(** This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/  Copyright (C) 2010-2013 Sylvie Boldo #<br /># Copyright (C) 2010-2013 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. *)
(** * Error of the rounded-to-nearest addition is representable. *)
Section Fprop_plus_error.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Section round_repr_same_exp.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem round_repr_same_exp :
forall m e,
exists m',
round beta fexp rnd (F2R (Float beta m e)) = F2R (Float beta m' e).