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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.
simple induction N; intro.
absurd (0 >= 2).
inversion H.
auto with arith real.
intro X; case (N_classic n); intro.
elim s; intro y.
rewrite y; intro; absurd (1 >= 2); inversion H; inversion H1;
auto with arith real.
rewrite y; intro; exists 1; split; auto with arith real.
split; auto with arith real.
intros; replace m with 1.
apply tech_total_order.
apply sym_equal; elim H0; intro.
inversion H1; auto with arith real.
intros.
elim (X g); intros.
cut
({(frac_part_n_alpha x <= frac_part_n_alpha n)%R} +
{~ (frac_part_n_alpha x <= frac_part_n_alpha n)%R}).
intro X0; elim X0; intro.
exists x; split.
elim p; intros; auto with arith real.
split.
elim p; intros.
elim H1; intros; auto with arith real.
intros; elim p; intros; elim H2; intros; cut (0 < m /\ m < n \/ m = n).
intro; elim H5; intro.
elim (H4 m H6); auto with arith real.
apply prop_alpha.
rewrite H6; unfold ordre_total in |- *; intro; auto with arith real.
elim H0; intros; cut (0 < m /\ m < n \/ 0 < m /\ m = n).
intro; elim H7; intros.
left; auto with arith real.
elim H8; intros; right; auto with arith real.
cut (0 < m /\ (m < n \/ m = n)).
intro; elim H7; intros; elim H9; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H6; inversion H6; auto with arith real.
exists n; split; auto with arith real.
split; auto with arith real.
intros; cut (0 < m /\ m < n \/ m = n).
intro; elim H1.
intro; elim p; intros; elim H4; intros; unfold ordre_total in |- *; intro.
cut (frac_part_n_alpha x <= frac_part_n_alpha m)%R.
cut (frac_part_n_alpha x > frac_part_n_alpha n)%R.
intros; unfold Rgt in H8; cut (frac_part_n_alpha n < frac_part_n_alpha m)%R.
intro; apply (Rlt_le (frac_part_n_alpha n) (frac_part_n_alpha m) H10).
apply
(Rlt_le_trans (frac_part_n_alpha n) (frac_part_n_alpha x)
(frac_part_n_alpha m) H8 H9).
apply (Rnot_le_lt (frac_part_n_alpha x) (frac_part_n_alpha n) b).
elim (H6 m H2).
intro; apply (Rlt_le (frac_part_n_alpha x) (frac_part_n_alpha m) H8).
intro; rewrite H8; unfold Rle in |- *; right; auto with arith real.
apply prop_alpha.
intro; rewrite H2; apply tech_total_order.
elim H0; intros.
cut (0 < m /\ m < n \/ 0 < m /\ m = n).
intro; elim H3; intro.
left; auto with arith real.
elim H4; intros.
right; auto with arith real.
cut (0 < m /\ (m < n \/ m = n)).
intro; elim H3; intros.
elim H5; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H2; inversion H2; auto with arith real.
apply (Rle_dec (frac_part_n_alpha x) (frac_part_n_alpha n)).
Qed.
simple induction N; intro.
absurd (0 >= 2).
inversion H.
auto with arith real.
intro X; case (N_classic n); intro.
elim s; intro y.
rewrite y; intro; absurd (1 >= 2); inversion H; inversion H1;
auto with arith real.
rewrite y; intro; exists 1; split; auto with arith real.
split; auto with arith real.
intros; replace m with 1.
apply tech_total_order.
apply sym_equal; elim H0; intro.
inversion H1; auto with arith real.
intros.
elim (X g); intros.
cut
({(frac_part_n_alpha x <= frac_part_n_alpha n)%R} +
{~ (frac_part_n_alpha x <= frac_part_n_alpha n)%R}).
intro X0; elim X0; intro.
exists n; split; auto with arith real.
split; auto with arith real.
intros; cut (0 < m /\ m < n \/ m = n).
intro; elim H1.
intro; elim p; intros; elim H4; intros; unfold ordre_total in |- *; intro.
cut (frac_part_n_alpha m <= frac_part_n_alpha x)%R.
intro;
apply
(Rle_trans (frac_part_n_alpha m) (frac_part_n_alpha x)
(frac_part_n_alpha n) H8 a).
elim (H6 m H2).
intro; apply (Rlt_le (frac_part_n_alpha m) (frac_part_n_alpha x) H8).
intro; rewrite H8; unfold Rle in |- *; right; auto with arith real.
apply prop_alpha.
intro; rewrite H2; apply tech_total_order.
elim H0; intros.
