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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.

Rounding to nearest, ties to even: existence, unicity...

Require Import Raux Defs Round_pred Generic_fmt Float_prop Ulp.

Notation ZnearestE := (Znearest (fun x => negb (Z.even x))).

Section Fcore_rnd_NE.

Variable beta : radix.

Notation bpow e := (bpow beta e).

Variable fexp : Z -> Z.

Context { valid_exp : Valid_exp fexp }.

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

Definition NE_prop (_ : R) f :=
  exists g : float beta, f = F2R g /\ canonical g /\ Z.even (Fnum g) = true.

Definition Rnd_NE_pt :=
  Rnd_NG_pt format NE_prop.

Definition DN_UP_parity_pos_prop :=
  forall x xd xu,
  (0 < x)%R ->
  ~ format x ->
  canonical xd ->
  canonical xu ->
  F2R xd = round beta fexp Zfloor x ->
  F2R xu = round beta fexp Zceil x ->
  Z.even (Fnum xu) = negb (Z.even (Fnum xd)).

Definition DN_UP_parity_prop :=
  forall x xd xu,
  ~ format x ->
  canonical xd ->
  canonical xu ->
  F2R xd = round beta fexp Zfloor x ->
  F2R xu = round beta fexp Zceil x ->
  Z.even (Fnum xu) = negb (Z.even (Fnum xd)).

Lemma DN_UP_parity_aux :
  DN_UP_parity_pos_prop ->
  DN_UP_parity_prop.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
DN_UP_parity_pos_prop -> DN_UP_parity_prop
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
DN_UP_parity_pos_prop -> DN_UP_parity_prop
intros Hpos x xd xu Hfx Hd Hu Hxd Hxu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct (total_order_T 0 x) as [[Hx|Hx]|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 < x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:0%R = x
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* . *)
exact (Hpos x xd xu Hx Hfx Hd Hu Hxd Hxu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:0%R = x
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
elim Hfx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:0%R = x
format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:0%R = x
format 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* . *)
assert (Hx': (0 < -x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Ropp_lt_cancel.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
(- - x < - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now rewrite Ropp_involutive, Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
xd, xu:float beta
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct xd as (md, ed).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed:Z
xu:float beta
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical xu
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (Fnum xu) =
negb (Z.even (Fnum {| Fnum := md; Fexp := ed |}))
destruct xu as (mu, eu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (Fnum {| Fnum := mu; Fexp := eu |}) =
negb (Z.even (Fnum {| Fnum := md; Fexp := ed |}))
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even mu = negb (Z.even md)
rewrite <- (Bool.negb_involutive (Z.even mu)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
negb (negb (Z.even mu)) = negb (Z.even md)
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
negb (Z.even mu) = Z.even md
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even md = negb (Z.even mu)
rewrite <- (Z.even_opp mu), <- (Z.even_opp md).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (- md) = negb (Z.even (- mu))
change (Z.even (Fnum (Float beta (-md) ed)) = negb (Z.even (Fnum (Float beta (-mu) eu)))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
Z.even (Fnum {| Fnum := - md; Fexp := ed |}) =
negb (Z.even (Fnum {| Fnum := - mu; Fexp := eu |}))
apply (Hpos (-x)%R _ _ Hx').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
~ format (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
H:format (- x)
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
apply Hfx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
H:format (- x)
format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
rewrite <- Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
H:format (- x)
format (- - x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
now apply canonical_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
canonical {| Fnum := - md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
now apply canonical_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - mu; Fexp := eu |} =
round beta fexp Zfloor (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
rewrite round_DN_opp, F2R_Zopp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
(- F2R {| Fnum := mu; Fexp := eu |})%R =
(- round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
now apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
F2R {| Fnum := - md; Fexp := ed |} =
round beta fexp Zceil (- x)
rewrite round_UP_opp, F2R_Zopp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hpos:DN_UP_parity_pos_prop
x:R
md, ed, mu, eu:Z
Hfx:~ format x
Hd:canonical {| Fnum := md; Fexp := ed |}
Hu:canonical {| Fnum := mu; Fexp := eu |}
Hxd:F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
Hxu:F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
Hx:(0 > x)%R
Hx':(0 < - x)%R
(- F2R {| Fnum := md; Fexp := ed |})%R =
(- round beta fexp Zfloor x)%R
now apply f_equal.
Qed.

Class Exists_NE :=
  exists_NE : Z.even beta = false \/ forall e,
  ((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\ ((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e).

Context { exists_NE_ : Exists_NE }.

