Built with Alectryon, running Coq+SerAPI v8.10.0+0.7.0. Coq sources are in this panel; goals and messages will appear in the other. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus.

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.

Remainder of the division and square root are in the FLX format

Require Import Psatz.
Require Import Core Operations Relative Sterbenz Mult_error.

Section Fprop_divsqrt_error.

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

Variable prec : Z.

Lemma generic_format_plus_prec :
  forall fexp, (forall e, (fexp e <= e - prec)%Z) ->
  forall x y (fx fy: float beta),
  (x = F2R fx)%R -> (y = F2R fy)%R -> (Rabs (x+y) < bpow (prec+Fexp fx))%R ->
  (Rabs (x+y) < bpow (prec+Fexp fy))%R ->
  generic_format beta fexp (x+y)%R.beta:radix
prec:Z
forall fexp : Z -> Z,
(forall e : Z, (fexp e <= e - prec)%Z) ->
forall (x y : R) (fx fy : float beta),
x = F2R fx ->
y = F2R fy ->
(Rabs (x + y) < bpow (prec + Fexp fx))%R ->
(Rabs (x + y) < bpow (prec + Fexp fy))%R ->
generic_format beta fexp (x + y)
Proof.beta:radix
prec:Z
forall fexp : Z -> Z,
(forall e : Z, (fexp e <= e - prec)%Z) ->
forall (x y : R) (fx fy : float beta),
x = F2R fx ->
y = F2R fy ->
(Rabs (x + y) < bpow (prec + Fexp fx))%R ->
(Rabs (x + y) < bpow (prec + Fexp fy))%R ->
generic_format beta fexp (x + y)
intros fexp Hfexp x y fx fy Hx Hy H1 H2.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
generic_format beta fexp (x + y)
case (Req_dec (x+y) 0); intros H.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R = 0%R
generic_format beta fexp (x + y)
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
generic_format beta fexp (x + y)
rewrite H; apply generic_format_0.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
generic_format beta fexp (x + y)
rewrite Hx, Hy, <- F2R_plus.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
generic_format beta fexp (F2R (Fplus fx fy))
apply generic_format_F2R.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
(let (Fnum, _) := Fplus fx fy in Fnum) <> 0%Z ->
(cexp beta fexp
   (F2R
      {|
      Fnum := let (Fnum, _) := Fplus fx fy in Fnum;
      Fexp := let (_, Fexp) := Fplus fx fy in Fexp |}) <=
 (let (_, Fexp) := Fplus fx fy in Fexp))%Z
intros _.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
(cexp beta fexp
   (F2R
      {|
      Fnum := let (Fnum, _) := Fplus fx fy in Fnum;
      Fexp := let (_, Fexp) := Fplus fx fy in Fexp |}) <=
 (let (_, Fexp) := Fplus fx fy in Fexp))%Z
case_eq (Fplus fx fy).beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
forall Fnum Fexp : Z,
Fplus fx fy = {| Fnum := Fnum; Fexp := Fexp |} ->
(cexp beta fexp (F2R {| Fnum := Fnum; Fexp := Fexp |}) <=
 Fexp)%Z
intros mz ez Hz.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(cexp beta fexp (F2R {| Fnum := mz; Fexp := ez |}) <=
 ez)%Z
rewrite <- Hz.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(cexp beta fexp (F2R (Fplus fx fy)) <= ez)%Z
apply Z.le_trans with (Z.min (Fexp fx) (Fexp fy)).beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(cexp beta fexp (F2R (Fplus fx fy)) <=
 Z.min (Fexp fx) (Fexp fy))%Z
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
rewrite F2R_plus, <- Hx, <- Hy.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(cexp beta fexp (x + y) <= Z.min (Fexp fx) (Fexp fy))%Z
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
unfold cexp.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(fexp (mag beta (x + y)) <= Z.min (Fexp fx) (Fexp fy))%Z
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
apply Z.le_trans with (1:=Hfexp _).beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(mag beta (x + y) - prec <= Z.min (Fexp fx) (Fexp fy))%Z
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
apply Zplus_le_reg_l with prec; ring_simplify.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(mag beta (x + y) <= prec + Z.min (Fexp fx) (Fexp fy))%Z
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
apply mag_le_bpow with (1 := H).beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Rabs (x + y) <
 bpow (prec + Z.min (Fexp fx) (Fexp fy)))%R
beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
now apply Z.min_case.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Z.min (Fexp fx) (Fexp fy) <= ez)%Z
rewrite <- Fexp_Fplus, Hz.beta:radix
prec:Z
fexp:Z -> Z
Hfexp:forall e : Z, (fexp e <= e - prec)%Z
x, y:R
fx, fy:float beta
Hx:x = F2R fx
Hy:y = F2R fy
H1:(Rabs (x + y) < bpow (prec + Fexp fx))%R
H2:(Rabs (x + y) < bpow (prec + Fexp fy))%R
H:(x + y)%R <> 0%R
mz, ez:Z
Hz:Fplus fx fy = {| Fnum := mz; Fexp := ez |}
(Fexp {| Fnum := mz; Fexp := ez |} <= ez)%Z
apply Z.le_refl.
Qed.

Context { prec_gt_0_ : Prec_gt_0 prec }.

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

Variable choice : Z -> bool.

Remainder of the division in FLX

Theorem div_error_FLX :
  forall rnd { Zrnd : Valid_rnd rnd } x y,
  format x -> format y ->
  format (x - round beta (FLX_exp prec) rnd (x/y) * y)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
format x ->
format y ->
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
Proof with auto with typeclass_instances.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
format x ->
format y ->
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
intros rnd Zrnd x y Hx Hy.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
destruct (Req_dec y 0) as [Zy|Zy].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y = 0%R
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
now rewrite Zy, Rmult_0_r, Rminus_0_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
destruct (Req_dec (round beta (FLX_exp prec) rnd (x/y)) 0) as [Hr|Hr].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) = 0%R
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
rewrite Hr; ring_simplify (x-0*y)%R; assumption.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
assert (Zx: x <> R0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
x <> R0
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
contradict Hr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:x = R0
round beta (FLX_exp prec) rnd (x / y) = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
rewrite Hr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:x = R0
round beta (FLX_exp prec) rnd (R0 / y) = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
unfold Rdiv.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:x = R0
round beta (FLX_exp prec) rnd (R0 * / y) = 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
now rewrite Rmult_0_l, round_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
destruct (canonical_generic_format _ _ x Hx) as (fx,(Hx1,Hx2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
destruct (canonical_generic_format _ _ y Hy) as (fy,(Hy1,Hy2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
destruct (canonical_generic_format beta (FLX_exp prec) (round beta (FLX_exp prec) rnd (x / y))) as (fr,(Hr1,Hr2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
format (round beta (FLX_exp prec) rnd (x / y))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
apply generic_format_round...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
format (x - round beta (FLX_exp prec) rnd (x / y) * y)
unfold Rminus; apply generic_format_plus_prec with fx (Fopp (Fmult fr fy)); trivial.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
forall e : Z, (FLX_exp prec e <= e - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(- (round beta (FLX_exp prec) rnd (x / y) * y))%R =
F2R (Fopp (Fmult fr fy))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
intros e; apply Z.le_refl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(- (round beta (FLX_exp prec) rnd (x / y) * y))%R =
F2R (Fopp (Fmult fr fy))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
now rewrite F2R_opp, F2R_mult, <- Hr1, <- Hy1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
(* *)
destruct (relative_error_FLX_ex beta prec (prec_gt_0 prec) rnd (x / y)%R) as (eps,(Heps1,Heps2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
rewrite Heps2.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs (x + - (x / y * (1 + eps) * y)) <
 bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
rewrite <- Rabs_Ropp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs (- (x + - (x / y * (1 + eps) * y))) <
 bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
replace (-(x + - (x / y * (1 + eps) * y)))%R with (x * eps)%R by now field.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs (x * eps) < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
rewrite Rabs_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * Rabs eps < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply Rlt_le_trans with (Rabs x * 1)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * Rabs eps < Rabs x * 1)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply Rmult_lt_compat_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(0 < Rabs x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs eps < 1)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
now apply Rabs_pos_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs eps < 1)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply Rlt_le_trans with (1 := Heps1).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(bpow (- prec + 1) <= 1)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
change 1%R with (bpow 0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(bpow (- prec + 1) <= bpow 0)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply bpow_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(- prec + 1 <= 0)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
generalize (prec_gt_0 prec).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(0 < prec)%Z -> (- prec + 1 <= 0)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
clear ; omega.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x * 1 <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
rewrite Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x <= bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
rewrite Hx2, <- Hx1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x <= bpow (prec + cexp x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
unfold cexp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
(Rabs x <= bpow (prec + FLX_exp prec (mag beta x)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
destruct (mag beta x) as (ex, Hex).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs x <=
 bpow
   (prec + FLX_exp prec (Build_mag_prop beta x ex Hex)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs x <= bpow (prec + FLX_exp prec ex))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
specialize (Hex Zx).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs x <= bpow (prec + FLX_exp prec ex))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply Rlt_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs x < bpow (prec + FLX_exp prec ex))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply Rlt_le_trans with (1 := proj2 Hex).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow ex <= bpow (prec + FLX_exp prec ex))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply bpow_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(ex <= prec + FLX_exp prec ex)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
unfold FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(ex <= prec + (ex - prec))%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
ring_simplify.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps < bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) rnd (x / y) =
(x / y * (1 + eps))%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(ex <= ex)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
apply Z.le_refl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + Fexp (Fopp (Fmult fr fy))))%R
(* *)
replace (Fexp (Fopp (Fmult fr fy))) with (Fexp fr + Fexp fy)%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + (Fexp fr + Fexp fy)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Fexp fr + Fexp fy)%Z = Fexp (Fopp (Fmult fr fy))
2: unfold Fopp, Fmult; destruct fr; destruct fy; now simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x + - (round beta (FLX_exp prec) rnd (x / y) * y)) <
 bpow (prec + (Fexp fr + Fexp fy)))%R
replace (x + - (round beta (FLX_exp prec) rnd (x / y) * y))%R with
  (y * (-(round beta (FLX_exp prec) rnd (x / y) - x/y)))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (y *
    - (round beta (FLX_exp prec) rnd (x / y) - x / y)) <
 bpow (prec + (Fexp fr + Fexp fy)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(y * - (round beta (FLX_exp prec) rnd (x / y) - x / y))%R =
(x + - (round beta (FLX_exp prec) rnd (x / y) * y))%R
2: field; assumption.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (y *
    - (round beta (FLX_exp prec) rnd (x / y) - x / y)) <
 bpow (prec + (Fexp fr + Fexp fy)))%R
rewrite Rabs_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y *
 Rabs
   (- (round beta (FLX_exp prec) rnd (x / y) - x / y)) <
 bpow (prec + (Fexp fr + Fexp fy)))%R
apply Rlt_le_trans with (Rabs y * bpow (Fexp fr))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y *
 Rabs
   (- (round beta (FLX_exp prec) rnd (x / y) - x / y)) <
 Rabs y * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
apply Rmult_lt_compat_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 < Rabs y)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (- (round beta (FLX_exp prec) rnd (x / y) - x / y)) <
 bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
now apply Rabs_pos_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (- (round beta (FLX_exp prec) rnd (x / y) - x / y)) <
 bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
rewrite Rabs_Ropp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (round beta (FLX_exp prec) rnd (x / y) - x / y) <
 bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
replace (bpow (Fexp fr)) with (ulp beta (FLX_exp prec) (F2R fr)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (round beta (FLX_exp prec) rnd (x / y) - x / y) <
 ulp beta (FLX_exp prec) (F2R fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
ulp beta (FLX_exp prec) (F2R fr) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
rewrite <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (round beta (FLX_exp prec) rnd (x / y) - x / y) <
 ulp beta (FLX_exp prec)
   (round beta (FLX_exp prec) rnd (x / y)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
ulp beta (FLX_exp prec) (F2R fr) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
apply error_lt_ulp_round...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x / y)%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
ulp beta (FLX_exp prec) (F2R fr) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
apply Rmult_integral_contrapositive_currified; try apply Rinv_neq_0_compat; assumption.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
ulp beta (FLX_exp prec) (F2R fr) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
rewrite ulp_neq_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
bpow (cexp (F2R fr)) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
F2R fr <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
2: now rewrite <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
bpow (cexp (F2R fr)) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
apply f_equal.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
cexp (F2R fr) = Fexp fr
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
now rewrite Hr2, <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + (Fexp fr + Fexp fy)))%R
replace (prec+(Fexp fr+Fexp fy))%Z with ((prec+Fexp fy)+Fexp fr)%Z by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + Fexp fy + Fexp fr))%R
rewrite bpow_plus.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y * bpow (Fexp fr) <=
 bpow (prec + Fexp fy) * bpow (Fexp fr))%R
apply Rmult_le_compat_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y <= bpow (prec + Fexp fy))%R
apply bpow_ge_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y <= bpow (prec + Fexp fy))%R
rewrite Hy2, <- Hy1 ; unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y <= bpow (prec + (mag beta y - prec)))%R
ring_simplify (prec + (mag beta y - prec))%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs y <= bpow (mag beta y))%R
destruct (mag beta y); simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
rnd:R -> Z
Zrnd:Valid_rnd rnd
x, y:R
Hx:format x
Hy:format y
Zy:y <> 0%R
Hr:round beta (FLX_exp prec) rnd (x / y) <> 0%R
Zx:x <> R0
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fy:float beta
Hy1:y = F2R fy
Hy2:canonical beta (FLX_exp prec) fy
fr:float beta
Hr1:round beta (FLX_exp prec) rnd (x / y) = F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:y <> 0%R ->
(bpow (mag_val - 1) <= Rabs y < bpow mag_val)%R
(Rabs y <= bpow mag_val)%R
left; now apply a.
Qed.