cut (0 < m /\ m < n \/ 0 < m /\ m = n).
intro; elim H3; intro.
left; auto with arith real.
elim H4; intros.
right; auto with arith real.
cut (0 < m /\ (m < n \/ m = n)).
intro; elim H3; intros.
elim H5; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H2; inversion H2; auto with arith real.
exists x; split.
elim p; intros; auto with arith real.
split.
elim p; intros.
elim H1; intros; auto with arith real.
intros; elim p; intros; elim H2; intros; cut (0 < m /\ m < n \/ m = n).
intro; elim H5; intro.
elim (H4 m H6); auto with arith real.
apply prop_alpha.
rewrite H6; unfold ordre_total in |- *; intro.
cut (frac_part_n_alpha n < frac_part_n_alpha x)%R.
intro; apply (Rlt_le (frac_part_n_alpha n) (frac_part_n_alpha x) H8).
apply (Rnot_le_lt (frac_part_n_alpha x) (frac_part_n_alpha n) b).
elim H0; intros; cut (0 < m /\ m < n \/ 0 < m /\ m = n).
intro; elim H7; intros.
left; auto with arith real.
elim H8; intros; right; auto with arith real.
cut (0 < m /\ (m < n \/ m = n)).
intro; elim H7; intros; elim H9; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H6; inversion H6; auto with arith real.
apply (Rle_dec (frac_part_n_alpha x) (frac_part_n_alpha n)).
Qed.
(**********)
Definition first (N : nat) :=
match N_classic N with
| inright p => match exist_first N p return nat with
| existT a b => a
end
| _ => 0
end.
(**********)
Definition last (N : nat) :=
match N_classic N with
| inright p => match exist_last N p return nat with
| existT a b => a
end
| _ => 0
end.
(*********************************************************)
(*       Definition of the next point                    *)
(*********************************************************)
simple induction N.
right; intros; absurd (m < 0); auto with arith real.
intros n X; elim X; intro y;
cut
({(0 <= M)%R /\ (M < frac_part_n_alpha n)%R} +
{(frac_part_n_alpha n <= M)%R /\ (M < 1)%R}).
intro X0; elim y; intros x y0.
left; elim X0; intro y1.
cut
({(frac_part_n_alpha n < frac_part_n_alpha x)%R} +
{(frac_part_n_alpha x <= frac_part_n_alpha n)%R}).
intro X1; elim X1; intro y2.
split with n; split.
elim y1; intros.
generalize (Rle_lt_trans 0 M (frac_part_n_alpha n) H1 H2).
intro; apply neq_O_lt.
cut (0%R <> frac_part_n_alpha n).
intro; unfold frac_part_n_alpha in H4;
generalize (R0_fp_O (INR n * alpha) H4); intro; generalize (sym_not_eq H5);
clear H5; intro; generalize (Rmult_neq_0_reg (INR n) alpha H5);
intro; elim H6; intros.
apply sym_not_equal.
apply (not_INR_O n); auto with arith real.
apply (Rlt_dichotomy_converse 0 (frac_part_n_alpha n)).
left; assumption.
split; auto with arith real.
split; elim y1; auto with arith real.
intros; cut (0 <= m /\ m < n \/ m = n).
intro; elim H6; intro.
elim H7; intros.
elim y0; intros.
elim H11; intros.
elim H13; intros.
generalize (H15 m H8 H9 H5); intro;
generalize (Rge_le (frac_part_n_alpha m) (frac_part_n_alpha x) H16);
intro;
generalize
(Rlt_le_trans (frac_part_n_alpha n) (frac_part_n_alpha x)
(frac_part_n_alpha m) y2 H17); intro;
generalize (Rlt_le (frac_part_n_alpha n) (frac_part_n_alpha m) H18);
intro; apply (Rle_ge (frac_part_n_alpha n) (frac_part_n_alpha m) H19).
rewrite H7; unfold Rge in |- *; right; auto with arith real.
cut (0 <= m /\ m < n \/ 0 <= m /\ m = n).
intro; elim H6; intros.
left; auto with arith real.
elim H7; intros; right; auto with arith real.
cut (0 <= m /\ (m < n \/ m = n)).
intro; elim H6; intros; elim H8; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H4; inversion H4; auto with arith real.
exists x; elim y0; intros.
split; auto with arith real.
split; elim H2; intros; auto with arith real.
split; elim H4; intros; auto with arith real.
cut (0 <= m /\ m < n \/ m = n).
intro; elim H10; intro.
elim H11; auto with arith real.
rewrite H11; apply Rle_ge; auto with arith real.