Theorem DN_UP_parity_generic_pos :
  DN_UP_parity_pos_prop.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
DN_UP_parity_pos_prop
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
DN_UP_parity_pos_prop
intros x xd xu H0x Hfx Hd Hu Hxd Hxu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct (mag beta x) as (ex, Hexa).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
specialize (Hexa (Rgt_not_eq _ _ H0x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
generalize Hexa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs x < bpow ex)%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd)) intros Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa, Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite (Rabs_pos_eq _ (Rlt_le _ _ H0x)) in Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct (Zle_or_lt ex (fexp ex)) as [Hxe|Hxe].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* small x *)
assert (Hd3 : Fnum xd = Z0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Fnum xd = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply eq_0_F2R with beta (Fexp xd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
F2R {| Fnum := Fnum xd; Fexp := Fexp xd |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
change (F2R xd = R0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
F2R xd = R0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hxd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
round beta fexp Zfloor x = R0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply round_DN_small_pos with (1 := Hex) (2 := Hxe).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
assert (Hu3 : xu = Float beta (1 * Zpower beta (fexp ex - fexp (fexp ex + 1))) (fexp (fexp ex + 1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
xu =
{|
Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply canonical_unique with (1 := Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
canonical
  {|
  Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
  Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
  Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply (f_equal fexp).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp ex + 1)%Z =
mag beta
  (F2R
     {|
     Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
     Fexp := fexp (fexp ex + 1) |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
  Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- F2R_change_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp ex + 1)%Z =
mag beta (F2R {| Fnum := 1; Fexp := fexp ex |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
  Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now rewrite F2R_bpow, mag_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
  Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
  Fexp := fexp (fexp ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- F2R_change_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu = F2R {| Fnum := 1; Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite F2R_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
F2R xu = bpow (fexp ex)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
bpow (fexp ex) = F2R xu
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hxu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
bpow (fexp ex) = round beta fexp Zceil x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
round beta fexp Zceil x = bpow (fexp ex)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply round_UP_small_pos with (1 := Hex) (2 := Hxe).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hd3, Hu3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even
  (Fnum
     {|
     Fnum := 1 * beta ^ (fexp ex - fexp (fexp ex + 1));
     Fexp := fexp (fexp ex + 1) |}) = negb (Z.even 0)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Zmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even
  (Fnum
     {|
     Fnum := beta ^ (fexp ex - fexp (fexp ex + 1));
     Fexp := fexp (fexp ex + 1) |}) = negb (Z.even 0)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
Z.even (beta ^ (fexp ex - fexp (fexp ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct exists_NE_ as [H|H].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:Z.even beta = false
Z.even (beta ^ (fexp ex - fexp (fexp ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (fexp ex - fexp (fexp ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Zeven_Zpower_odd with (2 := H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:Z.even beta = false
(0 <= fexp ex - fexp (fexp ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (fexp ex - fexp (fexp ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Zle_minus_le_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:Z.even beta = false
(fexp (fexp ex + 1) <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (fexp ex - fexp (fexp ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (fexp ex - fexp (fexp ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite (proj2 (H ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (fexp ex - fexp ex)) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
(ex <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now rewrite Zminus_diag.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(ex <= fexp ex)%Z
Hd3:Fnum xd = 0%Z
Hu3:xu =
{|
Fnum := 1 *
        beta ^ (fexp ex - fexp (fexp ex + 1));
Fexp := fexp (fexp ex + 1) |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
(ex <= fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
exact Hxe.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* large x *)
assert (Hd4: (bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= F2R xd < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hxd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply (round_DN_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
format (bpow (ex - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(fexp (ex - 1 + 1) <= ex - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
ring_simplify (ex - 1 + 1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(fexp ex <= ex - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(bpow (ex - 1) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Rle_lt_trans with (2 := proj2 Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(round beta fexp Zfloor x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply (round_DN_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= F2R xd)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hxd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply (round_DN_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
format 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
assert (Hxe2 : (fexp (ex + 1) <= ex)%Z) by now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
assert (Hud: (F2R xu = F2R xd + ulp beta fexp x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
F2R xu = (F2R xd + ulp beta fexp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hxu, Hxd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
round beta fexp Zceil x =
(round beta fexp Zfloor x + ulp beta fexp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply round_UP_DN_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct (total_order_T (bpow ex) (F2R xu)) as [[Hu2|Hu2]|Hu2].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex < F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* - xu > bpow ex  *)
elim (Rlt_not_le _ _ Hu2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex < F2R xu)%R
(F2R xu <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hxu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex < F2R xu)%R
(round beta fexp Zceil x <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply round_bounded_large_pos...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* - xu = bpow ex *)
assert (Hu3: xu = Float beta (1 * Zpower beta (ex - fexp (ex + 1))) (fexp (ex + 1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply canonical_unique with (1 := Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
canonical
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply (f_equal fexp).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
(ex + 1)%Z =
mag beta
  (F2R
     {|
     Fnum := 1 * beta ^ (ex - fexp (ex + 1));
     Fexp := fexp (ex + 1) |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- F2R_change_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
(ex + 1)%Z =
mag beta (F2R {| Fnum := 1; Fexp := ex |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
(fexp (ex + 1) <= ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now rewrite F2R_bpow, mag_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
(fexp (ex + 1) <= ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R xu =
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- Hu2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
bpow ex =
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R
  {|
  Fnum := 1 * beta ^ (ex - fexp (ex + 1));
  Fexp := fexp (ex + 1) |} = bpow ex
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- F2R_change_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
F2R {| Fnum := 1; Fexp := ex |} = bpow ex
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
(fexp (ex + 1) <= ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply F2R_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
(fexp (ex + 1) <= ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
exact Hxe2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
assert (Hd3: xd = Float beta (Zpower beta (ex - fexp ex) - 1) (fexp ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
assert (H: F2R xd = F2R (Float beta (Zpower beta (ex - fexp ex) - 1) (fexp ex))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(IZR
   (Fnum
      {|
      Fnum := beta ^ (ex - fexp ex) - 1;
      Fexp := fexp ex |}) *
 bpow
   (Fexp
      {|
      Fnum := beta ^ (ex - fexp ex) - 1;
      Fexp := fexp ex |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd)) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(IZR (beta ^ (ex - fexp ex) - 1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite minus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
((IZR (beta ^ (ex - fexp ex)) - 1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold Rminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
