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(* Copyright 2006 Milad Niqui                                           *)
(* This file is distributed under the terms of the                      *)
(* GNU Lesser General Public License Version 2.1                        *)
(* A copy of the license can be found at                                *)
(*                  <http://www.gnu.org/licenses/lgpl-2.1.html>         *)
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(* This file contains various properties of R that are not in the standard library. *)

(* Specifications: *)
Lemma Rlt_stepl:forall x y z, Rlt x y -> x=z -> Rlt z y.
Lemma Rle_zero_Rminus : forall r1 r2:R , 0 <= r1-r2  -> r2 <= r1.
Lemma Rle_Rminus_zero : forall r1 r2:R , r2 <= r1 -> 0 <= r1-r2.
Lemma Rlt_Rminus_zero : forall r1 r2:R , r2 < r1 -> 0 < r1-r2.
Lemma Rlt_not_eq': forall r1 r2 : R, r1 < r2 -> r2 <> r1.
Lemma Rmult_reg_nonzero_r: forall r1 r2 : R, r1 * r2 <> 0 -> r2 <> 0.
Lemma Rmult_reg_nonzero_l: forall r1 r2 : R, r1 * r2 <> 0 -> r1 <> 0.
Lemma Rlt_Ropp_pos: forall r : R, r < 0 -> 0 < - r.
Lemma Rlt_mult_neg_neg: forall r1 r2 : R, r1<0 -> r2<0 -> 0 < r1 * r2.
Definition Rinv_pos:= Rinv_0_lt_compat.