Qed.

(* Specifications: *)
(**********)
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)).
(**********)
Lemma P2 : forall n : nat, (frac_part_n_alpha n < 1)%R.
intro; unfold frac_part_n_alpha in |- *; elim (base_fp (INR n * alpha));
auto with arith real.
(**********)
Lemma en_plus : forall a : nat, a + 0 = a.
intro; omega.
(**********)
Lemma lt_plus_minus : forall n m p : nat, m < n -> n < m + p -> n - m < p.
intros; omega.
(**********)
Lemma le_plus_min : forall n m p : nat, n <= m + p -> n - m <= p.
intros; omega.
(**********)
Lemma lt_n_minus_plus : forall n m p : nat, n < m - p -> n + p < m.
intros; omega.
(**********)
Lemma lt_plus : forall n m : nat, 0 < m -> n < n + m.
intros; omega.
(**********)
Lemma lt_minus1 : forall n m : nat, n < m -> n - m = 0.
intros; omega.
(**********)
Lemma lt_minus2 : forall n m : nat, n < m -> 0 < m - n.
intros; omega.
(**********)
Lemma lt_minus_not : forall n m : nat, n < m -> m - n <> 0.
intros; omega.
(**********)
Lemma lt_O_plus : forall n m : nat, 0 < n -> 0 < n + m.
intros; omega.
(**********)
Lemma lt_O_plus_eq : forall n m : nat, 0 < n -> n + m <> 0.
intros; omega.
(**********)
Lemma lt_minus_p : forall n m p : nat, n < m -> n - p < m.
intros; omega.
(**********)
Lemma contra_lt_minus_p : forall n m p : nat, n < m - p -> n < m.
intros; omega.
(**********)
Lemma lt_not_eq : forall n m : nat, n < m -> n <> m.
intros; omega.
(**********)
Lemma le_minus_plus : forall n m p : nat, n - m <= p -> n <= p + m.
intros; omega.
(**********)
Lemma lt_minus_plus : forall n m p : nat, n - m < p -> n < p + m.
intros; omega.
(**********)
Lemma not_gt_le : forall n m : nat, ~ n > m -> n <= m.
intros; omega.
(**********)
Lemma tech_lt : forall n m p q : nat, n < m -> m - 1 + p < q -> n + p < q.
intros; omega.
(**********)
Lemma lt_reg_minus : forall n m p : nat, n < m -> p <= n -> n - p < m - p.
intros; omega.
(**********)
Lemma simpl_minus_plus :
forall a b c : nat, b <= a -> a <= c -> a - b + (c - a) = c - b.
(**********)
Lemma arith_2_0 : forall n : nat, n >= 2 -> 0 < n.
intros; omega.
(**********)
Lemma ge_trans : forall n m p : nat, n >= p -> p >= m -> n >= m.
intros; omega.
(**********)
Lemma tech_inter31b :
forall l n x m : nat, l < n -> n <= x -> x < m -> x - n + l < m.