(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \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)         *)
(************************************************************************)
Local Open Scope R_scope.
(** Tools *)
Definition pos_half_prf : 0 < /2.

(* The following two lemmas are general purposes about alternated series. They do not belong here. *)
Lemma Alt_first_term_bound :forall f l N n,
Un_decreasing f -> Un_cv f 0 ->
Un_cv (sum_f_R0 (tg_alt f)) l ->
(N <= n)%nat ->
R_dist (sum_f_R0 (tg_alt f) n) l <= f N.