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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 multiplication is in the FLX/FLT format

Require Import Core Operations Plus_error.

Section Fprop_mult_error.

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

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

Notation format := (generic_format beta (FLX_exp prec)).
Notation cexp := (cexp beta (FLX_exp prec)).

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

Auxiliary result that provides the exponent

Lemma mult_error_FLX_aux:
  forall x y,
  format x -> format y ->
  (round beta (FLX_exp prec) rnd (x * y) - (x * y) <> 0)%R ->
  exists f:float beta,
    (F2R f = round beta (FLX_exp prec) rnd (x * y) - (x * y))%R
    /\ (cexp (F2R f) <= Fexp f)%Z
    /\ (Fexp f = cexp x + cexp y)%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R ->
exists f : float beta,
  F2R f =
  (round beta (FLX_exp prec) rnd (x * y) - x * y)%R /\
  (cexp (F2R f) <= Fexp f)%Z /\
  Fexp f = (cexp x + cexp y)%Z
Proof with auto with typeclass_instances.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R ->
exists f : float beta,
  F2R f =
  (round beta (FLX_exp prec) rnd (x * y) - x * y)%R /\
  (cexp (F2R f) <= Fexp f)%Z /\
  Fexp f = (cexp x + cexp y)%Z
intros x y Hx Hy Hz.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
exists f : float beta,
  F2R f =
  (round beta (FLX_exp prec) rnd (x * y) - x * y)%R /\
  (cexp (F2R f) <= Fexp f)%Z /\
  Fexp f = (cexp x + cexp y)%Z
set (f := (round beta (FLX_exp prec) rnd (x * y))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
destruct (Req_dec (x * y) 0) as [Hxy0|Hxy0].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R = 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
contradict Hz.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R = 0%R
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R =
0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite Hxy0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R = 0%R
(round beta (FLX_exp prec) rnd 0 - 0)%R = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite round_0...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R = 0%R
(0 - 0)%R = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
destruct (mag beta (x * y)) as (exy, Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(x * y)%R <> 0%R ->
(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
specialize (Hexy Hxy0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
destruct (mag beta (f - x * y)) as (er, Her).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(f - x * y)%R <> 0%R ->
(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
specialize (Her Hz).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
destruct (mag beta x) as (ex, Hex).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
assert (Hx0: (x <> 0)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
x <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
contradict Hxy0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hxy0:x = 0%R
(x * y)%R = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
now rewrite Hxy0, Rmult_0_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
specialize (Hex Hx0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
destruct (mag beta y) as (ey, Hey).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
assert (Hy0: (y <> 0)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
y <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
contradict Hxy0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hxy0:y = 0%R
(x * y)%R = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
now rewrite Hxy0, Rmult_0_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
specialize (Hey Hy0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
(* *)
assert (Hc1: (cexp (x * y)%R - prec <= cexp x + cexp y)%Z).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(mag beta (x * y) - prec - prec <=
 mag beta x - prec + (mag beta y - prec))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hex).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(mag beta (x * y) - prec - prec <=
 ex - prec + (mag beta y - prec))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hey).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(mag beta (x * y) - prec - prec <=
 ex - prec + (ey - prec))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(exy - prec - prec <= ex - prec + (ey - prec))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
cut (exy - 1 < ex + ey)%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(exy - 1 < ex + ey)%Z ->
(exy - prec - prec <= ex - prec + (ey - prec))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(exy - 1 < ex + ey)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z omega.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(exy - 1 < ex + ey)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply (lt_bpow beta).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(bpow (exy - 1) < bpow (ex + ey))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Rle_lt_trans with (1 := proj1 Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs (x * y) < bpow (ex + ey))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite Rabs_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs x * Rabs y < bpow (ex + ey))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite bpow_plus.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs x * Rabs y < bpow ex * bpow ey)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Rmult_le_0_lt_compat.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(0 <= Rabs x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(0 <= Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs x < bpow ex)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs y < bpow ey)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Rabs_pos.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(0 <= Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs x < bpow ex)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs y < bpow ey)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Rabs_pos.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs x < bpow ex)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs y < bpow ey)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Hex.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
(Rabs y < bpow ey)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Hey.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
