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This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/

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.

Require Import Psatz.
Require Import Raux Defs Float_prop Generic_fmt.
Require Import FIX FLX FLT Ulp Operations.
Require Import Relative.


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 }.

Lemma round_repr_same_exp :
  forall m e,
  exists m',
  round beta fexp rnd (F2R (Float beta m e)) = F2R (Float beta m' e).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m e : Z,
exists m' : Z,
  round beta fexp rnd (F2R {| Fnum := m; Fexp := e |}) =
  F2R {| Fnum := m'; Fexp := e |}
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall m e : Z,
exists m' : Z,
  round beta fexp rnd (F2R {| Fnum := m; Fexp := e |}) =
  F2R {| Fnum := m'; Fexp := e |}
intros m e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
exists m' : Z,
  round beta fexp rnd (F2R {| Fnum := m; Fexp := e |}) =
  F2R {| Fnum := m'; Fexp := e |}
set (e' := cexp beta fexp (F2R (Float beta m e))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
exists m' : Z,
  round beta fexp rnd (F2R {| Fnum := m; Fexp := e |}) =
  F2R {| Fnum := m'; Fexp := e |}
unfold round, scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow
                 (-
                  cexp beta fexp
                    (F2R {| Fnum := m; Fexp := e |})));
    Fexp := cexp beta fexp
              (F2R {| Fnum := m; Fexp := e |}) |} =
  F2R {| Fnum := m'; Fexp := e |} fold e'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
destruct (Zle_or_lt e' e) as [He|He].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
exists m.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R
  {|
  Fnum := rnd
            (F2R {| Fnum := m; Fexp := e |} *
             bpow (- e'));
  Fexp := e' |} = F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
unfold F2R at 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R
  {|
  Fnum := rnd
            (IZR (Fnum {| Fnum := m; Fexp := e |}) *
             bpow (Fexp {| Fnum := m; Fexp := e |}) *
             bpow (- e'));
  Fexp := e' |} = F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |} simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R
  {|
  Fnum := rnd (IZR m * bpow e * bpow (- e'));
  Fexp := e' |} = F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite Rmult_assoc, <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R
  {|
  Fnum := rnd (IZR m * bpow (e + - e'));
  Fexp := e' |} = F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite <- IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R
  {|
  Fnum := rnd (IZR m * IZR (beta ^ (e + - e')));
  Fexp := e' |} = F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(0 <= e + - e')%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |} 2: omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R
  {|
  Fnum := rnd (IZR m * IZR (beta ^ (e + - e')));
  Fexp := e' |} = F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite <- mult_IZR, Zrnd_IZR...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
F2R {| Fnum := m * beta ^ (e + - e'); Fexp := e' |} =
F2R {| Fnum := m; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR
   (Fnum
      {| Fnum := m * beta ^ (e + - e'); Fexp := e' |}) *
 bpow
   (Fexp
      {| Fnum := m * beta ^ (e + - e'); Fexp := e' |}))%R =
(IZR (Fnum {| Fnum := m; Fexp := e |}) *
 bpow (Fexp {| Fnum := m; Fexp := e |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |} simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR (m * beta ^ (e + - e')) * bpow e')%R =
(IZR m * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite mult_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR m * IZR (beta ^ (e + - e')) * bpow e')%R =
(IZR m * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR m * (IZR (beta ^ (e + - e')) * bpow e'))%R =
(IZR m * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR m * (bpow (e + - e') * bpow e'))%R =
(IZR m * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(0 <= e + - e')%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |} 2: omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR m * (bpow (e + - e') * bpow e'))%R =
(IZR m * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(IZR m * bpow (e + - e' + e'))%R = (IZR m * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
apply (f_equal (fun v => IZR m * bpow v)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e' <= e)%Z
(e + - e' + e')%Z = e
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
exists m' : Z,
  F2R
    {|
    Fnum := rnd
              (F2R {| Fnum := m; Fexp := e |} *
               bpow (- e'));
    Fexp := e' |} = F2R {| Fnum := m'; Fexp := e |}
exists ((rnd (IZR m * bpow (e - e'))) * Zpower beta (e' - e))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
F2R
  {|
  Fnum := rnd
            (F2R {| Fnum := m; Fexp := e |} *
             bpow (- e'));
  Fexp := e' |} =
F2R
  {|
  Fnum := rnd (IZR m * bpow (e - e')) *
          beta ^ (e' - e);
  Fexp := e |}
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR
   (Fnum
      {|
      Fnum := rnd
                (IZR (Fnum {| Fnum := m; Fexp := e |}) *
                 bpow
                   (Fexp {| Fnum := m; Fexp := e |}) *
                 bpow (- e'));
      Fexp := e' |}) *
 bpow
   (Fexp
      {|
      Fnum := rnd
                (IZR (Fnum {| Fnum := m; Fexp := e |}) *
                 bpow
                   (Fexp {| Fnum := m; Fexp := e |}) *
                 bpow (- e'));
      Fexp := e' |}))%R =
(IZR
   (Fnum
      {|
      Fnum := rnd (IZR m * bpow (e - e')) *
              beta ^ (e' - e);
      Fexp := e |}) *
 bpow
   (Fexp
      {|
      Fnum := rnd (IZR m * bpow (e - e')) *
              beta ^ (e' - e);
      Fexp := e |}))%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR (rnd (IZR m * bpow e * bpow (- e'))) * bpow e')%R =
(IZR (rnd (IZR m * bpow (e - e')) * beta ^ (e' - e)) *
 bpow e)%R
rewrite mult_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR (rnd (IZR m * bpow e * bpow (- e'))) * bpow e')%R =
(IZR (rnd (IZR m * bpow (e - e'))) *
 IZR (beta ^ (e' - e)) * bpow e)%R
rewrite IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR (rnd (IZR m * bpow e * bpow (- e'))) * bpow e')%R =
(IZR (rnd (IZR m * bpow (e - e'))) * bpow (e' - e) *
 bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(0 <= e' - e)%Z 2: omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR (rnd (IZR m * bpow e * bpow (- e'))) * bpow e')%R =
(IZR (rnd (IZR m * bpow (e - e'))) * bpow (e' - e) *
 bpow e)%R
rewrite 2!Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR (rnd (IZR m * (bpow e * bpow (- e')))) * bpow e')%R =
(IZR (rnd (IZR m * bpow (e - e'))) *
 (bpow (e' - e) * bpow e))%R
rewrite <- 2!bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
(IZR (rnd (IZR m * bpow (e + - e'))) * bpow e')%R =
(IZR (rnd (IZR m * bpow (e - e'))) * bpow (e' - e + e))%R
apply (f_equal (fun v => _ * bpow v)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
m, e:Z
e':=cexp beta fexp (F2R {| Fnum := m; Fexp := e |}):Z
He:(e < e')%Z
e' = (e' - e + e)%Z
ring.
Qed.

End round_repr_same_exp.

Context { monotone_exp : Monotone_exp fexp }.
Notation format := (generic_format beta fexp).

Variable choice : Z -> bool.