((IZR (beta ^ (ex - fexp ex)) + - (1)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Rmult_plus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(IZR (beta ^ (ex - fexp ex)) * bpow (fexp ex) +
 - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite IZR_Zpower, <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(bpow (ex - fexp ex + fexp ex) +
 - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
ring_simplify (ex - fexp ex + fexp ex)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(bpow ex + - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hu2, Hud.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(F2R xd + ulp beta fexp x + - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(F2R xd + bpow (cexp beta fexp x) +
 - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(F2R xd + bpow (fexp (mag beta x)) +
 - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite mag_unique with beta x ex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(F2R xd + bpow (fexp ex) + - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(IZR (Fnum xd) * bpow (Fexp xd) + bpow (fexp ex) +
 - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(IZR (Fnum xd) * bpow (Fexp xd))%R =
(IZR (Fnum xd) * bpow (Fexp xd) + bpow (fexp ex) +
 - (1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd)) ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(bpow (ex - 1) <= x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
exact Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Zle_minus_le_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
(fexp ex <= ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply Zlt_le_weak.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply canonical_unique with (1 := Hd) (3 := H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
canonical
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply (f_equal fexp).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
ex =
mag beta
  (F2R
     {|
     Fnum := beta ^ (ex - fexp ex) - 1;
     Fexp := fexp ex |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite <- H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
ex = mag beta (F2R xd)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
H:F2R xd =
F2R
  {|
  Fnum := beta ^ (ex - fexp ex) - 1;
  Fexp := fexp ex |}
mag beta (F2R xd) = ex
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hd3, Hu3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even
  (Fnum
     {|
     Fnum := 1 * beta ^ (ex - fexp (ex + 1));
     Fexp := fexp (ex + 1) |}) =
negb
  (Z.even
     (Fnum
        {|
        Fnum := beta ^ (ex - fexp ex) - 1;
        Fexp := fexp ex |}))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold Fnum.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (1 * beta ^ (ex - fexp (ex + 1))) =
negb (Z.even (beta ^ (ex - fexp ex) - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Z.even_mul.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
(Z.even 1 || Z.even (beta ^ (ex - fexp (ex + 1))))%bool =
negb (Z.even (beta ^ (ex - fexp ex) - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd)) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
negb (Z.even (beta ^ (ex - fexp ex) - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold Zminus at 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
negb (Z.even (beta ^ (ex - fexp ex) + - (1)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Z.even_add.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
negb
  (Bool.eqb (Z.even (beta ^ (ex - fexp ex)))
     (Z.even (- (1))))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite eqb_sym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
negb
  (Bool.eqb (Z.even (- (1)))
     (Z.even (beta ^ (ex - fexp ex))))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd)) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
negb
  (if Z.even (beta ^ (ex - fexp ex))
   then false
   else true)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
fold (negb (Z.even (beta ^ (ex - fexp ex)))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
negb (negb (Z.even (beta ^ (ex - fexp ex))))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Bool.negb_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) =
Z.even (beta ^ (ex - fexp ex))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite (Z.even_pow beta (ex - fexp ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
(0 < ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd)) 2: omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
destruct exists_NE_.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:Z.even beta = false
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:Z.even beta = false
Z.even (beta ^ (ex - fexp (ex + 1))) = false
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Zeven_Zpower_odd with (2 := H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:Z.even beta = false
(0 <= ex - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
now apply Zle_minus_le_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (ex - fexp (ex + 1))) = Z.even beta
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
apply Z.even_pow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
(0 < ex - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
specialize (H ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:bpow ex = F2R xu
Hu3:xu =
{|
Fnum := 1 * beta ^ (ex - fexp (ex + 1));
Fexp := fexp (ex + 1) |}
Hd3:xd =
{|
Fnum := beta ^ (ex - fexp ex) - 1;
Fexp := fexp ex |}
H:((fexp ex < ex)%Z -> (fexp (ex + 1) < ex)%Z) /\
((ex <= fexp ex)%Z -> fexp (fexp ex + 1) = fexp ex)
(0 < ex - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hud:F2R xu = (F2R xd + ulp beta fexp x)%R
Hu2:(bpow ex > F2R xu)%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
(* - xu < bpow ex *)
revert Hud.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
F2R xu = (F2R xd + ulp beta fexp x)%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
F2R xu = (F2R xd + bpow (cexp beta fexp x))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(IZR (Fnum xu) * bpow (Fexp xu))%R =
(IZR (Fnum xd) * bpow (Fexp xd) +
 bpow (cexp beta fexp x))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite Hd, Hu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(IZR (Fnum xu) * bpow (cexp beta fexp (F2R xu)))%R =
(IZR (Fnum xd) * bpow (cexp beta fexp (F2R xd)) +
 bpow (cexp beta fexp x))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(IZR (Fnum xu) * bpow (fexp (mag beta (F2R xu))))%R =
(IZR (Fnum xd) * bpow (fexp (mag beta (F2R xd))) +
 bpow (fexp (mag beta x)))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
rewrite mag_unique with beta (F2R xu) ex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp (mag beta (F2R xd))) +
 bpow (fexp (mag beta x)))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
rewrite mag_unique with (1 := Hd4).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) +
 bpow (fexp (mag beta x)))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
rewrite mag_unique with (1 := Hexa).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R ->
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
Z.even (Fnum xu) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
replace (Fnum xu) with (Fnum xd + 1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
Z.even (Fnum xd + 1) = negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(Fnum xd + 1)%Z = Fnum xu
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
rewrite Z.even_add.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
Bool.eqb (Z.even (Fnum xd)) (Z.even 1) =
negb (Z.even (Fnum xd))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(Fnum xd + 1)%Z = Fnum xu
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
now apply eqb_sym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(Fnum xd + 1)%Z = Fnum xu
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
Fnum xu = (Fnum xd + 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
apply eq_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
IZR (Fnum xu) = IZR (Fnum xd + 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
rewrite plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
IZR (Fnum xu) = (IZR (Fnum xd) + 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
apply Rmult_eq_reg_r with (bpow (fexp ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(IZR (Fnum xu) * bpow (fexp ex))%R =
((IZR (Fnum xd) + 1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R =
((IZR (Fnum xd) + 1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R =
((IZR (Fnum xd) + 1) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
H:(IZR (Fnum xu) * bpow (fexp ex))%R =
(IZR (Fnum xd) * bpow (fexp ex) + bpow (fexp ex))%R
(bpow (fexp ex) > 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= Rabs (F2R xu) < bpow ex)%R
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= F2R xu < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 <= F2R xu)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(bpow (ex - 1) <= F2R xu)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(F2R xu < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 <= F2R xu)%R
apply Rle_trans with (1 := proj1 Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(x <= F2R xu)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(F2R xu < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 <= F2R xu)%R
rewrite Hxu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(F2R xu < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 <= F2R xu)%R
apply (round_UP_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(F2R xu < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 <= F2R xu)%R
exact Hu2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 <= F2R xu)%R
apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(0 < F2R xu)%R
apply Rlt_le_trans with (1 := H0x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(x <= F2R xu)%R
rewrite Hxu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
xd, xu:float beta
H0x:(0 < x)%R
Hfx:~ format x
Hd:canonical xd
Hu:canonical xu
Hxd:F2R xd = round beta fexp Zfloor x
Hxu:F2R xu = round beta fexp Zceil x
ex:Z
Hexa:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Hxe:(fexp ex < ex)%Z
Hd4:(bpow (ex - 1) <= Rabs (F2R xd) < bpow ex)%R
Hxe2:(fexp (ex + 1) <= ex)%Z
Hu2:(bpow ex > F2R xu)%R
(x <= round beta fexp Zceil x)%R
apply (round_UP_pt beta fexp x).
Qed.

