Library Flocq.Prop.Fprop_Sterbenz

This file is part of the Flocq formalization of floating-point arithmetic in Coq:

Copyright (C) 2010-2011 Sylvie Boldo
Copyright (C) 2010-2011 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

Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_generic_fmt.
Require Import Fcalc_ops.

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).

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.

End Fprop_Sterbenz.