(* *)
assert (Hc2: (cexp x + cexp y <= cexp (x * y)%R)%Z).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(cexp x + cexp y <= cexp (x * y))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(mag beta x - prec + (mag beta y - prec) <=
 mag beta (x * y) - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hex).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - prec + (mag beta y - prec) <=
 mag beta (x * y) - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hey).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - prec + (ey - prec) <= mag beta (x * y) - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - prec + (ey - prec) <= exy - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
cut ((ex - 1) + (ey - 1) < exy)%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - 1 + (ey - 1) < exy)%Z ->
(ex - prec + (ey - prec) <= exy - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - 1 + (ey - 1) < exy)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
generalize (prec_gt_0 prec).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(0 < prec)%Z ->
(ex - 1 + (ey - 1) < exy)%Z ->
(ex - prec + (ey - prec) <= exy - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - 1 + (ey - 1) < exy)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
clear ; omega.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(ex - 1 + (ey - 1) < exy)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply (lt_bpow beta).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1 + (ey - 1)) < bpow exy)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Rle_lt_trans with (2 := proj2 Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1 + (ey - 1)) <= Rabs (x * y))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite Rabs_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1 + (ey - 1)) <= Rabs x * Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite bpow_plus.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1) * bpow (ey - 1) <= Rabs x * Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Rmult_le_compat.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(0 <= bpow (ex - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(0 <= bpow (ey - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1) <= Rabs x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ey - 1) <= Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply bpow_ge_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(0 <= bpow (ey - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1) <= Rabs x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ey - 1) <= Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply bpow_ge_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ex - 1) <= Rabs x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ey - 1) <= Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Hex.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
(bpow (ey - 1) <= Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
apply Hey.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
(* *)
assert (Hr: ((F2R (Float beta (- (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) *
  Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) + rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) *
  beta ^ (cexp (x * y)%R - (cexp x + cexp y))) (cexp x + cexp y))) = f - x * y)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = (f - x * y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite Hx at 6.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
(f -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x);
   Fexp := cexp x |} * y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite Hy at 6.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
(f -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x);
   Fexp := cexp x |} *
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y);
   Fexp := cexp y |})%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite <- F2R_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
(f -
 F2R
   (Fmult
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta (FLX_exp prec) x);
      Fexp := cexp x |}
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta (FLX_exp prec) y);
      Fexp := cexp y |}))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
(f -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y);
   Fexp := cexp x + cexp y |})%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
unfold f, round, Rminus.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
(F2R
   {|
   Fnum := rnd
             (scaled_mantissa beta (FLX_exp prec)
                (x * y));
   Fexp := cexp (x * y) |} +
 -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y);
   Fexp := cexp x + cexp y |})%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
rewrite <- F2R_opp, Rplus_comm, <- F2R_plus.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
F2R
  (Fplus
     (Fopp
        {|
        Fnum := Ztrunc
                  (scaled_mantissa beta (FLX_exp prec)
                     x) *
                Ztrunc
                  (scaled_mantissa beta (FLX_exp prec)
                     y);
        Fexp := cexp x + cexp y |})
     {|
     Fnum := rnd
               (scaled_mantissa beta (FLX_exp prec)
                  (x * y));
     Fexp := cexp (x * y) |})
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
unfold Fplus.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
F2R
  (let
   '(m1, m2, e) :=
    Falign
      (Fopp
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLX_exp prec) x) *
                 Ztrunc
                   (scaled_mantissa beta
                      (FLX_exp prec) y);
         Fexp := cexp x + cexp y |})
      {|
      Fnum := rnd
                (scaled_mantissa beta (FLX_exp prec)
                   (x * y));
      Fexp := cexp (x * y) |} in
    {| Fnum := m1 + m2; Fexp := e |})
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} =
F2R
  (let
   '(m1, m2, e) :=
    if (cexp x + cexp y <=? cexp (x * y))%Z
    then
     ((-
       (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) *
        Ztrunc (scaled_mantissa beta (FLX_exp prec) y)))%Z,
     (rnd
        (scaled_mantissa beta (FLX_exp prec) (x * y)) *
      beta ^ (cexp (x * y) - (cexp x + cexp y)))%Z,
     (cexp x + cexp y)%Z)
    else
     ((-
       (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) *
        Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) *
