proposition: Lemma Ratan_CVU : CVU (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N) ps_atan 0 (mkposreal 1 Rlt_0_1).
proof: Proof. intros eps ep; destruct (Ratan_CVU' eps ep) as [N Pn]. exists N; intros n x nN b_y. case (Rtotal_order 0 x) as [xgt0 | [x0 | x0]]. - assert (Boule (/2) posreal_half x). { revert b_y; unfold Boule; simpl; intros b_y; apply Rabs_def2 in b_y. destruct b_y; unfold Boule; simpl; apply Rabs_def1; lra. } apply Pn; assumption. - rewrite <- x0, ps_atan0_0. rewrite <- (sum_eq (fun _ => 0)), sum_cte, Rmult_0_l, Rminus_0_r, Rabs_pos_eq. + assumption. + apply Rle_refl. + intros i _; unfold tg_alt, Ratan_seq, Rdiv; rewrite Nat.add_comm; simpl. solve[rewrite !Rmult_0_l, Rmult_0_r; auto]. - replace (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) with (-(ps_atan (-x) - sum_f_R0 (tg_alt (Ratan_seq (-x))) n)). { rewrite Rabs_Ropp. assert (Boule (/2) posreal_half (-x)). { revert b_y; unfold Boule; simpl; intros b_y; apply Rabs_def2 in b_y. destruct b_y; unfold Boule; simpl; apply Rabs_def1; lra. } apply Pn; assumption. } unfold Rminus; rewrite ps_atan_opp, Ropp_plus_distr, sum_Ratan_seq_opp. rewrite !Ropp_involutive; reflexivity. Qed.
imports: Lra Rbase PSeries_reg Rtrigo1 Rtrigo_facts Ranalysis_reg Rfunctions AltSeries Rseries SeqProp Ranalysis5 SeqSeries PartSum Lia