Remainder of the square in FLX (with p>1) and rounding to nearest

Variable Hp1 : Z.lt 1 prec.

Theorem sqrt_error_FLX_N :
  forall x, format x ->
  format (x - Rsqr (round beta (FLX_exp prec) (Znearest choice) (sqrt x)))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
forall x : R,
format x ->
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
Proof with auto with typeclass_instances.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
forall x : R,
format x ->
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
intros x Hx.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
destruct (total_order_T x 0) as [[Hxz|Hxz]|Hxz].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
unfold sqrt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      match Rcase_abs x with
      | left _ => 0
      | right a =>
          Rsqrt
            {|
            nonneg := x;
            cond_nonneg := Rge_le x 0 a |}
      end)²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
destruct (Rcase_abs x).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz, r:(x < 0)%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice) 0)²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
r:(x >= 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice)
      (Rsqrt
         {|
         nonneg := x;
         cond_nonneg := Rge_le x 0 r |}))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
rewrite round_0...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz, r:(x < 0)%R
format (x - 0²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
r:(x >= 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice)
      (Rsqrt
         {|
         nonneg := x;
         cond_nonneg := Rge_le x 0 r |}))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
unfold Rsqr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz, r:(x < 0)%R
format (x - 0 * 0)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
r:(x >= 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice)
      (Rsqrt
         {|
         nonneg := x;
         cond_nonneg := Rge_le x 0 r |}))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
now rewrite Rmult_0_l, Rminus_0_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
r:(x >= 0)%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (Rsqrt
         {|
         nonneg := x;
         cond_nonneg := Rge_le x 0 r |}))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
elim (Rlt_irrefl 0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x < 0)%R
r:(x >= 0)%R
(0 < 0)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
now apply Rgt_ge_trans with x.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
rewrite Hxz, sqrt_0, round_0...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format (0 - 0²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
unfold Rsqr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format (0 - 0 * 0)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
rewrite Rmult_0_l, Rminus_0_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:x = 0%R
format 0
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
apply generic_format_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
case (Req_dec (round beta (FLX_exp prec) (Znearest choice) (sqrt x)) 0); intros Hr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 0%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
rewrite Hr; unfold Rsqr; ring_simplify (x-0*0)%R; assumption.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
destruct (canonical_generic_format _ _ x Hx) as (fx,(Hx1,Hx2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
destruct (canonical_generic_format beta (FLX_exp prec) (round beta (FLX_exp prec) (Znearest choice) (sqrt x))) as (fr,(Hr1,Hr2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
format
  (round beta (FLX_exp prec) (Znearest choice)
     (sqrt x))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
format
  (x -
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x))²)
apply generic_format_round...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
format
  (x -
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x))²)
unfold Rminus; apply generic_format_plus_prec with fx (Fopp (Fmult fr fr)); trivial.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
forall e : Z, (FLX_exp prec e <= e - prec)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(-
 (round beta (FLX_exp prec) (Znearest choice) (sqrt x))²)%R =
F2R (Fopp (Fmult fr fr))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
intros e; apply Z.le_refl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(-
 (round beta (FLX_exp prec) (Znearest choice) (sqrt x))²)%R =
F2R (Fopp (Fmult fr fr))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
unfold Rsqr; now rewrite F2R_opp,F2R_mult, <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) (Znearest choice)
       (sqrt x))²) < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
(* *)
apply Rle_lt_trans with x.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) (Znearest choice)
       (sqrt x))²) <= x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rabs_minus_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <=
 (round beta (FLX_exp prec) (Znearest choice) (sqrt x))²)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
((round beta (FLX_exp prec) (Znearest choice) (sqrt x))² <=
 2 * x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rle_0_sqr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
((round beta (FLX_exp prec) (Znearest choice) (sqrt x))² <=
 2 * x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
destruct (relative_error_N_FLX_ex beta prec (prec_gt_0 prec) choice (sqrt x)) as (eps,(Heps1,Heps2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((round beta (FLX_exp prec) (Znearest choice) (sqrt x))² <=
 2 * x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Heps2.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((sqrt x * (1 + eps))² <= 2 * x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Rsqr_mult, Rsqr_sqrt, Rmult_comm.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((1 + eps)² * x <= 2 * x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R 2: now apply Rlt_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((1 + eps)² * x <= 2 * x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rmult_le_compat_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((1 + eps)² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply Rlt_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((1 + eps)² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rle_trans with (5²/4²)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((1 + eps)² <= 5² / 4²)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite <- Rsqr_div.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
((1 + eps)² <= (5 / 4)²)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rsqr_le_abs_1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(Rabs (1 + eps) <= Rabs (5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rle_trans with (1 := Rabs_triang _ _).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(Rabs 1 + Rabs eps <= Rabs (5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Rabs_R1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(1 + Rabs eps <= Rabs (5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rplus_le_reg_l with (-1)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(-1 + (1 + Rabs eps) <= -1 + Rabs (5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
replace (-1 + (1 + Rabs eps))%R with (Rabs eps) by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(Rabs eps <= -1 + Rabs (5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rle_trans with (1 := Heps1).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(/ 2 * bpow (- prec + 1) <= -1 + Rabs (5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Rabs_pos_eq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(/ 2 * bpow (- prec + 1) <= -1 + 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rmult_le_reg_l with 2%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 < 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 * (/ 2 * bpow (- prec + 1)) <= 2 * (-1 + 5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 * (/ 2 * bpow (- prec + 1)) <= 2 * (-1 + 5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite <- Rmult_assoc, Rinv_r, Rmult_1_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(bpow (- prec + 1) <= 2 * (-1 + 5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rle_trans with (bpow (-1)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(bpow (- prec + 1) <= bpow (-1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(bpow (-1) <= 2 * (-1 + 5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply bpow_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(- prec + 1 <= -1)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(bpow (-1) <= 2 * (-1 + 5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
omega.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(bpow (-1) <= 2 * (-1 + 5 / 4))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
replace (2 * (-1 + 5 / 4))%R with (/2)%R by field.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(bpow (-1) <= / 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rinv_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 < 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <= IZR (Z.pow_pos beta 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <= IZR (Z.pow_pos beta 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply IZR_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <= Z.pow_pos beta 1)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
unfold Zpower_pos.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <= Pos.iter (Z.mul beta) 1 1)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <= beta * 1)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Zmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <= beta)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Zle_bool_imp_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(2 <=? beta)%Z = true
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply beta.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
2%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_neq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
unfold Rdiv.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5 * / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rmult_le_pos.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= 5)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 <= / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rlt_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 < / 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rinv_0_lt_compat.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 < 4)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
4%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_neq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5² / 4² <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
unfold Rsqr.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(5 * 5 / (4 * 4) <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
replace (5 * 5 / (4 * 4))%R with (25 * /16)%R by field.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(25 * / 16 <= 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply Rmult_le_reg_r with 16%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(0 < 16)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(25 * / 16 * 16 <= 2 * 16)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(25 * / 16 * 16 <= 2 * 16)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Rmult_assoc, Rinv_l, Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
(25 <= 2 * 16)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
16%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply (IZR_le _ 32).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
eps:R
Heps1:(Rabs eps <= / 2 * bpow (- prec + 1))%R
Heps2:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + eps))%R
16%R <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply IZR_neq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + Fexp fx))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite Hx2, <- Hx1; unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (prec + (mag beta x - prec)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
ring_simplify (prec + (mag beta x - prec))%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(x < bpow (mag beta x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
destruct (mag beta x); simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:x <> 0%R ->
(bpow (mag_val - 1) <= Rabs x < bpow mag_val)%R
(x < bpow mag_val)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
rewrite <- (Rabs_right x).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:x <> 0%R ->
(bpow (mag_val - 1) <= Rabs x < bpow mag_val)%R
(Rabs x < bpow mag_val)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:x <> 0%R ->
(bpow (mag_val - 1) <= Rabs x < bpow mag_val)%R
(x >= 0)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
apply a.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:x <> 0%R ->
(bpow (mag_val - 1) <= Rabs x < bpow mag_val)%R
x <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:x <> 0%R ->
(bpow (mag_val - 1) <= Rabs x < bpow mag_val)%R
(x >= 0)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply Rgt_not_eq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:x <> 0%R ->
(bpow (mag_val - 1) <= Rabs x < bpow mag_val)%R
(x >= 0)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) 
       (Znearest choice) 
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
now apply Rgt_ge.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) (Znearest choice)
       (sqrt x))²) <
 bpow (prec + Fexp (Fopp (Fmult fr fr))))%R
(* *)
replace (Fexp (Fopp (Fmult fr fr))) with (Fexp fr + Fexp fr)%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) (Znearest choice)
       (sqrt x))²) < bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Fexp fr + Fexp fr)%Z = Fexp (Fopp (Fmult fr fr))
2: unfold Fopp, Fmult; destruct fr; now simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (x +
    -
    (round beta (FLX_exp prec) (Znearest choice)
       (sqrt x))²) < bpow (prec + (Fexp fr + Fexp fr)))%R
rewrite Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (x + - (F2R fr)²) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
replace (x + - Rsqr (F2R fr))%R with (-((F2R fr - sqrt x)*(F2R fr + sqrt x)))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (- ((F2R fr - sqrt x) * (F2R fr + sqrt x))) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(- ((F2R fr - sqrt x) * (F2R fr + sqrt x)))%R =
(x + - (F2R fr)²)%R
2: rewrite <- (sqrt_sqrt x) at 3; auto.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (- ((F2R fr - sqrt x) * (F2R fr + sqrt x))) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(- ((F2R fr - sqrt x) * (F2R fr + sqrt x)))%R =
(sqrt x * sqrt x + - (F2R fr)²)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
2: unfold Rsqr; ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (- ((F2R fr - sqrt x) * (F2R fr + sqrt x))) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite Rabs_Ropp, Rabs_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr - sqrt x) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rle_lt_trans with ((/2*bpow (Fexp fr))* Rabs (F2R fr + sqrt x))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr - sqrt x) * Rabs (F2R fr + sqrt x) <=
 / 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rmult_le_compat_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= Rabs (F2R fr + sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr - sqrt x) <= / 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rabs_pos.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr - sqrt x) <= / 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rle_trans with (/2*ulp beta  (FLX_exp prec) (F2R fr))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr - sqrt x) <=
 / 2 * ulp beta (FLX_exp prec) (F2R fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * ulp beta (FLX_exp prec) (F2R fr) <=
 / 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 / 2 *
 ulp beta (FLX_exp prec)
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * ulp beta (FLX_exp prec) (F2R fr) <=
 / 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply error_le_half_ulp_round...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * ulp beta (FLX_exp prec) (F2R fr) <=
 / 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
right; rewrite ulp_neq_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (cexp (F2R fr)))%R =
(/ 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
F2R fr <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
2: now rewrite <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (cexp (F2R fr)))%R =
(/ 2 * bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply f_equal.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
bpow (cexp (F2R fr)) = bpow (Fexp fr)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite Hr2, <- Hr1; trivial.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(/ 2 * bpow (Fexp fr) * Rabs (F2R fr + sqrt x) <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite Rmult_assoc, Rmult_comm.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(bpow (Fexp fr) * Rabs (F2R fr + sqrt x) * / 2 <
 bpow (prec + (Fexp fr + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
replace (prec+(Fexp fr+Fexp fr))%Z with (Fexp fr + (prec+Fexp fr))%Z by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(bpow (Fexp fr) * Rabs (F2R fr + sqrt x) * / 2 <
 bpow (Fexp fr + (prec + Fexp fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite bpow_plus, Rmult_assoc.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(bpow (Fexp fr) * (Rabs (F2R fr + sqrt x) * / 2) <
 bpow (Fexp fr) * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rmult_lt_compat_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 < bpow (Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr + sqrt x) * / 2 < bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply bpow_gt_0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr + sqrt x) * / 2 < bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rmult_lt_reg_l with (1 := Rlt_0_2).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(2 * (Rabs (F2R fr + sqrt x) * / 2) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rle_lt_trans with (Rabs (F2R fr + sqrt x)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(2 * (Rabs (F2R fr + sqrt x) * / 2) <=
 Rabs (F2R fr + sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr + sqrt x) < 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
right; field.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr + sqrt x) < 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rle_lt_trans with (1:=Rabs_triang _ _).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
(* . *)
assert (Rabs (F2R fr) < bpow (prec + Fexp fr))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite Hr2.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr) < bpow (prec + cexp (F2R fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr) <
 bpow (prec + (mag beta (F2R fr) - prec)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
ring_simplify (prec + (mag beta (F2R fr) - prec))%Z.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(Rabs (F2R fr) < bpow (mag beta (F2R fr)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
destruct (mag beta (F2R fr)); simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:F2R fr <> 0%R ->
(bpow (mag_val - 1) <= Rabs (F2R fr) <
 bpow mag_val)%R
(Rabs (F2R fr) < bpow mag_val)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply a.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
mag_val:Z
a:F2R fr <> 0%R ->
(bpow (mag_val - 1) <= Rabs (F2R fr) <
 bpow mag_val)%R
F2R fr <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite <- Hr1; auto.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
(* . *)
apply Rlt_le_trans with (bpow (prec + Fexp fr)+ Rabs (sqrt x))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (F2R fr) + Rabs (sqrt x) <
 bpow (prec + Fexp fr) + Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(bpow (prec + Fexp fr) + Rabs (sqrt x) <=
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
now apply Rplus_lt_compat_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(bpow (prec + Fexp fr) + Rabs (sqrt x) <=
 2 * bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
(* . *)
replace (2 * bpow (prec + Fexp fr))%R with (bpow (prec + Fexp fr) + bpow (prec + Fexp fr))%R by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(bpow (prec + Fexp fr) + Rabs (sqrt x) <=
 bpow (prec + Fexp fr) + bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rplus_le_compat_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(Rabs (sqrt x) <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
assert (sqrt x <> 0)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
sqrt x <> 0%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
(Rabs (sqrt x) <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rgt_not_eq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
(sqrt x > 0)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
(Rabs (sqrt x) <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
now apply sqrt_lt_R0.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
(Rabs (sqrt x) <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
destruct (mag beta (sqrt x)) as (es,Es).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:sqrt x <> 0%R ->
(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(Rabs (sqrt x) <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
specialize (Es H0).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(Rabs (sqrt x) <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rle_trans with (bpow es).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(Rabs (sqrt x) <= bpow es)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(bpow es <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
now apply Rlt_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(bpow es <= bpow (prec + Fexp fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply bpow_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(es <= prec + Fexp fr)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
case (Zle_or_lt es (prec + Fexp fr)) ; trivial.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
(prec + Fexp fr < es)%Z -> (es <= prec + Fexp fr)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
intros H1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(es <= prec + Fexp fr)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
absurd (Rabs (F2R fr) < bpow (es - 1))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
~ (Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rle_not_lt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(bpow (es - 1) <= Rabs (F2R fr))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
rewrite <- Hr1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(bpow (es - 1) <=
 Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x)))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply abs_round_ge_generic...beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
format (bpow (es - 1))
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(bpow (es - 1) <= Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply generic_format_bpow.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(FLX_exp prec (es - 1 + 1) <= es - 1)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(bpow (es - 1) <= Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
unfold FLX_exp; omega.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(bpow (es - 1) <= Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Es.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(Rabs (F2R fr) < bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply Rlt_le_trans with (1:=H).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(bpow (prec + Fexp fr) <= bpow (es - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
apply bpow_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
H:(Rabs (F2R fr) < bpow (prec + Fexp fr))%R
H0:sqrt x <> 0%R
es:Z
Es:(bpow (es - 1) <= Rabs (sqrt x) < bpow es)%R
H1:(prec + Fexp fr < es)%Z
(prec + Fexp fr <= es - 1)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
omega.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Hx:format x
Hxz:(x > 0)%R
Hr:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) <> 0%R
fx:float beta
Hx1:x = F2R fx
Hx2:canonical beta (FLX_exp prec) fx
fr:float beta
Hr1:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) = 
F2R fr
Hr2:canonical beta (FLX_exp prec) fr
(0 <= x)%R
now apply Rlt_le.
Qed.

