(** 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. *)
(** * Remainder of the division and square root are in the FLX format *)
Section Fprop_divsqrt_error.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable prec : Z.

(* Specifications: *)
Theorem generic_format_plus_prec:
forall fexp, (forall e, (fexp e  <= e - prec)%Z) ->
forall x y (fx fy: float beta),
(x = F2R fx)%R -> (y = F2R fy)%R -> (Rabs (x+y) < bpow (prec+Fexp fx))%R -> (Rabs (x+y) < bpow (prec+Fexp fy))%R
-> generic_format beta fexp (x+y)%R.
Theorem generic_format_plus_prec:
forall fexp, (forall e, (fexp e  <= e - prec)%Z) ->
forall x y (fx fy: float beta),
(x = F2R fx)%R -> (y = F2R fy)%R -> (Rabs (x+y) < bpow (prec+Fexp fx))%R -> (Rabs (x+y) < bpow (prec+Fexp fy))%R
-> generic_format beta fexp (x+y)%R.