       beta ^ (cexp x + cexp y - cexp (x * y)))%Z,
     rnd (scaled_mantissa beta (FLX_exp prec) (x * y)),
     cexp (x * y)) in {| Fnum := m1 + m2; Fexp := e |})
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
now rewrite Zle_imp_le_bool with (1 := Hc2).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
exists f0 : float beta,
  F2R f0 = (f - x * y)%R /\
  (cexp (F2R f0) <= Fexp f0)%Z /\
  Fexp f0 = (cexp x + cexp y)%Z
(* *)
exists (Float beta (- (Ztrunc (scaled_mantissa beta (FLX_exp prec) x) *
  Ztrunc (scaled_mantissa beta (FLX_exp prec) y)) + rnd (scaled_mantissa beta (FLX_exp prec) (x * y)) *
  beta ^ (cexp (x * y)%R - (cexp x + cexp y))) (cexp x + cexp y)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = (f - x * y)%R /\
(cexp
   (F2R
      {|
      Fnum := -
              (Ztrunc
                 (scaled_mantissa beta (FLX_exp prec)
                    x) *
               Ztrunc
                 (scaled_mantissa beta (FLX_exp prec)
                    y)) +
              rnd
                (scaled_mantissa beta (FLX_exp prec)
                   (x * y)) *
              beta
              ^ (cexp (x * y) - (cexp x + cexp y));
      Fexp := cexp x + cexp y |}) <=
 Fexp
   {|
   Fnum := -
           (Ztrunc
              (scaled_mantissa beta (FLX_exp prec) x) *
            Ztrunc
              (scaled_mantissa beta (FLX_exp prec) y)) +
           rnd
             (scaled_mantissa beta (FLX_exp prec)
                (x * y)) *
           beta ^ (cexp (x * y) - (cexp x + cexp y));
   Fexp := cexp x + cexp y |})%Z /\
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = (cexp x + cexp y)%Z
split;[assumption|split].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
(cexp
   (F2R
      {|
      Fnum := -
              (Ztrunc
                 (scaled_mantissa beta (FLX_exp prec)
                    x) *
               Ztrunc
                 (scaled_mantissa beta (FLX_exp prec)
                    y)) +
              rnd
                (scaled_mantissa beta (FLX_exp prec)
                   (x * y)) *
              beta
              ^ (cexp (x * y) - (cexp x + cexp y));
      Fexp := cexp x + cexp y |}) <=
 Fexp
   {|
   Fnum := -
           (Ztrunc
              (scaled_mantissa beta (FLX_exp prec) x) *
            Ztrunc
              (scaled_mantissa beta (FLX_exp prec) y)) +
           rnd
             (scaled_mantissa beta (FLX_exp prec)
                (x * y)) *
           beta ^ (cexp (x * y) - (cexp x + cexp y));
   Fexp := cexp x + cexp y |})%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
rewrite Hr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
(cexp (f - x * y) <=
 Fexp
   {|
   Fnum := -
           (Ztrunc
              (scaled_mantissa beta (FLX_exp prec) x) *
            Ztrunc
              (scaled_mantissa beta (FLX_exp prec) y)) +
           rnd
             (scaled_mantissa beta (FLX_exp prec)
                (x * y)) *
           beta ^ (cexp (x * y) - (cexp x + cexp y));
   Fexp := cexp x + cexp y |})%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
(cexp (f - x * y) <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
clear Hr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (f - x * y) <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
apply Z.le_trans with (cexp (x * y)%R - prec)%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (f - x * y) <= cexp (x * y) - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(mag beta (f - x * y) - prec <=
 mag beta (x * y) - prec - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
apply Zplus_le_compat_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(mag beta (f - x * y) <= mag beta (x * y) - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
rewrite mag_unique with (1 := Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(mag beta (f - x * y) <= exy - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
apply mag_le_bpow with (1 := Hz).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(Rabs (round beta (FLX_exp prec) rnd (x * y) - x * y) <
 bpow (exy - prec))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
replace (bpow (exy - prec)) with (ulp beta (FLX_exp prec) (x * y)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(Rabs (round beta (FLX_exp prec) rnd (x * y) - x * y) <
 ulp beta (FLX_exp prec) (x * y))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
ulp beta (FLX_exp prec) (x * y) = bpow (exy - prec)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
apply error_lt_ulp...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
ulp beta (FLX_exp prec) (x * y) = bpow (exy - prec)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
rewrite ulp_neq_0; trivial.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
bpow (cexp (x * y)) = bpow (exy - prec)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
unfold cexp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
bpow (FLX_exp prec (mag beta (x * y))) =
bpow (exy - prec)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
now rewrite mag_unique with (1 := Hexy).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
(cexp (x * y) - prec <= cexp x + cexp y)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(cexp x + cexp y)%Z
apply Hc1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hz:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
f:=round beta (FLX_exp prec) rnd (x * y):R
Hxy0:(x * y)%R <> 0%R
exy:Z
Hexy:(bpow (exy - 1) <= Rabs (x * y) < bpow exy)%R
er:Z
Her:(bpow (er - 1) <= Rabs (f - x * y) < bpow er)%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx0:x <> 0%R
ey:Z
Hey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy0:y <> 0%R
Hc1:(cexp (x * y) - prec <= cexp x + cexp y)%Z
Hc2:(cexp x + cexp y <= cexp (x * y))%Z
Hr:F2R
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta 
                (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta 
               (FLX_exp prec) 
               (x * y)) *
          beta
          ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = 
(f - x * y)%R
Fexp
  {|
  Fnum := -
          (Ztrunc
             (scaled_mantissa beta (FLX_exp prec) x) *
           Ztrunc
             (scaled_mantissa beta (FLX_exp prec) y)) +
          rnd
            (scaled_mantissa beta (FLX_exp prec)
               (x * y)) *
          beta ^ (cexp (x * y) - (cexp x + cexp y));
  Fexp := cexp x + cexp y |} = (cexp x + cexp y)%Z
reflexivity.
Qed.