Lemma sqrt_error_N_FLX_aux1 x (Fx : format x) (Px : (0 < x)%R) :
  exists (mu : R) (e : Z), (format mu /\ x = mu * bpow (2 * e) :> R
                            /\ 1 <= mu < bpow 2)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
exists (mu : R) (e : Z),
  format mu /\
  x = (mu * bpow (2 * e))%R /\ (1 <= mu < bpow 2)%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
exists (mu : R) (e : Z),
  format mu /\
  x = (mu * bpow (2 * e))%R /\ (1 <= mu < bpow 2)%R
set (e := ((mag beta x - 1) / 2)%Z).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
exists (mu : R) (e0 : Z),
  format mu /\
  x = (mu * bpow (2 * e0))%R /\ (1 <= mu < bpow 2)%R
set (mu := (x * bpow (-2 * e)%Z)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
exists (mu0 : R) (e0 : Z),
  format mu0 /\
  x = (mu0 * bpow (2 * e0))%R /\ (1 <= mu0 < bpow 2)%R
assert (Hbe : (bpow (-2 * e) * bpow (2 * e) = 1)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
(bpow (-2 * e) * bpow (2 * e))%R = 1%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
exists (mu0 : R) (e0 : Z),
  format mu0 /\
  x = (mu0 * bpow (2 * e0))%R /\ (1 <= mu0 < bpow 2)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
(bpow (-2 * e) * bpow (2 * e))%R = 1%R now rewrite <- bpow_plus; case e; simpl; [reflexivity| |]; intro p;
    rewrite Z.pos_sub_diag. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
exists (mu0 : R) (e0 : Z),
  format mu0 /\
  x = (mu0 * bpow (2 * e0))%R /\ (1 <= mu0 < bpow 2)%R
assert (Fmu : format mu); [now apply mult_bpow_exact_FLX|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
exists (mu0 : R) (e0 : Z),
  format mu0 /\
  x = (mu0 * bpow (2 * e0))%R /\ (1 <= mu0 < bpow 2)%R
exists mu, e; split; [exact Fmu|split; [|split]].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
x = (mu * bpow (2 * e))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(1 <= mu)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(mu < bpow 2)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
x = (mu * bpow (2 * e))%R set (e2 := (2 * e)%Z); simpl; unfold mu; rewrite Rmult_assoc.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
x = (x * (bpow (-2 * e) * bpow e2))%R
  now unfold e2; rewrite Hbe, Rmult_1_r. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(1 <= mu)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(mu < bpow 2)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(1 <= mu)%R apply (Rmult_le_reg_r (bpow (2 * e))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(0 < bpow (2 * e))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(1 * bpow (2 * e) <= mu * bpow (2 * e))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(0 < bpow (2 * e))%R apply bpow_gt_0. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(1 * bpow (2 * e) <= mu * bpow (2 * e))%R
  rewrite Rmult_1_l; set (e2 := (2 * e)%Z); simpl; unfold mu.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
(bpow e2 <= x * bpow (-2 * e) * bpow e2)%R
  unfold e2; rewrite Rmult_assoc, Hbe, Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
(bpow (2 * e) <= x)%R
  apply (Rle_trans _ (bpow (mag beta x - 1))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
(bpow (2 * e) <= bpow (mag beta x - 1))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
(bpow (mag beta x - 1) <= x)%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
(bpow (2 * e) <= bpow (mag beta x - 1))%R now apply bpow_le; unfold e; apply Z_mult_div_ge. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
(bpow (mag beta x - 1) <= x)%R
  set (l := mag _ _); rewrite <- (Rabs_pos_eq _ (Rlt_le _ _ Px)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
l:=mag beta x:mag_prop beta x
(bpow (l - 1) <= Rabs x)%R
  unfold l; apply bpow_mag_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
e2:=(2 * e)%Z:Z
l:=mag beta x:mag_prop beta x
x <> 0%R
  intro Hx; revert Px; rewrite Hx; apply Rlt_irrefl. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(mu < bpow 2)%R
simpl; unfold mu; change (IZR _) with (bpow 2).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(x * bpow (-2 * e) < bpow 2)%R
apply (Rmult_lt_reg_r (bpow (2 * e))); [now apply bpow_gt_0|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(x * bpow (-2 * e) * bpow (2 * e) <
 bpow 2 * bpow (2 * e))%R
rewrite Rmult_assoc, Hbe, Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(x < bpow 2 * bpow (2 * e))%R
apply (Rlt_le_trans _ (bpow (mag beta x))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(x < bpow (mag beta x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(bpow (mag beta x) <= bpow 2 * bpow (2 * e))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(x < bpow (mag beta x))%R rewrite <- (Rabs_pos_eq _ (Rlt_le _ _ Px)) at 1; apply bpow_mag_gt. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
(bpow (mag beta x) <= bpow 2 * bpow (2 * e))%R
rewrite <- bpow_plus; apply bpow_le; unfold e; set (mxm1 := (_ - 1)%Z).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
mxm1:=(mag beta x - 1)%Z:Z
(mag beta x <= 2 + 2 * (mxm1 / 2))%Z
replace (_ * _)%Z with (2 * (mxm1 / 2) + mxm1 mod 2 - mxm1 mod 2)%Z by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
mxm1:=(mag beta x - 1)%Z:Z
(mag beta x <=
 2 + (2 * (mxm1 / 2) + mxm1 mod 2 - mxm1 mod 2))%Z
rewrite <- Z.div_mod; [|now simpl].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
mxm1:=(mag beta x - 1)%Z:Z
(mag beta x <= 2 + (mxm1 - mxm1 mod 2))%Z
apply (Zplus_le_reg_r _ _ (mxm1 mod 2 - mag beta x)%Z).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Px:(0 < x)%R
e:=((mag beta x - 1) / 2)%Z:Z
mu:=(x * bpow (-2 * e))%R:R
Hbe:(bpow (-2 * e) * bpow (2 * e))%R = 1%R
Fmu:format mu
mxm1:=(mag beta x - 1)%Z:Z
(mag beta x + (mxm1 mod 2 - mag beta x) <=
 2 + (mxm1 - mxm1 mod 2) + (mxm1 mod 2 - mag beta x))%Z
unfold mxm1; destruct (Z.mod_bound_or (mag beta x - 1) 2); omega.
Qed.

Notation u_ro := (u_ro beta prec).