Lemma plus_error_aux :
  forall x y,
  (cexp beta fexp x <= cexp beta fexp y)%Z ->
  format x -> format y ->
  format (round beta fexp (Znearest choice) (x + y) - (x + y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
forall x y : R,
(cexp beta fexp x <= cexp beta fexp y)%Z ->
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
forall x y : R,
(cexp beta fexp x <= cexp beta fexp y)%Z ->
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
intros x y.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
(cexp beta fexp x <= cexp beta fexp y)%Z ->
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
set (ex := cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
(ex <= cexp beta fexp y)%Z ->
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
set (ey := cexp beta fexp y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
(ex <= ey)%Z ->
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
intros He Hx Hy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
destruct (Req_dec (round beta fexp (Znearest choice) (x + y) - (x + y)) R0) as [H0|H0].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R = R0
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
rewrite H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R = R0
format R0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
set (mx := Ztrunc (scaled_mantissa beta fexp x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
set (my := Ztrunc (scaled_mantissa beta fexp y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
(* *)
assert (Hxy: (x + y)%R = F2R (Float beta (mx + my * beta ^ (ey - ex)) ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
(x + y)%R =
F2R
  {| Fnum := mx + my * beta ^ (ey - ex); Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
rewrite Hx, Hy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp y);
   Fexp := cexp beta fexp y |})%R =
F2R
  {| Fnum := mx + my * beta ^ (ey - ex); Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
fold mx my ex ey.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
(F2R {| Fnum := mx; Fexp := ex |} +
 F2R {| Fnum := my; Fexp := ey |})%R =
F2R
  {| Fnum := mx + my * beta ^ (ey - ex); Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
rewrite <- F2R_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
F2R
  (Fplus {| Fnum := mx; Fexp := ex |}
     {| Fnum := my; Fexp := ey |}) =
F2R
  {| Fnum := mx + my * beta ^ (ey - ex); Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
unfold Fplus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
F2R
  (let
   '(m1, m2, e) :=
    Falign {| Fnum := mx; Fexp := ex |}
      {| Fnum := my; Fexp := ey |} in
    {| Fnum := m1 + m2; Fexp := e |}) =
F2R
  {| Fnum := mx + my * beta ^ (ey - ex); Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y)) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
F2R
  (let
   '(m1, m2, e) :=
    if (ex <=? ey)%Z
    then (mx, (my * beta ^ (ey - ex))%Z, ex)
    else ((mx * beta ^ (ex - ey))%Z, my, ey) in
    {| Fnum := m1 + m2; Fexp := e |}) =
F2R
  {| Fnum := mx + my * beta ^ (ey - ex); Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
now rewrite Zle_imp_le_bool with (1 := He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
(* *)
rewrite Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
format
  (round beta fexp (Znearest choice)
     (F2R
        {|
        Fnum := mx + my * beta ^ (ey - ex);
        Fexp := ex |}) -
   F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |})
destruct (round_repr_same_exp (Znearest choice) (mx + my * beta ^ (ey - ex)) ex) as (mxy, Hxy').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
format
  (round beta fexp (Znearest choice)
     (F2R
        {|
        Fnum := mx + my * beta ^ (ey - ex);
        Fexp := ex |}) -
   F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |})
rewrite Hxy'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
format
  (F2R {| Fnum := mxy; Fexp := ex |} -
   F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |})
assert (H: (F2R (Float beta mxy ex) - F2R (Float beta (mx + my * beta ^ (ey - ex)) ex))%R =
  F2R (Float beta (mxy - (mx + my * beta ^ (ey - ex))) ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
format
  (F2R {| Fnum := mxy; Fexp := ex |} -
   F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |})
now rewrite <- F2R_minus, Fminus_same_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
format
  (F2R {| Fnum := mxy; Fexp := ex |} -
   F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |})
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
format
  (F2R
     {|
     Fnum := mxy - (mx + my * beta ^ (ey - ex));
     Fexp := ex |})
apply generic_format_F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(mxy - (mx + my * beta ^ (ey - ex)))%Z <> 0%Z ->
(cexp beta fexp
   (F2R
      {|
      Fnum := mxy - (mx + my * beta ^ (ey - ex));
      Fexp := ex |}) <= ex)%Z
intros _.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(cexp beta fexp
   (F2R
      {|
      Fnum := mxy - (mx + my * beta ^ (ey - ex));
      Fexp := ex |}) <= ex)%Z
apply monotone_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(mag beta
   (F2R
      {|
      Fnum := mxy - (mx + my * beta ^ (ey - ex));
      Fexp := ex |}) <= mag beta x)%Z
rewrite <- H, <- Hxy', <- Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(mag beta
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= mag beta x)%Z
apply mag_le_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(round beta fexp (Znearest choice) (x + y) - (x + y))%R <>
0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs x)%R
exact H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs x)%R
pattern x at 3 ; replace x with (-(y - (x + y)))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs (- (y - (x + y))))%R
rewrite Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
He:(ex <= ey)%Z
Hx:format x
Hy:format y
H0:(round beta fexp (Znearest choice) (x + y) -
 (x + y))%R <> R0
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hxy:(x + y)%R =
F2R
  {|
  Fnum := mx + my * beta ^ (ey - ex);
  Fexp := ex |}
mxy:Z
Hxy':round beta fexp (Znearest choice)
  (F2R
     {|
     Fnum := mx + my * beta ^ (ey - ex);
     Fexp := ex |}) =
F2R {| Fnum := mxy; Fexp := ex |}
H:(F2R {| Fnum := mxy; Fexp := ex |} -
 F2R
   {|
   Fnum := mx + my * beta ^ (ey - ex);
   Fexp := ex |})%R =
F2R
  {|
  Fnum := mxy - (mx + my * beta ^ (ey - ex));
  Fexp := ex |}
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs (y - (x + y)))%R
now apply (round_N_pt beta _ choice (x + y)).
Qed.

Error of the addition

Theorem plus_error :
  forall x y,
  format x -> format y ->
  format (round beta fexp (Znearest choice) (x + y) - (x + y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
forall x y : R,
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
forall x y : R,
format x ->
format y ->
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
intros x y Hx Hy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Hx:format x
Hy:format y
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
destruct (Zle_or_lt (cexp beta fexp x) (cexp beta fexp y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Hx:format x
Hy:format y
H:(cexp beta fexp x <= cexp beta fexp y)%Z
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Hx:format x
Hy:format y
H:(cexp beta fexp y < cexp beta fexp x)%Z
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
now apply plus_error_aux.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Hx:format x
Hy:format y
H:(cexp beta fexp y < cexp beta fexp x)%Z
format
  (round beta fexp (Znearest choice) (x + y) - (x + y))
rewrite Rplus_comm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Hx:format x
Hy:format y
H:(cexp beta fexp y < cexp beta fexp x)%Z
format
  (round beta fexp (Znearest choice) (y + x) - (y + x))
apply plus_error_aux ; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Hx:format x
Hy:format y
H:(cexp beta fexp y < cexp beta fexp x)%Z
(cexp beta fexp y <= cexp beta fexp x)%Z
now apply Zlt_le_weak.
Qed.

End Fprop_plus_error.

Section Fprop_plus_zero.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { exp_not_FTZ : Exp_not_FTZ fexp }.
Notation format := (generic_format beta fexp).

