Library Stdlib.Reals.Cos_plus

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Require Import Cos_rel.
Require Import Lia Lra.
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).

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.