Error of the multiplication in FLX

Theorem mult_error_FLX :
  forall x y,
  format x -> format y ->
  format (round beta (FLX_exp prec) rnd (x * y) - (x * y))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
intros x y Hx Hy.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
destruct (Req_dec (round beta (FLX_exp prec) rnd (x * y) - x * y) 0) as [Hr0|Hr0].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R =
0%R
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
rewrite Hr0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R =
0%R
format 0
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
apply generic_format_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
destruct (mult_error_FLX_aux x y Hx Hy Hr0) as ((m,e),(H1,(H2,H3))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(cexp (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(cexp x + cexp y)%Z
format (round beta (FLX_exp prec) rnd (x * y) - x * y)
rewrite <- H1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hr0:(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(cexp (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(cexp x + cexp y)%Z
format (F2R {| Fnum := m; Fexp := e |})
now apply generic_format_F2R.
Qed.

Lemma mult_bpow_exact_FLX :
  forall x e,
  format x ->
  format (x * bpow e)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x -> format (x * bpow e)
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x -> format (x * bpow e)
intros x e Fx.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
format (x * bpow e)
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Zx:x = 0%R
format (x * bpow e)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
format (x * bpow e)
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Zx:x = 0%R
format (x * bpow e) rewrite Zx, Rmult_0_l; apply generic_format_0. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
format (x * bpow e)
rewrite Fx.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
format
  (F2R
     {|
     Fnum := Ztrunc
               (scaled_mantissa beta (FLX_exp prec) x);
     Fexp := cexp x |} * bpow e)
set (mx := Ztrunc _); set (ex := cexp _).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
format (F2R {| Fnum := mx; Fexp := ex |} * bpow e)
pose (f := {| Fnum := mx; Fexp := ex + e |} : float beta).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
format (F2R {| Fnum := mx; Fexp := ex |} * bpow e)
apply (generic_format_F2R' _ _ _ f).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
F2R f = (F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <> 0%R ->
(cexp (F2R {| Fnum := mx; Fexp := ex |} * bpow e) <=
 Fexp f)%Z
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
F2R f = (F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R now unfold F2R; simpl; rewrite bpow_plus, Rmult_assoc. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <> 0%R ->
(cexp (F2R {| Fnum := mx; Fexp := ex |} * bpow e) <=
 Fexp f)%Z
intro Nzmx; unfold mx, ex; rewrite <- Fx.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(cexp (x * bpow e) <= Fexp f)%Z
unfold f, ex; simpl; unfold cexp; rewrite (mag_mult_bpow _ _ _ Nzx).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
Nzx:x <> 0%R
mx:=Ztrunc (scaled_mantissa beta (FLX_exp prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(FLX_exp prec (mag beta x + e) <=
 FLX_exp prec (mag beta x) + e)%Z
unfold FLX_exp; omega.
Qed.

End Fprop_mult_error.

Section Fprop_mult_error_FLT.

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

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

Notation format := (generic_format beta (FLT_exp emin prec)).
Notation cexp := (cexp beta (FLT_exp emin prec)).