Theorem DN_UP_parity_generic :
  DN_UP_parity_prop.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
DN_UP_parity_prop
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
DN_UP_parity_prop
apply DN_UP_parity_aux.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
DN_UP_parity_pos_prop
apply DN_UP_parity_generic_pos.
Qed.

Theorem Rnd_NE_pt_total :
  round_pred_total Rnd_NE_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_total Rnd_NE_pt
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_total Rnd_NE_pt
apply satisfies_any_imp_NG.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
satisfies_any format
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
NG_existence_prop format NE_prop
now apply generic_format_satisfies_any.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
NG_existence_prop format NE_prop
intros x d u Hf Hd Hu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
NE_prop x u \/ NE_prop x d
generalize (proj1 Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
format d -> NE_prop x u \/ NE_prop x d
unfold generic_format.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
d =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp d);
  Fexp := cexp beta fexp d |} ->
NE_prop x u \/ NE_prop x d
set (ed := cexp beta fexp d).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
d =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp d);
  Fexp := ed |} -> NE_prop x u \/ NE_prop x d
set (md := Ztrunc (scaled_mantissa beta fexp d)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
d = F2R {| Fnum := md; Fexp := ed |} ->
NE_prop x u \/ NE_prop x d
intros Hd1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
NE_prop x u \/ NE_prop x d
case_eq (Z.even md) ; [ intros He | intros Ho ].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
He:Z.even md = true
NE_prop x u \/ NE_prop x d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u \/ NE_prop x d
right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
He:Z.even md = true
NE_prop x d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u \/ NE_prop x d
exists (Float beta md ed).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
He:Z.even md = true
d = F2R {| Fnum := md; Fexp := ed |} /\
canonical {| Fnum := md; Fexp := ed |} /\
Z.even (Fnum {| Fnum := md; Fexp := ed |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u \/ NE_prop x d
unfold Generic_fmt.canonical.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
He:Z.even md = true
d = F2R {| Fnum := md; Fexp := ed |} /\
Fexp {| Fnum := md; Fexp := ed |} =
cexp beta fexp (F2R {| Fnum := md; Fexp := ed |}) /\
Z.even (Fnum {| Fnum := md; Fexp := ed |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u \/ NE_prop x d
rewrite <- Hd1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
He:Z.even md = true
d = d /\
Fexp {| Fnum := md; Fexp := ed |} = cexp beta fexp d /\
Z.even (Fnum {| Fnum := md; Fexp := ed |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u \/ NE_prop x d
now repeat split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u \/ NE_prop x d
left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
NE_prop x u
generalize (proj1 Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
format u -> NE_prop x u
unfold generic_format.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
u =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp u);
  Fexp := cexp beta fexp u |} -> NE_prop x u
set (eu := cexp beta fexp u).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
u =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp u);
  Fexp := eu |} -> NE_prop x u
set (mu := Ztrunc (scaled_mantissa beta fexp u)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
u = F2R {| Fnum := mu; Fexp := eu |} -> NE_prop x u
intros Hu1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
NE_prop x u
rewrite Hu1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
NE_prop x (F2R {| Fnum := mu; Fexp := eu |})
eexists ; repeat split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Z.even (Fnum {| Fnum := mu; Fexp := eu |}) = true
unfold Generic_fmt.canonical.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Fexp {| Fnum := mu; Fexp := eu |} =
cexp beta fexp (F2R {| Fnum := mu; Fexp := eu |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Z.even (Fnum {| Fnum := mu; Fexp := eu |}) = true
now rewrite <- Hu1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Z.even (Fnum {| Fnum := mu; Fexp := eu |}) = true
rewrite (DN_UP_parity_generic x (Float beta md ed) (Float beta mu eu)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
negb (Z.even (Fnum {| Fnum := md; Fexp := ed |})) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
~ format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
negb (Z.even md) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
~ format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
now rewrite Ho.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
~ format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
exact Hf.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := md; Fexp := ed |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
unfold Generic_fmt.canonical.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Fexp {| Fnum := md; Fexp := ed |} =
cexp beta fexp (F2R {| Fnum := md; Fexp := ed |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
now rewrite <- Hd1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
canonical {| Fnum := mu; Fexp := eu |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
unfold Generic_fmt.canonical.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Fexp {| Fnum := mu; Fexp := eu |} =
cexp beta fexp (F2R {| Fnum := mu; Fexp := eu |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
now rewrite <- Hu1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := md; Fexp := ed |} =
round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
rewrite <- Hd1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
d = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
apply Rnd_DN_pt_unique with (1 := Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Rnd_DN_pt format x (round beta fexp Zfloor x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
now apply round_DN_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
F2R {| Fnum := mu; Fexp := eu |} =
round beta fexp Zceil x
rewrite <- Hu1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
u = round beta fexp Zceil x
apply Rnd_UP_pt_unique with (1 := Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hf:~ format x
Hd:Rnd_DN_pt format x d
Hu:Rnd_UP_pt format x u
ed:=cexp beta fexp d:Z
md:=Ztrunc (scaled_mantissa beta fexp d):Z
Hd1:d = F2R {| Fnum := md; Fexp := ed |}
Ho:Z.even md = false
eu:=cexp beta fexp u:Z
mu:=Ztrunc (scaled_mantissa beta fexp u):Z
Hu1:u = F2R {| Fnum := mu; Fexp := eu |}
Rnd_UP_pt format x (round beta fexp Zceil x)
now apply round_UP_pt.
Qed.

Theorem Rnd_NE_pt_monotone :
  round_pred_monotone Rnd_NE_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_monotone Rnd_NE_pt
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_monotone Rnd_NE_pt
apply Rnd_NG_pt_monotone.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
Rnd_NG_pt_unique_prop format NE_prop
intros x d u Hd Hdn Hu Hun (cd, (Hd1, Hd2)) (cu, (Hu1, Hu2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
d = u
destruct (Req_dec x d) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x = d
d = u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = u
rewrite <- Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x = d
x = u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = u
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x = d
u = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = u
apply Rnd_UP_pt_idempotent with (1 := Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x = d
format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = u
rewrite Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x = d
format d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = u
apply Hd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = u
rewrite (DN_UP_parity_aux DN_UP_parity_generic_pos x cd cu) in Hu2 ; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ negb (Z.even (Fnum cd)) = true
Hx:x <> d
d = u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
~ format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cd = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
now rewrite (proj2 Hd2) in Hu2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
~ format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cd = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
intros Hf.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
Hf:format x
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cd = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
apply Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
Hf:format x
x = d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cd = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
Hf:format x
d = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cd = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
now apply Rnd_DN_pt_idempotent with (1 := Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cd = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
rewrite <- Hd1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
d = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
apply Rnd_DN_pt_unique with (1 := Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
Rnd_DN_pt format x (round beta fexp Zfloor x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
now apply round_DN_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
F2R cu = round beta fexp Zceil x
rewrite <- Hu1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
u = round beta fexp Zceil x
apply Rnd_UP_pt_unique with (1 := Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x, d, u:R
Hd:Rnd_DN_pt format x d
Hdn:Rnd_N_pt format x d
Hu:Rnd_UP_pt format x u
Hun:Rnd_N_pt format x u
cd:float beta
Hd1:d = F2R cd
Hd2:canonical cd /\ Z.even (Fnum cd) = true
cu:float beta
Hu1:u = F2R cu
Hu2:canonical cu /\ Z.even (Fnum cu) = true
Hx:x <> d
Rnd_UP_pt format x (round beta fexp Zceil x)
now apply round_UP_pt.
Qed.