Section round_plus_eq_zero_aux.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Lemma round_plus_neq_0_aux :
  forall x y,
  (cexp beta fexp x <= cexp beta fexp y)%Z ->
  format x -> format y ->
  (0 < x + y)%R ->
  round beta fexp rnd (x + y) <> 0%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
(cexp beta fexp x <= cexp beta fexp y)%Z ->
format x ->
format y ->
(0 < x + y)%R -> round beta fexp rnd (x + y) <> 0%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
(cexp beta fexp x <= cexp beta fexp y)%Z ->
format x ->
format y ->
(0 < x + y)%R -> round beta fexp rnd (x + y) <> 0%R
intros x y He Hx Hy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
round beta fexp rnd (x + y) <> 0%R
destruct (mag beta (x + y)) as (exy, Hexy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(x + y)%R <> 0%R ->
(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
round beta fexp rnd (x + y) <> 0%R
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(x + y)%R <> 0%R ->
(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
round beta fexp rnd (x + y) <> 0%R
specialize (Hexy (Rgt_not_eq _ _ Hxy)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
round beta fexp rnd (x + y) <> 0%R
destruct (Zle_or_lt exy (fexp exy)) as [He'|He'].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
(* . *)
assert (H: (x + y)%R = F2R (Float beta (Ztrunc (x * bpow (- fexp exy)) +
  Ztrunc (y * bpow (- fexp exy))) (fexp exy))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite (subnormal_exponent beta fexp exy x He' Hx) at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
(F2R
   {|
   Fnum := Ztrunc (x * bpow (- fexp exy));
   Fexp := fexp exy |} + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite (subnormal_exponent beta fexp exy y He' Hy) at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
(F2R
   {|
   Fnum := Ztrunc (x * bpow (- fexp exy));
   Fexp := fexp exy |} +
 F2R
   {|
   Fnum := Ztrunc (y * bpow (- fexp exy));
   Fexp := fexp exy |})%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
now rewrite <- F2R_plus, Fplus_same_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
round beta fexp rnd
  (F2R
     {|
     Fnum := Ztrunc (x * bpow (- fexp exy)) +
             Ztrunc (y * bpow (- fexp exy));
     Fexp := fexp exy |}) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |} <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
format
  (F2R
     {|
     Fnum := Ztrunc (x * bpow (- fexp exy)) +
             Ztrunc (y * bpow (- fexp exy));
     Fexp := fexp exy |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite <- H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
(x + y)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
format
  (F2R
     {|
     Fnum := Ztrunc (x * bpow (- fexp exy)) +
             Ztrunc (y * bpow (- fexp exy));
     Fexp := fexp exy |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
format
  (F2R
     {|
     Fnum := Ztrunc (x * bpow (- fexp exy)) +
             Ztrunc (y * bpow (- fexp exy));
     Fexp := fexp exy |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
apply generic_format_F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
(Ztrunc (x * bpow (- fexp exy)) +
 Ztrunc (y * bpow (- fexp exy)))%Z <> 0%Z ->
(cexp beta fexp
   (F2R
      {|
      Fnum := Ztrunc (x * bpow (- fexp exy)) +
              Ztrunc (y * bpow (- fexp exy));
      Fexp := fexp exy |}) <= fexp exy)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
intros _.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
(cexp beta fexp
   (F2R
      {|
      Fnum := Ztrunc (x * bpow (- fexp exy)) +
              Ztrunc (y * bpow (- fexp exy));
      Fexp := fexp exy |}) <= fexp exy)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite <- H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
(cexp beta fexp (x + y) <= fexp exy)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
(fexp (mag beta (x + y)) <= fexp exy)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
rewrite mag_unique with (1 := Hexy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(exy <= fexp exy)%Z
H:(x + y)%R =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- fexp exy)) +
          Ztrunc (y * bpow (- fexp exy));
  Fexp := fexp exy |}
(fexp exy <= fexp exy)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
apply Z.le_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
round beta fexp rnd (x + y) <> 0%R
(* . *)
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
False
elim Rle_not_lt with (1 := round_le beta _ rnd _ _ (proj1 Hexy)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
(round beta fexp rnd (Rabs (x + y)) <
 round beta fexp rnd (bpow (exy - 1)))%R
rewrite (Rabs_pos_eq _ (Rlt_le _ _ Hxy)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
(round beta fexp rnd (x + y) <
 round beta fexp rnd (bpow (exy - 1)))%R
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
(0 < round beta fexp rnd (bpow (exy - 1)))%R
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
(0 < bpow (exy - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
format (bpow (exy - 1))
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
format (bpow (exy - 1))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
(fexp (exy - 1 + 1) <= exy - 1)%Z
apply Zlt_succ_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
He:(cexp beta fexp x <= cexp beta fexp y)%Z
Hx:format x
Hy:format y
Hxy:(0 < x + y)%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x + y) < bpow exy)%R
He':(fexp exy < exy)%Z
H:round beta fexp rnd (x + y) = 0%R
(fexp (exy - 1 + 1) < Z.succ (exy - 1))%Z
now rewrite (Zsucc_pred exy) in He'.
Qed.

End round_plus_eq_zero_aux.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

rnd(x+y)=0 -> x+y = 0 provided this is not a FTZ format

Theorem round_plus_neq_0 :
  forall x y,
  format x -> format y ->
  (x + y <> 0)%R ->
  round beta fexp rnd (x + y) <> 0%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
(x + y)%R <> 0%R -> round beta fexp rnd (x + y) <> 0%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
(x + y)%R <> 0%R -> round beta fexp rnd (x + y) <> 0%R
intros x y Hx Hy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
round beta fexp rnd (x + y) <> 0%R
destruct (Rle_or_lt 0 (x + y)) as [H1|H1].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
(* . *)
destruct (Zle_or_lt (cexp beta fexp x) (cexp beta fexp y)) as [H2|H2].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp x <= cexp beta fexp y)%Z
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
apply round_plus_neq_0_aux...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp x <= cexp beta fexp y)%Z
(0 < x + y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
rewrite Rplus_comm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
round beta fexp rnd (y + x) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
apply round_plus_neq_0_aux ; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
(cexp beta fexp y <= cexp beta fexp x)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
(0 < y + x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
now apply Zlt_le_weak.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(0 <= x + y)%R
H2:(cexp beta fexp y < cexp beta fexp x)%Z
(0 < y + x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (x + y) <> 0%R
(* . *)
rewrite <- (Ropp_involutive (x + y)), Ropp_plus_distr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp rnd (- (- x + - y)) <> 0%R
rewrite round_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
(- round beta fexp (Zrnd_opp rnd) (- x + - y))%R <>
0%R
apply Ropp_neq_0_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
round beta fexp (Zrnd_opp rnd) (- x + - y) <> 0%R
destruct (Zle_or_lt (cexp beta fexp (-x)) (cexp beta fexp (-y))) as [H2|H2].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- x) <= cexp beta fexp (- y))%Z
round beta fexp (Zrnd_opp rnd) (- x + - y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
round beta fexp (Zrnd_opp rnd) (- x + - y) <> 0%R
apply round_plus_neq_0_aux; try apply generic_format_opp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- x) <= cexp beta fexp (- y))%Z
(0 < - x + - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
round beta fexp (Zrnd_opp rnd) (- x + - y) <> 0%R
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
round beta fexp (Zrnd_opp rnd) (- x + - y) <> 0%R
rewrite Rplus_comm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
round beta fexp (Zrnd_opp rnd) (- y + - x) <> 0%R
apply round_plus_neq_0_aux; try apply generic_format_opp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
(cexp beta fexp (- y) <= cexp beta fexp (- x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
(0 < - y + - x)%R
now apply Zlt_le_weak.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x + y)%R <> 0%R
H1:(x + y < 0)%R
H2:(cexp beta fexp (- y) < cexp beta fexp (- x))%Z
(0 < - y + - x)%R
lra.
Qed.

