(** 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. *)
(** * Sterbenz conditions for exact subtraction *)
Section Fprop_Sterbenz.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { monotone_exp : Monotone_exp fexp }.
Notation format := (generic_format beta fexp).

(* Specifications: *)
Theorem generic_format_plus :
forall x y,
format x -> format y ->
(Rabs (x + y) < bpow (Zmin (ln_beta beta x) (ln_beta beta y)))%R ->
format (x + y)%R.
Theorem generic_format_plus_weak :
forall x y,
format x -> format y ->
(Rabs (x + y) <= Rmin (Rabs x) (Rabs y))%R ->
format (x + y)%R.
Lemma sterbenz_aux :
forall x y, format x -> format y ->
(y <= x <= 2 * y)%R ->
format (x - y)%R.
Theorem sterbenz :
forall x y, format x -> format y ->
(y / 2 <= x <= 2 * y)%R ->
format (x - y)%R.
Theorem generic_format_plus :
forall x y,
format x -> format y ->
(Rabs (x + y) < bpow (Zmin (ln_beta beta x) (ln_beta beta y)))%R ->
format (x + y)%R.
Theorem generic_format_plus_weak :
forall x y,
format x -> format y ->
(Rabs (x + y) <= Rmin (Rabs x) (Rabs y))%R ->
format (x + y)%R.
Lemma sterbenz_aux :
forall x y, format x -> format y ->
(y <= x <= 2 * y)%R ->
format (x - y)%R.
Theorem sterbenz :
forall x y, format x -> format y ->
(y / 2 <= x <= 2 * y)%R ->
format (x - y)%R.