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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 nat_scope.
Local Open Scope R_scope.
Definition Majxy (x y:R) (n:nat) : R :=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n).

(* Specifications: *)
Lemma Majxy_cv_R0 : forall x y:R, Un_cv (Majxy x y) 0.
Lemma reste1_maj :
forall (x y:R) (N:nat),
(0 < N)%nat -> Rabs (Reste1 x y N) <= Majxy x y (pred N).
Lemma reste2_maj :
forall (x y:R) (N:nat), (0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N.
Lemma reste1_cv_R0 : forall x y:R, Un_cv (Reste1 x y) 0.
Lemma reste2_cv_R0 : forall x y:R, Un_cv (Reste2 x y) 0.
Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0.
Theorem cos_plus : forall x y:R, cos (x + y) = cos x * cos y - sin x * sin y.