Theorem round_plus_eq_0 :
  forall x y,
  format x -> format y ->
  round beta fexp rnd (x + y) = 0%R ->
  (x + y = 0)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
round beta fexp rnd (x + y) = 0%R -> (x + y)%R = 0%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
round beta fexp rnd (x + y) = 0%R -> (x + y)%R = 0%R
intros x y Fx Fy H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
H:round beta fexp rnd (x + y) = 0%R
(x + y)%R = 0%R
destruct (Req_dec (x + y) 0) as [H'|H'].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
H:round beta fexp rnd (x + y) = 0%R
H':(x + y)%R = 0%R
(x + y)%R = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
H:round beta fexp rnd (x + y) = 0%R
H':(x + y)%R <> 0%R
(x + y)%R = 0%R
exact H'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
H:round beta fexp rnd (x + y) = 0%R
H':(x + y)%R <> 0%R
(x + y)%R = 0%R
contradict H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exp_not_FTZ:Exp_not_FTZ fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
H':(x + y)%R <> 0%R
round beta fexp rnd (x + y) <> 0%R
now apply round_plus_neq_0.
Qed.

End Fprop_plus_zero.

Section Fprop_plus_FLT.
Variable beta : radix.

Notation bpow e := (bpow beta e).

Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.

Theorem FLT_format_plus_small: forall x y,
  generic_format beta (FLT_exp emin prec) x ->
  generic_format beta (FLT_exp emin prec) y ->
   (Rabs (x+y) <= bpow (prec+emin))%R ->
    generic_format beta (FLT_exp emin prec) (x+y).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall x y : R,
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
(Rabs (x + y) <= bpow (prec + emin))%R ->
generic_format beta (FLT_exp emin prec) (x + y)
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall x y : R,
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
(Rabs (x + y) <= bpow (prec + emin))%R ->
generic_format beta (FLT_exp emin prec) (x + y)
intros x y Fx Fy H.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FLT_exp emin prec) (x + y)
apply generic_format_FLT_FIX...beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H:(Rabs (x + y) <= bpow (prec + emin))%R
(Rabs (x + y) <= bpow (emin + prec))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FIX_exp emin) (x + y)
rewrite Zplus_comm; assumption.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FIX_exp emin) (x + y)
apply generic_format_FIX_FLT, FIX_format_generic in Fx.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
Fx:FIX_format beta emin x
Fy:generic_format beta (FLT_exp emin prec) y
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FIX_exp emin) (x + y)
apply generic_format_FIX_FLT, FIX_format_generic in Fy.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
Fx:FIX_format beta emin x
Fy:FIX_format beta emin y
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FIX_exp emin) (x + y)
destruct Fx as [nx H1x H2x].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
Fy:FIX_format beta emin y
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FIX_exp emin) (x + y)
destruct Fy as [ny H1y H2y].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
generic_format beta (FIX_exp emin) (x + y)
apply generic_format_FIX.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
FIX_format beta emin (x + y)
exists (Float beta (Fnum nx+Fnum ny)%Z emin).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
(x + y)%R =
F2R {| Fnum := Fnum nx + Fnum ny; Fexp := emin |}
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
Fexp {| Fnum := Fnum nx + Fnum ny; Fexp := emin |} =
emin
rewrite H1x,H1y; unfold F2R; simpl.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
(IZR (Fnum nx) * bpow (Fexp nx) +
 IZR (Fnum ny) * bpow (Fexp ny))%R =
(IZR (Fnum nx + Fnum ny) * bpow emin)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
Fexp {| Fnum := Fnum nx + Fnum ny; Fexp := emin |} =
emin
rewrite H2x, H2y.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
(IZR (Fnum nx) * bpow emin + IZR (Fnum ny) * bpow emin)%R =
(IZR (Fnum nx + Fnum ny) * bpow emin)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
Fexp {| Fnum := Fnum nx + Fnum ny; Fexp := emin |} =
emin
rewrite plus_IZR; ring.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
nx:float beta
H1x:x = F2R nx
H2x:Fexp nx = emin
ny:float beta
H1y:y = F2R ny
H2y:Fexp ny = emin
H:(Rabs (x + y) <= bpow (prec + emin))%R
Fexp {| Fnum := Fnum nx + Fnum ny; Fexp := emin |} =
emin
easy.
Qed.

Variable choice : Z -> bool.

Lemma FLT_plus_error_N_ex : forall x y,
  generic_format beta (FLT_exp emin prec) x ->
  generic_format beta (FLT_exp emin prec) y ->
  exists eps,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) (x + y)
  = ((x + y) * (1 + eps))%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
forall x y : R,
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
Proof.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
forall x y : R,
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
intros x y Fx Fy.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
assert (Pb := u_rod1pu_ro_pos beta prec).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
destruct (Rle_or_lt (bpow (emin + prec - 1)) (Rabs (x + y))) as [M|M].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(bpow (emin + prec - 1) <= Rabs (x + y))%R
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(Rabs (x + y) < bpow (emin + prec - 1))%R
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) 
    (Znearest choice) (x + y) =
  ((x + y) * (1 + eps))%R
{beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(bpow (emin + prec - 1) <= Rabs (x + y))%R
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R destruct (relative_error_N_FLX'_ex beta prec prec_gt_0_ choice (x + y))
    as (d, (Bd, Hd)).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(bpow (emin + prec - 1) <= Rabs (x + y))%R
d:R
Bd:(Rabs d <= u_ro beta prec / (1 + u_ro beta prec))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (x + y) =
((x + y) * (1 + d))%R
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
  now exists d; split; [exact Bd|]; rewrite <- Hd; apply round_FLT_FLX. }beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(Rabs (x + y) < bpow (emin + prec - 1))%R
exists eps : R,
  (Rabs eps <= u_ro beta prec / (1 + u_ro beta prec))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (x + y) = ((x + y) * (1 + eps))%R
exists 0%R; rewrite Rabs_R0; split; [exact Pb|]; rewrite Rplus_0_r, Rmult_1_r.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(Rabs (x + y) < bpow (emin + prec - 1))%R
round beta (FLT_exp emin prec) (Znearest choice)
  (x + y) = (x + y)%R
apply round_generic; [apply valid_rnd_N|].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(Rabs (x + y) < bpow (emin + prec - 1))%R
generic_format beta (FLT_exp emin prec) (x + y)
apply FLT_format_plus_small; [exact Fx|exact Fy|].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
Pb:(0 <= u_ro beta prec / (1 + u_ro beta prec))%R
M:(Rabs (x + y) < bpow (emin + prec - 1))%R
(Rabs (x + y) <= bpow (prec + emin))%R
apply Rlt_le, (Rlt_le_trans _ _ _ M), bpow_le; lia.
Qed.