Theorem Rnd_NE_pt_round :
  round_pred Rnd_NE_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred Rnd_NE_pt
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred Rnd_NE_pt
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_total Rnd_NE_pt
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_monotone Rnd_NE_pt
apply Rnd_NE_pt_total.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
round_pred_monotone Rnd_NE_pt
apply Rnd_NE_pt_monotone.
Qed.

Lemma round_NE_pt_pos :
  forall x,
  (0 < x)%R ->
  Rnd_NE_pt x (round beta fexp ZnearestE x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
(0 < x)%R -> Rnd_NE_pt x (round beta fexp ZnearestE x)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
(0 < x)%R -> Rnd_NE_pt x (round beta fexp ZnearestE x)
intros x Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
Rnd_N_pt format x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
NE_prop x (round beta fexp ZnearestE x) \/
(forall f2 : R,
 Rnd_N_pt format x f2 ->
 f2 = round beta fexp ZnearestE x)
now apply round_N_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
NE_prop x (round beta fexp ZnearestE x) \/
(forall f2 : R,
 Rnd_N_pt format x f2 ->
 f2 = round beta fexp ZnearestE x)
unfold NE_prop.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
(exists g : float beta,
   round beta fexp ZnearestE x = F2R g /\
   canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R,
 Rnd_N_pt format x f2 ->
 f2 = round beta fexp ZnearestE x)
set (mx := scaled_mantissa beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
(exists g : float beta,
   round beta fexp ZnearestE x = F2R g /\
   canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R,
 Rnd_N_pt format x f2 ->
 f2 = round beta fexp ZnearestE x)
set (xr := round beta fexp ZnearestE x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct (Req_dec (mx - IZR (Zfloor mx)) (/2)) as [Hm|Hm].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* midpoint *)
left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
exists g : float beta,
  xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
exists (Float beta (Ztrunc (scaled_mantissa beta fexp xr)) (cexp beta fexp xr)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
xr =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |} /\
canonical
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |} /\
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
xr =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
canonical
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |} /\
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply round_N_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
canonical
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |} /\
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
canonical
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold Generic_fmt.canonical.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Fexp
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |} =
cexp beta fexp
  (F2R
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
cexp beta fexp xr =
cexp beta fexp
  (F2R
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
xr =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp xr);
  Fexp := cexp beta fexp xr |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply round_N_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Fnum
     {|
     Fnum := Ztrunc (scaled_mantissa beta fexp xr);
     Fexp := cexp beta fexp xr |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even (Ztrunc (scaled_mantissa beta fexp xr)) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold xr, round, Znearest.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := match
                     Rcompare
                       (scaled_mantissa beta fexp x -
                        IZR
                          (Zfloor
                             (scaled_mantissa beta
                                fexp x))) (/ 2)
                   with
                   | Eq =>
                       if
                        negb
                          (Z.even
                             (Zfloor
                                (scaled_mantissa beta
                                   fexp x)))
                       then
                        Zceil
                          (scaled_mantissa beta fexp x)
                       else
                        Zfloor
                          (scaled_mantissa beta fexp x)
                   | Lt =>
                       Zfloor
                         (scaled_mantissa beta fexp x)
                   | Gt =>
                       Zceil
                         (scaled_mantissa beta fexp x)
                   end;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
fold mx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := match
                     Rcompare (mx - IZR (Zfloor mx))
                       (/ 2)
                   with
                   | Eq =>
                       if negb (Z.even (Zfloor mx))
                       then Zceil mx
                       else Zfloor mx
                   | Lt => Zfloor mx
                   | Gt => Zceil mx
                   end;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := match Rcompare (/ 2) (/ 2) with
                   | Eq =>
                       if negb (Z.even (Zfloor mx))
                       then Zceil mx
                       else Zfloor mx
                   | Lt => Zfloor mx
                   | Gt => Zceil mx
                   end;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Rcompare_Eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb (Z.even (Zfloor mx))
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
(/ 2)%R = (/ 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr) 2: apply refl_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb (Z.even (Zfloor mx))
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
case_eq (Z.even (Zfloor mx)) ; intros Hmx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb true
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* . even floor *)
change (Z.even (Ztrunc (scaled_mantissa beta fexp (round beta fexp Zfloor x))) = true).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct (Rle_or_lt (round beta fexp Zfloor x) 0) as [Hr|Hr].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite (Rle_antisym _ _ Hr).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
Z.even (Ztrunc (scaled_mantissa beta fexp 0)) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
Z.even (Ztrunc (0 * bpow (- cexp beta fexp 0))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Rmult_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
Z.even (Ztrunc 0) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite <- (round_0 beta fexp Zfloor).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
(round beta fexp Zfloor 0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply round_le...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(round beta fexp Zfloor x <= 0)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zfloor x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite scaled_mantissa_DN...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = true
Hr:(0 < round beta fexp Zfloor x)%R
Z.even
  (Ztrunc (IZR (Zfloor (scaled_mantissa beta fexp x)))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (F2R
           {|
           Fnum := if negb false
                   then Zceil mx
                   else Zfloor mx;
           Fexp := cexp beta fexp x |}))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* . odd floor *)
change (Z.even (Ztrunc (scaled_mantissa beta fexp (round beta fexp Zceil x))) = true).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct (mag beta x) as (ex, Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
specialize (Hex (Rgt_not_eq _ _ Hx)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite (Rabs_pos_eq _ (Rlt_le _ _ Hx)) in Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct (Z_lt_le_dec (fexp ex) ex) as [He|He].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* .. large pos *)
assert (Hu := round_bounded_large_pos _ _ Zceil _ _ He Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
assert (Hfc: Zceil mx = (Zfloor mx + 1)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Zceil mx = (Zfloor mx + 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply Zceil_floor_neq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
IZR (Zfloor mx) <> mx
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite H in Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - mx)%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold Rminus in Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx + - mx)%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Rplus_opp_r in Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:0%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
elim (Rlt_irrefl 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:0%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
(0 < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Hm at 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:0%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
(0 < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:0%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
H:IZR (Zfloor mx) = mx
(0 < 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now apply IZR_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct (proj2 Hu) as [Hu'|Hu'].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* ... u <> bpow *)
unfold scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc
     (round beta fexp Zceil x *
      bpow
        (- cexp beta fexp (round beta fexp Zceil x)))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite cexp_fexp_pos with (1 := conj (proj1 Hu) Hu').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc (round beta fexp Zceil x * bpow (- fexp ex))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold round, F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc
     (IZR
        (Fnum
           {|
           Fnum := Zceil (scaled_mantissa beta fexp x);
           Fexp := cexp beta fexp x |}) *
      bpow
        (Fexp
           {|
           Fnum := Zceil (scaled_mantissa beta fexp x);
           Fexp := cexp beta fexp x |}) *
      bpow (- fexp ex))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc
     (IZR (Zceil (scaled_mantissa beta fexp x)) *
      bpow (cexp beta fexp x) * bpow (- fexp ex))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite cexp_fexp_pos with (1 := Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc
     (IZR (Zceil (scaled_mantissa beta fexp x)) *
      bpow (fexp ex) * bpow (- fexp ex))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_r, Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even
  (Ztrunc (IZR (Zceil (scaled_mantissa beta fexp x)))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even (Zceil (scaled_mantissa beta fexp x)) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
fold mx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even (Zceil mx) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Hfc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':(round beta fexp Zceil x < bpow ex)%R
Z.even (Zfloor mx + 1) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now rewrite Z.even_add, Hmx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* ... u = bpow *)
rewrite Hu'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even (Ztrunc (scaled_mantissa beta fexp (bpow ex))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold scaled_mantissa, cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even
  (Ztrunc