Lemma sqrt_error_N_FLX_aux2 x (Fx : format x) :
  (1 <= x)%R ->
  (x = 1 :> R \/ x = 1 + 2 * u_ro :> R \/ 1 + 4 * u_ro <= x)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
(1 <= x)%R ->
x = 1%R \/
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
(1 <= x)%R ->
x = 1%R \/
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
intro HxGe1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
x = 1%R \/
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
assert (Pu_ro : (0 <= u_ro)%R); [apply Rmult_le_pos; [lra|apply bpow_ge_0]|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
x = 1%R \/
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
destruct (Rle_or_lt x 1) as [HxLe1|HxGt1]; [now left; apply Rle_antisym|right].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
assert (F1 : format 1); [now apply generic_format_FLX_1|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
assert (H2eps : (2 * u_ro = bpow (-prec + 1))%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
(2 * u_ro)%R = bpow (- prec + 1)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
(2 * u_ro)%R = bpow (- prec + 1) unfold u_ro; rewrite bpow_plus; field. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
assert (HmuGe1p2eps : (1 + 2 * u_ro <= x)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
(1 + 2 * u_ro <= x)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
(1 + 2 * u_ro <= x)%R rewrite H2eps, <- succ_FLX_1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
(succ beta (FLX_exp prec) 1 <= x)%R
  now apply succ_le_lt; [now apply FLX_exp_valid| | |]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
x = (1 + 2 * u_ro)%R \/ (1 + 4 * u_ro <= x)%R
destruct (Rle_or_lt x (1 + 2 * u_ro)) as [HxLe1p2eps|HxGt1p2eps];
  [now left; apply Rle_antisym|right].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(1 + 4 * u_ro <= x)%R
assert (Hulp1p2eps : (ulp beta (FLX_exp prec) (1 + 2 * u_ro) = 2 * u_ro)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
ulp beta (FLX_exp prec) (1 + 2 * u_ro) = (2 * u_ro)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
(1 + 4 * u_ro <= x)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
ulp beta (FLX_exp prec) (1 + 2 * u_ro) = (2 * u_ro)%R destruct (ulp_succ_pos _ _ _ F1 Rlt_0_1) as [Hsucc|Hsucc].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hsucc:ulp beta (FLX_exp prec)
  (succ beta (FLX_exp prec) 1) =
ulp beta (FLX_exp prec) 1
ulp beta (FLX_exp prec) (1 + 2 * u_ro) = (2 * u_ro)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hsucc:succ beta (FLX_exp prec) 1 = bpow (mag beta 1)
ulp beta (FLX_exp prec) (1 + 2 * u_ro) = (2 * u_ro)%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hsucc:ulp beta (FLX_exp prec)
  (succ beta (FLX_exp prec) 1) =
ulp beta (FLX_exp prec) 1
ulp beta (FLX_exp prec) (1 + 2 * u_ro) = (2 * u_ro)%R now rewrite H2eps, <- succ_FLX_1, <- ulp_FLX_1. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hsucc:succ beta (FLX_exp prec) 1 = bpow (mag beta 1)
ulp beta (FLX_exp prec) (1 + 2 * u_ro) = (2 * u_ro)%R
  exfalso; revert Hsucc; apply Rlt_not_eq.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(succ beta (FLX_exp prec) 1 < bpow (mag beta 1))%R
  rewrite succ_FLX_1, mag_1, bpow_1, <- H2eps; simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(1 + 2 * u_ro < IZR beta)%R
  apply (Rlt_le_trans _ 2); [apply Rplus_lt_compat_l|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(2 * u_ro < R1)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(2 <= IZR beta)%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(2 * u_ro < R1)%R unfold u_ro; rewrite <-Rmult_assoc, Rinv_r, Rmult_1_l; [|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(bpow (- prec + 1) < R1)%R
    change R1 with (bpow 0); apply bpow_lt; omega. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
(2 <= IZR beta)%R
  apply IZR_le, Zle_bool_imp_le, radix_prop. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
(1 + 4 * u_ro <= x)%R
assert (Hsucc1p2eps :
          (succ beta (FLX_exp prec) (1 + 2 * u_ro) = 1 + 4 * u_ro)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
Hsucc1p2eps:succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R
(1 + 4 * u_ro <= x)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R unfold succ; rewrite Rle_bool_true; [rewrite Hulp1p2eps; ring|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
(0 <= 1 + 2 * u_ro)%R
  apply Rplus_le_le_0_compat; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
Hsucc1p2eps:succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R
(1 + 4 * u_ro <= x)%R
rewrite <- Hsucc1p2eps.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
Hsucc1p2eps:succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R
(succ beta (FLX_exp prec) (1 + 2 * u_ro) <= x)%R
apply succ_le_lt; [now apply FLX_exp_valid| |exact Fx|now simpl].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
Hsucc1p2eps:succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R
format (1 + 2 * u_ro)
rewrite H2eps, <- succ_FLX_1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
HxGe1:(1 <= x)%R
Pu_ro:(0 <= u_ro)%R
HxGt1:(1 < x)%R
F1:format 1
H2eps:(2 * u_ro)%R = bpow (- prec + 1)
HmuGe1p2eps:(1 + 2 * u_ro <= x)%R
HxGt1p2eps:(1 + 2 * u_ro < x)%R
Hulp1p2eps:ulp beta (FLX_exp prec) (1 + 2 * u_ro) =
(2 * u_ro)%R
Hsucc1p2eps:succ beta (FLX_exp prec) (1 + 2 * u_ro) =
(1 + 4 * u_ro)%R
format (succ beta (FLX_exp prec) 1)
now apply generic_format_succ; [apply FLX_exp_valid|].
Qed.

Lemma sqrt_error_N_FLX_aux3 :
  (u_ro / sqrt (1 + 4 * u_ro) <= 1 - 1 / sqrt (1 + 2 * u_ro))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(u_ro / sqrt (1 + 4 * u_ro) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(u_ro / sqrt (1 + 4 * u_ro) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
assert (Pu_ro : (0 <= u_ro)%R); [apply Rmult_le_pos; [lra|apply bpow_ge_0]|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro / sqrt (1 + 4 * u_ro) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
unfold Rdiv; apply (Rplus_le_reg_r (/ sqrt (1 + 2 * u_ro))); ring_simplify.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * / sqrt (1 + 4 * u_ro) + / sqrt (1 + 2 * u_ro) <=
 1)%R
apply (Rmult_le_reg_r (sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 < sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((u_ro * / sqrt (1 + 4 * u_ro) + / sqrt (1 + 2 * u_ro)) *
 (sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro)) <=
 1 * (sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro)))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 < sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro))%R apply Rmult_lt_0_compat; apply sqrt_lt_R0; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((u_ro * / sqrt (1 + 4 * u_ro) + / sqrt (1 + 2 * u_ro)) *
 (sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro)) <=
 1 * (sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro)))%R
field_simplify; [|split; apply Rgt_not_eq, Rlt_gt, sqrt_lt_R0; lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * sqrt (1 + 2 * u_ro) + sqrt (1 + 4 * u_ro) <=
 sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro))%R
try unfold Rdiv; rewrite ?Rinv_1, ?Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * sqrt (1 + 2 * u_ro) + sqrt (1 + 4 * u_ro) <=
 sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro))%R
apply Rsqr_incr_0_var; [|now apply Rmult_le_pos; apply sqrt_pos].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((u_ro * sqrt (1 + 2 * u_ro) + sqrt (1 + 4 * u_ro))² <=
 (sqrt (1 + 4 * u_ro) * sqrt (1 + 2 * u_ro))²)%R
rewrite <-sqrt_mult; [|lra|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((u_ro * sqrt (1 + 2 * u_ro) + sqrt (1 + 4 * u_ro))² <=
 (sqrt ((1 + 4 * u_ro) * (1 + 2 * u_ro)))²)%R
rewrite Rsqr_sqrt; [|apply Rmult_le_pos; lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((u_ro * sqrt (1 + 2 * u_ro) + sqrt (1 + 4 * u_ro))² <=
 (1 + 4 * u_ro) * (1 + 2 * u_ro))%R
unfold Rsqr; ring_simplify; unfold pow; rewrite !Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * u_ro *
 (sqrt (1 + 2 * u_ro) * sqrt (1 + 2 * u_ro)) +
 2 * u_ro * sqrt (1 + 2 * u_ro) * sqrt (1 + 4 * u_ro) +
 sqrt (1 + 4 * u_ro) * sqrt (1 + 4 * u_ro) <=
 8 * (u_ro * u_ro) + 6 * u_ro + 1)%R
rewrite !sqrt_def; [|lra|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * u_ro * (1 + 2 * u_ro) +
 2 * u_ro * sqrt (1 + 2 * u_ro) * sqrt (1 + 4 * u_ro) +
 (1 + 4 * u_ro) <= 8 * (u_ro * u_ro) + 6 * u_ro + 1)%R
apply (Rplus_le_reg_r (-u_ro * u_ro - 1 -4 * u_ro - 2 * u_ro ^ 3)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * u_ro * (1 + 2 * u_ro) +
 2 * u_ro * sqrt (1 + 2 * u_ro) * sqrt (1 + 4 * u_ro) +
 (1 + 4 * u_ro) +
 (- u_ro * u_ro - 1 - 4 * u_ro - 2 * u_ro ^ 3) <=
 8 * (u_ro * u_ro) + 6 * u_ro + 1 +
 (- u_ro * u_ro - 1 - 4 * u_ro - 2 * u_ro ^ 3))%R
ring_simplify; apply Rsqr_incr_0_var.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((2 * u_ro * sqrt (1 + 2 * u_ro) * sqrt (1 + 4 * u_ro))² <=
 (-2 * u_ro ^ 3 + 7 * u_ro ^ 2 + 2 * u_ro)²)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 <= -2 * u_ro ^ 3 + 7 * u_ro ^ 2 + 2 * u_ro)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((2 * u_ro * sqrt (1 + 2 * u_ro) * sqrt (1 + 4 * u_ro))² <=
 (-2 * u_ro ^ 3 + 7 * u_ro ^ 2 + 2 * u_ro)²)%R unfold Rsqr; ring_simplify.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(4 * u_ro ^ 2 * sqrt (1 + 2 * u_ro) ^ 2 *
 sqrt (1 + 4 * u_ro) ^ 2 <=
 4 * u_ro ^ 6 - 28 * u_ro ^ 5 + 41 * u_ro ^ 4 +
 28 * u_ro ^ 3 + 4 * u_ro ^ 2)%R
  unfold pow; rewrite !Rmult_1_r, !sqrt_def; [|lra|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(4 * (u_ro * u_ro) * (1 + 2 * u_ro) * (1 + 4 * u_ro) <=
 4 * (u_ro * (u_ro * (u_ro * (u_ro * (u_ro * u_ro))))) -
 28 * (u_ro * (u_ro * (u_ro * (u_ro * u_ro)))) +
 41 * (u_ro * (u_ro * (u_ro * u_ro))) +
 28 * (u_ro * (u_ro * u_ro)) + 4 * (u_ro * u_ro))%R
  apply (Rplus_le_reg_r (-32 * u_ro ^ 4 - 24 * u_ro ^ 3 - 4 * u_ro ^ 2)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(4 * (u_ro * u_ro) * (1 + 2 * u_ro) * (1 + 4 * u_ro) +
 (-32 * u_ro ^ 4 - 24 * u_ro ^ 3 - 4 * u_ro ^ 2) <=
 4 * (u_ro * (u_ro * (u_ro * (u_ro * (u_ro * u_ro))))) -
 28 * (u_ro * (u_ro * (u_ro * (u_ro * u_ro)))) +
 41 * (u_ro * (u_ro * (u_ro * u_ro))) +
 28 * (u_ro * (u_ro * u_ro)) + 4 * (u_ro * u_ro) +
 (-32 * u_ro ^ 4 - 24 * u_ro ^ 3 - 4 * u_ro ^ 2))%R
  ring_simplify.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 <=
 4 * u_ro ^ 6 - 28 * u_ro ^ 5 + 9 * u_ro ^ 4 +
 4 * u_ro ^ 3)%R
  replace (_ + _)%R
    with (((4 * u_ro ^ 2 - 28 * u_ro + 9) * u_ro + 4) * u_ro ^ 3)%R by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 <=
 ((4 * u_ro ^ 2 - 28 * u_ro + 9) * u_ro + 4) *
 u_ro ^ 3)%R
  apply Rmult_le_pos; [|now apply pow_le].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 <= (4 * u_ro ^ 2 - 28 * u_ro + 9) * u_ro + 4)%R
  assert (Heps_le_half : (u_ro <= 1 / 2)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro <= 1 / 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
Heps_le_half:(u_ro <= 1 / 2)%R
(0 <= (4 * u_ro ^ 2 - 28 * u_ro + 9) * u_ro + 4)%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro <= 1 / 2)%R unfold u_ro, Rdiv; rewrite Rmult_comm; apply Rmult_le_compat_r; [lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(bpow (- prec + 1) <= 1)%R
    change 1%R with (bpow 0); apply bpow_le; omega. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
Heps_le_half:(u_ro <= 1 / 2)%R
(0 <= (4 * u_ro ^ 2 - 28 * u_ro + 9) * u_ro + 4)%R
  apply (Rle_trans _ (-8 * u_ro + 4)); [lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
Heps_le_half:(u_ro <= 1 / 2)%R
(-8 * u_ro + 4 <=
 (4 * u_ro ^ 2 - 28 * u_ro + 9) * u_ro + 4)%R
  apply Rplus_le_compat_r, Rmult_le_compat_r; [apply Pu_ro|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
Heps_le_half:(u_ro <= 1 / 2)%R
(-8 <= 4 * u_ro ^ 2 - 28 * u_ro + 9)%R
  now assert (H : (0 <= u_ro ^ 2)%R); [apply pow2_ge_0|lra]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 <= -2 * u_ro ^ 3 + 7 * u_ro ^ 2 + 2 * u_ro)%R
assert (H : (u_ro ^ 3 <= u_ro ^ 2)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro ^ 3 <= u_ro ^ 2)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
H:(u_ro ^ 3 <= u_ro ^ 2)%R
(0 <= -2 * u_ro ^ 3 + 7 * u_ro ^ 2 + 2 * u_ro)%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro ^ 3 <= u_ro ^ 2)%R unfold pow; rewrite <-!Rmult_assoc, Rmult_1_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro * u_ro * u_ro <= u_ro * u_ro * 1)%R
  apply Rmult_le_compat_l; [now apply Rmult_le_pos; apply Pu_ro|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(u_ro <= 1)%R
  now apply Rlt_le, u_ro_lt_1. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
H:(u_ro ^ 3 <= u_ro ^ 2)%R
(0 <= -2 * u_ro ^ 3 + 7 * u_ro ^ 2 + 2 * u_ro)%R
now assert (H' : (0 <= u_ro ^ 2)%R); [apply pow2_ge_0|lra].
Qed.

Lemma om1ds1p2u_ro_pos : (0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
unfold Rdiv; rewrite Rmult_1_l, <-Rinv_1 at 1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(0 <= / 1 - / sqrt (1 + 2 * u_ro))%R
apply Rle_0_minus, Rinv_le; [lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(1 <= sqrt (1 + 2 * u_ro))%R
rewrite <- sqrt_1 at 1; apply sqrt_le_1_alt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(1 <= 1 + 2 * u_ro)%R
assert (H := u_ro_pos beta prec); lra.
Qed.