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

Error of the multiplication in FLT with underflow requirements

Theorem mult_error_FLT :
  forall x y,
  format x -> format y ->
  (x * y <> 0 -> bpow (emin + 2*prec - 1) <= Rabs (x * y))%R ->
  format (round beta (FLT_exp emin prec) rnd (x * y) - (x * y))%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
((x * y)%R <> 0%R ->
 (bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R) ->
format
  (round beta (FLT_exp emin prec) rnd (x * y) - x * y)
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
((x * y)%R <> 0%R ->
 (bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R) ->
format
  (round beta (FLT_exp emin prec) rnd (x * y) - x * y)
intros x y Hx Hy Hxy.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
format
  (round beta (FLT_exp emin prec) rnd (x * y) - x * y)
set (f := (round beta (FLT_exp emin prec) rnd (x * y))).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
format (f - x * y)
destruct (Req_dec (f - x * y) 0) as [Hr0|Hr0].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R = 0%R
format (f - x * y)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
format (f - x * y)
rewrite Hr0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R = 0%R
format 0
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
format (f - x * y)
apply generic_format_0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
format (f - x * y)
destruct (Req_dec (x * y) 0) as [Hxy'|Hxy'].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R = 0%R
format (f - x * y)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
unfold f.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R = 0%R
format
  (round beta (FLT_exp emin prec) rnd (x * y) - x * y)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
rewrite Hxy'.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R = 0%R
format (round beta (FLT_exp emin prec) rnd 0 - 0)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
rewrite round_0...beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R = 0%R
format (0 - 0)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
ring_simplify (0 - 0)%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R = 0%R
format 0
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
apply generic_format_0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(x * y)%R <> 0%R ->
(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
specialize (Hxy Hxy').beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
format (f - x * y)
destruct (mult_error_FLX_aux beta prec rnd x y) as ((m,e),(H1,(H2,H3))).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
generic_format beta (FLX_exp prec) x
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
generic_format beta (FLX_exp prec) y
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
now apply generic_format_FLX_FLT with emin.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
generic_format beta (FLX_exp prec) y
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
now apply generic_format_FLX_FLT with emin.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R <>
0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
rewrite <- (round_FLT_FLX beta emin).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(round beta (FLT_exp emin prec) rnd (x * y) - x * y)%R <>
0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(bpow (emin + prec - 1) <= Rabs (x * y))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
assumption.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(bpow (emin + prec - 1) <= Rabs (x * y))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
apply Rle_trans with (2:=Hxy).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(bpow (emin + prec - 1) <= bpow (emin + 2 * prec - 1))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
apply bpow_le.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(emin + prec - 1 <= emin + 2 * prec - 1)%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
generalize (prec_gt_0 prec).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
(0 < prec)%Z ->
(emin + prec - 1 <= emin + 2 * prec - 1)%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
clear ; omega.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
rewrite <- (round_FLT_FLX beta emin) in H1.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
(bpow (emin + prec - 1) <= Rabs (x * y))%R
2:apply Rle_trans with (2:=Hxy).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLX_exp prec) rnd (x * y) - x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
(bpow (emin + prec - 1) <= bpow (emin + 2 * prec - 1))%R
2:apply bpow_le ; generalize (prec_gt_0 prec) ; clear ; omega.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (f - x * y)
unfold f; rewrite <- H1.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
format (F2R {| Fnum := m; Fexp := e |})
apply generic_format_F2R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
m <> 0%Z ->
(cexp (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
intros _.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <=
 Fexp {| Fnum := m; Fexp := e |})%Z
H3:Fexp {| Fnum := m; Fexp := e |} =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
(cexp (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
simpl in H2, H3.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
(cexp (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
unfold cexp, FLT_exp.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
(Z.max
   (mag beta (F2R {| Fnum := m; Fexp := e |}) - prec)
   emin <= e)%Z
case (Zmax_spec (mag beta (F2R (Float beta m e)) - prec) emin);
  intros (M1,M2); rewrite M2.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec >=
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin =
(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec)%Z
(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <= e)%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(emin <= e)%Z
apply Z.le_trans with (2:=H2).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec >=
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin =
(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec)%Z
(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <=
 Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}))%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(emin <= e)%Z
unfold cexp, FLX_exp.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec >=
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin =
(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec)%Z
(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <=
 mag beta (F2R {| Fnum := m; Fexp := e |}) - prec)%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(emin <= e)%Z
apply Z.le_refl.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(emin <= e)%Z
rewrite H3.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(emin <=
 Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
unfold cexp, FLX_exp.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(emin <= mag beta x - prec + (mag beta y - prec))%Z
assert (Hxy0:(x*y <> 0)%R).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
(x * y)%R <> 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
contradict Hr0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hr0:(x * y)%R = 0%R
(f - x * y)%R = 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
unfold f.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hr0:(x * y)%R = 0%R
(round beta (FLT_exp emin prec) rnd (x * y) - x * y)%R =
0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
rewrite Hr0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hr0:(x * y)%R = 0%R
(round beta (FLT_exp emin prec) rnd 0 - 0)%R = 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
rewrite round_0...beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hr0:(x * y)%R = 0%R
(0 - 0)%R = 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
ring.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
assert (Hx0: (x <> 0)%R).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
x <> 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
contradict Hxy0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:x = 0%R
(x * y)%R = 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
now rewrite Hxy0, Rmult_0_l.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
assert (Hy0: (y <> 0)%R).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
y <> 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
contradict Hxy0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hx0:x <> 0%R
Hxy0:y = 0%R
(x * y)%R = 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
now rewrite Hxy0, Rmult_0_r.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
(emin <= mag beta x - prec + (mag beta y - prec))%Z
destruct (mag beta x) as (ex,Ex) ; simpl.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(emin <= ex - prec + (mag beta y - prec))%Z
specialize (Ex Hx0).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(emin <= ex - prec + (mag beta y - prec))%Z
destruct (mag beta y) as (ey,Ey) ; simpl.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(emin <= ex - prec + (ey - prec))%Z
specialize (Ey Hy0).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(emin <= ex - prec + (ey - prec))%Z
assert (emin + 2 * prec -1 < ex + ey)%Z.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(emin + 2 * prec - 1 < ex + ey)%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
H:(emin + 2 * prec - 1 < ex + ey)%Z
(emin <= ex - prec + (ey - prec))%Z
2: omega.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(emin + 2 * prec - 1 < ex + ey)%Z
apply (lt_bpow beta).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(bpow (emin + 2 * prec - 1) < bpow (ex + ey))%R
apply Rle_lt_trans with (1:=Hxy).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs (x * y) < bpow (ex + ey))%R
rewrite Rabs_mult, bpow_plus.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs x * Rabs y < bpow ex * bpow ey)%R
apply Rmult_le_0_lt_compat; try apply Rabs_pos.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs x < bpow ex)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
apply Ex.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Hxy:(bpow (emin + 2 * prec - 1) <= Rabs (x * y))%R
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Hr0:(f - x * y)%R <> 0%R
Hxy':(x * y)%R <> 0%R
m, e:Z
H1:F2R {| Fnum := m; Fexp := e |} =
(round beta (FLT_exp emin prec) rnd (x * y) -
 x * y)%R
H2:(Generic_fmt.cexp beta (FLX_exp prec)
   (F2R {| Fnum := m; Fexp := e |}) <= e)%Z
H3:e =
(Generic_fmt.cexp beta (FLX_exp prec) x +
 Generic_fmt.cexp beta (FLX_exp prec) y)%Z
M1:(mag beta (F2R {| Fnum := m; Fexp := e |}) - prec <
 emin)%Z
M2:Z.max
  (mag beta (F2R {| Fnum := m; Fexp := e |}) -
   prec) emin = emin
Hxy0:(x * y)%R <> 0%R
Hx0:x <> 0%R
Hy0:y <> 0%R
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Ey:(bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
apply Ey.
Qed.