cut (0 <= m /\ m < n \/ 0 <= m /\ m = n).
intro; elim H10; intros.
left; auto with arith real.
elim H11; intros; right; auto with arith real.
cut (0 <= m /\ (m < n \/ m = n)).
intro; elim H10; intros; elim H12; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H8; inversion H8; auto with arith real.
generalize (Rle_dec (frac_part_n_alpha x) (frac_part_n_alpha n)); intro X1;
elim X1; intro y2.
right; auto with arith real.
left; generalize (Rnot_le_lt (frac_part_n_alpha x) (frac_part_n_alpha n) y2);
unfold Rgt in |- *; auto with arith real.
exists x; elim y0; intros.
split; auto with arith real.
split; elim H2; intros; auto with arith real.
split; elim H4; intros; auto with arith real.
cut (0 <= m /\ m < n \/ m = n).
intro; elim H10; intro.
elim H11; intros.
apply (H6 m H7 H13 H9).
elim y1; intros; rewrite H11 in H9; absurd (frac_part_n_alpha n > M)%R.
inversion H12.
unfold Rgt in |- *; apply (Rlt_asym (frac_part_n_alpha n) M);
auto with arith real.
rewrite H14; unfold Rgt in |- *; apply Rlt_irrefl.
auto with arith real.
cut (0 <= m /\ m < n \/ 0 <= m /\ m = n).
intro; elim H10; intros.
left; auto with arith real.
elim H11; intros; right; auto with arith real.
cut (0 <= m /\ (m < n \/ m = n)).
intro; elim H10; intros; elim H12; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H8; inversion H8; auto with arith real.
apply (inser_trans_R 0 M 1 (frac_part_n_alpha n)); auto with arith real.
intro X0; elim X0; intro y0.
left; exists n; split.
elim y0; intros.
generalize (Rle_lt_trans 0 M (frac_part_n_alpha n) H1 H2).
intro; apply neq_O_lt.
cut (0%R <> frac_part_n_alpha n).
intro; unfold frac_part_n_alpha in H4;
generalize (R0_fp_O (INR n * alpha) H4); intro; generalize (sym_not_eq H5);
clear H5; intro; generalize (Rmult_neq_0_reg (INR n) alpha H5);
intro; elim H6; intros.
apply sym_not_equal.
apply (not_INR_O n); auto with arith real.
apply (Rlt_dichotomy_converse 0 (frac_part_n_alpha n)).
left; assumption.
split; auto with arith real.
split; elim y0; auto with arith real.
intros; cut (0 <= m /\ m < n \/ m = n).
intro; elim H6; intro.
elim H7; intros.
absurd (frac_part_n_alpha m > M)%R.
generalize (y m H8 H9); intro; elim H10; intros.
inversion H12.
unfold Rgt in |- *; apply (Rlt_asym (frac_part_n_alpha m) M);
auto with arith real.
rewrite H13; unfold Rgt in |- *; apply Rlt_irrefl.
auto with arith real.
rewrite H7; unfold Rge in |- *; right; auto with arith real.
cut (0 <= m /\ m < n \/ 0 <= m /\ m = n).
intro; elim H6; intros.
left; auto with arith real.
elim H7; intros; right; auto with arith real.
cut (0 <= m /\ (m < n \/ m = n)).
intro; elim H6; intros; elim H8; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H4; inversion H4; auto with arith real.
right; intros; cut (0 <= m /\ m < n \/ m = n).
intro; elim H3; intro.
elim H4; intros.
apply (y m H5 H6).
split.
unfold frac_part_n_alpha in |- *; generalize (base_fp (INR m * alpha)); intro;
elim H5; intros.
apply (Rge_le (frac_part (INR m * alpha)) 0 H6).
rewrite H4; elim y0; auto with arith real.
cut (0 <= m /\ m < n \/ 0 <= m /\ m = n).
intro; elim H3; intros.
left; auto with arith real.
elim H4; intros; right; auto with arith real.
cut (0 <= m /\ (m < n \/ m = n)).
intro; elim H3; intros; elim H5; intro.
left; auto with arith real.
right; auto with arith real.
split; auto with arith real.
unfold lt in H2; inversion H2; auto with arith real.
apply (inser_trans_R 0 M 1 (frac_part_n_alpha n)); auto with arith real.
Qed.
intro; unfold frac_part_n_alpha in |- *; elim (base_fp (INR n * alpha));
intros; apply (Rge_le (frac_part (INR n * alpha)) 0 H).
Qed.

(**********)
Lemma P2 : forall n : nat, (frac_part_n_alpha n < 1)%R.