Lemma om1ds1p2u_ro_le_u_rod1pu_ro :
  (1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
assert (Pu_ro := u_ro_pos beta prec).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
apply (Rmult_le_reg_r (sqrt (1 + 2 * u_ro) * (1 + u_ro))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 < sqrt (1 + 2 * u_ro) * (1 + u_ro))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((1 - 1 / sqrt (1 + 2 * u_ro)) *
 (sqrt (1 + 2 * u_ro) * (1 + u_ro)) <=
 u_ro / (1 + u_ro) *
 (sqrt (1 + 2 * u_ro) * (1 + u_ro)))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(0 < sqrt (1 + 2 * u_ro) * (1 + u_ro))%R apply Rmult_lt_0_compat; [apply sqrt_lt_R0|]; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
((1 - 1 / sqrt (1 + 2 * u_ro)) *
 (sqrt (1 + 2 * u_ro) * (1 + u_ro)) <=
 u_ro / (1 + u_ro) *
 (sqrt (1 + 2 * u_ro) * (1 + u_ro)))%R
field_simplify; [|lra|intro H; apply sqrt_eq_0 in H; lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(sqrt (1 + 2 * u_ro) * u_ro + sqrt (1 + 2 * u_ro) -
 u_ro - 1 <= sqrt (1 + 2 * u_ro) * u_ro)%R
try unfold Rdiv; unfold Rminus; rewrite ?Rinv_1, ?Rmult_1_r, !Rplus_assoc.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(sqrt (1 + 2 * u_ro) * u_ro +
 (sqrt (1 + 2 * u_ro) + (- u_ro + - (1))) <=
 sqrt (1 + 2 * u_ro) * u_ro)%R
rewrite <-(Rplus_0_r (sqrt _ * _)) at 2; apply Rplus_le_compat_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(sqrt (1 + 2 * u_ro) + (- u_ro + - (1)) <= 0)%R
apply (Rplus_le_reg_r (1 + u_ro)); ring_simplify.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(sqrt (1 + 2 * u_ro) <= u_ro + 1)%R
rewrite <-(sqrt_square (_ + 1)); [|lra]; apply sqrt_le_1_alt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
Pu_ro:(0 <= u_ro)%R
(1 + 2 * u_ro <= (u_ro + 1) * (u_ro + 1))%R
assert (H : (0 <= u_ro * u_ro)%R); [apply Rmult_le_pos|]; lra.
Qed.

Lemma s1p2u_rom1_pos : (0 <= sqrt (1 + 2 * u_ro) - 1)%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(0 <= sqrt (1 + 2 * u_ro) - 1)%R
apply (Rplus_le_reg_r 1); ring_simplify.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(1 <= sqrt (1 + 2 * u_ro))%R
rewrite <-sqrt_1 at 1; apply sqrt_le_1_alt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
(1 <= 1 + 2 * u_ro)%R
assert (H := u_ro_pos beta prec); lra.
Qed.

Theorem sqrt_error_N_FLX x (Fx : format x) :
  (Rabs (round beta (FLX_exp prec) (Znearest choice) (sqrt x) - sqrt x)
   <= (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Peps := u_ro_pos beta prec).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Peps' : (0 < u_ro)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
(0 < u_ro)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
(Rabs
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x) - 
    sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
(0 < u_ro)%R unfold u_ro; apply Rmult_lt_0_compat; [lra|apply bpow_gt_0]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Pb := om1ds1p2u_ro_pos).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Pb' := s1p2u_rom1_pos).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
destruct (Rle_or_lt x 0) as [Nx|Px].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nx:(x <= 0)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
(Rabs
   (round beta (FLX_exp prec) 
      (Znearest choice) (sqrt x) - 
    sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nx:(x <= 0)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R rewrite (sqrt_neg _ Nx), round_0, Rabs_R0, Rmult_0_r; [|apply valid_rnd_N].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nx:(x <= 0)%R
(Rabs (0 - 0) <= 0)%R
  now unfold Rminus; rewrite Rplus_0_l, Rabs_Ropp, Rabs_R0; right. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
destruct (sqrt_error_N_FLX_aux1 _ Fx Px)
  as (mu, (e, (Fmu, (Hmu, (HmuGe1, HmuLtsqradix))))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
pose (t := sqrt x).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (sqrt x) - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
set (rt := round _ _ _ _).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Ht : (t = sqrt mu * bpow e)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
t = (sqrt mu * bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
t = (sqrt mu * bpow e)%R unfold t; rewrite Hmu, sqrt_mult_alt; [|now apply (Rle_trans _ _ _ Rle_0_1)].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
(sqrt mu * sqrt (bpow (2 * e)))%R =
(sqrt mu * bpow e)%R
  now rewrite sqrt_bpow. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
destruct (sqrt_error_N_FLX_aux2 _ Fmu HmuGe1) as [Hmu'|[Hmu'|Hmu']].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R unfold rt; fold t; rewrite Ht, Hmu', sqrt_1, Rmult_1_l.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(Rabs
   (round beta (FLX_exp prec) (Znearest choice)
      (bpow e) - bpow e) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (bpow e))%R
  rewrite round_generic; [|now apply valid_rnd_N|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(Rabs (bpow e - bpow e) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (bpow e))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
format (bpow e)
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(Rabs (bpow e - bpow e) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (bpow e))%R rewrite Rminus_diag_eq, Rabs_R0; [|now simpl].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(0 <= (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (bpow e))%R
    now apply Rmult_le_pos; [|apply Rabs_pos]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
format (bpow e)
  apply generic_format_bpow'; [now apply FLX_exp_valid|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = 1%R
(FLX_exp prec e <= e)%Z
  unfold FLX_exp; omega. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R assert (Hsqrtmu : (1 <= sqrt mu < 1 + u_ro)%R); [rewrite Hmu'; split|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(1 <= sqrt (1 + 2 * u_ro))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(sqrt (1 + 2 * u_ro) < 1 + u_ro)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(1 <= sqrt (1 + 2 * u_ro))%R rewrite <- sqrt_1 at 1; apply sqrt_le_1_alt; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(sqrt (1 + 2 * u_ro) < 1 + u_ro)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(sqrt (1 + 2 * u_ro) < 1 + u_ro)%R rewrite <- sqrt_square; [|lra]; apply sqrt_lt_1_alt; split; [lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
(1 + 2 * u_ro < (1 + u_ro) * (1 + u_ro))%R
    ring_simplify; assert (0 < u_ro ^ 2)%R; [apply pow_lt|]; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  assert (Fbpowe : generic_format beta (FLX_exp prec) (bpow e)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
format (bpow e)
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
format (bpow e) apply generic_format_bpow; unfold FLX_exp; omega. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  assert (Hrt : rt = bpow e :> R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
rt = bpow e
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
Hrt:rt = bpow e
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
rt = bpow e unfold rt; fold t; rewrite Ht; simpl; apply Rle_antisym.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(round beta (FLX_exp prec) (Znearest choice)
   (sqrt mu * bpow e) <= bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(bpow e <=
 round beta (FLX_exp prec) 
   (Znearest choice) (sqrt mu * bpow e))%R
    {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(round beta (FLX_exp prec) (Znearest choice)
   (sqrt mu * bpow e) <= bpow e)%R apply round_N_le_midp; [now apply FLX_exp_valid|exact Fbpowe|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(sqrt mu * bpow e <
 (bpow e + succ beta (FLX_exp prec) (bpow e)) / 2)%R
      apply (Rlt_le_trans _ ((1 + u_ro) * bpow e)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(sqrt mu * bpow e < (1 + u_ro) * bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
((1 + u_ro) * bpow e <=
 (bpow e + succ beta (FLX_exp prec) (bpow e)) / 2)%R
      {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(sqrt mu * bpow e < (1 + u_ro) * bpow e)%R now apply Rmult_lt_compat_r; [apply bpow_gt_0|]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
((1 + u_ro) * bpow e <=
 (bpow e + succ beta (FLX_exp prec) (bpow e)) / 2)%R
      unfold succ; rewrite Rle_bool_true; [|now apply bpow_ge_0].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
((1 + u_ro) * bpow e <=
 (bpow e + (bpow e + ulp beta (FLX_exp prec) (bpow e))) /
 2)%R
      rewrite ulp_bpow; unfold FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
((1 + u_ro) * bpow e <=
 (bpow e + (bpow e + bpow (e + 1 - prec))) / 2)%R
      unfold Z.sub, u_ro; rewrite !bpow_plus; right; field. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(bpow e <=
 round beta (FLX_exp prec) (Znearest choice)
   (sqrt mu * bpow e))%R
    apply round_ge_generic;
      [now apply FLX_exp_valid|now apply valid_rnd_N|exact Fbpowe|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(bpow e <= sqrt mu * bpow e)%R
    rewrite <- (Rmult_1_l (bpow _)) at 1.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
(1 * bpow e <= sqrt mu * bpow e)%R
    now apply Rmult_le_compat_r; [apply bpow_ge_0|]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
Hrt:rt = bpow e
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
  fold t; rewrite Hrt, Ht, Hmu', <-(Rabs_pos_eq _ Pb), <-Rabs_mult.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':mu = (1 + 2 * u_ro)%R
Hsqrtmu:(1 <= sqrt mu < 1 + u_ro)%R
Fbpowe:format (bpow e)
Hrt:rt = bpow e
(Rabs (bpow e - sqrt (1 + 2 * u_ro) * bpow e) <=
 Rabs
   ((1 - 1 / sqrt (1 + 2 * u_ro)) *
    (sqrt (1 + 2 * u_ro) * bpow e)))%R
  rewrite Rabs_minus_sym; right; f_equal; field; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Hsqrtmu : (1 + u_ro < sqrt mu)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
(1 + u_ro < sqrt mu)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
(1 + u_ro < sqrt mu)%R apply (Rlt_le_trans _ (sqrt (1 + 4 * u_ro))); [|now apply sqrt_le_1_alt].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
(1 + u_ro < sqrt (1 + 4 * u_ro))%R
  assert (P1peps : (0 <= 1 + u_ro)%R)
    by now apply Rplus_le_le_0_compat; [lra|apply Peps].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
P1peps:(0 <= 1 + u_ro)%R
(1 + u_ro < sqrt (1 + 4 * u_ro))%R
  rewrite <- (sqrt_square (1 + u_ro)); [|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
P1peps:(0 <= 1 + u_ro)%R
(sqrt ((1 + u_ro) * (1 + u_ro)) < sqrt (1 + 4 * u_ro))%R
  apply sqrt_lt_1_alt; split; [now apply Rmult_le_pos|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
P1peps:(0 <= 1 + u_ro)%R
((1 + u_ro) * (1 + u_ro) < 1 + 4 * u_ro)%R
  apply (Rplus_lt_reg_r (-1 - 2 * u_ro)); ring_simplify; simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
P1peps:(0 <= 1 + u_ro)%R
(u_ro * (u_ro * 1) < 2 * u_ro)%R
  rewrite Rmult_1_r; apply Rmult_lt_compat_r; [apply Peps'|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
P1peps:(0 <= 1 + u_ro)%R
(u_ro < 2)%R
  now apply (Rlt_le_trans _ 1); [apply u_ro_lt_1|lra]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Hulpt : (ulp beta (FLX_exp prec) t = 2 * u_ro * bpow e)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
ulp beta (FLX_exp prec) t = (2 * u_ro * bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
ulp beta (FLX_exp prec) t = (2 * u_ro * bpow e)%R unfold ulp; rewrite Req_bool_false; [|apply Rgt_not_eq, Rlt_gt].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
bpow (cexp t) = (2 * u_ro * bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 < t)%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
bpow (cexp t) = (2 * u_ro * bpow e)%R unfold u_ro; rewrite <-Rmult_assoc, Rinv_r, Rmult_1_l, <-bpow_plus; [|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
bpow (cexp t) = bpow (- prec + 1 + e)
    f_equal; unfold cexp, FLX_exp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(mag beta t - prec)%Z = (- prec + 1 + e)%Z
    assert (Hmagt : (mag beta t = 1 + e :> Z)%Z).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
mag beta t = (1 + e)%Z
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hmagt:mag beta t = (1 + e)%Z
(mag beta t - prec)%Z = (- prec + 1 + e)%Z
    {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
mag beta t = (1 + e)%Z apply mag_unique.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(bpow (1 + e - 1) <= Rabs t < bpow (1 + e))%R
      unfold t; rewrite (Rabs_pos_eq _ (Rlt_le _ _ (sqrt_lt_R0 _ Px))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(bpow (1 + e - 1) <= sqrt x < bpow (1 + e))%R
      fold t; split.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(bpow (1 + e - 1) <= t)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(t < bpow (1 + e))%R
      {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(bpow (1 + e - 1) <= t)%R rewrite Ht; replace (_ - _)%Z with e by ring.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(bpow e <= sqrt mu * bpow e)%R
        rewrite <- (Rmult_1_l (bpow _)) at 1; apply Rmult_le_compat_r.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 <= bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(1 <= sqrt mu)%R
        {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 <= bpow e)%R apply bpow_ge_0. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(1 <= sqrt mu)%R
        now rewrite <- sqrt_1; apply sqrt_le_1_alt. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(t < bpow (1 + e))%R
      rewrite bpow_plus, bpow_1, Ht; simpl.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(sqrt mu * bpow e < IZR beta * bpow e)%R
      apply Rmult_lt_compat_r; [now apply bpow_gt_0|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(sqrt mu < IZR beta)%R
      rewrite <- sqrt_square.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(sqrt mu < sqrt (IZR beta * IZR beta))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 <= IZR beta)%R
      {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(sqrt mu < sqrt (IZR beta * IZR beta))%R apply sqrt_lt_1_alt; split; [lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(mu < IZR beta * IZR beta)%R
        apply (Rlt_le_trans _ _ _ HmuLtsqradix); right.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
bpow 2 = (IZR beta * IZR beta)%R
        now unfold bpow, Z.pow_pos; simpl; rewrite Zmult_1_r, mult_IZR. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 <= IZR beta)%R
      apply IZR_le, (Z.le_trans _ 2), Zle_bool_imp_le, radix_prop; omega. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hmagt:mag beta t = (1 + e)%Z
(mag beta t - prec)%Z = (- prec + 1 + e)%Z
    rewrite Hmagt; ring. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 < t)%R
  rewrite Ht; apply Rmult_lt_0_compat; [|now apply bpow_gt_0].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
(0 < sqrt mu)%R
  now apply (Rlt_le_trans _ 1); [lra|rewrite <- sqrt_1; apply sqrt_le_1_alt]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (Pt : (0 < t)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
(0 < t)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
(0 < t)%R rewrite Ht; apply Rmult_lt_0_compat; [lra|apply bpow_gt_0]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
assert (H : (Rabs ((rt - sqrt x) / sqrt x)
             <= 1 - 1 / sqrt (1 + 2 * u_ro))%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs ((rt - sqrt x) / sqrt x) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
H:(Rabs ((rt - sqrt x) / sqrt x) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs ((rt - sqrt x) / sqrt x) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R unfold Rdiv; rewrite Rabs_mult, (Rabs_pos_eq (/ _));
    [|now left; apply Rinv_0_lt_compat].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs (rt - sqrt x) * / sqrt x <=
 1 - 1 * / sqrt (1 + 2 * u_ro))%R
  apply (Rle_trans _ ((u_ro * bpow e) / t)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs (rt - sqrt x) * / sqrt x <= u_ro * bpow e / t)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / t <= 1 - 1 * / sqrt (1 + 2 * u_ro))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs (rt - sqrt x) * / sqrt x <= u_ro * bpow e / t)%R unfold Rdiv; apply Rmult_le_compat_r; [now left; apply Rinv_0_lt_compat|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(Rabs (rt - sqrt x) <= u_ro * bpow e)%R
    apply (Rle_trans _ _ _ (error_le_half_ulp _ _ _ _)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(/ 2 * ulp beta (FLX_exp prec) (sqrt x) <=
 u_ro * bpow e)%R
    fold t; rewrite Hulpt; right; field. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / t <= 1 - 1 * / sqrt (1 + 2 * u_ro))%R
  apply (Rle_trans _ (u_ro / sqrt (1 + 4 * u_ro))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / t <= u_ro / sqrt (1 + 4 * u_ro))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro / sqrt (1 + 4 * u_ro) <=
 1 - 1 * / sqrt (1 + 2 * u_ro))%R
  {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / t <= u_ro / sqrt (1 + 4 * u_ro))%R apply (Rle_trans _ (u_ro * bpow e / (sqrt (1 + 4 * u_ro) * bpow e))).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / t <=
 u_ro * bpow e / (sqrt (1 + 4 * u_ro) * bpow e))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / (sqrt (1 + 4 * u_ro) * bpow e) <=
 u_ro / sqrt (1 + 4 * u_ro))%R
    {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / t <=
 u_ro * bpow e / (sqrt (1 + 4 * u_ro) * bpow e))%R unfold Rdiv; apply Rmult_le_compat_l;
        [now apply Rmult_le_pos; [apply Peps|apply bpow_ge_0]|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(/ t <= / (sqrt (1 + 4 * u_ro) * bpow e))%R
      apply Rinv_le.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(0 < sqrt (1 + 4 * u_ro) * bpow e)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(sqrt (1 + 4 * u_ro) * bpow e <= t)%R
      {beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(0 < sqrt (1 + 4 * u_ro) * bpow e)%R apply Rmult_lt_0_compat; [apply sqrt_lt_R0; lra|apply bpow_gt_0]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(sqrt (1 + 4 * u_ro) * bpow e <= t)%R
      now rewrite Ht; apply Rmult_le_compat_r;
        [apply bpow_ge_0|apply sqrt_le_1_alt]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro * bpow e / (sqrt (1 + 4 * u_ro) * bpow e) <=
 u_ro / sqrt (1 + 4 * u_ro))%R
    right; field; split; apply Rgt_not_eq, Rlt_gt;
      [apply sqrt_lt_R0; lra|apply bpow_gt_0]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
(u_ro / sqrt (1 + 4 * u_ro) <=
 1 - 1 * / sqrt (1 + 2 * u_ro))%R
  apply sqrt_error_N_FLX_aux3. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
H:(Rabs ((rt - sqrt x) / sqrt x) <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
(Rabs (rt - sqrt x) <=
 (1 - 1 / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
revert H; unfold Rdiv; rewrite Rabs_mult, Rabs_Rinv; [|fold t; lra]; intro H.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
H:(Rabs (rt - sqrt x) * / Rabs (sqrt x) <=
 1 - 1 * / sqrt (1 + 2 * u_ro))%R
(Rabs (rt - sqrt x) <=
 (1 - 1 * / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x))%R
apply (Rmult_le_reg_r (/ Rabs (sqrt x)));
  [apply Rinv_0_lt_compat, Rabs_pos_lt; fold t; lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
Peps:(0 <= u_ro)%R
Peps':(0 < u_ro)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Pb':(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Px:(0 < x)%R
mu:R
e:Z
Fmu:format mu
Hmu:x = (mu * bpow (2 * e))%R
HmuGe1:(1 <= mu)%R
HmuLtsqradix:(mu < bpow 2)%R
t:=sqrt x:R
rt:=round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x):R
Ht:t = (sqrt mu * bpow e)%R
Hmu':(1 + 4 * u_ro <= mu)%R
Hsqrtmu:(1 + u_ro < sqrt mu)%R
Hulpt:ulp beta (FLX_exp prec) t =
(2 * u_ro * bpow e)%R
Pt:(0 < t)%R
H:(Rabs (rt - sqrt x) * / Rabs (sqrt x) <=
 1 - 1 * / sqrt (1 + 2 * u_ro))%R
(Rabs (rt - sqrt x) * / Rabs (sqrt x) <=
 (1 - 1 * / sqrt (1 + 2 * u_ro)) * Rabs (sqrt x) *
 / Rabs (sqrt x))%R
apply (Rle_trans _ _ _ H); right; field; split; [apply Rabs_no_R0;fold t|]; lra.
Qed.