Lemma F2R_ge: forall (y:float beta),
   (F2R y <> 0)%R -> (bpow (Fexp y) <= Rabs (F2R y))%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall y : float beta,
F2R y <> 0%R -> (bpow (Fexp y) <= Rabs (F2R y))%R
Proof.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall y : float beta,
F2R y <> 0%R -> (bpow (Fexp y) <= Rabs (F2R y))%R
intros (ny,ey).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
F2R {| Fnum := ny; Fexp := ey |} <> 0%R ->
(bpow (Fexp {| Fnum := ny; Fexp := ey |}) <=
 Rabs (F2R {| Fnum := ny; Fexp := ey |}))%R
rewrite <- F2R_Zabs; unfold F2R; simpl.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
case (Zle_lt_or_eq 0 (Z.abs ny)).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
(0 <= Z.abs ny)%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
(0 < Z.abs ny)%Z ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
apply Z.abs_nonneg.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
(0 < Z.abs ny)%Z ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
intros Hy _.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
Hy:(0 < Z.abs ny)%Z
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
rewrite <- (Rmult_1_l (bpow _)) at 1.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
Hy:(0 < Z.abs ny)%Z
(1 * bpow ey <= IZR (Z.abs ny) * bpow ey)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
apply Rmult_le_compat_r.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
Hy:(0 < Z.abs ny)%Z
(0 <= bpow ey)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
Hy:(0 < Z.abs ny)%Z
(1 <= IZR (Z.abs ny))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
apply bpow_ge_0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
Hy:(0 < Z.abs ny)%Z
(1 <= IZR (Z.abs ny))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
apply IZR_le; omega.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
0%Z = Z.abs ny ->
(IZR ny * bpow ey)%R <> 0%R ->
(bpow ey <= IZR (Z.abs ny) * bpow ey)%R
intros H1 H2; contradict H2.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
H1:0%Z = Z.abs ny
(IZR ny * bpow ey)%R = 0%R
replace ny with 0%Z.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
H1:0%Z = Z.abs ny
(0 * bpow ey)%R = 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
H1:0%Z = Z.abs ny
0%Z = ny
simpl; ring.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
ny, ey:Z
H1:0%Z = Z.abs ny
0%Z = ny
now apply sym_eq, Z.abs_0_iff, sym_eq.
Qed.