Lemma FLT_plus_error_N_round_ex : forall x y,
  generic_format beta (FLT_exp emin prec) x ->
  generic_format beta (FLT_exp emin prec) y ->
  exists eps,
  (Rabs eps <= u_ro beta prec)%R /\
  (x + y
   = round beta (FLT_exp emin prec) (Znearest choice) (x + y) * (1 + eps))%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
forall x y : R,
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
exists eps : R,
  (Rabs eps <= u_ro beta prec)%R /\
  (x + y)%R =
  (round beta (FLT_exp emin prec) (Znearest choice)
     (x + y) * (1 + eps))%R
Proof.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
forall x y : R,
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
exists eps : R,
  (Rabs eps <= u_ro beta prec)%R /\
  (x + y)%R =
  (round beta (FLT_exp emin prec) (Znearest choice)
     (x + y) * (1 + eps))%R
intros x y Fx Fy.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
x, y:R
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
exists eps : R,
  (Rabs eps <= u_ro beta prec)%R /\
  (x + y)%R =
  (round beta (FLT_exp emin prec) (Znearest choice)
     (x + y) * (1 + eps))%R
now apply relative_error_N_round_ex_derive, FLT_plus_error_N_ex.
Qed.

End Fprop_plus_FLT.

Section Fprop_plus_mult_ulp.

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 }.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Notation format := (generic_format beta fexp).
Notation cexp := (cexp beta fexp).

Lemma ex_shift :
  forall x e, format x -> (e <= cexp x)%Z ->
  exists m, (x = IZR m * bpow e)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x ->
(e <= cexp x)%Z ->
exists m : Z, x = (IZR m * bpow e)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x ->
(e <= cexp x)%Z ->
exists m : Z, x = (IZR m * bpow e)%R
intros x e Fx He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
exists m : Z, x = (IZR m * bpow e)%R
exists (Ztrunc (scaled_mantissa beta fexp x)*Zpower beta (cexp x -e))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
x =
(IZR
   (Ztrunc (scaled_mantissa beta fexp x) *
    beta ^ (cexp x - e)) * bpow e)%R
rewrite Fx at 1; unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp x))%R =
(IZR
   (Ztrunc (scaled_mantissa beta fexp x) *
    beta ^ (cexp x - e)) * bpow e)%R
rewrite mult_IZR, Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp x))%R =
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 (IZR (beta ^ (cexp x - e)) * bpow e))%R
f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
bpow (cexp x) = (IZR (beta ^ (cexp x - e)) * bpow e)%R
rewrite IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
bpow (cexp x) = (bpow (cexp x - e) * bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
(0 <= cexp x - e)%Z
2: omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(e <= cexp x)%Z
bpow (cexp x) = (bpow (cexp x - e) * bpow e)%R
rewrite <- bpow_plus; f_equal; ring.
Qed.

Lemma mag_minus1 :
  forall z, z <> 0%R ->
  (mag beta z - 1)%Z = mag beta (z / IZR beta).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall z : R,
z <> 0%R ->
(mag beta z - 1)%Z = mag beta (z / IZR beta)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall z : R,
z <> 0%R ->
(mag beta z - 1)%Z = mag beta (z / IZR beta)
intros z Hz.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
z:R
Hz:z <> 0%R
(mag beta z - 1)%Z = mag beta (z / IZR beta)
unfold Zminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
z:R
Hz:z <> 0%R
(mag beta z + - (1))%Z = mag beta (z / IZR beta)
rewrite <- mag_mult_bpow by easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
z:R
Hz:z <> 0%R
mag beta (z * bpow (- (1))) = mag beta (z / IZR beta)
now rewrite bpow_opp, bpow_1.
Qed.