     (bpow ex * bpow (- fexp (mag beta (bpow ex))))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite mag_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even (Ztrunc (bpow ex * bpow (- fexp (ex + 1)))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite <- bpow_plus, <- IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even (Ztrunc (IZR (beta ^ (ex + - fexp (ex + 1))))) =
true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
case_eq (Z.even beta) ; intros Hr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct exists_NE_ as [Hs|Hs].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now rewrite Hs in Hr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
destruct (Hs ex) as (H,_).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
H:(fexp ex < ex)%Z -> (fexp (ex + 1) < ex)%Z
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Z.even_pow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
H:(fexp ex < ex)%Z -> (fexp (ex + 1) < ex)%Z
Z.even beta = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
H:(fexp ex < ex)%Z -> (fexp (ex + 1) < ex)%Z
(0 < ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
exact Hr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = true
Hs:forall e : Z,
((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\
((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e)
H:(fexp ex < ex)%Z -> (fexp (ex + 1) < ex)%Z
(0 < ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
assert (Z.even (Zfloor mx) = true).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (Zfloor mx) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
H:Z.even (Zfloor mx) = true
Z.even (beta ^ (ex + - fexp (ex + 1))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr) 2: now rewrite H in Hmx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (Zfloor mx) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
replace (Zfloor mx) with (Zceil mx + -1)%Z by omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (Zceil mx + -1) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Z.even_add.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Bool.eqb (Z.even (Zceil mx)) (Z.even (-1)) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply eqb_true.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (Zceil mx) = Z.even (-1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold mx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (Zceil (scaled_mantissa beta fexp x)) =
Z.even (-1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
replace (Zceil (scaled_mantissa beta fexp x)) with (Zpower beta (ex - fexp ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
Z.even (beta ^ (ex - fexp ex)) = Z.even (-1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(beta ^ (ex - fexp ex))%Z =
Zceil (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Zeven_Zpower_odd with (2 := Hr).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
false = Z.even (-1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(beta ^ (ex - fexp ex))%Z =
Zceil (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(beta ^ (ex - fexp ex))%Z =
Zceil (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(beta ^ (ex - fexp ex))%Z =
Zceil (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply eq_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
IZR (beta ^ (ex - fexp ex)) =
IZR (Zceil (scaled_mantissa beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (ex - fexp ex) =
IZR (Zceil (scaled_mantissa beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(0 <= ex - fexp ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr) 2: omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (ex - fexp ex) =
IZR (Zceil (scaled_mantissa beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply Rmult_eq_reg_r with (bpow (fexp ex)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(bpow (ex - fexp ex) * bpow (fexp ex))%R =
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold Zminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(bpow (ex + - fexp ex) * bpow (fexp ex))%R =
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(bpow ex * bpow (- fexp ex) * bpow (fexp ex))%R =
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite Rmult_assoc, <- bpow_plus, Zplus_opp_l, Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow ex =
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
pattern (fexp ex) ; rewrite <- cexp_fexp_pos with (1 := Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow ex =
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
bpow (fexp ex) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
Hr:Z.even beta = false
(bpow (fexp ex) > 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
generalize (proj1 (valid_exp ex) He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(fexp ex < ex)%Z
Hu:(bpow (ex - 1) <= round beta fexp Zceil x <=
 bpow ex)%R
Hfc:Zceil mx = (Zfloor mx + 1)%Z
Hu':round beta fexp Zceil x = bpow ex
(fexp (ex + 1) <= ex)%Z ->
(0 <= ex + - fexp (ex + 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* .. small pos *)
assert (Z.even (Zfloor mx) = true).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even (Zfloor mx) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
H:Z.even (Zfloor mx) = true
Z.even
  (Ztrunc
     (scaled_mantissa beta fexp
        (round beta fexp Zceil x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr) 2: now rewrite H in Hmx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even (Zfloor mx) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
unfold mx, scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even (Zfloor (x * bpow (- cexp beta fexp x))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
rewrite cexp_fexp_pos with (1 := Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R = (/ 2)%R
Hmx:Z.even (Zfloor mx) = false
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
He:(ex <= fexp ex)%Z
Z.even (Zfloor (x * bpow (- fexp ex))) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
now rewrite mantissa_DN_small_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
(exists g : float beta,
   xr = F2R g /\ canonical g /\ Z.even (Fnum g) = true) \/
(forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr)
(* not midpoint *)
right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
forall f2 : R, Rnd_N_pt format x f2 -> f2 = xr
intros g Hg.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
g = xr
destruct (Req_dec x g) as [Hxg|Hxg].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x = g
g = xr
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
g = xr
rewrite <- Hxg.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x = g
x = xr
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
g = xr
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x = g
xr = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
g = xr
apply round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x = g
format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
g = xr
rewrite Hxg.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x = g
format g
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
g = xr
apply Hg.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
g = xr
set (d := round beta fexp Zfloor x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
g = xr
set (u := round beta fexp Zceil x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
g = xr
apply Rnd_N_pt_unique with (d := d) (u := u) (4 := Hg).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
Rnd_DN_pt format x d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
Rnd_UP_pt format x u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(x - d)%R <> (u - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
Rnd_N_pt format x xr
now apply round_DN_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
Rnd_UP_pt format x u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(x - d)%R <> (u - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
Rnd_N_pt format x xr
now apply round_UP_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(x - d)%R <> (u - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
Rnd_N_pt format x xr
2: now apply round_N_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(x - d)%R <> (u - x)%R
rewrite <- (scaled_mantissa_mult_bpow beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(scaled_mantissa beta fexp x * bpow (cexp beta fexp x) -
 d)%R <>
(u -
 scaled_mantissa beta fexp x * bpow (cexp beta fexp x))%R
unfold d, u, round, F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(scaled_mantissa beta fexp x * bpow (cexp beta fexp x) -
 IZR
   (Fnum
      {|
      Fnum := Zfloor (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Zfloor (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}))%R <>
(IZR
   (Fnum
      {|
      Fnum := Zceil (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Zceil (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) -
 scaled_mantissa beta fexp x * bpow (cexp beta fexp x))%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(scaled_mantissa beta fexp x * bpow (cexp beta fexp x) -
 IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R <>
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) -
 scaled_mantissa beta fexp x * bpow (cexp beta fexp x))%R fold mx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
(mx * bpow (cexp beta fexp x) -
 IZR (Zfloor mx) * bpow (cexp beta fexp x))%R <>
(IZR (Zceil mx) * bpow (cexp beta fexp x) -
 mx * bpow (cexp beta fexp x))%R
rewrite <- 2!Rmult_minus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R <>
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
False
apply Rmult_eq_reg_r in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
apply Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
(mx - IZR (Zfloor mx))%R = (/ 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
apply Rcompare_Eq_inv.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Rcompare (mx - IZR (Zfloor mx)) (/ 2) = Eq
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
rewrite Rcompare_floor_ceil_middle.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Rcompare (mx - IZR (Zfloor mx)) (IZR (Zceil mx) - mx) =
Eq
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
IZR (Zfloor mx) <> mx
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
now apply Rcompare_Eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
IZR (Zfloor mx) <> mx
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
contradict Hxg.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
x = g
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
g = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
apply Rnd_N_pt_idempotent with (1 := Hg).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
format x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
rewrite <- (scaled_mantissa_mult_bpow beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
format
  (scaled_mantissa beta fexp x *
   bpow (cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
fold mx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
format (mx * bpow (cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
rewrite <- Hxg.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
format (IZR (Zfloor mx) * bpow (cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
change (IZR (Zfloor mx) * bpow (cexp beta fexp x))%R with d.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(mx - IZR (Zfloor mx))%R = (IZR (Zceil mx) - mx)%R
Hxg:IZR (Zfloor mx) = mx
format d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
now eapply round_DN_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
bpow (cexp beta fexp x) <> 0%R
apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(0 < x)%R
mx:=scaled_mantissa beta fexp x:R
xr:=round beta fexp ZnearestE x:R
Hm:(mx - IZR (Zfloor mx))%R <> (/ 2)%R
g:R
Hg:Rnd_N_pt format x g
Hxg:x <> g
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((mx - IZR (Zfloor mx)) * bpow (cexp beta fexp x))%R =
((IZR (Zceil mx) - mx) * bpow (cexp beta fexp x))%R
(bpow (cexp beta fexp x) > 0)%R
apply bpow_gt_0.
Qed.