Theorem sqrt_error_N_FLX_ex x (Fx : format x) :
  exists eps,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) (sqrt x)
  = (sqrt x * (1 + eps))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) (sqrt x) =
  (sqrt x * (1 + eps))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x:R
Fx:format x
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLX_exp prec) (Znearest choice) (sqrt x) =
  (sqrt x * (1 + eps))%R
now apply relative_error_le_conversion;
  [apply valid_rnd_N|apply om1ds1p2u_ro_pos|apply sqrt_error_N_FLX].
Qed.

Lemma sqrt_error_N_round_ex_derive :
  forall x rx,
  (exists eps,
   (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\ rx = (x * (1 + eps))%R) ->
  exists eps,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\ x = (rx * (1 + eps))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
forall x rx : R,
(exists eps : R,
   (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
   rx = (x * (1 + eps))%R) ->
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
forall x rx : R,
(exists eps : R,
   (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
   rx = (x * (1 + eps))%R) ->
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
intros x rx (d, (Bd, Hd)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
assert (H := Rabs_le_inv _ _ Bd).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
assert (H' := om1ds1p2u_ro_le_u_rod1pu_ro).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
assert (H'' := u_rod1pu_ro_le_u_ro beta prec).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
assert (H''' := u_ro_lt_1 beta prec prec_gt_0_).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
assert (Hpos := s1p2u_rom1_pos).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
destruct (Req_dec rx 0) as [Zfx|Nzfx].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Zfx:rx = 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Zfx:rx = 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R exists 0%R; split; [now rewrite Rabs_R0|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Zfx:rx = 0%R
x = (rx * (1 + 0))%R
  rewrite Rplus_0_r, Rmult_1_r, Zfx; rewrite Zfx in Hd.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:0%R = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Zfx:rx = 0%R
x = 0%R
  destruct (Rmult_integral _ _ (sym_eq Hd)); lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Zx:x = 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Zx:x = 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R now exfalso; revert Hd; rewrite Zx; rewrite Rmult_0_l. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
set (d' := ((x - rx) / rx)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
assert (Hd' : (Rabs d' <= sqrt (1 + 2 * u_ro) - 1)%R).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d' <= sqrt (1 + 2 * u_ro) - 1)%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
Hd':(Rabs d' <= sqrt (1 + 2 * u_ro) - 1)%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d' <= sqrt (1 + 2 * u_ro) - 1)%R unfold d'; rewrite Hd.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs ((x - x * (1 + d)) / (x * (1 + d))) <=
 sqrt (1 + 2 * u_ro) - 1)%R
  replace (_ / _)%R with (- d / (1 + d))%R; [|now field; split; lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs (- d / (1 + d)) <= sqrt (1 + 2 * u_ro) - 1)%R
  unfold Rdiv; rewrite Rabs_mult, Rabs_Ropp.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d * Rabs (/ (1 + d)) <= sqrt (1 + 2 * u_ro) - 1)%R
  rewrite (Rabs_pos_eq (/ _)); [|apply Rlt_le, Rinv_0_lt_compat; lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d * / (1 + d) <= sqrt (1 + 2 * u_ro) - 1)%R
  apply (Rmult_le_reg_r (1 + d)); [lra|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d * / (1 + d) * (1 + d) <=
 (sqrt (1 + 2 * u_ro) - 1) * (1 + d))%R
  rewrite Rmult_assoc, Rinv_l, Rmult_1_r; [|lra].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(Rabs d <= (sqrt (1 + 2 * u_ro) - 1) * (1 + d))%R
  apply (Rle_trans _ _ _ Bd).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
(1 - 1 / sqrt (1 + 2 * u_ro) <=
 (sqrt (1 + 2 * u_ro) - 1) * (1 + d))%R
  apply (Rle_trans _ ((sqrt (1 + 2 * u_ro) - 1) * (1/sqrt (1 + 2 * u_ro))));
    [right; field|apply Rmult_le_compat_l]; lra. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
x, rx, d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:rx = (x * (1 + d))%R
H:(- (1 - 1 / sqrt (1 + 2 * u_ro)) <= d <=
 1 - 1 / sqrt (1 + 2 * u_ro))%R
H':(1 - 1 / sqrt (1 + 2 * u_ro) <= u_ro / (1 + u_ro))%R
H'':(u_ro / (1 + u_ro) <= u_ro)%R
H''':(u_ro < 1)%R
Hpos:(0 <= sqrt (1 + 2 * u_ro) - 1)%R
Nzfx:rx <> 0%R
Nzx:x <> 0%R
d':=((x - rx) / rx)%R:R
Hd':(Rabs d' <= sqrt (1 + 2 * u_ro) - 1)%R
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  x = (rx * (1 + eps))%R
now exists d'; split; [exact Hd'|]; unfold d'; field.
Qed.

sqrt(1 + 2 u_ro) - 1 <= u_ro

Theorem sqrt_error_N_FLX_round_ex :
  forall x,
  format x ->
  exists eps,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  sqrt x = (round beta (FLX_exp prec) (Znearest choice) (sqrt x) * (1 + eps))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
forall x : R,
format x ->
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  sqrt x =
  (round beta (FLX_exp prec) (Znearest choice)
     (sqrt x) * (1 + eps))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
forall x : R,
format x ->
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  sqrt x =
  (round beta (FLX_exp prec) (Znearest choice)
     (sqrt x) * (1 + eps))%R
now intros x Fx; apply sqrt_error_N_round_ex_derive, sqrt_error_N_FLX_ex.
Qed.

Variable emin : Z.
Hypothesis Hemin : (emin <= 2 * (1 - prec))%Z.

