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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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(**          Definition of the limit                     *)
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Local Open Scope R_scope.

(* Specifications: *)
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(** *   Calculus               *)
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Lemma eps2_Rgt_R0 : forall eps:R, eps > 0 -> eps * / 2 > 0.
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Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps.
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Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2.
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Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps.
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Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps.
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Lemma prop_eps : forall r:R, (forall eps:R, eps > 0 -> r < eps) -> r <= 0.
Definition mul_factor (l l':R) := / (1 + (Rabs l + Rabs l')).