Theorem round_plus_F2R :
  forall x y, format x -> format y -> (x <> 0)%R ->
  exists m,
  round beta fexp rnd (x+y) = F2R (Float beta m (cexp (x / IZR beta))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
x <> 0%R ->
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
x <> 0%R ->
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
intros x y Fx Fy Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
case (Zle_or_lt (mag beta (x/IZR beta)) (mag beta y)); intros H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
pose (e:=cexp (x / IZR beta)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
destruct (ex_shift x e) as (nx, Hnx); try exact Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
(e <= cexp x)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
apply monotone_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
(mag beta (x / IZR beta) <= mag beta x)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
rewrite <- (mag_minus1 x Zx); omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
destruct (ex_shift y e) as (ny, Hny); try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
(e <= cexp y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
ny:Z
Hny:y = (IZR ny * bpow e)%R
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
apply monotone_exp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
ny:Z
Hny:y = (IZR ny * bpow e)%R
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
destruct (round_repr_same_exp beta fexp rnd (nx+ny) e) as (n,Hn).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
ny:Z
Hny:y = (IZR ny * bpow e)%R
n:Z
Hn:round beta fexp rnd
  (F2R {| Fnum := nx + ny; Fexp := e |}) =
F2R {| Fnum := n; Fexp := e |}
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
exists n.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
ny:Z
Hny:y = (IZR ny * bpow e)%R
n:Z
Hn:round beta fexp rnd
  (F2R {| Fnum := nx + ny; Fexp := e |}) =
F2R {| Fnum := n; Fexp := e |}
round beta fexp rnd (x + y) =
F2R {| Fnum := n; Fexp := cexp (x / IZR beta) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
fold e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
ny:Z
Hny:y = (IZR ny * bpow e)%R
n:Z
Hn:round beta fexp rnd
  (F2R {| Fnum := nx + ny; Fexp := e |}) =
F2R {| Fnum := n; Fexp := e |}
round beta fexp rnd (x + y) =
F2R {| Fnum := n; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
rewrite <- Hn; f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta (x / IZR beta) <= mag beta y)%Z
e:=cexp (x / IZR beta):Z
nx:Z
Hnx:x = (IZR nx * bpow e)%R
ny:Z
Hny:y = (IZR ny * bpow e)%R
n:Z
Hn:round beta fexp rnd
  (F2R {| Fnum := nx + ny; Fexp := e |}) =
F2R {| Fnum := n; Fexp := e |}
(x + y)%R = F2R {| Fnum := nx + ny; Fexp := e |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
rewrite Hnx, Hny; unfold F2R; simpl; rewrite plus_IZR; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
(* *)
destruct (ex_shift (round beta fexp rnd (x + y)) (cexp (x/IZR beta))) as (n,Hn).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
format (round beta fexp rnd (x + y))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x / IZR beta) <=
 cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply generic_format_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x / IZR beta) <=
 cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply Z.le_trans with (cexp (x+y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x / IZR beta) <= cexp (x + y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply monotone_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(mag beta (x / IZR beta) <= mag beta (x + y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite <- mag_minus1 by easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(mag beta x - 1 <= mag beta (x + y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite <- (mag_abs beta (x+y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
(* . *)
assert (U: (Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y) = Rabs x - Rabs y)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
assert (V: forall x y, (Rabs y <= Rabs x)%R ->
   (Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y) = Rabs x - Rabs y)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\ Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
clear; intros x y.x, y:R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
case (Rle_or_lt 0 y); intros Hy.x, y:R
Hy:(0 <= y)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
case Hy; intros Hy'.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
case (Rle_or_lt 0 x); intros Hx.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(0 <= x)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
intros _; rewrite (Rabs_pos_eq y) by easy.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(0 <= x)%R
Rabs (x + y) = (Rabs x + y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_pos_eq x) by easy.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(0 <= x)%R
Rabs (x + y) = (x + y)%R \/
y <> 0%R /\ Rabs (x + y) = (x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
left; apply Rabs_pos_eq.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(0 <= x)%R
(0 <= x + y)%R
x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rplus_le_le_0_compat.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_pos_eq y) by easy.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_left x) by easy.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
(y <= - x)%R ->
Rabs (x + y) = (- x + y)%R \/
y <> 0%R /\ Rabs (x + y) = (- x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
intros H; right; split.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
H:(y <= - x)%R
y <> 0%R
x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
H:(y <= - x)%R
Rabs (x + y) = (- x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rgt_not_eq.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
H:(y <= - x)%R
Rabs (x + y) = (- x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite Rabs_left1.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
H:(y <= - x)%R
(- (x + y))%R = (- x - y)%R
x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
H:(y <= - x)%R
(x + y <= 0)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
ring.x, y:R
Hy:(0 <= y)%R
Hy':(0 < y)%R
Hx:(x < 0)%R
H:(y <= - x)%R
(x + y <= 0)%R
x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply Rplus_le_reg_l with (-x)%R; ring_simplify; assumption.x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
intros _; left.x, y:R
Hy:(0 <= y)%R
Hy':0%R = y
Rabs (x + y) = (Rabs x + Rabs y)%R
x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now rewrite <- Hy', Rabs_R0, 2!Rplus_0_r.x, y:R
Hy:(y < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
case (Rle_or_lt 0 x); intros Hx.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_left y) by easy.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
(- y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + - y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - - y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_pos_eq x) by easy.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
(- y <= x)%R ->
Rabs (x + y) = (x + - y)%R \/
y <> 0%R /\ Rabs (x + y) = (x - - y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
intros H; right; split.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
H:(- y <= x)%R
y <> 0%R
x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
H:(- y <= x)%R
Rabs (x + y) = (x - - y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rlt_not_eq.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
H:(- y <= x)%R
Rabs (x + y) = (x - - y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite Rabs_pos_eq.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
H:(- y <= x)%R
(x + y)%R = (x - - y)%R
x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
H:(- y <= x)%R
(0 <= x + y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
ring.x, y:R
Hy:(y < 0)%R
Hx:(0 <= x)%R
H:(- y <= x)%R
(0 <= x + y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply Rplus_le_reg_l with (-y)%R; ring_simplify; assumption.x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(Rabs y <= Rabs x)%R ->
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
intros _; left.x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_left y) by easy.x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
Rabs (x + y) = (Rabs x + - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite (Rabs_left x) by easy.x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
Rabs (x + y) = (- x + - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite Rabs_left1.x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(- (x + y))%R = (- x + - y)%R
x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(x + y <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
ring.x, y:R
Hy:(y < 0)%R
Hx:(x < 0)%R
(x + y <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply V; left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
(Rabs y < Rabs x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply lt_mag with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
(0 < Rabs x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
(mag beta (Rabs y) < mag beta (Rabs x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
(mag beta (Rabs y) < mag beta (Rabs x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite <- mag_minus1 in H1; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta x - 1)%Z
V:forall x0 y0 : R,
(Rabs y0 <= Rabs x0)%R ->
Rabs (x0 + y0) = (Rabs x0 + Rabs y0)%R \/
y0 <> 0%R /\
Rabs (x0 + y0) = (Rabs x0 - Rabs y0)%R
(mag beta (Rabs y) < mag beta (Rabs x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite 2!mag_abs; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R \/
y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
(* . *)
destruct U as [U|U].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite U; apply Z.le_trans with (mag beta x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(mag beta x - 1 <= mag beta x)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(mag beta x <= mag beta (Rabs x + Rabs y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(mag beta x <= mag beta (Rabs x + Rabs y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite <- mag_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(mag beta (Rabs x) <= mag beta (Rabs x + Rabs y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply mag_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(0 < Rabs x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(Rabs x <= Rabs x + Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(Rabs x <= Rabs x + Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply Rplus_le_reg_l with (-Rabs x)%R; ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:Rabs (x + y) = (Rabs x + Rabs y)%R
(0 <= Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply Rabs_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U:y <> 0%R /\ Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
destruct U as (U',U); rewrite U.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta x - 1 <= mag beta (Rabs x - Rabs y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite <- mag_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta (Rabs x) - 1 <= mag beta (Rabs x - Rabs y))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply mag_minus_lb.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(0 < Rabs x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(0 < Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta (Rabs y) <= mag beta (Rabs x) - 2)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(0 < Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta (Rabs y) <= mag beta (Rabs x) - 2)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta (Rabs y) <= mag beta (Rabs x) - 2)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite 2!mag_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta y <= mag beta x - 2)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
assert (mag beta y < mag beta x - 1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
(mag beta y < mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
H:(mag beta y < mag beta x - 1)%Z
(mag beta y <= mag beta x - 2)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now rewrite (mag_minus1 x Zx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
U':y <> 0%R
U:Rabs (x + y) = (Rabs x - Rabs y)%R
H:(mag beta y < mag beta x - 1)%Z
(mag beta y <= mag beta x - 2)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(cexp (x + y) <= cexp (round beta fexp rnd (x + y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply cexp_round_ge...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
round beta fexp rnd (x + y) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
apply round_plus_neq_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
(x + y)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
contradict H1; apply Zle_not_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(x + y)%R = 0%R
(mag beta (x / IZR beta) <= mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite <- (mag_minus1 x Zx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(x + y)%R = 0%R
(mag beta x - 1 <= mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
replace y with (-x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(x + y)%R = 0%R
(mag beta x - 1 <= mag beta (- x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(x + y)%R = 0%R
(- x)%R = y
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
rewrite mag_opp; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(x + y)%R = 0%R
(- x)%R = y
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
H1:(mag beta y < mag beta (x / IZR beta))%Z
n:Z
Hn:round beta fexp rnd (x + y) =
(IZR n * bpow (cexp (x / IZR beta)))%R
exists m : Z,
  round beta fexp rnd (x + y) =
  (IZR m * bpow (cexp (x / IZR beta)))%R
now exists n.
Qed.

Context {exp_not_FTZ : Exp_not_FTZ fexp}.

