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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope Z_scope.
Local Open Scope R_scope.

(* Specifications: *)
Lemma minus_neq_O : forall n i:nat, (i < n)%nat -> (n - i)%nat <> 0%nat.
Lemma le_minusni_n : forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat.
Lemma lt_minus_O_lt : forall m n:nat, (m < n)%nat -> (0 < n - m)%nat.
Lemma even_odd_cor :
forall n:nat, exists p : nat, n = (2 * p)%nat \/ n = S (2 * p).
(* 2m <= 2n => m<=n *)
Lemma le_double : forall m n:nat, (2 * m <= 2 * n)%nat -> (m <= n)%nat.
(** Here, we have the euclidian division *)
(** This lemma is used in the proof of sin_eq_0 : (sin x)=0<->x=kPI *)
Lemma euclidian_division :
forall x y:R,
y <> 0 ->
exists k : Z, (exists r : R, x = IZR k * y + r /\ 0 <= r < Rabs y).
Lemma tech8 : forall n i:nat, (n <= S n + i)%nat.