N; auto with arith real.
intros; elim H; intro y.
elim y; intro y0.
left; right; auto with arith real.
right; rewrite <- y0; auto with arith real.
right; auto with arith real.
Qed.
intro; unfold ordre_total in |- *; intro; unfold Rle in |- *; right;
auto with arith real.
Qed.

(**********)
Lemma exist_first :
forall N : nat,
N >= 2 ->
sigT
(fun n : nat =>
0 < n /\ n < N /\ (forall m : nat, 0 < m /\ m < N -> ordre_total n m)).