Theorem round_plus_ge_ulp :
  forall x y, format x -> format y ->
  round beta fexp rnd (x+y) <> 0%R ->
  (ulp beta fexp (x/IZR beta) <= Rabs (round beta fexp rnd (x+y)))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
forall x y : R,
format x ->
format y ->
round beta fexp rnd (x + y) <> 0%R ->
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
forall x y : R,
format x ->
format y ->
round beta fexp rnd (x + y) <> 0%R ->
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
intros x y Fx Fy KK.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
case (Req_dec x 0); intros Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
(* *)
rewrite Zx, Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp (0 / IZR beta) <=
 Rabs (round beta fexp rnd y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp (0 / IZR beta) <= Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
unfold Rdiv; rewrite Rmult_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp 0 <= Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite Fy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp 0 <=
 Rabs
   (F2R
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp y);
      Fexp := cexp y |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
unfold F2R; simpl; rewrite Rabs_mult.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 Rabs (bpow (cexp y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite (Rabs_pos_eq (bpow _)) by apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
case (Z.eq_dec (Ztrunc (scaled_mantissa beta fexp y)) 0); intros Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) = 0%Z
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
contradict KK.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) = 0%Z
round beta fexp rnd (x + y) = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite Zx, Fy, Hm; unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) = 0%Z
round beta fexp rnd (0 + 0 * bpow (cexp y)) = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite Rplus_0_l, Rmult_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) = 0%Z
round beta fexp rnd 0 = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply round_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply Rle_trans with (1*bpow (cexp y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <= 1 * bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <= bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite <- ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(ulp beta fexp 0 <= ulp beta fexp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply ulp_ge_ulp_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
intros K; apply Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
K:y = 0%R
Ztrunc (scaled_mantissa beta fexp y) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite K, scaled_mantissa_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
K:y = 0%R
Ztrunc 0 = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 * bpow (cexp y) <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))) *
 bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(0 <= bpow (cexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 <=
 Rabs (IZR (Ztrunc (scaled_mantissa beta fexp y))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite <- abs_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 <=
 IZR (Z.abs (Ztrunc (scaled_mantissa beta fexp y))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply IZR_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(1 <= Z.abs (Ztrunc (scaled_mantissa beta fexp y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply (Zlt_le_succ 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x = 0%R
Hm:Ztrunc (scaled_mantissa beta fexp y) <> 0%Z
(0 < Z.abs (Ztrunc (scaled_mantissa beta fexp y)))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
now apply Z.abs_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
(* *)
destruct (round_plus_F2R x y Fx Fy Zx) as (m,Hm).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
case (Z.eq_dec m 0); intros Zm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m = 0%Z
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
contradict KK.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m = 0%Z
round beta fexp rnd (x + y) = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite Hm, Zm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m = 0%Z
F2R {| Fnum := 0; Fexp := cexp (x / IZR beta) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
apply F2R_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(ulp beta fexp (x / IZR beta) <=
 Rabs (round beta fexp rnd (x + y)))%R
rewrite Hm, <- F2R_Zabs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(ulp beta fexp (x / IZR beta) <=
 F2R
   {| Fnum := Z.abs m; Fexp := cexp (x / IZR beta) |})%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(bpow (cexp (x / IZR beta)) <=
 F2R
   {| Fnum := Z.abs m; Fexp := cexp (x / IZR beta) |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
rewrite <- (Rmult_1_l (bpow _)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(1 * bpow (cexp (x / IZR beta)) <=
 F2R
   {| Fnum := Z.abs m; Fexp := cexp (x / IZR beta) |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(0 <=
 bpow
   (Fexp
      {|
      Fnum := Z.abs m;
      Fexp := cexp (x / IZR beta) |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(1 <=
 IZR
   (Fnum
      {|
      Fnum := Z.abs m;
      Fexp := cexp (x / IZR beta) |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(1 <=
 IZR
   (Fnum
      {|
      Fnum := Z.abs m;
      Fexp := cexp (x / IZR beta) |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
apply IZR_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(1 <=
 Fnum
   {| Fnum := Z.abs m; Fexp := cexp (x / IZR beta) |})%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
apply (Zlt_le_succ 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(0 <
 Fnum
   {| Fnum := Z.abs m; Fexp := cexp (x / IZR beta) |})%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
now apply Z.abs_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(x / IZR beta)%R <> 0%R
apply Rmult_integral_contrapositive_currified with (1 := Zx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
(/ IZR beta)%R <> 0%R
apply Rinv_neq_0_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
exp_not_FTZ:Exp_not_FTZ fexp
x, y:R
Fx:format x
Fy:format y
KK:round beta fexp rnd (x + y) <> 0%R
Zx:x <> 0%R
m:Z
Hm:round beta fexp rnd (x + y) =
F2R {| Fnum := m; Fexp := cexp (x / IZR beta) |}
Zm:m <> 0%Z
IZR beta <> 0%R
apply Rgt_not_eq, radix_pos.
Qed.

End Fprop_plus_mult_ulp.

Section Fprop_plus_ge_ulp.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.

Theorem round_FLT_plus_ge :
  forall x y e,
  generic_format beta (FLT_exp emin prec) x -> generic_format beta (FLT_exp emin prec) y ->
  (bpow (e + prec) <= Rabs x)%R ->
  round beta (FLT_exp emin prec) rnd (x + y) <> 0%R ->
  (bpow e <= Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall (x y : R) (e : Z),
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
(bpow (e + prec) <= Rabs x)%R ->
round beta (FLT_exp emin prec) rnd (x + y) <> 0%R ->
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
Proof with auto with typeclass_instances.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall (x y : R) (e : Z),
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
(bpow (e + prec) <= Rabs x)%R ->
round beta (FLT_exp emin prec) rnd (x + y) <> 0%R ->
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
intros x y e Fx Fy He KK.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
assert (Zx: x <> 0%R).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
x <> 0%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
  contradict He.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
He:x = 0%R
~ (bpow (e + prec) <= Rabs x)%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
  apply Rlt_not_le; rewrite He, Rabs_R0.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
He:x = 0%R
(0 < bpow (e + prec))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
  apply bpow_gt_0.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
apply Rle_trans with (ulp beta (FLT_exp emin prec) (x/IZR beta)).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <= ulp beta (FLT_exp emin prec) (x / IZR beta))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta (FLT_exp emin prec) (x / IZR beta) <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
2: apply round_plus_ge_ulp...beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <= ulp beta (FLT_exp emin prec) (x / IZR beta))%R
rewrite ulp_neq_0.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 bpow (cexp beta (FLT_exp emin prec) (x / IZR beta)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
unfold cexp.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 bpow (FLT_exp emin prec (mag beta (x / IZR beta))))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
rewrite <- mag_minus1; try assumption.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <= bpow (FLT_exp emin prec (mag beta x - 1)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
unfold FLT_exp; apply bpow_le.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(e <= Z.max (mag beta x - 1 - prec) emin)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply Z.le_trans with (2:=Z.le_max_l _ _).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(e <= mag beta x - 1 - prec)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
destruct (mag beta x) as (n,Hn); simpl.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(e <= n - 1 - prec)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
assert (e + prec < n)%Z; try omega.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(e + prec < n)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply lt_bpow with beta.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(bpow (e + prec) < bpow n)%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply Rle_lt_trans with (1:=He).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(Rabs x < bpow n)%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
now apply Hn.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply Rmult_integral_contrapositive_currified; try assumption.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(/ IZR beta)%R <> 0%R
apply Rinv_neq_0_compat.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
IZR beta <> 0%R
apply Rgt_not_eq.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(IZR beta > 0)%R
apply radix_pos.
Qed.