Theorem mult_error_FLT_ge_bpow :
  forall x y e,
  format x -> format y ->
  (bpow (e+2*prec-1) <= Rabs (x * y))%R ->
  (round beta (FLT_exp emin prec) rnd (x * y) - (x * y) <> 0)%R ->
  (bpow e <= Rabs (round beta (FLT_exp emin prec) rnd (x * y) - (x * y)))%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x y : R) (e : Z),
format x ->
format y ->
(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R ->
(round beta (FLT_exp emin prec) rnd (x * y) - x * y)%R <>
0%R ->
(bpow e <=
 Rabs
   (round beta (FLT_exp emin prec) rnd (x * y) - x * y))%R
Proof with auto with typeclass_instances.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x y : R) (e : Z),
format x ->
format y ->
(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R ->
(round beta (FLT_exp emin prec) rnd (x * y) - x * y)%R <>
0%R ->
(bpow e <=
 Rabs
   (round beta (FLT_exp emin prec) rnd (x * y) - x * y))%R
intros x y e.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
format x ->
format y ->
(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R ->
(round beta (FLT_exp emin prec) rnd (x * y) - x * y)%R <>
0%R ->
(bpow e <=
 Rabs
   (round beta (FLT_exp emin prec) rnd (x * y) - x * y))%R
set (f := (round beta (FLT_exp emin prec) rnd (x * y))).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
format x ->
format y ->
(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R ->
(f - x * y)%R <> 0%R -> (bpow e <= Rabs (f - x * y))%R
intros Fx Fy H1.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
(f - x * y)%R <> 0%R -> (bpow e <= Rabs (f - x * y))%R
unfold f; rewrite Fx, Fy, <- F2R_mult.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
(round beta (FLT_exp emin prec) rnd
   (F2R
      (Fmult
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) x);
         Fexp := cexp x |}
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) y);
         Fexp := cexp y |})) -
 F2R
   (Fmult
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x);
      Fexp := cexp x |}
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp y |}))%R <> 0%R ->
(bpow e <=
 Rabs
   (round beta (FLT_exp emin prec) rnd
      (F2R
         (Fmult
            {|
            Fnum := Ztrunc
                      (scaled_mantissa beta
                         (FLT_exp emin prec) x);
            Fexp := cexp x |}
            {|
            Fnum := Ztrunc
                      (scaled_mantissa beta
                         (FLT_exp emin prec) y);
            Fexp := cexp y |})) -
    F2R
      (Fmult
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) x);
         Fexp := cexp x |}
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) y);
         Fexp := cexp y |})))%R
simpl Fmult.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
(round beta (FLT_exp emin prec) rnd
   (F2R
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}) -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLT_exp emin prec)
                x) *
           Ztrunc
             (scaled_mantissa beta (FLT_exp emin prec)
                y);
   Fexp := cexp x + cexp y |})%R <> 0%R ->
(bpow e <=
 Rabs
   (round beta (FLT_exp emin prec) rnd
      (F2R
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) x) *
                 Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) y);
         Fexp := cexp x + cexp y |}) -
    F2R
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}))%R
destruct (round_repr_same_exp beta (FLT_exp emin prec)
 rnd (Ztrunc (scaled_mantissa beta (FLT_exp emin prec) x) *
             Ztrunc (scaled_mantissa beta (FLT_exp emin prec) y))
      (cexp x + cexp y)) as (n,Hn).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
Hn:round beta (FLT_exp emin prec) rnd
  (F2R
     {|
     Fnum := Ztrunc
               (scaled_mantissa beta
                  (FLT_exp emin prec) x) *
             Ztrunc
               (scaled_mantissa beta
                  (FLT_exp emin prec) y);
     Fexp := cexp x + cexp y |}) =
F2R {| Fnum := n; Fexp := cexp x + cexp y |}
(round beta (FLT_exp emin prec) rnd
   (F2R
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}) -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLT_exp emin prec)
                x) *
           Ztrunc
             (scaled_mantissa beta (FLT_exp emin prec)
                y);
   Fexp := cexp x + cexp y |})%R <> 0%R ->
(bpow e <=
 Rabs
   (round beta (FLT_exp emin prec) rnd
      (F2R
         {|
         Fnum := Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) x) *
                 Ztrunc
                   (scaled_mantissa beta
                      (FLT_exp emin prec) y);
         Fexp := cexp x + cexp y |}) -
    F2R
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}))%R
rewrite Hn; clear Hn.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
(F2R {| Fnum := n; Fexp := cexp x + cexp y |} -
 F2R
   {|
   Fnum := Ztrunc
             (scaled_mantissa beta (FLT_exp emin prec)
                x) *
           Ztrunc
             (scaled_mantissa beta (FLT_exp emin prec)
                y);
   Fexp := cexp x + cexp y |})%R <> 0%R ->
(bpow e <=
 Rabs
   (F2R {| Fnum := n; Fexp := cexp x + cexp y |} -
    F2R
      {|
      Fnum := Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}))%R
rewrite <- F2R_minus, Fminus_same_exp.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta (FLT_exp emin prec)
               x) *
          Ztrunc
            (scaled_mantissa beta (FLT_exp emin prec)
               y);
  Fexp := cexp x + cexp y |} <> 0%R ->
(bpow e <=
 Rabs
   (F2R
      {|
      Fnum := n -
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}))%R
intros K.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
(bpow e <=
 Rabs
   (F2R
      {|
      Fnum := n -
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}))%R
eapply Rle_trans with (2:=F2R_ge _ K).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
(bpow e <=
 bpow
   (Fexp
      {|
      Fnum := n -
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) x) *
              Ztrunc
                (scaled_mantissa beta
                   (FLT_exp emin prec) y);
      Fexp := cexp x + cexp y |}))%R
simpl (Fexp _).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
(bpow e <= bpow (cexp x + cexp y))%R
apply bpow_le.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
(e <= cexp x + cexp y)%Z
unfold cexp, FLT_exp.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
(e <=
 Z.max (mag beta x - prec) emin +
 Z.max (mag beta y - prec) emin)%Z
destruct (mag beta x) as (ex,Hx).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
(e <=
 Z.max (Build_mag_prop beta x ex Hx - prec) emin +
 Z.max (mag beta y - prec) emin)%Z
destruct (mag beta y) as (ey,Hy).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(e <=
 Z.max (Build_mag_prop beta x ex Hx - prec) emin +
 Z.max (Build_mag_prop beta y ey Hy - prec) emin)%Z
simpl; apply Z.le_trans with ((ex-prec)+(ey-prec))%Z.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(e <= ex - prec + (ey - prec))%Z
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(ex - prec + (ey - prec) <=
 Z.max (ex - prec) emin + Z.max (ey - prec) emin)%Z
2: apply Zplus_le_compat; apply Z.le_max_l.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(e <= ex - prec + (ey - prec))%Z
assert (e + 2*prec -1< ex+ey)%Z;[idtac|omega].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(e + 2 * prec - 1 < ex + ey)%Z
apply lt_bpow with beta.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(bpow (e + 2 * prec - 1) < bpow (ex + ey))%R
apply Rle_lt_trans with (1:=H1).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs (x * y) < bpow (ex + ey))%R
rewrite Rabs_mult, bpow_plus.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs x * Rabs y < bpow ex * bpow ey)%R
apply Rmult_lt_compat.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(0 <= Rabs x)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(0 <= Rabs y)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs x < bpow ex)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
apply Rabs_pos.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(0 <= Rabs y)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs x < bpow ex)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
apply Rabs_pos.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs x < bpow ex)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
apply Hx.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
x <> 0%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
intros K'; contradict H1; apply Rlt_not_le.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
K':x = 0%R
(Rabs (x * y) < bpow (e + 2 * prec - 1))%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
rewrite K', Rmult_0_l, Rabs_R0; apply bpow_gt_0.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
(Rabs y < bpow ey)%R
apply Hy.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
H1:(bpow (e + 2 * prec - 1) <= Rabs (x * y))%R
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
y <> 0%R
intros K'; contradict H1; apply Rlt_not_le.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
e:Z
f:=round beta (FLT_exp emin prec) rnd (x * y):R
Fx:format x
Fy:format y
n:Z
K:F2R
  {|
  Fnum := n -
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) x) *
          Ztrunc
            (scaled_mantissa beta
               (FLT_exp emin prec) y);
  Fexp := cexp x + cexp y |} <> 0%R
ex:Z
Hx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
ey:Z
Hy:y <> 0%R -> (bpow (ey - 1) <= Rabs y < bpow ey)%R
K':y = 0%R
(Rabs (x * y) < bpow (e + 2 * prec - 1))%R
rewrite K', Rmult_0_r, Rabs_R0; apply bpow_gt_0.
Qed.

