(** This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/  Copyright (C) 2011-2015 Sylvie Boldo #<br /># Copyright (C) 2011-2015 Catherine Lelay #<br /># Copyright (C) 2011-2015 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
(** This file describes compactness results: finite covering of opens, finite covering based on a gauge function, specific instances for 1D and 2D spaces. *)
(* This enables a compatibility with Coq 8.4, which didn't have the right lemmas Rplus_lt_reg_l and Rplus_lt_reg_r.  So this Import should disappear in the future.  A better solution would be to have the fix in a specific compatibility file, rather than in Rcomplements.v. *)
Open Scope R_scope.

Lemma completeness_any :
forall P : R -> Prop,
( forall x y, x <= y -> P y -> P x ) ->
forall He : (exists x, P x),
forall Hb : (bound P),
forall x, x < proj1_sig (completeness _ Hb He) ->
~ ~ P x.