Lemma round_FLT_plus_ge' :
  forall x y e,
  generic_format beta (FLT_exp emin prec) x -> generic_format beta (FLT_exp emin prec) y ->
  (x <> 0%R -> (bpow (e+prec) <= Rabs x)%R) ->
  (x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R) ->
  round beta (FLT_exp emin prec) rnd (x+y) <> 0%R ->
  (bpow e <= Rabs (round beta (FLT_exp emin prec) rnd (x+y)))%R.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall (x y : R) (e : Z),
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
(x <> 0%R -> (bpow (e + prec) <= Rabs x)%R) ->
(x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R) ->
round beta (FLT_exp emin prec) rnd (x + y) <> 0%R ->
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
Proof with auto with typeclass_instances.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall (x y : R) (e : Z),
generic_format beta (FLT_exp emin prec) x ->
generic_format beta (FLT_exp emin prec) y ->
(x <> 0%R -> (bpow (e + prec) <= Rabs x)%R) ->
(x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R) ->
round beta (FLT_exp emin prec) rnd (x + y) <> 0%R ->
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
intros x y e Fx Fy H1 H2 H3.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
case (Req_dec x 0); intros H4.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x = 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
case (Req_dec y 0); intros H5.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x = 0%R
H5:y = 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x = 0%R
H5:y <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
contradict H3.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H4:x = 0%R
H5:y = 0%R
round beta (FLT_exp emin prec) rnd (x + y) = 0%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x = 0%R
H5:y <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
rewrite H4, H5, Rplus_0_l; apply round_0...beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x = 0%R
H5:y <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
rewrite H4, Rplus_0_l.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x = 0%R
H5:y <> 0%R
(bpow e <= Rabs (round beta (FLT_exp emin prec) rnd y))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
rewrite round_generic...beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow e <=
 Rabs (round beta (FLT_exp emin prec) rnd (x + y)))%R
apply round_FLT_plus_ge; try easy.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLT_exp emin prec) x
Fy:generic_format beta (FLT_exp emin prec) y
H1:x <> 0%R -> (bpow (e + prec) <= Rabs x)%R
H2:x = 0%R -> y <> 0%R -> (bpow e <= Rabs y)%R
H3:round beta (FLT_exp emin prec) rnd (x + y) <> 0%R
H4:x <> 0%R
(bpow (e + prec) <= Rabs x)%R
now apply H1.
Qed.

Theorem round_FLX_plus_ge :
  forall x y e,
  generic_format beta (FLX_exp prec) x -> generic_format beta (FLX_exp prec) y ->
  (bpow (e+prec) <= Rabs x)%R ->
  (round beta (FLX_exp prec) rnd (x+y) <> 0)%R ->
  (bpow e <= Rabs (round beta (FLX_exp prec) rnd (x+y)))%R.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall (x y : R) (e : Z),
generic_format beta (FLX_exp prec) x ->
generic_format beta (FLX_exp prec) y ->
(bpow (e + prec) <= Rabs x)%R ->
round beta (FLX_exp prec) rnd (x + y) <> 0%R ->
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
Proof with auto with typeclass_instances.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
forall (x y : R) (e : Z),
generic_format beta (FLX_exp prec) x ->
generic_format beta (FLX_exp prec) y ->
(bpow (e + prec) <= Rabs x)%R ->
round beta (FLX_exp prec) rnd (x + y) <> 0%R ->
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
intros x y e Fx Fy He KK.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
assert (Zx: x <> 0%R).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
x <> 0%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
  contradict He.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
He:x = 0%R
~ (bpow (e + prec) <= Rabs x)%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
  apply Rlt_not_le; rewrite He, Rabs_R0.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
He:x = 0%R
(0 < bpow (e + prec))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
  apply bpow_gt_0.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
apply Rle_trans with (ulp beta (FLX_exp prec) (x/IZR beta)).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <= ulp beta (FLX_exp prec) (x / IZR beta))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(ulp beta (FLX_exp prec) (x / IZR beta) <=
 Rabs (round beta (FLX_exp prec) rnd (x + y)))%R
2: apply round_plus_ge_ulp...beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <= ulp beta (FLX_exp prec) (x / IZR beta))%R
rewrite ulp_neq_0.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 bpow (cexp beta (FLX_exp prec) (x / IZR beta)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
unfold cexp.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <=
 bpow (FLX_exp prec (mag beta (x / IZR beta))))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
rewrite <- mag_minus1 by easy.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(bpow e <= bpow (FLX_exp prec (mag beta x - 1)))%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
unfold FLX_exp; apply bpow_le.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(e <= mag beta x - 1 - prec)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
destruct (mag beta x) as (n,Hn); simpl.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(e <= n - 1 - prec)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
assert (e + prec < n)%Z; try omega.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(e + prec < n)%Z
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply lt_bpow with beta.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(bpow (e + prec) < bpow n)%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply Rle_lt_trans with (1:=He).beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
n:Z
Hn:x <> 0%R -> (bpow (n - 1) <= Rabs x < bpow n)%R
(Rabs x < bpow n)%R
beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
now apply Hn.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(x / IZR beta)%R <> 0%R
apply Rmult_integral_contrapositive_currified; try assumption.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(/ IZR beta)%R <> 0%R
apply Rinv_neq_0_compat.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
IZR beta <> 0%R
apply Rgt_not_eq.beta:radix
rnd:R -> Z
valid_rnd:Valid_rnd rnd
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
x, y:R
e:Z
Fx:generic_format beta (FLX_exp prec) x
Fy:generic_format beta (FLX_exp prec) y
He:(bpow (e + prec) <= Rabs x)%R
KK:round beta (FLX_exp prec) rnd (x + y) <> 0%R
Zx:x <> 0%R
(IZR beta > 0)%R
apply radix_pos.
Qed.

End Fprop_plus_ge_ulp.

Section Fprop_plus_le_ops.

Variable beta : radix.
Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Variable choice : Z -> bool.

Lemma plus_error_le_l :
  forall x y,
  generic_format beta fexp x -> generic_format beta fexp y ->
  (Rabs (round beta fexp (Znearest choice) (x + y) - (x + y)) <= Rabs x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
forall x y : R,
generic_format beta fexp x ->
generic_format beta fexp y ->
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
forall x y : R,
generic_format beta fexp x ->
generic_format beta fexp y ->
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs x)%R
intros x y Fx Fy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, y:R
Fx:generic_format beta fexp x
Fy:generic_format beta fexp y
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs x)%R
apply (Rle_trans _ (Rabs (y - (x + y)))); [now apply round_N_pt|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, y:R
Fx:generic_format beta fexp x
Fy:generic_format beta fexp y
(Rabs (y - (x + y)) <= Rabs x)%R
rewrite Rabs_minus_sym; right; f_equal; ring.
Qed.

Lemma plus_error_le_r :
  forall x y,
  generic_format beta fexp x -> generic_format beta fexp y ->
  (Rabs (round beta fexp (Znearest choice) (x + y) - (x + y)) <= Rabs y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
forall x y : R,
generic_format beta fexp x ->
generic_format beta fexp y ->
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
forall x y : R,
generic_format beta fexp x ->
generic_format beta fexp y ->
(Rabs
   (round beta fexp (Znearest choice) (x + y) -
    (x + y)) <= Rabs y)%R now intros x y Fx Fy; rewrite Rplus_comm; apply plus_error_le_l. Qed.

End Fprop_plus_le_ops.