Lemma mult_bpow_exact_FLT :
  forall x e,
  format x ->
  (emin + prec - mag beta x <= e)%Z ->
  format (x * bpow e)%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x ->
(emin + prec - mag beta x <= e)%Z ->
format (x * bpow e)
Proof.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x ->
(emin + prec - mag beta x <= e)%Z ->
format (x * bpow e)
intros x e Fx He.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
format (x * bpow e)
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Zx:x = 0%R
format (x * bpow e)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
format (x * bpow e)
{beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Zx:x = 0%R
format (x * bpow e) rewrite Zx, Rmult_0_l; apply generic_format_0. }beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
format (x * bpow e)
rewrite Fx.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
format
  (F2R
     {|
     Fnum := Ztrunc
               (scaled_mantissa beta
                  (FLT_exp emin prec) x);
     Fexp := cexp x |} * bpow e)
set (mx := Ztrunc _); set (ex := cexp _).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
format (F2R {| Fnum := mx; Fexp := ex |} * bpow e)
pose (f := {| Fnum := mx; Fexp := ex + e |} : float beta).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
format (F2R {| Fnum := mx; Fexp := ex |} * bpow e)
apply (generic_format_F2R' _ _ _ f).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
F2R f = (F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <> 0%R ->
(cexp (F2R {| Fnum := mx; Fexp := ex |} * bpow e) <=
 Fexp f)%Z
{beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
F2R f = (F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R now unfold F2R; simpl; rewrite bpow_plus, Rmult_assoc. }beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <> 0%R ->
(cexp (F2R {| Fnum := mx; Fexp := ex |} * bpow e) <=
 Fexp f)%Z
intro Nzmx; unfold mx, ex; rewrite <- Fx.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(cexp (x * bpow e) <= Fexp f)%Z
unfold f, ex; simpl; unfold cexp; rewrite (mag_mult_bpow _ _ _ Nzx).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(FLT_exp emin prec (mag beta x + e) <=
 FLT_exp emin prec (mag beta x) + e)%Z
unfold FLT_exp; rewrite Z.max_l; [|omega]; rewrite <- Z.add_max_distr_r.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(mag beta x + e - prec <=
 Z.max (mag beta x - prec + e) (emin + e))%Z
set (n := (_ - _ + _)%Z); apply (Z.le_trans _ n); [unfold n; omega|].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(emin + prec - mag beta x <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
n:=(mag beta x - prec + e)%Z:Z
(n <= Z.max n (emin + e))%Z
apply Z.le_max_l.
Qed.

Lemma mult_bpow_pos_exact_FLT :
  forall x e,
  format x ->
  (0 <= e)%Z ->
  format (x * bpow e)%R.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x -> (0 <= e)%Z -> format (x * bpow e)
Proof.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall (x : R) (e : Z),
format x -> (0 <= e)%Z -> format (x * bpow e)
intros x e Fx He.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
format (x * bpow e)
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Zx:x = 0%R
format (x * bpow e)
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
format (x * bpow e)
{beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Zx:x = 0%R
format (x * bpow e) rewrite Zx, Rmult_0_l; apply generic_format_0. }beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
format (x * bpow e)
rewrite Fx.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
format
  (F2R
     {|
     Fnum := Ztrunc
               (scaled_mantissa beta
                  (FLT_exp emin prec) x);
     Fexp := cexp x |} * bpow e)
set (mx := Ztrunc _); set (ex := cexp _).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
format (F2R {| Fnum := mx; Fexp := ex |} * bpow e)
pose (f := {| Fnum := mx; Fexp := ex + e |} : float beta).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
format (F2R {| Fnum := mx; Fexp := ex |} * bpow e)
apply (generic_format_F2R' _ _ _ f).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
F2R f = (F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R
beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <> 0%R ->
(cexp (F2R {| Fnum := mx; Fexp := ex |} * bpow e) <=
 Fexp f)%Z
{beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
F2R f = (F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R now unfold F2R; simpl; rewrite bpow_plus, Rmult_assoc. }beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <> 0%R ->
(cexp (F2R {| Fnum := mx; Fexp := ex |} * bpow e) <=
 Fexp f)%Z
intro Nzmx; unfold mx, ex; rewrite <- Fx.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(cexp (x * bpow e) <= Fexp f)%Z
unfold f, ex; simpl; unfold cexp; rewrite (mag_mult_bpow _ _ _ Nzx).beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(FLT_exp emin prec (mag beta x + e) <=
 FLT_exp emin prec (mag beta x) + e)%Z
unfold FLT_exp; rewrite <-Z.add_max_distr_r.beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(Z.max (mag beta x + e - prec) emin <=
 Z.max (mag beta x - prec + e) (emin + e))%Z
replace (_ - _ + e)%Z with (mag beta x + e - prec)%Z; [ |ring].beta:radix
emin, prec:Z
prec_gt_0_:Prec_gt_0 prec
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x:R
e:Z
Fx:format x
He:(0 <= e)%Z
Nzx:x <> 0%R
mx:=Ztrunc
  (scaled_mantissa beta (FLT_exp emin prec) x):Z
ex:=cexp x:Z
f:={| Fnum := mx; Fexp := ex + e |} : float beta:float beta
Nzmx:(F2R {| Fnum := mx; Fexp := ex |} * bpow e)%R <>
0%R
(Z.max (mag beta x + e - prec) emin <=
 Z.max (mag beta x + e - prec) (emin + e))%Z
apply Z.max_le_compat_l; omega.
Qed.

End Fprop_mult_error_FLT.