Theorem sqrt_error_N_FLT_ex :
  forall x,
  generic_format beta (FLT_exp emin prec) x ->
  exists eps,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice) (sqrt x)
  = (sqrt x * (1 + eps))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
forall x : R,
generic_format beta (FLT_exp emin prec) x ->
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
forall x : R,
generic_format beta (FLT_exp emin prec) x ->
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
intros x Fx.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
assert (Heps := u_ro_pos).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
assert (Pb := om1ds1p2u_ro_pos).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
destruct (Rle_or_lt x 0) as [Nx|Px].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Nx:(x <= 0)%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) 
    (Znearest choice) (sqrt x) =
  (sqrt x * (1 + eps))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Nx:(x <= 0)%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R exists 0%R; split; [now rewrite Rabs_R0|].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Nx:(x <= 0)%R
round beta (FLT_exp emin prec) (Znearest choice)
  (sqrt x) = (sqrt x * (1 + 0))%R
  now rewrite (sqrt_neg x Nx), round_0, Rmult_0_l; [|apply valid_rnd_N]. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
assert (Fx' := generic_format_FLX_FLT _ _ _ _ Fx).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
destruct (sqrt_error_N_FLX_ex _ Fx') as (d, (Bd, Hd)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
exists eps : R,
  (Rabs eps <= 1 - 1 / sqrt (1 + 2 * u_ro))%R /\
  round beta (FLT_exp emin prec) (Znearest choice)
    (sqrt x) = (sqrt x * (1 + eps))%R
exists d; split; [exact Bd|]; rewrite <-Hd; apply round_FLT_FLX.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(bpow (emin + prec - 1) <= Rabs (sqrt x))%R
apply (Rle_trans _ (bpow (emin / 2)%Z)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(bpow (emin + prec - 1) <= bpow (emin / 2))%R
beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(bpow (emin / 2) <= Rabs (sqrt x))%R
{beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(bpow (emin + prec - 1) <= bpow (emin / 2))%R apply bpow_le, Z.div_le_lower_bound; lia. }beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(bpow (emin / 2) <= Rabs (sqrt x))%R
apply (Rle_trans _ _ _ (sqrt_bpow_ge _ _)).beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(sqrt (bpow emin) <= Rabs (sqrt x))%R
rewrite Rabs_pos_eq; [|now apply sqrt_pos]; apply sqrt_le_1_alt.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Fx:generic_format beta (FLT_exp emin prec) x
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
(bpow emin <= x)%R
revert Fx; apply generic_format_ge_bpow; [|exact Px].beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
x:R
Heps:forall (beta0 : radix) (prec0 : Z),
(0 <= Relative.u_ro beta0 prec0)%R
Pb:(0 <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Px:(0 < x)%R
Fx':format x
d:R
Bd:(Rabs d <= 1 - 1 / sqrt (1 + 2 * u_ro))%R
Hd:round beta (FLX_exp prec) 
  (Znearest choice) (sqrt x) =
(sqrt x * (1 + d))%R
forall e : Z, (emin <= FLT_exp emin prec e)%Z
intro e; unfold FLT_exp; apply Z.le_max_r.
Qed.

sqrt(1 + 2 u_ro) - 1 <= u_ro

Theorem sqrt_error_N_FLT_round_ex :
  forall x,
  generic_format beta (FLT_exp emin prec) x ->
  exists eps,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  sqrt x
  = (round beta (FLT_exp emin prec) (Znearest choice) (sqrt x) * (1 + eps))%R.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
forall x : R,
generic_format beta (FLT_exp emin prec) x ->
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  sqrt x =
  (round beta (FLT_exp emin prec) (Znearest choice)
     (sqrt x) * (1 + eps))%R
Proof.beta:radix
prec:Z
prec_gt_0_:Prec_gt_0 prec
choice:Z -> bool
Hp1:(1 < prec)%Z
emin:Z
Hemin:(emin <= 2 * (1 - prec))%Z
forall x : R,
generic_format beta (FLT_exp emin prec) x ->
exists eps : R,
  (Rabs eps <= sqrt (1 + 2 * u_ro) - 1)%R /\
  sqrt x =
  (round beta (FLT_exp emin prec) (Znearest choice)
     (sqrt x) * (1 + eps))%R
now intros x Fx; apply sqrt_error_N_round_ex_derive, sqrt_error_N_FLT_ex.
Qed.

End Fprop_divsqrt_error.

Section format_REM_aux.

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

Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { monotone_exp : Monotone_exp fexp }.

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

Notation format := (generic_format beta fexp).

Lemma format_REM_aux:
  forall x y : R,
  format x -> format y -> (0 <= x)%R -> (0 < y)%R ->
  ((0 < x/y < /2)%R -> rnd (x/y) = 0%Z) ->
  format (x - IZR (rnd (x/y))*y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
(0 <= x)%R ->
(0 < y)%R ->
((0 < x / y < / 2)%R -> rnd (x / y) = 0%Z) ->
format (x - IZR (rnd (x / y)) * y)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
forall x y : R,
format x ->
format y ->
(0 <= x)%R ->
(0 < y)%R ->
((0 < x / y < / 2)%R -> rnd (x / y) = 0%Z) ->
format (x - IZR (rnd (x / y)) * y)
intros x y Fx Fy Hx Hy rnd_small.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
format (x - IZR (rnd (x / y)) * y)
pose (n:=rnd (x / y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
format (x - IZR (rnd (x / y)) * y)
assert (Hn:(IZR n = round beta (FIX_exp 0) rnd (x/y))%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
IZR n = round beta (FIX_exp 0) rnd (x / y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
format (x - IZR (rnd (x / y)) * y)
unfold round, FIX_exp, cexp, scaled_mantissa, F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
IZR n = (IZR (rnd (x / y * 1)) * 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
format (x - IZR (rnd (x / y)) * y)
now rewrite 2!Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
format (x - IZR (rnd (x / y)) * y)
assert (H:(0 <= n)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
(0 <= n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
format (x - IZR (rnd (x / y)) * y)
apply le_IZR; rewrite Hn; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
(0 <= round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
format (x - IZR (rnd (x / y)) * y)
apply Rle_trans with (round beta (FIX_exp 0) rnd 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
(0 <= round beta (FIX_exp 0) rnd 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
(round beta (FIX_exp 0) rnd 0 <=
 round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
format (x - IZR (rnd (x / y)) * y)
right; apply sym_eq, round_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
(round beta (FIX_exp 0) rnd 0 <=
 round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
format (x - IZR (rnd (x / y)) * y)
apply round_le...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
(0 <= x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
format (x - IZR (rnd (x / y)) * y)
apply Fourier_util.Rle_mult_inv_pos; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
format (x - IZR (rnd (x / y)) * y)
case (Zle_lt_or_eq 0 n); try exact H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
(0 < n)%Z -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
clear H; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
case (Zle_lt_or_eq 1 n).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
(1 <= n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
(1 < n)%Z -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
(1 < n)%Z -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
clear H; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
set (ex := cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
set (ey := cexp beta fexp y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
set (mx := Ztrunc (scaled_mantissa beta fexp x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
set (my := Ztrunc (scaled_mantissa beta fexp y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
case (Zle_or_lt ey ex); intros Hexy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
(* ey <= ex *)
assert (H0:(x-IZR n *y = F2R (Float beta (mx*beta^(ex-ey) - n*my) ey))%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold Rminus; rewrite Rplus_comm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(- (IZR n * y) + x)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
replace (IZR n) with (F2R (Float beta n 0)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(- (F2R {| Fnum := n; Fexp := 0 |} * y) + x)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Fx, Fy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(-
 (F2R {| Fnum := n; Fexp := 0 |} *
  F2R
    {|
    Fnum := Ztrunc (scaled_mantissa beta fexp y);
    Fexp := cexp beta fexp y |}) +
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
fold mx my ex ey.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(-
 (F2R {| Fnum := n; Fexp := 0 |} *
  F2R {| Fnum := my; Fexp := ey |}) +
 F2R {| Fnum := mx; Fexp := ex |})%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- F2R_mult.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(-
 F2R
   (Fmult {| Fnum := n; Fexp := 0 |}
      {| Fnum := my; Fexp := ey |}) +
 F2R {| Fnum := mx; Fexp := ex |})%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- F2R_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
(F2R
   (Fopp
      (Fmult {| Fnum := n; Fexp := 0 |}
         {| Fnum := my; Fexp := ey |})) +
 F2R {| Fnum := mx; Fexp := ex |})%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- F2R_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R
  (Fplus
     (Fopp
        (Fmult {| Fnum := n; Fexp := 0 |}
           {| Fnum := my; Fexp := ey |}))
     {| Fnum := mx; Fexp := ex |}) =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold Fplus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R
  (let
   '(m1, m2, e) :=
    Falign
      (Fopp
         (Fmult {| Fnum := n; Fexp := 0 |}
            {| Fnum := my; Fexp := ey |}))
      {| Fnum := mx; Fexp := ex |} in
    {| Fnum := m1 + m2; Fexp := e |}) =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y) simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R
  (let
   '(m1, m2, e) :=
    if (ey <=? ex)%Z
    then
     ((- (n * my))%Z, (mx * beta ^ (ex - ey))%Z, ey)
    else ((- (n * my) * beta ^ (ey - ex))%Z, mx, ex)
    in {| Fnum := m1 + m2; Fexp := e |}) =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Zle_imp_le_bool with (1 := Hexy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R
  {|
  Fnum := - (n * my) + mx * beta ^ (ex - ey);
  Fexp := ey |} =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
f_equal; f_equal; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
F2R {| Fnum := n; Fexp := 0 |} = IZR n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold F2R; simpl; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
fold n; rewrite H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
format
  (F2R
     {|
     Fnum := mx * beta ^ (ex - ey) - n * my;
     Fexp := ey |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply generic_format_F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z ->
(cexp beta fexp
   (F2R
      {|
      Fnum := mx * beta ^ (ex - ey) - n * my;
      Fexp := ey |}) <= ey)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- H0; intros H3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(cexp beta fexp (x - IZR n * y) <= ey)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply monotone_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(mag beta (x - IZR n * y) <= mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply mag_le_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(x - IZR n * y)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (x - IZR n * y) <= Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite H0; apply F2R_neq_0; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (x - IZR n * y) <= Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rmult_le_reg_l with (/Rabs y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(0 < / Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= / Rabs y * Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(0 < Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= / Rabs y * Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= / Rabs y * Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= / Rabs y * Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Rinv_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
Rabs y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
2: apply Rgt_not_eq, Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(/ Rabs y * Rabs (x - IZR n * y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- Rabs_Rinv.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (/ y) * Rabs (x - IZR n * y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (/ y) * Rabs (x - IZR n * y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- Rabs_mult.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (/ y * (x - IZR n * y)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
replace (/y * (x - IZR n *y))%R with (-(IZR n - x/y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (- (IZR n - x / y)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(- (IZR n - x / y))%R = (/ y * (x - IZR n * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (IZR n - x / y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(- (IZR n - x / y))%R = (/ y * (x - IZR n * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Hn.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(Rabs (round beta (FIX_exp 0) rnd (x / y) - x / y) <=
 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(- (IZR n - x / y))%R = (/ y * (x - IZR n * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rle_trans with (1:= error_le_ulp beta (FIX_exp 0) _ _).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(ulp beta (FIX_exp 0) (x / y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(- (IZR n - x / y))%R = (/ y * (x - IZR n * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite ulp_FIX.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(bpow 0 <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(- (IZR n - x / y))%R = (/ y * (x - IZR n * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
simpl; apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
(- (IZR n - x / y))%R = (/ y * (x - IZR n * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ey <= ex)%Z
H0:(x - IZR n * y)%R =
F2R
  {|
  Fnum := mx * beta ^ (ex - ey) - n * my;
  Fexp := ey |}
H3:(mx * beta ^ (ex - ey) - n * my)%Z <> 0%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
(* ex < ey: impossible as 1 < n *)
absurd (1 < n)%Z; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
~ (1 < n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Zle_not_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(n <= 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply le_IZR; simpl; rewrite Hn.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(round beta (FIX_exp 0) rnd (x / y) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply round_le_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
generic_format beta (FIX_exp 0) 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x / y <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply generic_format_FIX.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
FIX_format beta 0 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x / y <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
exists (Float beta 1 0); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
1%R = F2R {| Fnum := 1; Fexp := 0 |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x / y <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold F2R; simpl; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x / y <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rmult_le_reg_r with y; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x / y * y <= 1 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold Rdiv; rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x * (/ y * y) <= 1 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Rinv_l, Rmult_1_r, Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
assert (mag beta x < mag beta y)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(mag beta x < mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
H0:(mag beta x < mag beta y)%Z
(x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
case (Zle_or_lt (mag beta y) (mag beta x)); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
(mag beta y <= mag beta x)%Z ->
(mag beta x < mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
H0:(mag beta x < mag beta y)%Z
(x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
intros J; apply monotone_exp in J; clear -J Hexy.beta:radix
fexp:Z -> Z
x, y:R
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
Hexy:(ex < ey)%Z
J:(fexp (mag beta y) <= fexp (mag beta x))%Z
(mag beta x < mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
H0:(mag beta x < mag beta y)%Z
(x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold ex, ey, cexp in Hexy; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(1 < n)%Z
ex:=cexp beta fexp x:Z
ey:=cexp beta fexp y:Z
mx:=Ztrunc (scaled_mantissa beta fexp x):Z
my:=Ztrunc (scaled_mantissa beta fexp y):Z
Hexy:(ex < ey)%Z
H0:(mag beta x < mag beta y)%Z
(x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
left; apply lt_mag with beta; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
1%Z = n -> format (x - IZR (rnd (x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
(* n = 1 -> Sterbenz + rnd_small *)
intros Hn'; fold n; rewrite <- Hn'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
format (x - 1 * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
case Hx; intros Hx'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
assert (J:(0 < x/y)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
(0 < x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Fourier_util.Rlt_mult_inv_pos; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply sterbenz...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
assert (H0:(Rabs (1  - x/y) < 1)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(Rabs (1 - x / y) < 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Hn', Hn.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(Rabs (round beta (FIX_exp 0) rnd (x / y) - x / y) <
 round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rlt_le_trans with (ulp beta (FIX_exp 0) (round beta (FIX_exp 0) rnd (x / y)))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(Rabs (round beta (FIX_exp 0) rnd (x / y) - x / y) <
 ulp beta (FIX_exp 0)
   (round beta (FIX_exp 0) rnd (x / y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(ulp beta (FIX_exp 0)
   (round beta (FIX_exp 0) rnd (x / y)) <=
 round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply error_lt_ulp_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(x / y)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(ulp beta (FIX_exp 0)
   (round beta (FIX_exp 0) rnd (x / y)) <=
 round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(ulp beta (FIX_exp 0)
   (round beta (FIX_exp 0) rnd (x / y)) <=
 round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite ulp_FIX.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(bpow 0 <= round beta (FIX_exp 0) rnd (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- Hn, <- Hn'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
(bpow 0 <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(Rabs (1 - x / y) < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rabs_lt_inv in H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(y / 2 <= x <= 2 * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
split; apply Rmult_le_reg_l with (/y)%R; try now apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * (y / 2) <= / y * x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
unfold Rdiv; rewrite <- Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * y * / 2 <= / y * x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Rinv_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 * / 2 <= / y * x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 * / 2 <= / y * x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite Rmult_1_l, Rmult_comm; fold (x/y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ 2 <= x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
case (Rle_or_lt (/2) (x/y)); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(x / y < / 2)%R -> (/ 2 <= x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
K:(x / y < / 2)%R
(/ 2 <= x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
elim Zlt_not_le with (1 := H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
K:(x / y < / 2)%R
(n <= 0)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Zeq_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
K:(x / y < / 2)%R
n = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply rnd_small.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
K:(x / y < / 2)%R
(0 < x / y < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
now split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(/ y * x <= / y * (2 * y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Ropp_le_cancel; apply Rplus_le_reg_l with 1%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 + - (/ y * (2 * y)) <= 1 + - (/ y * x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
apply Rle_trans with (1-x/y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 + - (/ y * (2 * y)) <= 1 - x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 - x / y <= 1 + - (/ y * x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
2: right; unfold Rdiv; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 + - (/ y * (2 * y)) <= 1 - x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
left; apply Rle_lt_trans with (2:=proj1 H0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
(1 + - (/ y * (2 * y)) <= - (1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
right; field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':(0 < x)%R
J:(0 < x / y)%R
H0:(- (1) < 1 - x / y < 1)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (x - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
rewrite <- Hx', Rminus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 < n)%Z
Hn':1%Z = n
Hx':0%R = x
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:(0 <= n)%Z
0%Z = n -> format (x - IZR (rnd (x / y)) * y)
(* n = 0 *)
clear H; intros H; fold n; rewrite <- H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Fx:format x
Fy:format y
Hx:(0 <= x)%R
Hy:(0 < y)%R
rnd_small:(0 < x / y < / 2)%R -> rnd (x / y) = 0%Z
n:=rnd (x / y):Z
Hn:IZR n = round beta (FIX_exp 0) rnd (x / y)
H:0%Z = n
format (x - 0 * y)
now rewrite Rmult_0_l, Rminus_0_r.
Qed.