Theorem round_NE_opp :
  forall x,
  round beta fexp ZnearestE (-x) = (- round beta fexp ZnearestE x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
round beta fexp ZnearestE (- x) =
(- round beta fexp ZnearestE x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
round beta fexp ZnearestE (- x) =
(- round beta fexp ZnearestE x)%R
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
round beta fexp ZnearestE (- x) =
(- round beta fexp ZnearestE x)%R
unfold round.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
F2R
  {|
  Fnum := ZnearestE (scaled_mantissa beta fexp (- x));
  Fexp := cexp beta fexp (- x) |} =
(-
 F2R
   {|
   Fnum := ZnearestE (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
F2R
  {|
  Fnum := ZnearestE (scaled_mantissa beta fexp (- x));
  Fexp := cexp beta fexp (- x) |} =
(-
 F2R
   {|
   Fnum := ZnearestE (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
rewrite scaled_mantissa_opp, cexp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
F2R
  {|
  Fnum := ZnearestE (- scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |} =
(-
 F2R
   {|
   Fnum := ZnearestE (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
rewrite Znearest_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
F2R
  {|
  Fnum := -
          Znearest
            (fun t : Z =>
             negb (negb (Z.even (- (t + 1)))))
            (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |} =
(-
 F2R
   {|
   Fnum := ZnearestE (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
rewrite <- F2R_Zopp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
F2R
  {|
  Fnum := -
          Znearest
            (fun t : Z =>
             negb (negb (Z.even (- (t + 1)))))
            (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |} =
F2R
  {|
  Fnum := - ZnearestE (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |}
apply (f_equal (fun v => F2R (Float beta (-v) _))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Znearest
  (fun t : Z => negb (negb (Z.even (- (t + 1)))))
  (scaled_mantissa beta fexp x) =
ZnearestE (scaled_mantissa beta fexp x)
set (m := scaled_mantissa beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
Znearest
  (fun t : Z => negb (negb (Z.even (- (t + 1))))) m =
ZnearestE m
unfold Znearest.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
match Rcompare (m - IZR (Zfloor m)) (/ 2) with
| Eq =>
    if negb (negb (Z.even (- (Zfloor m + 1))))
    then Zceil m
    else Zfloor m
| Lt => Zfloor m
| Gt => Zceil m
end =
match Rcompare (m - IZR (Zfloor m)) (/ 2) with
| Eq =>
    if negb (Z.even (Zfloor m))
    then Zceil m
    else Zfloor m
| Lt => Zfloor m
| Gt => Zceil m
end
case Rcompare ; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
(if negb (negb (Z.even (- (Zfloor m + 1))))
 then Zceil m
 else Zfloor m) =
(if negb (Z.even (Zfloor m))
 then Zceil m
 else Zfloor m)
apply (f_equal (fun (b : bool) => if b then Zceil m else Zfloor m)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
negb (negb (Z.even (- (Zfloor m + 1)))) =
negb (Z.even (Zfloor m))
rewrite Bool.negb_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
Z.even (- (Zfloor m + 1)) = negb (Z.even (Zfloor m))
rewrite Z.even_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
Z.even (Zfloor m + 1) = negb (Z.even (Zfloor m))
rewrite Z.even_add.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
m:=scaled_mantissa beta fexp x:R
Bool.eqb (Z.even (Zfloor m)) (Z.even 1) =
negb (Z.even (Zfloor m))
now rewrite eqb_sym.
Qed.

