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(* 02110-1301 USA                                                     *)
(****************************************************************************)
(*                                                                          *)
(*  Proof of the three gap theorem.                                         *)
(*                                                                          *)
(*  Micaela Mayero (INRIA-Rocquencourt)                                     *)
(*  September 1998                                                          *)
(*                                                                          *)
(****************************************************************************)
(****************************************************************************)
(*                               prop_elem.v                                *)
(****************************************************************************)
(*********************************************************)
(*          Elementary properties first last             *)
(*                                                       *)
(*********************************************************)
Require Export tools.
Unset Standard Proposition Elimination Names.
(*********************************************************)
intros; unfold first in |- *; case (N_classic N); intro.
absurd (N >= 2).
elim s; intro y; rewrite y; unfold ge in |- *; auto with arith real.
exact H.
case (exist_first N g); intros; elim a; intros; elim H1; intros;
auto with arith real.
Qed.
intros; unfold last in |- *; case (N_classic N); intro.
absurd (N >= 2).
elim s; intro y; rewrite y; unfold ge in |- *; auto with arith real.
exact H.
case (exist_last N g); intros; elim a; intros; elim H1; intros;
auto with arith real.
Qed.
intros; apply max_case2.
apply (first_N N H).
apply (last_N N H).
Qed.
intros; unfold first in |- *; case (N_classic N); intro; auto with arith real.
case (exist_first N g); intros; elim a; intros; elim H0; intros;
auto with arith real.
Qed.
intros; unfold last in |- *; case (N_classic N); intro; auto with arith real.
case (exist_last N g); intros; elim a; intros; elim H0; intros;
auto with arith real.
Qed.
intros; unfold first in |- *; case (N_classic N); intro.
absurd (N >= 2).
elim s; intro y; rewrite y; unfold ge in |- *; auto with arith real.
exact H.
case (exist_first N g); intros; elim a; intros; auto with arith real.
Qed.
intros; unfold last in |- *; case (N_classic N); intro.
absurd (N >= 2).
elim s; intro y; rewrite y; unfold ge in |- *; auto with arith real.
exact H.
case (exist_last N g); intros; elim a; intros; auto with arith real.
Qed.

(**********)
Lemma first_n :
forall N n : nat,
N >= 2 ->
0 < n ->
n < N ->
(0 < alpha)%R /\ (alpha < 1)%R ->
(frac_part_n_alpha (first N) <= frac_part_n_alpha n)%R.