End format_REM_aux.

Section format_REM.

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

Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { monotone_exp : Monotone_exp fexp }.

Notation format := (generic_format beta fexp).

Theorem format_REM :
  forall rnd : R -> Z, Valid_rnd rnd ->
  forall x y : R,
  ((Rabs (x/y) < /2)%R -> rnd (x/y)%R = 0%Z) ->
  format x -> format y ->
  format (x - IZR (rnd (x/y)%R) * y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
(* assume 0 < y *)
assert (H: forall rnd : R -> Z, Valid_rnd rnd ->
  forall x y : R,
  ((Rabs (x/y) < /2)%R -> rnd (x/y)%R = 0%Z) ->
  format x -> format y -> (0 < y)%R ->
  format (x - IZR (rnd (x/y)%R) * y)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
intros rnd valid_rnd x y Hrnd Fx Fy Hy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
case (Rle_or_lt 0 x); intros Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply format_REM_aux; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
(0 < x / y < / 2)%R -> rnd (x / y)%R = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
K:(0 < x / y < / 2)%R
rnd (x / y)%R = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply Hrnd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
K:(0 < x / y < / 2)%R
(Rabs (x / y) < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
K:(0 < x / y < / 2)%R
(x / y < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
K:(0 < x / y < / 2)%R
(0 <= x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(0 <= x)%R
K:(0 < x / y < / 2)%R
(0 <= x / y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply Rlt_le, K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
replace (x - IZR (rnd (x/y)) * y)%R with
   (- (-x - IZR (Zrnd_opp rnd (-x/y)) * y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (- x - IZR (Zrnd_opp rnd (- x / y)) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply format_REM_aux; try easy...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
format (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(0 <= - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(0 < - x / y < / 2)%R -> Zrnd_opp rnd (- x / y) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(0 <= - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(0 < - x / y < / 2)%R -> Zrnd_opp rnd (- x / y) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply Ropp_le_cancel; rewrite Ropp_0, Ropp_involutive; now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(0 < - x / y < / 2)%R -> Zrnd_opp rnd (- x / y) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
replace (- x / y)%R with (- (x/y))%R by (unfold Rdiv; ring).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(0 < - (x / y) < / 2)%R ->
Zrnd_opp rnd (- (x / y)) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
Zrnd_opp rnd (- (x / y)) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
unfold Zrnd_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(- rnd (- - (x / y))%R)%Z = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
rewrite Ropp_involutive, Hrnd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(- 0)%Z = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(Rabs (x / y) < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(Rabs (x / y) < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
rewrite Rabs_left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(- (x / y) < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(x / y < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(x / y < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
apply Ropp_lt_cancel.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
K:(0 < - (x / y) < / 2)%R
(- 0 < - (x / y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
now rewrite Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (Zrnd_opp rnd (- x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
unfold Zrnd_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (- rnd (- (- x / y))) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
replace (- (- x / y))%R with (x / y)%R by (unfold Rdiv; ring).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - IZR (- rnd (x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
rewrite opp_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
Hx:(x < 0)%R
(- (- x - - IZR (rnd (x / y)) * y))%R =
(x - IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y ->
(0 < y)%R -> format (x - IZR (rnd (x / y)%R) * y)
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x y : R,
((Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z) ->
format x ->
format y -> format (x - IZR (rnd (x / y)%R) * y)
(* *)
intros rnd valid_rnd x y Hrnd Fx Fy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
format (x - IZR (rnd (x / y)%R) * y)
case (Rle_or_lt 0 y); intros Hy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 <= y)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
destruct Hy as [Hy|Hy].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(0 < y)%R
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:0%R = y
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
now apply H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:0%R = y
format (x - IZR (rnd (x / y)%R) * y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
now rewrite <- Hy, Rmult_0_r, Rminus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (x - IZR (rnd (x / y)%R) * y)
replace (IZR (rnd (x/y)) * y)%R with
  (IZR ((Zrnd_opp rnd) ((x / -y))) * -y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (x - IZR (Zrnd_opp rnd (x / - y)) * - y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
apply H; try easy...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(Rabs (x / - y) < / 2)%R ->
Zrnd_opp rnd (x / - y) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
replace (x / - y)%R with (- (x/y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(Rabs (- (x / y)) < / 2)%R ->
Zrnd_opp rnd (- (x / y)) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- (x / y))%R = (x / - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
K:(Rabs (- (x / y)) < / 2)%R
Zrnd_opp rnd (- (x / y)) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- (x / y))%R = (x / - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
unfold Zrnd_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
K:(Rabs (- (x / y)) < / 2)%R
(- rnd (- - (x / y))%R)%Z = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- (x / y))%R = (x / - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
rewrite Ropp_involutive, Hrnd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
K:(Rabs (- (x / y)) < / 2)%R
(- 0)%Z = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
K:(Rabs (- (x / y)) < / 2)%R
(Rabs (x / y) < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- (x / y))%R = (x / - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
K:(Rabs (- (x / y)) < / 2)%R
(Rabs (x / y) < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- (x / y))%R = (x / - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
now rewrite <- Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- (x / y))%R = (x / - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
field; now apply Rlt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
format (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(0 < - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
apply Ropp_lt_cancel; now rewrite Ropp_0, Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (Zrnd_opp rnd (x / - y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
unfold Zrnd_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (- rnd (- (x / - y))) * - y)%R =
(IZR (rnd (x / y)) * y)%R
replace (- (x / - y))%R with (x/y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(IZR (- rnd (x / y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(x / y)%R = (- (x / - y))%R
rewrite opp_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(- IZR (rnd (x / y)) * - y)%R =
(IZR (rnd (x / y)) * y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(x / y)%R = (- (x / - y))%R
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
H:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 y0 : R,
((Rabs (x0 / y0) < / 2)%R ->
 rnd0 (x0 / y0)%R = 0%Z) ->
format x0 ->
format y0 ->
(0 < y0)%R ->
format (x0 - IZR (rnd0 (x0 / y0)%R) * y0)
rnd:R -> Z
valid_rnd:Valid_rnd rnd
x, y:R
Hrnd:(Rabs (x / y) < / 2)%R -> rnd (x / y)%R = 0%Z
Fx:format x
Fy:format y
Hy:(y < 0)%R
(x / y)%R = (- (x / - y))%R
field; now apply Rlt_not_eq.
Qed.

Theorem format_REM_ZR:
  forall x y : R,
  format x -> format y ->
  format (x - IZR (Ztrunc (x/y)) * y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall x y : R,
format x ->
format y -> format (x - IZR (Ztrunc (x / y)) * y)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall x y : R,
format x ->
format y -> format (x - IZR (Ztrunc (x / y)) * y)
intros x y Fx Fy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
format (x - IZR (Ztrunc (x / y)) * y)
apply format_REM; try easy...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
(Rabs (x / y) < / 2)%R -> Ztrunc (x / y) = 0%Z
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Ztrunc (x / y) = 0%Z
apply Z.abs_0_iff.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Z.abs (Ztrunc (x / y)) = 0%Z
rewrite <- Ztrunc_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Ztrunc (Rabs (x / y)) = 0%Z
rewrite Ztrunc_floor by apply Rabs_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Zfloor (Rabs (x / y)) = 0%Z
apply Zle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(Zfloor (Rabs (x / y)) <= 0)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Zfloor (Rabs (x / y)))%Z
replace 0%Z with (Zfloor (/2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(Zfloor (Rabs (x / y)) <= Zfloor (/ 2))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Zfloor (/ 2) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Zfloor (Rabs (x / y)))%Z
apply Zfloor_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(Rabs (x / y) <= / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Zfloor (/ 2) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Zfloor (Rabs (x / y)))%Z
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Zfloor (/ 2) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Zfloor (Rabs (x / y)))%Z
apply Zfloor_imp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= / 2 < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Zfloor (Rabs (x / y)))%Z
simpl ; lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Zfloor (Rabs (x / y)))%Z
apply Zfloor_lub.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(0 <= Rabs (x / y))%R
apply Rabs_pos.
Qed.

Theorem format_REM_N :
  forall choice,
  forall x y : R,
  format x -> format y ->
  format (x - IZR (Znearest choice (x/y)) * y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall (choice : Z -> bool) (x y : R),
format x ->
format y ->
format (x - IZR (Znearest choice (x / y)) * y)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
forall (choice : Z -> bool) (x y : R),
format x ->
format y ->
format (x - IZR (Znearest choice (x / y)) * y)
intros choice x y Fx Fy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Fx:format x
Fy:format y
format (x - IZR (Znearest choice (x / y)) * y)
apply format_REM; try easy...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Fx:format x
Fy:format y
(Rabs (x / y) < / 2)%R ->
Znearest choice (x / y) = 0%Z
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
Znearest choice (x / y) = 0%Z
apply Znearest_imp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
monotone_exp:Monotone_exp fexp
choice:Z -> bool
x, y:R
Fx:format x
Fy:format y
K:(Rabs (x / y) < / 2)%R
(Rabs (x / y - 0) < / 2)%R
now rewrite Rminus_0_r.
Qed.

End format_REM.