Lemma round_NE_abs:
  forall x : R,
  round beta fexp ZnearestE (Rabs x) = Rabs (round beta fexp ZnearestE x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
round beta fexp ZnearestE (Rabs x) =
Rabs (round beta fexp ZnearestE x)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
round beta fexp ZnearestE (Rabs x) =
Rabs (round beta fexp ZnearestE x)
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
round beta fexp ZnearestE (Rabs x) =
Rabs (round beta fexp ZnearestE x)
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE (Rabs x)
unfold Rabs at 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE
  (if Rcase_abs x then (- x)%R else x)
destruct (Rcase_abs x) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE x
rewrite round_NE_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rabs (round beta fexp ZnearestE x) =
(- round beta fexp ZnearestE x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE x
apply Rabs_left1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
(round beta fexp ZnearestE x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE x
rewrite <- (round_0 beta fexp ZnearestE).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
(round beta fexp ZnearestE x <=
 round beta fexp ZnearestE 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE x
apply round_le...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
(x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE x
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
Rabs (round beta fexp ZnearestE x) =
round beta fexp ZnearestE x
apply Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
(0 <= round beta fexp ZnearestE x)%R
rewrite <- (round_0 beta fexp ZnearestE).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
(round beta fexp ZnearestE 0 <=
 round beta fexp ZnearestE x)%R
apply round_le...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x >= 0)%R
(0 <= x)%R
now apply Rge_le.
Qed.

Theorem round_NE_pt :
  forall x,
  Rnd_NE_pt x (round beta fexp ZnearestE x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
Rnd_NE_pt x (round beta fexp ZnearestE x)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
forall x : R,
Rnd_NE_pt x (round beta fexp ZnearestE x)
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Rnd_NE_pt x (round beta fexp ZnearestE x)
destruct (total_order_T x 0) as [[Hx|Hx]|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
apply Rnd_NG_pt_opp_inv.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
forall x0 : R, format x0 -> format (- x0)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
forall x0 f : R, NE_prop x0 f -> NE_prop (- x0) (- f)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
forall x0 f : R, NE_prop x0 f -> NE_prop (- x0) (- f)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
unfold NE_prop.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
R ->
forall f : R,
(exists g : float beta,
   f = F2R g /\ canonical g /\ Z.even (Fnum g) = true) ->
exists g : float beta,
  (- f)%R = F2R g /\
  canonical g /\ Z.even (Fnum g) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
intros _ f ((mg,eg),(H1,(H2,H3))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
exists g : float beta,
  (- f)%R = F2R g /\
  canonical g /\ Z.even (Fnum g) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
exists (Float beta (- mg) eg).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
(- f)%R = F2R {| Fnum := - mg; Fexp := eg |} /\
canonical {| Fnum := - mg; Fexp := eg |} /\
Z.even (Fnum {| Fnum := - mg; Fexp := eg |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
repeat split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
(- f)%R = F2R {| Fnum := - mg; Fexp := eg |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
canonical {| Fnum := - mg; Fexp := eg |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
Z.even (Fnum {| Fnum := - mg; Fexp := eg |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
(- F2R {| Fnum := mg; Fexp := eg |})%R =
F2R {| Fnum := - mg; Fexp := eg |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
canonical {| Fnum := - mg; Fexp := eg |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
Z.even (Fnum {| Fnum := - mg; Fexp := eg |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
now rewrite F2R_Zopp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
canonical {| Fnum := - mg; Fexp := eg |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
Z.even (Fnum {| Fnum := - mg; Fexp := eg |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
now apply canonical_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
Z.even (Fnum {| Fnum := - mg; Fexp := eg |}) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
f:R
mg, eg:Z
H1:f = F2R {| Fnum := mg; Fexp := eg |}
H2:canonical {| Fnum := mg; Fexp := eg |}
H3:Z.even (Fnum {| Fnum := mg; Fexp := eg |}) = true
Z.even (- mg) = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop 
  (- x) (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
now rewrite Z.even_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop (- x)
  (- round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
rewrite <- round_NE_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
Rnd_NG_pt format NE_prop (- x)
  (round beta fexp ZnearestE (- x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
apply round_NE_pt_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
rewrite Hx, round_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
Rnd_NE_pt 0 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
apply Rnd_NG_pt_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:x = 0%R
format 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
exists_NE_:Exists_NE
x:R
Hx:(x > 0)%R
Rnd_NE_pt x (round beta fexp ZnearestE x)
now apply round_NE_pt_pos.
Qed.

End Fcore_rnd_NE.

Notations for backward-compatibility with Flocq 1.4.

Notation rndNE := ZnearestE (only parsing).