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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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(*                               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.
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); unfold ordre_total in |- *; intros; elim a; intros.
elim H4; intros.
generalize (H6 n); intros.
elim H7.
intro; apply Rlt_le; auto with arith real.
unfold Rle in |- *; auto with arith real.
split; auto with arith real.
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); unfold ordre_total in |- *; intros; elim a; intros.
elim H4; intros.
generalize (H6 n); intros.
elim H7.
intro; apply Rlt_le; auto with arith real.
unfold Rle in |- *; auto with arith real.
split; auto with arith real.
auto with arith real.
Qed.
intros; split; unfold ordre_total in |- *; intro.
apply (first_n N k H H0 H1 H2).
apply (last_n N k H H0 H1 H2).
Qed.

(**********)
Lemma le_first_last :
forall N : nat, N >= 2 -> ordre_total (first N) (last N).