proposition: Lemma cos_ge_0_3PI2 : forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x.
proof: Proof. intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1); unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x) by ring. generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1; generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1; intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1). rewrite Rplus_opp_r. intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3; generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3; intro H3; generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3). replace (2 * PI + - (3 * (PI / 2))) with (PI / 2) by field. intro H4; apply (cos_ge_0 (2 * PI - x) (Rlt_le (- (PI / 2)) (2 * PI - x) (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4). Qed.
imports: Rbase Rfunctions SeqSeries Rtrigo_fun Rtrigo_def Rtrigo_alt Cos_rel Cos_plus ZArith_base Zcomplements Lia Lra Ranalysis1 Rsqrt_def PSeries_reg